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Stability of the generalized polar decomposition method for the approximation of the matrix exponential. (English) Zbl 1216.65052

Summary: The generalized polar decomposition method (or briefly GPD method) has been introduced by H. Z. Munthe-Kaas and A. Zanna [SIAM J. Matrix Anal. Appl. 23, No. 3, 840–862 (2002; Zbl 0999.65060)] 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 \(\exp(Z)v\) for a vector \(v\). 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 \(v\).

MSC:

65G50 Roundoff error
65L07 Numerical investigation of stability of solutions to ordinary differential equations
65F30 Other matrix algorithms (MSC2010)

Citations:

Zbl 0999.65060
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