Algebraic Computations of Scaled Padé Fractions

Abstract

Two companion algorithms are developed for constructing Pade fractions along an off-diagonal path of the Padé table for a function ${{ - A(z)} / {B(z)}}$, where $A(z)$ and $B(z)$ are formal power series over a field. One of the algorithms computes the first n Padé fractions along the off-diagonal in time $O(n^2 )$. When $A(z)$ and $B(z)$ are finite power series (i.e., polynomials), it is shown that the algorithm is equivalent to Euclid’s extended algorithm for computing greatest common divisors. The other algorithm, a generalization of the first, proceeds along the off-diagonal in quadratic steps, and is of complexity $O(n\log ^2 n)$. When $A(z)$ and $B(z)$ are polynomials, the second algorithm becomes a fast Euclid’s extended algorithm for computing greatest common divisors. The algorithm is of the same complexity as other fast greatest common divisor methods, but its iterative nature provides a practical advantage during implementation. The algorithms may also be used for computing Padé fractions along an anti-diagonal path of the Padé table. The fast algorithm is of the same complexity as other fast algorithms for anti-diagonal computations. However, it has the advantage of being able to determine easily any specific Padé fraction along the anti-diagonal.