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G Rational Padé Approximations

  The (p,q) Padé approximant to the matrix exponential eX is, by definition, the unique (p,q) rational function [100]
which matches the Taylor series expansion of eX through terms to the power p+q. Its coefficients are therefore determined by solving the algebraic equations
The result is
Choosing p=q one obtains the diagonal Padé approximation. This choice is to prefer, because it yields a higher order approximation with the same amount of computation.
\equiv \sum_{j=0}^p c_j\boldsymbol{X}^j,
where the coefficients cj can be conveniently constructed by means of the recursion
\begin{gather}c_0=1,\qquad c_j=c_{j-1}\frac{p+1-j}{j(2p+1-j)}.
The computation of the polynomials are best done with a Horner scheme. C. Moler and C. Van Loan [87] showed that if $\Vert\boldsymbol{X}\Vert _2/2^m\leq 1/2$, then
0.34\cdot 10^{-15} (p=6)

Christoph Wasshuber