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.

$\mathrm{with}\left(\mathrm{numapprox}\right)\:$

$\mathrm{hermite\_pade}\left(\left[\sin \left(x\right)\,\cos \left(x\right)\,\exp \left(x\right)\right]\,x=0\,\left[3\,2\,5\right]\right)$

$\left[5x^3+45x^2-255x-1275\,75x^2+495x+120\,x^5-20x^4+160x^3-600x^2+900x-120\right]$

$\mathrm{hermite\_pade}\left(\left[\sin \left(x\right)\,\cos \left(x\right)\right]\,x=\mathrm{\pi }\,7\right)$

$\left[-6{\left(x-\mathrm{\pi }\right)}^2+15\,{\left(x-\mathrm{\pi }\right)}^3-15x+15\mathrm{\pi }\right]$

$\mathrm{ff}≔\left[\cos \left(2x\right)\left(x+1\right)+3\,{\cos \left(x\right)}^2+x\cos \left(x\right)+1\,\cos \left(2x\right)+1\,\cos \left(x\right)\right]\:$

$\mathrm{gg}≔\mathrm{hermite\_pade}\left(\mathrm{ff}\,x=0\,20\right)$

$\mathrm{gg}≔\left[2\,2x-4\,-3x\,-2x^2+4x\right]$

$\mathrm{simplify}\left(\mathrm{ff}\left[1\right]\mathrm{gg}\left[1\right]+\mathrm{ff}\left[2\right]\mathrm{gg}\left[2\right]+\mathrm{ff}\left[3\right]\mathrm{gg}\left[3\right]+\mathrm{ff}\left[4\right]\mathrm{gg}\left[4\right]\right)$

$0$

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.