Abstract

. The purpose of this paper is two-fold: to analyse the behaviour of inverse iteration for computing a single eigenvector of a complex, square matrix; and to review Jim Wilkinson's contributions to the development of the method. In the process we derive several new results regarding the convergence of inverse iteration in exact arithmetic. In the case of normal matrices we show that residual norms decrease strictly monotonically. For eighty percent of the starting vectors a single iteration is enough. In the case of non-normal matrices, we show that the iterates converge asymptotically to an invariant subspace. However the residual norms may not converge. The growth in residual norms from one iteration to the next can exceed the departure of the matrix from normality. We present an example where the residual growth is exponential in the departure of the matrix from normality. We also explain the often significant regress of the residuals after the first iteration: it occurs when the no...

Citations

3200	Topics in Matrix Analysis - Horn, Johnson - 1994
1305	Perturbation theory for linear operators - Kato - 1985
640	The Symmetric Eigenvalue Problem - Parlett - 1998
348	Numerical Methods for Large Eigenvalue Problems - Saad - 2011
217	Fundamental algorithms - Knuth - 1973
94	Error and perturbation bounds for subspaces associated with certain eigenvalue problems - Stewart - 1973
77	On condition numbers and the distance to the nearest ill-posed problem - Demmel - 1987
72	The rotation of eigenvectors by a perturbation - Davis, Kahan - 1970
42	Ill-conditioned eigensystems and the computation of the Jordan canonical form - Golub, Wilkinson - 1976
36	Average-case stability of Gaussian elimination - Trefethen, Schreiber - 1990
33	The Theory of Matrices Second Edition - Lancaster, Tismenetsky - 1985
32	Numerical linear algebra - Kahan - 1966
29	Pseudospectra of matrices, in Numerical Analysis - Trefethen - 1991
26	Bounds for iterates, inverses, spectral variation and fields of values of nonnormal matrices - Henrici - 1962
25	Improve the accuracy of inverse iteration - Jessup - 1992
23	Norms and exclusion theorems - Bauer, Fike - 1960
22	On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors, IV. Generalized Rayleigh quotient for nonlinear elementary divisors - Ostrowski - 1959
20	Residual bounds on approximate eigensystems of nonnormal matrices - Kahan, Parlett, et al. - 1982
18	The calculation of eigenvectors of codiagonal matrices - Wilkinson - 1958
13	Gaussian Elimination with Partial Pivoting Can Fail in Practice - Foster - 1994
11	Is nonnormality a serious difficulty - Chaitin-Chatelin - 1994
11	The condition numbers of the matrix eigenvalue problem - Smith - 1967
7	Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices - Eisenstat, Ipsen - 1998
7	The calculation of specified eigenvectors by inverse iteration, contribution II/18, volume - Peters, Wilkinson - 1971
6	A geometric theory for the QR, LU , and power iterations - Parlett, Poole - 1973
5	On measures of nonnormality of matrices - Elsner, Paardekooper - 1987
5	Matrix powers in finite precision arithmetic - Higham, Knight - 1995
5	Inverse iteration, Newton’s method and nonlinear eigenvalue problems. In: The contributions of - Osborne - 1978
5	Beitrage zur mathematischen Behandlung komplexer Eigenwertprobleme, Teil III - Wielandt - 1944
4	When is a Linear System Ill-Conditioned - Chandrasekaran - 1994
4	Ill-conditioned eigenproblems - Chatelin - 1986
4	On Fernando's method to find the most redundant equation in a tridiagonal system. Linear Algebra and its Applications - Parlett, Dhillon - 1997
2	Discussion of a paper by K.K - e, Wilson - 1973
2	Backward errors for eigenvalue and singular value decompositions - Chandrasekaran, Ipsen - 1994
2	Propres de Matrices - Valeurs - 1986
2	Eigenproblem solution by a combined Sturm sequence and inverse iteration technique - Gupta - 1973
2	of a unified numerical procedure for free vibration analysis of structures - Development - 1981
2	Helmut Wielandt's contributions to the numerical solution of complex eigenvalue problems - Ipsen
2	The solution of large symmetric eigenvalue problems by sectioning - Jensen - 1972
2	iteration, ill-conditioned equations and Newton's method - Inverse - 1979
2	Frequency modification using Newton's method and inverse iteration eigenvector updating - Smith, Huttin - 1992
2	The calculation of the eigenvectors of a general complex matrix by inverse iteration - Varah - 1968
1	Moments and characteristic roots - Bauer, Householder - 1960
1	Bounds for the departure from normality and the Frobenius norm of matrix eigenvalues - Lee - 1994
1	On perturbation bounds for eigenvalues of a matrix. Unpublished manuscript - Li - 1985
1	error bounds for computed eigensystems - Rigorous - 1961