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phipade

phipade - Evaluate phi functions using (6,6)-Pad� approximations. SYNOPSIS: phi_k = phipade(z, k); [phi_1, phi_2, ..., phi_k] = phipade(z, k); DESCRIPTION: This function evaluates phi functions needed in exponential integrators using diagonal Pad� approximants (currently (6,6)-Pad�). We define the phi functions according to the integral representation \phi_k(z) = \frac{1}{(k - 1)!} \int_0^1 e^{z (1-x)} x^{k-1} dx for k=1, 2, ... PARAMETERS: z - Evaluation point. Assumed to be one of - 1D vector, treated as the main diagonal of a diagonal matrix - sparse diagonal matrix - full or sparse matrix k - Which phi function(s) to evaluate. Index (integer) of the (highest) phi function needed. RETURNS: phi_k = \phi_k(z) [phi_1, phi_2, ..., phi_k] = DEAL(\phi_1(z), \phi_2(z), ..., \phi_k(z)) NOTES: When computing more than one phi function, it is the caller's responsibility to provide enough output arguments to hold all of the \phi_k function values. For efficiency reasons, phipade caches recently computed function values. The caching behaviour is contingent on the WANTCACHE function and may be toggled on or off as needed.