Some tools for forming polynomial or rational approximations of the inverse
of a function. polyapprox depends on mpmath.
Two of the functions in this library are inverse_taylor(f, x0, n)
and inverse_pade(f, x0, m, n).
inverse_taylor(f, x0, n) computes the Taylor polynomial coefficients of the
inverse of f.
Given a callable f, and a point x0, it finds the Taylor polynomial of
degree n of the inverse of f at x0.
If y0 = f(x0), and if the inverse of f is g, this function returns
the Taylor polynomial coefficients of g(y) at y0.
f'(x0) must be nonzero.
For example,
>>> from mpmath import mp
>>> mp.dps = 40
>>> from polyapprox import inverse_taylor
Compute the Taylor coefficients of the inverse of the sine function
sin(x) at x=1.
>>> inverse_taylor(mp.sin, 1, 5)
[mpf('1.0'),
mpf('1.850815717680925617911753241398650193470396'),
mpf('2.667464736243829370645086306803786566557799'),
mpf('8.745566949501434796799480049601499630239969'),
mpf('34.55691117453807764026147509020588920253199'),
mpf('152.9343377104818039879748855586655382173672')]
Compare that to computing the Taylor polynomial coefficients of the arcsine function directly:
>>> mp.taylor(mp.asin, mp.sin(1), 5)
[mpf('1.0'),
mpf('1.850815717680925617911753241398650193470396'),
mpf('2.667464736243829370645086306803786566557799'),
mpf('8.745566949501434796799480049601499630240153'),
mpf('34.55691117453807764026147509020588920253199'),
mpf('152.9343377104818039879748855586655382173702')]
inverse_pade(f, x0, m, n) Padé approximant coefficients of the inverse of f.
Given a callable f, and a point x0, it finds the Padé approximant of degree
(m, n) of the inverse of f at x0.
If y0 = f(x0), and if the inverse of f is g, this function returns
the Padé approximant coefficients of g(y) at y0.
f'(x0) must be nonzero.
For example,
>>> from mpmath import mp
>>> mp.dps = 40
>>> from polyapprox import inverse_pade
Compute the Padé approximant to the inverse of sin(x) at x=1.
>>> inverse_pade(mp.sin, 1, 5, 4)
([mpf('1.0'),
mpf('-5.428836087225345782152614868037223199487785'),
mpf('-14.59025586448337482707922792297134121701713'),
mpf('76.66727054306441691994858675947043862200347'),
mpf('20.92630843471146736348587129663693571301545'),
mpf('-91.95538065543221755259217919809770565490541')],
[mpf('1.0'),
mpf('-7.279651804906271400064368109435873392958164'),
mpf('-3.784426620962357975179374311522526358975659'),
mpf('94.34419434777145262733458338059694153109452'),
mpf('-114.4848137234397209897780633142520390436954')])
Compare that to computing the Padé approximant of the arcsine function directly.
>>> c = mp.taylor(mp.asin, mp.sin(1), 10)
>>> mp.pade(c, 5, 4)
([mpf('1.0'),
mpf('-5.428836087225345782152614868037223199022362'),
mpf('-14.59025586448337482707922792297134122127003'),
mpf('76.66727054306441691994858675947043863057723'),
mpf('20.92630843471146736348587129663693571903839'),
mpf('-91.95538065543221755259217919809770566742443')],
[mpf('1.0'),
mpf('-7.279651804906271400064368109435873392492793'),
mpf('-3.784426620962357975179374311522526364090111'),
mpf('94.34419434777145262733458338059694154789217'),
mpf('-114.4848137234397209897780633142520390591904')])