Error analysis of Padé iterations for computing matrix invariant subspaces

Abstract

The method of Padématrix iteration is commonly used for computing matrix sign function and invariant subspaces of a real or complex matrix. In this paper, a detailed rounding error analysis is given for two classical schemes of the Padé matrix iteration, using basic matrix floating point arithmetics. Error estimations of computing invariant subspaces by the Padé sign iteration are also provided. Numerical experiments are given to show the numerical behaviors of the Pad´e iterations and the corresponding subspace computation.

Keywords

Invariant subspace / matrix sign function / rounding error / Padéiteration / subspace approximation

References

[1] Bamberger A, Engquist B, Halpern L, Joly P. Higher order paraxial wave equation approximations in heterogeneous media. SIAM J Appl Math, 1988, 48(1): 129-154

[2] Benner P, Quintana-Ort E S. Solving stable generalized Lyapunov equations with the matrix sign function. Numerical Algorithms, 1999, 20: 75-100

[3] Byers R, He C Y, Mehrmann V. On the matrix sign function method for the computation of invariant subspaces. Proceedings of the 1996 IEEE International Symposium on Computer-Aided Control System Design, Dearborn, MI, September 15-18, 1996

[4] Demman E, Beavers N. The matrix sign function and computation in systems. Appl Math Comput, 1976, 2: 63-94

[5] Golub G H, Van Loan C F. Matrix Computations. 3rd ed. Baltimore: Johns Hopkins University Press, 1996

[6] Higham N J. Stable iterations for the matrix square root. Numerical Algorithms, 1997, 15: 227-242

[7] Higham N J. Accuracy and Stability of Numerical Algorithms. 2nd ed. Phil: SIAM, 2002

[8] Kenney C, Laub A J. Rational iterative methods for the matrix sign function. SIAM J Matrix Anal Appl, 1991, 12(2): 273-291

[9] Kenney C, Laub A J. The matrix sign function. IEEE Trans Automat Control, 1995, 40(8): 1330-1348

[10] Roberts J D. Linear model reduction and solution of the algebraic Riccati equation. Inter J Control, 1980, 32: 677-687 (Reprint of Technical Report No TR-13, CUED/B-Control, Engineering Department, Cambridge University, 1971)

[11] Zhang Z Y, Zha H Y. Structure and perturbation analysis of truncated SVD for column-partitioned matrices. SIAM J Matrix Anal Appl, 2001, 22(4): 1245-1262

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