numpy.
poly
Find the coefficients of a polynomial with the given sequence of roots.
Returns the coefficients of the polynomial whose leading coefficient is one for the given sequence of zeros (multiple roots must be included in the sequence as many times as their multiplicity; see Examples). A square matrix (or array, which will be treated as a matrix) can also be given, in which case the coefficients of the characteristic polynomial of the matrix are returned.
A sequence of polynomial roots, or a square array or matrix object.
1D array of polynomial coefficients from highest to lowest degree:
c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N] where c[0] always equals 1.
c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]
If input is the wrong shape (the input must be a 1-D or square 2-D array).
See also
polyval
Compute polynomial values.
roots
Return the roots of a polynomial.
polyfit
Least squares polynomial fit.
poly1d
A one-dimensional polynomial class.
Notes
Specifying the roots of a polynomial still leaves one degree of freedom, typically represented by an undetermined leading coefficient. [1] In the case of this function, that coefficient - the first one in the returned array - is always taken as one. (If for some reason you have one other point, the only automatic way presently to leverage that information is to use polyfit.)
The characteristic polynomial,
System Message: WARNING/2 (p_a(t))
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System Message: WARNING/2 (p_a(t) = \mathrm{det}(t\, \mathbf{I} - \mathbf{A})) latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.18> and hyphenation patterns for 3 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h 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/base/utf8.def (/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu) (/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu) (/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu))) (/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)) ! LaTeX Error: File `anyfontsize.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.8 \usepackage {bm}^^M No pages of output. Transcript written on math.log. ,
System Message: WARNING/2 (p_a(t) = \mathrm{det}(t\, \mathbf{I} - \mathbf{A}))
where I is the n-by-n identity matrix. [2]
References
M. Sullivan and M. Sullivan, III, “Algebra and Trignometry, Enhanced With Graphing Utilities,” Prentice-Hall, pg. 318, 1996.
G. Strang, “Linear Algebra and Its Applications, 2nd Edition,” Academic Press, pg. 182, 1980.
Examples
Given a sequence of a polynomial’s zeros:
>>> np.poly((0, 0, 0)) # Multiple root example array([1., 0., 0., 0.])
The line above represents z**3 + 0*z**2 + 0*z + 0.
>>> np.poly((-1./2, 0, 1./2)) array([ 1. , 0. , -0.25, 0. ])
The line above represents z**3 - z/4
>>> np.poly((np.random.random(1)[0], 0, np.random.random(1)[0])) array([ 1. , -0.77086955, 0.08618131, 0. ]) # random
Given a square array object:
>>> P = np.array([[0, 1./3], [-1./2, 0]]) >>> np.poly(P) array([1. , 0. , 0.16666667])
Note how in all cases the leading coefficient is always 1.
