Generalized polar decomposition method (or briefly GPD method) has been introduced by Munthe-Kaas and Zanna to approximate the matrix exponential. In this paper, we investigate the numerical stability of that method with respect to roundoff propagation. The numerical GPD method includes two parts: splitting of a matrix $Z\in \mathfrak{g}$, a Lie algebra of matrices and computing $\exp(Z)\mathbf{v}$ for a vector $\mathbf{v}$. We show that the former is stable provided that $\|Z\|$ is not so large, while the latter is not stable in general except with some restrictions on the entries of the matrix Z and the vector $\mathbf{v}$.
References
N. J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR1368629 N. J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR1368629
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, Springer-Verlag, New York, 2nd ed., 1999. MR1723696 J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, Springer-Verlag, New York, 2nd ed., 1999. MR1723696
C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, SIAM Rev., 20 (1978), pp. 801–836. MR508383 10.1137/1020098 C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, SIAM Rev., 20 (1978), pp. 801–836. MR508383 10.1137/1020098
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, vol. 12 of Texts in Applied Mathematics, Springer-Verlag, New York, 2nd ed., 1993. MR1295246 J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, vol. 12 of Texts in Applied Mathematics, Springer-Verlag, New York, 2nd ed., 1993. MR1295246
A. Zanna and H. Z. Munthe-Kaas, Generalized polar decompositions for the approximation of the matrix exponential, SIAM J. Matrix Anal. Appl., 23 (2001/02), pp. 840–862. MR1896821 0999.65060 10.1137/S0895479800377551 A. Zanna and H. Z. Munthe-Kaas, Generalized polar decompositions for the approximation of the matrix exponential, SIAM J. Matrix Anal. Appl., 23 (2001/02), pp. 840–862. MR1896821 0999.65060 10.1137/S0895479800377551
