The Pad\'e iterations for the matrix sign function and their reciprocals are optimal

TL;DR

This paper proves that among rational iterations converging to the matrix sign function, Padé iterations and their reciprocals are optimal in minimizing the sum of numerator and denominator degrees.

Contribution

It establishes the uniqueness and optimality of Padé iterations and their reciprocals for the matrix sign function among locally convergent rational methods.

Findings

Padé iterations are uniquely optimal among rational methods.

Reciprocals of Padé iterations share this optimality.

The result applies to locally convergent rational iterations with order s>1.

Abstract

It is proved that among the rational iterations locally converging with order s>1 to the sign function, the Pad\'e iterations and their reciprocals are the unique rationals with the lowest sum of the degrees of numerator and denominator.