polynomial.laguerre.
lagfromroots
生成具有给定根的拉盖尔级数。
函数返回多项式的系数
System Message: WARNING/2 (P(x)=(x-r_0) * (x - r_1) * …*(X-RYN),)
latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2018-12-01> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2018/09/03 v1.4i Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} LaTeX Warning: Command \textellipsis invalid in math mode on input line 14. ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character , (U+FF0C) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} LaTeX Warning: Command \textellipsis invalid in math mode on input line 14. ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} ! Package inputenc Error: Unicode character , (U+FF0C) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ... (x - r_1) * …*(X-RYN),\end{split} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 468 bytes). Transcript written on math.log.
以拉盖尔形式,其中 r_n 是否在中指定了根 roots . 如果一个零有多重性n,那么它必须出现在 roots n次。例如,如果2是重数3的根,而3是重数2的根,那么 roots 看起来像 [2, 2, 2,3, 3] . 根可以以任何顺序出现。
如果返回系数为 c 然后
System Message: WARNING/2 (P(x)=C_0+C_1 * L_1(x) + ... + c_n * Lnn(x))
latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2018-12-01> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2018/09/03 v1.4i Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ..._1(x) + ... + c_n * Lnn(x)\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ..._1(x) + ... + c_n * Lnn(x)\end{split} ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ..._1(x) + ... + c_n * Lnn(x)\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ..._1(x) + ... + c_n * Lnn(x)\end{split} ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ..._1(x) + ... + c_n * Lnn(x)\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ..._1(x) + ... + c_n * Lnn(x)\end{split} ! Package inputenc Error: Unicode character ( (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ..._1(x) + ... + c_n * Lnn(x)\end{split} ! Package inputenc Error: Unicode character ) (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ..._1(x) + ... + c_n * Lnn(x)\end{split} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 504 bytes). Transcript written on math.log.
对于拉盖尔形式的Monic多项式,最后一项的系数一般不是1。
包含根的序列。
一维系数数组。如果所有的根都是真的那么 out 是一个实数数组,如果某些根是复杂的,则 out 即使结果中的所有系数都是真实的,也是复杂的(参见下面的示例)。
参见
numpy.polynomial.polynomial.polyfromroots
numpy.polynomial.legendre.legfromroots
numpy.polynomial.chebyshev.chebfromroots
numpy.polynomial.hermite.hermfromroots
numpy.polynomial.hermite_e.hermefromroots
实例
>>> from numpy.polynomial.laguerre import lagfromroots, lagval >>> coef = lagfromroots((-1, 0, 1)) >>> lagval((-1, 0, 1), coef) array([0., 0., 0.]) >>> coef = lagfromroots((-1j, 1j)) >>> lagval((-1j, 1j), coef) array([0.+0.j, 0.+0.j])