mmf.math.integrate.integrate_1d.quadrature¶
| Decay | Represents information about the asymptotic form of indefinite integrals. |
| PowerDecay(x, n[, x0]) | Represents a power-law decay. |
| ExpDecay(x, b[, x0]) | Represents an exponential decay. |
| IntegrationError([mesg, res, err]) | Error during integration. |
| get_weights(x, roots[, cond]) | Return the list of weights wx for quadratures of increasing |
| get_weights_y(x, y, wy) | Return the weights wx for a quadrature of order at least n >= len(y) for the abscissa x give the n exact quadrature abscissa y and weights wy defining a quadrature of order 2n. |
| integrate(f, a, b[, abs_tol, rel_tol, ...]) | Return (res, err) where res is the integral of the function f(x) from a to b and err is an estimate of the absolute error. |
| quad(f, a, b, *varargin, **kwargs) | An improved version of integrate.quad that does some argument checking and deals with points properly. |
| mquad(f, a, b[, abs_tol, verbosity, fa, fb, ...]) | Return (res, err) where res is the numerically evaluated |
| h_roots(n) | Return the abscissa and weights for the Gaussian quadrature of |
Inheritance diagram for mmf.math.integrate.integrate_1d.quadrature:
Adaptive quadrature routines for one-dimensional integrals.
This module provides access to several adaptive quadrature routines for performing one-dimensional integrals. The main routine is integrate(), which performs a one-dimensional integral. It allows the user to specify properties of the integrand to facilitate an accurate computation.
In particular, the nature of the decay at inf can be described by using a subclass of Decay such as PowerDecay or ExpDecay to effect a transformation for dealing with these tails.
Orthogonal Polynomials¶
Here we review some properties of orthogonal polynomials and define notations that are useful for quadratures. We consider polynomials orthogonal to some measure
System Message: WARNING/2 (\d{\alpha(x)})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\braket{f, g} = \int_{a}^{b} f(x)g(x)\d{\alpha(x)} = \int_{a}^{b} f(x)g(x) W(x)\d{x} )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.Formally we shall work with orthonormal polynomials
System Message: WARNING/2 (p_n(x))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\braket{p_m, p_n} = \delta_{mn})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (P_n(x))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\braket{P_m, P_n} = \delta_{mn}\sqrt{h_m h_n}. )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.The polynomials are characterized in terms of the coefficients
System Message: WARNING/2 (a_n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (b_n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (c_n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (h_n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (e_n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\lambda_n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (W(x))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (Q(x))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (L(x))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (x_{n;i})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (p_n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (w_{n;i})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (&P_{n+1} = (a_n x + b_n) P_n - c_n P_{n-1},\\ &h_n = \braket{P_n,P_n},\\ &Q(x)P_n'' + L(x)P_n' + \lambda_n P_n = 0,\\ &P_n = \frac{1}{e_n W(x)}\diff[n]{}{x}[W(x)Q^{n}(x)],\\ &p_n(x_{n;i}) = 0,\\ &\int_{a}^{b} f(x)W(x)\d{x} = \sum_i w_{n;i} f(x_{n;i}) )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.where the last integral is exact for polynomials
System Message: WARNING/2 (f(x))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (2n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (w_{n;i})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\lambda_{n;i})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (&\begin{aligned} P_n &= k_n x^n + k'_n x^{n-1} + \cdots, & a_n &= \frac{k_{n+1}}{k_{n}}, & b_n &= a_n\left(\frac{k'_{n+1}}{k_{n+1}} - \frac{k'_{n}}{k_{n}}\right), & c_n &= a_n\frac{k_{n-1}h_{n}}{k_{n}h_{n-1}} \end{aligned},\\ &R(x) = W(x)Q(x) = \exp\left(\int\frac{L(x)}{Q(x)}\d{x}\right),\\ &\sum_{i=0}^{n-1} w_{n;i}p_j(x_{n;i})p_k(x_{n;i}) = \delta_{jk} \quad \forall j,k < n,\\ &w_{n;i} = \int_{a}^{b} \left[\frac{p_n(x)}{(x-x_{n;i}) p'_n(x)}\right]^2\d{\alpha(x)} = -\frac{h_n}{p_{n+1}(x_i)p'_n(x_i)} )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.Todo
Check formulae for the weights with h_roots() which seems to work.
Here we summarize some of the known and useful analytic results:
| Name | System Message: WARNING/2 ((a,b)) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (W(x)) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (a_n) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (b_n) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (c_n) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (h_n) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (e_n) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (\lambda_n) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (Q(x)) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (L(x)) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
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| Hermite | System Message: WARNING/2 ((-\infty,\infty)) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (e^{-x^2}) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
2 | 0 | 2n | System Message: WARNING/2 (2^n n!\sqrt{\pi}) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 ((-1)^n) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
2n | 1 | -2x |
| Laguerre | System Message: WARNING/2 ((0,\infty)) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (x^{\beta}e^{-x}) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (\frac{-1}{n+1}) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (\frac{2n+1+\beta}{n+1}) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (\frac{n+\beta}{n+1}) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (\frac{\Gamma(n+\beta+1)}{n!}) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
System Message: WARNING/2 (n!) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
n | x | System Message: WARNING/2 (\beta+1-x) latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log. |
Quadrature Formulae¶
The general idea of a quadrature is to express an integral as
System Message: WARNING/2 (\int_{a}^{b} f(x)\d{\alpha(x)} = \int_{a}^{b} f(x) W(x)\d{x} \approx \sum_{i} w_i f(x_i) )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.where
System Message: WARNING/2 (W(x))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (w_i)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (x_i)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (2n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (f(x))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (2n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (2n)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\sum_{n=0}^{n-1} w_n P_m(x) = \int_{a}^{b} \d{\alpha(x)}\; P_m(x) \Biggr|_{m=0}^{2n-1} = m_m )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.where
System Message: WARNING/2 (m_m)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (P_m(x))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (P_m(x) = x^m)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.If the abscissa are given, we can determine the weights to satisfy
System Message: WARNING/2 (m)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\sigma = \sum_i\abs{w_i}/\abs{\sum_i w_i} \geq 1)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (w = \mat{V}_x(\mat{V}_x^T\mat{V}_x)^{-1}m )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.where
System Message: WARNING/2 (m)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (V_x)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (x)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\begin{aligned} \mat{V}_x &= \mat{V}_p(x) = \begin{pmatrix} P_0(x_0) & P_1(x_0) & \cdots & P_{m-1}(x_0)\\ P_0(x_1) & P_1(x_1) & \cdots & P_{m-1}(x_1)\\ \vdots & \vdots & \ddots & \vdots\\ P_0(x_{n-1}) & P_1(x_{n-1}) & \cdots & P_{m-1}(x_{n-1})\\ \end{pmatrix}, & m &= \begin{pmatrix} \int_{a}^{b} P_0(x) \d{\alpha(x)}\\ \int_{a}^{b} P_1(x) \d{\alpha(x)}\\ \vdots\\ \int_{a}^{b} P_{m-1}(x) \d{\alpha(x)}\\ \end{pmatrix}. \end{aligned} )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.If there is an all positive solution, then this will be all positive. To see this, consider the singular value decomposition of mat{X}:
System Message: WARNING/2 (\mat{V}_x^{T} = \mat{U} \cdot \begin{pmatrix} \mat{D} & \mat{0} \end{pmatrix}\cdot \begin{pmatrix} \mat{V}^T\\ \mat{A}^T \end{pmatrix} = \mat{U}\cdot\mat{D}\cdot\mat{V}^T )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.We may write the solution as
System Message: WARNING/2 (w = \mat{V}\cdot\mat{D}^{-1}\cdot\mat{U}^T\cdot\vect{m} +\mat{A}\cdot\alpha )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.where the projection of w in the subspace spanned by
System Message: WARNING/2 (\mat{V})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\mat{A})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\alpha)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\alpha=0)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.Note
This assumes some properties about the nature of the set of solutions that are typically true for the integration problem with reasonable (positive) weights. This should be clarified at some point.
This formula is acceptable only for low orders, otherwise the Vandermonde matrix quickly becomes ill-conditioned. A better solution is to use a robust algorithm such as that in Demmel and Koev, “Accurate SVDs of polynomial Vandermonde matrices involving orthonormal polynomials”, Linear Algebra and its Applications 411 (2006) 382–396. There they point out that the Vandermonde matrix we need
System Message: WARNING/2 (\mat{V}_x = \mat{E}\cdot\mat{V}_y)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\mat{V}_y = \mat{V}_{p}(y))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (y_i)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (P_n(y_i)=0)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\mat{E}_{ij} = \prod^{n-1}_{\substack{ k=0\\ k\neq j}} \frac{x_i - y_k}{y_j - y_k}. )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.For classical orthogonal polynomials, the roots
System Message: WARNING/2 (y_i)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (w_x = \mat{E}\left(\mat{E}^T\mat{E}\right)^{-1} w_y. )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.This can also be directly computed from the SVD or QR decompositions of
System Message: WARNING/2 (\mat{E}^T)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (L_\infty)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (E = P\cdot Q\cdot R\\ w_x = P\cdot Q \cdot (R^{T})^{-1}\cdot w_y )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.This requires the following knowledge:
The Vandermonde matrix should be formed from orthogonal polynomials
:System Message: WARNING/2 (P_n(y))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
System Message: WARNING/2 (V_p(x) = \begin{pmatrix} P_0(x_0) & P_1(x_0) & \cdots & P_{m-1}(x_0)\\ P_0(x_1) & P_1(x_1) & \cdots & P_{m-1}(x_1)\\ \vdots & \vdots & \ddots & \vdots\\ P_0(x_{n-1}) & P_1(x_{n-1}) & \cdots & P_{m-1}(x_{n-1})\\ \end{pmatrix}. )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
Note that one can scale out the weights if needed, so this is not an arduous requirement in most cases.
The algorithm requires the roots
of the highest orderSystem Message: WARNING/2 (y_i)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
:System Message: WARNING/2 (P_m(y))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
System Message: WARNING/2 (P_m(y_i) = 0. )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
The algorithm requires the Christoffel numbers
:System Message: WARNING/2 (\lambda_i)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
System Message: WARNING/2 (\sum_i \lambda_i P_j(y_i)P_k(y_i) = \delta_{jk}. )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
Here is an example of abscissa for integrals between 0 and 1.
>>> def m_(n):
... return 1./(np.arange(n).reshape((n,1))+1)
>>> def X_(x,n):
... return x**np.arange(n).reshape((n,1))
>>> def w(x,n):
... X = np.mat(X_(x,n))
... l = np.linalg.solve(X*X.T, m_(n))
... return X.T*l
As an example, let’s consider order two quadratures over the interval [0,1] containing the endpoints and one interior point. An analysis of Simpsons rule shows that positive weights exist if the midpoint lies between 1/3 and 2/3:
>>> np.min(w(np.array([0,0.9/3,1]),3))
-0.0555555555...
>>> round(np.min(w(np.array([0,1/3,1]),3)),14)
0.0
>>> np.min(w(np.array([0,1/2,1]),3))
0.1666666...
>>> round(np.min(w(np.array([0,2/3,1]),3)),14)
0.0
>>> np.min(w(np.array([0,2.1/3,1]),3))
-0.055555555555...
Now, suppose we have
- class mmf.math.integrate.integrate_1d.quadrature.Decay[source]¶
Bases: float
Represents information about the asymptotic form of indefinite integrals.
Methods
as_integer_ratio() -> (int, int) Returns a pair of integers, whose ratio is exactly equal to the original float and with a positive denominator. conjugate Returns self, the complex conjugate of any float. fromhex((string) -> float) Create a floating-point number from a hexadecimal string. hex float.hex() -> string is_integer Returns True if the float is an integer. - __init__()¶
x.__init__(...) initializes x; see help(type(x)) for signature
- class mmf.math.integrate.integrate_1d.quadrature.PowerDecay(x, n, x0=0)[source]¶
Bases: mmf.math.integrate.integrate_1d.quadrature.Decay
Represents a power-law decay.
The function should behave as
System Message: WARNING/2 (\lim_{x\rightarrow \pm\infty} = \frac{a}{\abs{x}^{n}} )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.where this behaviour becomes dominant for
.System Message: WARNING/2 (\abs{x}>\abs{x_{0}})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.Parameters : n : float
Power of decay
System Message: WARNING/2 (x^{-n})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
x0 : float
Point beyond which decay is the dominant behaviour. This must have the same sign as the value (and must not be zero).
Notes
The transformation used maps
toSystem Message: WARNING/2 (\abs{x}\in [x_0,\infty))
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latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (x &= x_{0}(1-y)^{\frac{1}{1-n}} = x_{0}e^{\frac{\ln(1-y)}{1-n}},\\ y &= 1 - \left(\frac{x}{x_{0}}\right)^{1-n},\\ \int_{x_{0}}^{\infty}\d{x}\; f(x) &= \int_{0}^{1}\d{y}\; g(y),\\ g(y) &= f\Bigl(x(y)\Bigr)\frac{x_{0}}{n-1}e^{\frac{n\ln(1-y)}{1-n}}. )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.If
asSystem Message: WARNING/2 (f(x)\rightarrow ax^{-n})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log., thenSystem Message: WARNING/2 (x\rightarrow \infty)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.approaches a constant asSystem Message: WARNING/2 (g(y))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.which is very easy to integrate.System Message: WARNING/2 (y\rightarrow 1)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.Examples
>>> f = lambda p:1./(p**2 + 1) >>> a = PowerDecay(-np.inf, x0=-1.0, n=2) >>> b = PowerDecay(np.inf, x0=1.0, n=2) >>> a.y_x(a.x_y(0.5)) 0.5 >>> a.y_x(-1.0) 0.0 >>> b.y_x(1.0) 0.0 >>> quad(f, -np.inf, -1.0) (0.785398163397448..., 1.2888...e-10) >>> quad(a.get_g(f), a.y_x(-np.inf), a.y_x(-1.0)) (0.785398163397448..., 8.7196...e-15) >>> quad(f, 1.0, np.inf) (0.785398163397448..., 1.2888...e-10) >>> quad(b.get_g(f), b.y_x(1.0), b.y_x(np.inf)) (0.785398163397448..., 8.71967...e-15)
This is what integrate does automatically:
>>> integrate(f, a, -1.0) (0.785398163397448..., 8.71967...e-15) >>> integrate(f, 1.0, b) (0.785398163397448..., 8.71967...e-15)
Methods
as_integer_ratio() -> (int, int) Returns a pair of integers, whose ratio is exactly equal to the original float and with a positive denominator. conjugate Returns self, the complex conjugate of any float. fromhex((string) -> float) Create a floating-point number from a hexadecimal string. get_g(f) Return the function g(y) which, when integrated over hex float.hex() -> string is_integer Returns True if the float is an integer. x_y(y) Return x(y) where andSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.y_x(x) Return y(x) where andSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.- get_g(f)[source]¶
Return the function g(y) which, when integrated over
is the same as integrating the original function f(x) overSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log..System Message: WARNING/2 (x\in[x_0,\infty))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
- x_y(y)[source]¶
Return x(y) where
andSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log..System Message: WARNING/2 (x\in[x_0,\infty))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
- y_x(x)[source]¶
Return y(x) where
andSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log..System Message: WARNING/2 (x\in[x_0,\infty))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
- class mmf.math.integrate.integrate_1d.quadrature.ExpDecay(x, b, x0=None)[source]¶
Bases: mmf.math.integrate.integrate_1d.quadrature.Decay
Represents an exponential decay.
The function should behave as
System Message: WARNING/2 (\lim_{x\rightarrow \pm\infty} = a e^{-(b\abs{x})^{n}} )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.where this behaviour becomes dominant for
.System Message: WARNING/2 (\abs{x}>\abs{x_{0}}\sim b^{-1})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.Parameters : b : float
Length-scale: i.e. decay of form
System Message: WARNING/2 (\exp(-bx^n))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
x0 : float
Point beyond which decay is the dominant behaviour. This must have the same sign as the value. If not specified, then it is set to 1/b
Notes
The transformation used maps
to yin[0,1).System Message: WARNING/2 (\abs{x}\in [x_0,\infty))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (x &= x_{0}-b^{-1}\ln(1-y),\\ y &= 1 - e^{b(x_{0}-x)},\\ \int_{x_{0}}^{\infty}\d{x}\; f(x) &= \int_{0}^{1}\d{y}\; g(y),\\ g(y) &= \frac{f\Bigl(x(y)\Bigr)}{b(1-y)}. )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.Examples
>>> f = lambda p:3/np.sqrt(np.pi)/(np.exp((3.0*p)**2)) >>> a = ExpDecay(-np.inf, b=3) >>> b = ExpDecay(np.inf, b=3) >>> a.y_x(a.x_y(0.5)) 0.5... >>> a.y_x(-3.0) 0.0 >>> b.y_x(3.0) 0.0 >>> quad(f, -np.inf, 0) (0.49999999..., 2.2113658...e-09) >>> quad(a.get_g(f), a.y_x(-np.inf), a.y_x(0)) (0.5000000..., 1.177...e-09)
This is what integrate does automatically:
>>> integrate(f, a, 0) (0.5, 4.8318...e-13)
Methods
as_integer_ratio() -> (int, int) Returns a pair of integers, whose ratio is exactly equal to the original float and with a positive denominator. conjugate Returns self, the complex conjugate of any float. fromhex((string) -> float) Create a floating-point number from a hexadecimal string. get_g(f) Return the function g(y) which, when integrated over hex float.hex() -> string is_integer Returns True if the float is an integer. x_y(y) Return x(y) where andSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.y_x(x) Return y(x) where andSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.- get_g(f)[source]¶
Return the function g(y) which, when integrated over
is the same as integrating the original function f(x) overSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log..System Message: WARNING/2 (x\in[x_0,\infty))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
- x_y(y)[source]¶
Return x(y) where
andSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log..System Message: WARNING/2 (x\in[x_0,\infty))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
- y_x(x)[source]¶
Return y(x) where
andSystem Message: WARNING/2 (y \in [0,1))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log..System Message: WARNING/2 (x\in[x_0,\infty))
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
- class mmf.math.integrate.integrate_1d.quadrature.IntegrationError(mesg=None, res=None, err=None)¶
Bases: exceptions.Exception
Error during integration.
- __init__(mesg=None, res=None, err=None)¶
- mmf.math.integrate.integrate_1d.quadrature.get_weights(x, roots, cond=1)[source]¶
Return the list of weights wx for quadratures of increasing order with positive weights for the abscissa x give (y, wy) = roots(n) being n and weights for some other (Gaussian) quadrature.
Note
The formulation is based on the assumption that the weights and abscissa are quadratures for polynomials (not including the weight). The weight is implicit.
Parameters : x : array
Abscissa at which quadratures weights wx are desired.
roots : function
(y, wy) = roots(n) should be the abscissa and weights for a Gaussian quadrature. I.e., the roots of the n‘th orthogonal polynomial
. This should be computed to machine precision.System Message: WARNING/2 (P_n(y_i) = 0)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
cond : float
- The abscissa are chosen so that :math:`sum abs{w_i}/abs{sum
w_i}leq` cond. Must have 1 <= cond.
See also
- Uses
- func:get_weights_y.
Examples
>>> x = np.linspace(-1, 1, 102)[1:-1] >>> wx = get_weights(x, sp.special.orthogonal.p_roots) >>> print len(wx) 18 >>> np.allclose(np.dot(wx[-1], x**16), 2./17) True >>> np.allclose(np.dot(wx[-1], x**17), 0) True
- mmf.math.integrate.integrate_1d.quadrature.get_weights_y(x, y, wy)[source]¶
Return the weights wx for a quadrature of order at least n >= len(y) for the abscissa x give the n exact quadrature abscissa y and weights wy defining a quadrature of order 2n.
Parameters : x : array
Abscissa at which quadratures weights wx are desired.
y : array
Abscissa for the Gaussian quadrature. I.e., the roots of the n‘th orthogonal polynomial
. This should be computed to machine precision.System Message: WARNING/2 (P_n(y_i) = 0)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.
wy : array
Weights (Christoffel numbers) corresponding to the abscissa y for the Gaussian quadrature. These should be computed to machine precision.
Notes
Forms the matrix
System Message: WARNING/2 (E)
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (\mat{E}_{ij} = \prod_{\substack{ k=0\\ k\neq j}^{n-1}} \frac{x_i - y_k}{y_j - y_k}. )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.that interpolates from y to x using the Lagrange interpolation formula. The forms the QR factorization using row-sorting:
System Message: WARNING/2 (\mat{E} = P\cdot\mat{Q}\cdot\mat{R} )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.where
is square. The interpolated coefficients are:System Message: WARNING/2 (\mat{R})
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.System Message: WARNING/2 (w_x = P\cdot Q \cdot (R^{T})^{-1}\cdot w_y )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.We use sp.linalg.lu_solve() to solve
but this expects to have arguments R^T = L*U where L has unit diagonals, so we need to rescale R slightly.System Message: WARNING/2 (R^T x = w_y)
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>>> x = np.linspace(-1, 1, 102)[1:-1] >>> wx_ = [2] >>> n = 0 >>> while min(wx_) >= 0: ... # Find maximum order with positive weights. ... n += 1 ... y, wy = sp.special.orthogonal.p_roots(n) ... wx = wx_ ... wx_ = get_weights_y(x, y, wy) >>> n = n - 1 >>> print n 18 >>> np.allclose(np.dot(wx, x**16), 2./17) True >>> np.allclose(np.dot(wx, x**17), 0) True
- mmf.math.integrate.integrate_1d.quadrature.integrate(f, a, b, abs_tol=None, rel_tol=None, points=None, singular_points=None, **kwargs)[source]¶
Return (res, err) where res is the integral of the function f(x) from a to b and err is an estimate of the absolute error.
System Message: WARNING/2 (\int_{a}^{b}f(x)\d{x} )
latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1 9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian , estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) ! LaTeX Error: File `mmfmath.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.12 ^^M No pages of output. Transcript written on math.log.Parameters : a, b : float
Limits of integration. These can be finite or infinite.
abs_tol, rel_tol : float
Desired tolerance goals.
points : list of floats
Points where the integrand is significant i.e. where does the integrand change behaviour. If there is a sharp feature, then it should be located by these points (the adaptive routine might otherwise miss it.). These points will be explicitly included in the integral so they should not be singular points.
singular_points : list of floats
List of singular points of the integrand. These will be excluded from the integral but approached with caution.
- mmf.math.integrate.integrate_1d.quadrature.quad(f, a, b, *varargin, **kwargs)[source]¶
An improved version of integrate.quad that does some argument checking and deals with points properly.
Return (ans, err).
Examples
>>> def f(x): return 1./x**2 >>> (ans, err) = quad(f, 1, np.inf, points=[]) >>> abs(ans - 1.0) < err True >>> (ans, err) = quad(f, 1, np.inf, points=[3.0, 2.0]) >>> abs(ans - 1.0) < err True
- mmf.math.integrate.integrate_1d.quadrature.mquad(f, a, b, abs_tol=1e-12, verbosity=0, fa=None, fb=None, save_fx=False, res_dict=None, max_fcnt=10000, min_step_size=None, norm=<function <lambda> at 0x52b9500>, points=[])¶
Return (res, err) where res is the numerically evaluated integral using adaptive Simpson quadrature.
mquad tries to approximate the integral of function f from a to b to within an error of abs_tol using recursive adaptive Simpson quadrature. mquad allows the function y = f(x) to be array-valued. In the matrix valued case, the infinity norm of the matrix is used as it’s “absolute value”.
Parameters : f : function
Possibly array valued function to integrate. If this emits a NaN, then an AssertionError is raised to allow the user to optimize this check away (as it exists in the core of the loops)
a, b : float
Integration range (a, b)
fa, fb : float
f(a) and f(b) respectively (if already computed)
abs_tol : float
Approximate absolute tolerance on integral
verbosity : int
Display info if greater than zero. Shows the values of [fcnt a b-a Q] during the iteration.
save_fx : bool
If True, then save the abscissa and function values in res_dict.
res_dict : dict
Details are stored here. Pass a dictionary to access these. The dictionary will be modified.
max_fcnt : int
Maximum number of function evaluations.
min_step_size : float
Smallest step size limiting termination
norm : function
Function to compute the norm of the values of f(x). This allows the use to use matrices, vectors, etc. This norm is used to determine convergence.
points : list of floats
List of special points. The integral will be subdivided into these regions.
Notes
The basic idea here is to use a nested quadrature rule to assess the value of the integrand over a given integral, and then subdivide if tolerances are not met.
Based on “adaptsim” by Walter Gander. Ref: W. Gander and W. Gautschi, “Adaptive Quadrature Revisited”, 1998. http://www.inf.ethz.ch/personal/gander
Examples
Orthogonality of planewaves on [0, 2pi]
>>> def f(x): ... v = np.exp(1j*np.array([[1.0, 2.0, 3.0]])*x) ... return v.T.conj()*v/2.0/np.pi >>> ans = mquad(f, 0, 2*np.pi) >>> abs(ans - np.eye(ans.shape[0])).max() < _abs_tol True
>>> res_dict = {} >>> def f(x): return x**2 >>> ans = mquad(f, -2, 1, res_dict=res_dict, save_fx=True) >>> abs(ans - 3.0) < _abs_tol True >>> x = np.array([xy[0] for xy in res_dict['xy']]) >>> y = np.array([xy[1] for xy in res_dict['xy']]) >>> abs(y - f(x)).max() 0.0
Handling singularities.
>>> def f(x): return 1.0/np.sqrt(x) + 1.0/np.sqrt(1.0-x) >>> abs(mquad(f, 0, 1, abs_tol=1e-8) - 4.0) < 2e-7 True
Handling discontinuities.
>>> def f(x): ... if x < 0: ... return 0.0 ... else: ... return 1.0 >>> abs(mquad(f, -2.0, 1.0) - 1.0) < 1e-10 True
>>> def f(x): return 1./x >>> mquad(f, 1, np.inf) Traceback (most recent call last): ... ValueError: Infinite endpoints not supported.
- mmf.math.integrate.integrate_1d.quadrature.h_roots(n)[source]¶
Return the abscissa and weights for the Gaussian quadrature of order n in the Hermite polynomials.
Notes
We use the scipy routines for the roots, but our own custom formula for the weights because the scipy routine screws up these for large n.
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