Integrate a polynomial.
Returns the polynomial cs, integrated m times from lbnd to x.
At each iteration the resulting series is multiplied by scl and
an integration constant, k, is added. The scaling factor is for use
in a linear change of variable. (“Buyer beware”: note that, depending
on what one is doing, one may want scl to be the reciprocal of what
one might expect; for more information, see the Notes section below.)
The argument cs is a sequence of coefficients, from lowest order
term to highest, e.g., [1,2,3] represents the polynomial
1 + 2*x + 3*x**2.
| Parameters : | cs : array_like
1-d array of polynomial coefficients, ordered from low to high.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at zero
is the first value in the list, the value of the second integral
at zero is the second value, etc. If k == [] (the default),
all constants are set to zero. If m == 1, a single scalar can
be given instead of a list.
lbnd : scalar, optional
The lower bound of the integral. (Default: 0)
scl : scalar, optional
Following each integration the result is multiplied by scl
before the integration constant is added. (Default: 1)
|
|---|
| Returns : | S : ndarray
Coefficients of the integral.
|
|---|
| Raises : | ValueError :
|
|---|
Notes
Note that the result of each integration is multiplied by scl.
Why is this important to note? Say one is making a linear change of
variable
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kurmanji, lao, latin, latvian, lithuanian, mongolian, mongolianlmc, bokmal, nyn
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in an integral relative to x. Then
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This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/TeX Live for SUSE Linux)
restricted \write18 enabled.
entering extended mode
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LaTeX2e <2011/06/27>
Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, ge
rman-x-2011-07-01, ngerman-x-2011-07-01, afrikaans, ancientgreek, ibycus, arabi
c, armenian, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danis
h, dutch, ukenglish, usenglishmax, esperanto, estonian, ethiopic, farsi, finnis
h, french, friulan, galician, german, ngerman, swissgerman, monogreek, greek, h
ungarian, icelandic, assamese, bengali, gujarati, hindi, kannada, malayalam, ma
rathi, oriya, panjabi, tamil, telugu, indonesian, interlingua, irish, italian,
kurmanji, lao, latin, latvian, lithuanian, mongolian, mongolianlmc, bokmal, nyn
orsk, polish, portuguese, romanian, romansh, russian, sanskrit, serbian, serbia
nc, slovak, slovenian, spanish, swedish, turkish, turkmen, ukrainian, uppersorb
ian, welsh, 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
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, so one will need to set scl equal to
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This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/TeX Live for SUSE Linux)
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entering extended mode
(./math.tex
LaTeX2e <2011/06/27>
Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, ge
rman-x-2011-07-01, ngerman-x-2011-07-01, afrikaans, ancientgreek, ibycus, arabi
c, armenian, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danis
h, dutch, ukenglish, usenglishmax, esperanto, estonian, ethiopic, farsi, finnis
h, french, friulan, galician, german, ngerman, swissgerman, monogreek, greek, h
ungarian, icelandic, assamese, bengali, gujarati, hindi, kannada, malayalam, ma
rathi, oriya, panjabi, tamil, telugu, indonesian, interlingua, irish, italian,
kurmanji, lao, latin, latvian, lithuanian, mongolian, mongolianlmc, bokmal, nyn
orsk, polish, portuguese, romanian, romansh, russian, sanskrit, serbian, serbia
nc, slovak, slovenian, spanish, swedish, turkish, turkmen, ukrainian, uppersorb
ian, welsh, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
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(/usr/share/texmf/tex/latex/base/size12.clo))
(/usr/share/texmf/tex/latex/base/inputenc.sty
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- perhaps not what one would have first thought.
Examples
>>> from numpy import polynomial as P
>>> cs = (1,2,3)
>>> P.polyint(cs) # should return array([0, 1, 1, 1])
array([ 0., 1., 1., 1.])
>>> P.polyint(cs,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20])
array([ 0. , 0. , 0. , 0.16666667, 0.08333333,
0.05 ])
>>> P.polyint(cs,k=3) # should return array([3, 1, 1, 1])
array([ 3., 1., 1., 1.])
>>> P.polyint(cs,lbnd=-2) # should return array([6, 1, 1, 1])
array([ 6., 1., 1., 1.])
>>> P.polyint(cs,scl=-2) # should return array([0, -2, -2, -2])
array([ 0., -2., -2., -2.])
