Computing the polar decomposition with applications
SIAM Journal on Scientific and Statistical Computing
Topics in matrix analysis
Newton's method for the matrix square root
Mathematics of Computation
On scaling Newton's method for polar decomposition and the matrix sign function
SIAM Journal on Matrix Analysis and Applications
Compututational Techniques for Real Logarithms of Matrices
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
Approximating the Logarithm of a Matrix to Specified Accuracy
SIAM Journal on Matrix Analysis and Applications
Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices
Journal of Computational and Applied Mathematics
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A real square matrix A is P-orthogonal if AT 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 PT = P-1, P2 = ±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.