numapprox - Maple Programming Help

numapprox

 Calling Sequence hermite_pade([f1, f2,..., fn], x, N) hermite_pade([f1, f2,..., fn], x, [d1, d2,..., dn]) hermite_pade([f1, f2,..., fn], x=a, N) hermite_pade([f1, f2,..., fn], x=a, [d1, d2,..., dn])

Parameters

 f1, ..., fn - expressions representing the functions to be approximated x - the variable appearing in the f_i's a - the point about which to expand in a series N - non-negative integer d1, ..., dn - degree bounds

Description

 • The function hermite_pade computes a Hermite-Pade approximation of degree (d1,..., dn) for the functions f1,..., fn with respect to the variable x. When the degrees are not specified, but rather the order N is given, then an approximation of minimal degree is computed.
 • Specifically, f1,..., fn are expanded in Taylor series about the point $x=a$ (if a is not specified then the expansion is about the point $x=0$), to order $\mathrm{d1}+...+\mathrm{dn}+n-1$, and then the Hermite-Pade rational approximation is computed.
 • The (d1,..., dn) Hermite-Pade approximation is defined to be the list of polynomials $\mathrm{p1}\left(x\right),...,\mathrm{pn}\left(x\right)$ with $\mathrm{deg}\left(p[i]\left(x\right)\right)\le {d}_{i}$ such that the Taylor series expansion of $\mathrm{p1}\left(x\right)\mathrm{f1}\left(x\right)+...+\mathrm{pn}\left(x\right)\mathrm{fn}\left(x\right)$ has maximal valuation at $x=a$.
 • Various levels of user information will be displayed during the computation if infolevel[hermite_pade] is assigned values between 1 and 3.
 • This code is based on a procedure by H. Derksen in previous versions of the share library.
 • The command with(numapprox,hermite_pade) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{numapprox}\right):$
 > $\mathrm{hermite_pade}\left(\left[\mathrm{sin}\left(x\right),\mathrm{cos}\left(x\right),{ⅇ}^{x}\right],x=0,\left[3,2,5\right]\right)$
 $\left[{5}{}{{x}}^{{3}}{+}{45}{}{{x}}^{{2}}{-}{255}{}{x}{-}{1275}{,}{75}{}{{x}}^{{2}}{+}{495}{}{x}{+}{120}{,}{{x}}^{{5}}{-}{20}{}{{x}}^{{4}}{+}{160}{}{{x}}^{{3}}{-}{600}{}{{x}}^{{2}}{+}{900}{}{x}{-}{120}\right]$ (1)
 > $\mathrm{hermite_pade}\left(\left[\mathrm{sin}\left(x\right),\mathrm{cos}\left(x\right)\right],x=\mathrm{π},7\right)$
 $\left[{-}{6}{}{\left({x}{-}{\mathrm{π}}\right)}^{{2}}{+}{15}{,}{\left({x}{-}{\mathrm{π}}\right)}^{{3}}{-}{15}{}{x}{+}{15}{}{\mathrm{π}}\right]$ (2)
 > $\mathrm{ff}≔\left[\mathrm{cos}\left(2x\right)\left(x+1\right)+3,{\mathrm{cos}\left(x\right)}^{2}+x\mathrm{cos}\left(x\right)+1,\mathrm{cos}\left(2x\right)+1,\mathrm{cos}\left(x\right)\right]:$
 > $\mathrm{gg}≔\mathrm{hermite_pade}\left(\mathrm{ff},x=0,20\right)$
 ${\mathrm{gg}}{:=}\left[{2}{,}{2}{}{x}{-}{4}{,}{-}{3}{}{x}{,}{-}{2}{}{{x}}^{{2}}{+}{4}{}{x}\right]$ (3)
 > $\mathrm{simplify}\left({\mathrm{ff}}_{1}{\mathrm{gg}}_{1}+{\mathrm{ff}}_{2}{\mathrm{gg}}_{2}+{\mathrm{ff}}_{3}{\mathrm{gg}}_{3}+{\mathrm{ff}}_{4}{\mathrm{gg}}_{4}\right)$
 ${0}$ (4)

References

 Beckermann, B., and Labahn, G. "A uniform approach for Hermite Pade and simultaneous Pade approximants and their matrix-type generalizations." Numerical Algorithms, Vol. 3, (1992): 45-54.
 Beckermann, B., and Labahn, G. "A uniform approach for the fast computation of matrix-type Pade approximants." SIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 3, (1994): 804-823.
 Derksen,H. An algorithm to compute generalized Pade-Hermite forms. 1994. Available at http://www.math.lsa.umich.edu/~hderksen/preprints/pade.dvi.