Padé Approximants via Matrix Elementary Row Operations: Algorithm, Error Bounds, and Optimality
Year: 2026
Type: article
Abstract: We develop a matrix elementary row operations (ERO) framework for computing Padé approximants and analyzing their accuracy, extending an integer augmented-matrix device to the whole Padé table. The [ m / n ] approximant of a formal power series is obtained by row-reducing an augmented Padé–Sylvester matrix, and a determinant criterion certifies nondegeneracy. For the exponential learning function L ( t ) = 1 − e − a t , a transfer lemma proves that the lower triangle m ≥ n of the Padé table is nondegenerate, and an exact Beta-integral remainder gives a uniform error bound whose diagonal constant K n , n = 1 / ( 2 n n ) is independent of a . The Padé-to-Taylor error ratio has an exact quotient-of-integrals representation; it decreases strictly, stays below K n , n , and behaves like ( 2 n ) ! / ( a t ) 2 n as t → ∞ . We also give a shape analysis of the Padé curve, an order-selection theorem, and a numerical stability study. For optimal time allocation, the Padé surrogate recovers the exact optimum within a certified bound.
Authors: Wannatida Yonwilad, Ekkawit Lampai
Institutions: Kalasin University, Rajabhat Maha Sarakham University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7225707887