Computing the square root and logarithm of a real P-orthogonal matrix

Abstract

A real square matrix A is P-orthogonal if A T PA= P; P is a fixed real nonsingular matrix, but most of the results in this work require that it is symmetric positive definite or P T =P −1, P 2=±I . The class of P-orthogonal matrices includes, for instance, orthogonal and symplectic matrices as particular cases. We present an efficient iterative method for computing the P-orthogonal factor in the generalized polar decomposition, which generalizes the well-known Newton's method for the standard polar decomposition. A connection between Newton's method for the matrix square root and polar iterates brings out a new iterative method for computing the principal square root of a P-orthogonal matrix. One important feature of this method is that, when P is symmetric positive definite, it allows us to restore the P-orthogonal property of the exact square root by computing the nearest P-orthogonal matrix. We also analyse the problem of finding the nearest P-symmetric and P-skew-symmetric matrices. New bounds and new estimates for the Padé error of the matrix logarithm are given in order to improve the existing Briggs–Padé algorithms and adapt them to P-orthogonal matrices. Special attention will be paid to the orthogonal case.