mnPadéApproximation

mnPadéApproximation refers to a Padé approximation of order m/n, seeking a rational function P_m(z)/Q_n(z) with deg P ≤ m and deg Q ≤ n that best matches a target function near z = 0. The coefficients are chosen so that the Taylor expansion of P_m(z)/Q_n(z) agrees with the series of the target function through the term z^{m+n}, i.e., as many initial coefficients as possible.

With a function given by f(z) = sum_{k≥0} a_k z^k, the [m/n] Padé approximant satisfies Q_n(z) f(z) − P_m(z) = O(z^{m+n+1}). A common algorithm fixes Q_0 = 1 and solves for the remaining q_1,...,q_n from a linear Toeplitz system derived from the coefficients a_k, after which P_m is computed by convolution of the a_k with the q's. The resulting rational function can provide accurate approximations beyond the original radius of convergence and can reveal poles of the function.

Applications include analytic continuation, evaluation of special functions, and model order reduction in control and signal processing. mnPadéApproximation extends to vector- or matrix-valued functions and, in multivariate settings, to degree vectors (m1, m2, ...).

Limitations include possible non-uniqueness in degenerate cases, sensitivity to noisy coefficients, and the emergence of spurious poles. The choice of m and n significantly affects accuracy and stability; heuristic or problem-specific criteria are used to select them. Related topics include Padé tables, continued fractions, and other rational approximation methods.

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