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#

G Rational Padé Approximations

The (*p*,*q*) Padé approximant to the matrix exponential
*e*^{X} is,
by definition, the unique (*p*,*q*) rational function [100]

which matches the Taylor series expansion of
*e*^{X} 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.

where the coefficients *c*_{j} can be conveniently constructed by means of the
recursion

The computation of the polynomials are best done with a Horner scheme. C. Moler
and C. Van Loan [87] showed that if
,
then

*Christoph Wasshuber*