Generate a monic polynomial with given roots.
Return the coefficients of the polynomial
System Message: WARNING/2 (p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
)
latex exited with error
[stdout]
This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2011/06/27>
Babel <3.9f> and hyphenation patterns for 78 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.131 \endinput
^^M
No pages of output.
Transcript written on math.log.
where the r_n are the roots specified in roots. If a zero has
multiplicity n, then it must appear in roots n times. For instance,
if 2 is a root of multiplicity three and 3 is a root of multiplicity 2,
then roots looks something like [2, 2, 2, 3, 3]. The roots can appear
in any order.
If the returned coefficients are c, then
System Message: WARNING/2 (p(x) = c_0 + c_1 * x + ... + x^n
)
latex exited with error
[stdout]
This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2011/06/27>
Babel <3.9f> and hyphenation patterns for 78 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.131 \endinput
^^M
No pages of output.
Transcript written on math.log.
The coefficient of the last term is 1 for monic polynomials in this
form.
Parameters: | roots : array_like
Sequence containing the roots.
|
Returns: | out : ndarray
1-D array of the polynomial’s coefficients If all the roots are
real, then out is also real, otherwise it is complex. (see
Examples below).
|
See also
chebfromroots, legfromroots, lagfromroots, hermfromroots, hermefromroots
Notes
The coefficients are determined by multiplying together linear factors
of the form (x - r_i), i.e.
System Message: WARNING/2 (p(x) = (x - r_0) (x - r_1) ... (x - r_n)
)
latex exited with error
[stdout]
This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2011/06/27>
Babel <3.9f> and hyphenation patterns for 78 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.131 \endinput
^^M
No pages of output.
Transcript written on math.log.
where n == len(roots) - 1; note that this implies that 1 is always
returned for
System Message: WARNING/2 (a_n)
latex exited with error
[stdout]
This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux)
restricted \write18 enabled.
entering extended mode
(./math.tex
LaTeX2e <2011/06/27>
Babel <3.9f> and hyphenation patterns for 78 languages loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
! LaTeX Error: File `utf8x.def’ not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: def)
Enter file name:
! Emergency stop.
<read *>
l.131 \endinput
^^M
No pages of output.
Transcript written on math.log.
.
Examples
>>> from numpy.polynomial import polynomial as P
>>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x
array([ 0., -1., 0., 1.])
>>> j = complex(0,1)
>>> P.polyfromroots((-j,j)) # complex returned, though values are real
array([ 1.+0.j, 0.+0.j, 1.+0.j])