polynomial.legendre.
legvander2d
给定度数的伪范德蒙矩阵。
返回度的伪范德蒙矩阵 deg 和采样点 (x, y) . 伪范德蒙矩阵的定义如下：
System Message: WARNING/2 (V […，（DEG〔1〕] + 1） *i + j] = L_i(x) * LYJ（Y），)
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) LaTeX Warning: Command \textellipsis invalid in math mode on input line 14. ! 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 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\end{split} ! Package inputenc Error: Unicode character 〔 (U+3014) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...i + j] = L_i(x) * LYJ（Y），\end{split} ! Package inputenc Error: Unicode character 〕 (U+3015) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\end{split} LaTeX Warning: Command \textellipsis invalid in math mode on input line 14. ! 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 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\end{split} ! Package inputenc Error: Unicode character 〔 (U+3014) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...i + j] = L_i(x) * LYJ（Y），\end{split} ! Package inputenc Error: Unicode character 〕 (U+3015) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\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 ...i + j] = L_i(x) * LYJ（Y），\end{split} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 460 bytes). Transcript written on math.log.
在哪里？ 0 <= i <= deg[0] 和 0 <= j <= deg[1] . 主要指标 V 索引点 (x, y) 最后一个索引编码勒让德多项式的度数。
如果 V = legvander2d(x, y, [xdeg, ydeg]) ，然后是 V 对应于二维系数数组的元素 c 形状（xdeg+1，ydeg+1）的顺序
V = legvander2d(x, y, [xdeg, ydeg])
System Message: WARNING/2 (C 00，C 01，C 02…，C_10，C_11，C_12…)
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+FF0C) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...C 02…，C_10，C_11，C_12…\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 ...C 02…，C_10，C_11，C_12…\end{split} LaTeX Warning: Command \textellipsis invalid in math mode on input line 14. ! 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 ...C 02…，C_10，C_11，C_12…\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 ...C 02…，C_10，C_11，C_12…\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 ...C 02…，C_10，C_11，C_12…\end{split} LaTeX Warning: Command \textellipsis invalid in math mode on input line 14. ! 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 ...C 02…，C_10，C_11，C_12…\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 ...C 02…，C_10，C_11，C_12…\end{split} LaTeX Warning: Command \textellipsis invalid in math mode on input line 14. ! 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 ...C 02…，C_10，C_11，C_12…\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 ...C 02…，C_10，C_11，C_12…\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 ...C 02…，C_10，C_11，C_12…\end{split} LaTeX Warning: Command \textellipsis invalid in math mode on input line 14. [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 396 bytes). Transcript written on math.log.
和 np.dot(V, c.flat) 和 legval2d(x, y, c) 在环形交叉路口也一样。这种等效性对于最小二乘拟合和大量相同度数和采样点的二维勒让德级数的计算都是有用的。
np.dot(V, c.flat)
legval2d(x, y, c)
点坐标数组，所有形状相同。根据元素是否复杂，数据类型将转换为float64或complex128。标量转换为一维数组。
窗体的最大度数列表 [x_deg, y_deg] .
返回矩阵的形状为 x.shape + (order,) 在哪里 . 数据类型将与转换后的类型相同 x 和 y .
x.shape + (order,)
参见
legvander
legvander3d
legval2d
legval3d
笔记
1.7.0 新版功能.