An Algorithm of Ordered Schur Factorization For Real Nonsymmetric Matrix

In this paper, we present an algorithm for computing a Schur factorization of a real nonsymmetric matrix with ordered diagonal blocks such that upper left blocks contains the largest magnitude eigenvalues. Especially in case of multiple eigenvalues, when matrix is non diagonalizable, we construct an invariant subspaces with few additional tricks which are heuristic and numerical results shows the stability and accuracy of the algorithm.

Authors:

• Lokendra K. Balyan

Keywords:

• Schur Factorization
• Eigenvalues of nonsymmetric matrix
• Orthoganal matrix.

References:

[1] Z. Bai, J.W. Demmel, On swapping diagonal blocks in real Schur form,
Lin. Alg. appli., Vol. 186, P. 73-95, 1993.
[2] P.V. Dooren, A generalized eigenvalue approch for solving Riccati equations,
SIAM Sci. and Stat. Comp, Vol. 2, P. 121-135, 1981.
[3] D.L. Harrer, A block Arnoldi method for large nonsymmetric eigenvalue
problem, Centre for Mathematics and its applications, School of Mathematical
Science, Australian National University, Canberra, ACT 0200.
[4] Z. Bai, J Demmel, J. Dongarra, A. Ruhe, H. Vorst, Templates for the
solustion of algebraic eigenvalue problems, A practical guide, SIAM,
Philadelphia, (2000).
[5] J.J. Du Croz, S.J. Hamnarling, Eigenvalue Problems, in Numerical
algorithm, Oxford Uni. Press, Newyork, P. 29-66.
[6] G.H. Golub and F. Van Loan, Matrix Computations (3rd Edition) Jhons
Hopkins Uni. Press, Baltimore 1996.
[7] B.S. Garbow, J.M. Boyle, J.J. Dongarra, C.B. Moler, Matrix Eigensystem
Routines - EISPACK Guide Extension, Lecture Notes in Computer
Science, Springer-Verlag, Vol. 51, 1977.
[8] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra,
J. Du Croz, A. Greenbaum, S. Hammarling, A. Mckenney, D. Sorensen,
LAPACK User-s Guide. SIAM, Philadelphia, 3rd Edition, 1999.