gfun[guessrelation] guess a polynomial-coefficient relation between series

gfun[hermite_pade] compute a Hermite-Pade approximant

gfun[approximantbasis] modular approximant basis

gfun[gfun_pade] rational approximant from coefficient data

gfun[pade2] historical compatibility entry point

Calling Sequence

guessrelation([s1,s2,...,sn], z, N)

guessrelation([s1,s2,...,sn], z, [d1,d2,...,dn])

guessrelation([s1,s2,...,sn], z=a, N)

guessrelation([s1,s2,...,sn], z=a, [d1,d2,...,dn])

hermite_pade([s1,s2,...,sn], z, N)

hermite_pade([s1,s2,...,sn], z, [d1,d2,...,dn])

hermite_pade([s1,s2,...,sn], z=a, N)

hermite_pade([s1,s2,...,sn], z=a, [d1,d2,...,dn])

approximantbasis([s1,s2,...,sn], z, N, modulo=p, <options>)

approximantbasis([s1,s2,...,sn], z=a, [d1,d2,...,dn], modulo=p, <options>)

gfun_pade(L, z)

pade2([s1,s2,...,sn], z, N)

pade2([s1,s2,...,sn], z, [d1,d2,...,dn])

pade2([s1,s2,...,sn], z=a, N)

pade2([s1,s2,...,sn], z=a, [d1,d2,...,dn])

Parameters

[s1,...,sn] - list of functions or power series; the series to be approximated z - name; variable in the functions or power series N - non-negative integer; order of approximation

[d1,d2,...,dn] - list of non-negative integers; degree bounds

a - algebraic expression; expansion point

p - prime; nonzero modulus required by approximantbasis

L - list; coefficients of a power series in ascending order

options - equations controlling mode, checkterms, and the column shift

Description

The function guessrelation finds one linear relation with polynomial coefficients between the supplied power series and returns the coefficients of that relation as a list, or FAIL if it finds none. It defaults to mode=guessing: the terms selected by checkterms are held back from fitting and used to validate the candidate relation.
hermite_pade performs the corresponding Hermite-Pade approximation and defaults to mode=approximant. In this mode all available terms are used to construct the relation. It is a fast alternative to numapprox[hermite_pade].
Both guessrelation and hermite_pade accept an explicit mode=guessing or mode=approximant option, which overrides their respective defaults.
pade2 is the historical compatibility entry point. It retains its existing default, mode=guessing. New code should normally use guessrelation for validated relation discovery or hermite_pade for approximation.
approximantbasis solves the corresponding basis problem over the prime field selected by modulo=p. Instead of choosing one relation, it returns a Matrix whose rows are polynomial-coefficient relations. In mode=approximant it returns the full square basis; in mode=guessing it returns the rows that satisfy the available series data.
Use approximantbasis when the whole modular relation space is needed. It currently requires a nonzero prime modulus. Its shift option (also accepted as columnshift or column_shift) supplies one integer column shift per input series.
gfun_pade is the lower-level rational case. Given the coefficient list L, it returns a rational function whose expansion agrees with those coefficients, or FAIL. For conversions that also consider different generating-function types, use gfun[listtoratpoly].
If either single-relation command returns [P1,...,Pn], then P1*s1+...+Pn*sn vanishes to the fitted precision and, in mode=guessing, also passes the requested held-out validation.
Use the specialized listto* and seriesto* commands when a recurrence, rational function, algebraic equation, or differential equation is wanted in its usual formatted form.
The guessing commands share relation-finding algorithms based mainly on Hermite-Pade approximation and matrix rational interpolation.
These commands are part of the gfun package. After with(gfun) they may be called by their short names; the long forms gfun[guessrelation](..), gfun[hermite_pade](..), gfun[approximantbasis](..), gfun[gfun_pade](..), and the historical gfun[pade2](..) are always available.

Examples

> with(gfun):

> H := hermite_pade([1, exp(z)], z=0, [2,2]);

\[[-z^{2}-6 z -12, z^{2}-6 z +12]\]

By default, hermite_pade uses all supplied terms to construct one approximant relation.

> A := approximantbasis([1, exp(z)], z=0, 5, modulo=1009);

\[\left[\begin{array}{cc} z^{3}+6 z^{2}+18 z +24 & 6 z +985 \\ 1008 z^{2}+1003 z +997 & z^{2}+1003 z +12 \end{array}\right]\]

> gfun_pade([1, 1, 1, 1, 1], z);

\[-\frac{1}{-1+z}\]

The first example is simply a Pad\303\251 approximant:

> S:=series(add(combinat[fibonacci](i)*z^i,i=0..10),z,10);

\[z +z^{2}+2 z^{3}+3 z^{4}+5 z^{5}+8 z^{6}+13 z^{7}+21 z^{8}+34 z^{9}+\mathrm{O}\! \left(z^{10}\right)\]

> guessrelation([1,S],z,10);

\[[z, z^{2}+z -1]\]

This means that [formula] coincides with the generating function of the Fibonacci numbers, at least up to order 10.

> series(-%[1]/%[2],z,10);

\[z +z^{2}+2 z^{3}+3 z^{4}+5 z^{5}+8 z^{6}+13 z^{7}+21 z^{8}+34 z^{9}+\mathrm{O}\! \left(z^{10}\right)\]

An example with powers of a series:

> S:=series(add(1/(i+1)*binomial(2*i,i)*z^i,i=0..10),z,10);

\[1+z +2 z^{2}+5 z^{3}+14 z^{4}+42 z^{5}+132 z^{6}+429 z^{7}+1430 z^{8}+4862 z^{9}+\mathrm{O}\! \left(z^{10}\right)\]

> guessrelation([1,S,S^2],z,10);

\[[1, -1, z]\]

This means that the Catalan numbers [formula]have a generating function that is a solution of the simple polynomial:

> add(%[i]*Y^(i-1),i=1..nops(%));

\[z \,Y^{2}-Y +1\]

Given a power series, one can look for various types of equations that it may satisfy. This is used extensively by gfun for linear differential equations (by taking a series S and its derivatives) or non-linear ones in guessode.