Computing the square root and logarithm of a real \(P\)-orthogonal matrix (Q1398682)

The authors consider \(P\)-orthogonal square matrices A such that \(A^TPA=P\) for a fixed real nonsingular matrix \(P\). But most of the results require that \(P\) be symmetric or \(P^T=P^{-1}\) and \(P^2=\pm I\). They present an efficient iterative method for computing the \(P\)-orthogonal factor in the generalized polar decomposition which generalizes Newton's method for polar decomposition. A new iterative method is given for computing the principal square root of a \(P\)-orthogonal matrix. When \(P\) is symmetric positive definite, this method restores the \(P\)-orthogonal properties of the exact square root by computing the nearest \(P\)-orthogonal matrix. New bounds and new estimates for the Padé error of the matrix logarithm improve the existing Briggs-Padé algorithms and adapt them to \(P\)-orthogonal matrices.