Inverse Iteration for Large Defective Eigensystems

Abstract

The solutions of large eigensystems having cluster, multiple, or defective eigenvalues by inverse iteration using one trial vector at a time are of interest. The Rayleigh quotient is used as shift. New formulas for each particular case are derived for definite convergence. Six numerical examples, including three-multiple defective eigenvalues, are given.

Keywords

Inverse iteration defective eigensystems computational method

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References

Beattie, C. and Fox, D.W. , 1989, "Local criteria and containment for Rayleigh quotient iteration," SIAM Journal on Matrix Analysis and Applications 10, 80-93.

Collar, A.R. and Simpson, A. , 1987, Matrices and Engineering Dynamics, Ellis Horwood, Chichester, UK.

Golub, G.H. and Wilkinson, J.H. , 1976, "Ill-conditioned eigensystems and the computation of the Jordan canonical form," SIAM Review 18, 578-619.

Leung, A.Y.T. , 1987, "Inverse iteration for damped natural vibration ," Journal of Sound and Vibration 118, 193-198.

Leung, A.Y.T. , 1988, "Inverse iteration for the quadratic eigenvalue problem ," Journal of Sound and Vibration 124, 249-267.

Parlett, B.N. , 1980, The Symmetric Eigenvalue Problem, Prentice Hall, Englewood Cliffs, NJ.

Reddy, J.N. , 1986, Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill, New York .

Szyld, D.B. , 1988, "Criteria for combining inverse and Rayleigh quotient iteration," SIAM Journal on Numerical Analysis 25, 1369-1375.

Watkins, D.S. , 1991, Fundamentals of Matrix Computations, John Wiley, New York.

10.

Wilkinson, J.H. , 1965, Algebraic Eigenvalue Problem, Oxford University Press, Oxford, UK.

11.

Wilkinson, J. H, and Reinsch, C. , 1971, Handbookfor Automatic Computation: Vol. 2. Linear Algebra , Springer- Verlag, Berlin .