The computation of eigenvalues and eigenvectors of very large sparse matrices

Several methods are available for computing eigenvalues and eigenvectors of large sparse matrices, but as yet no outstandingly good algorithm is generally known. For the symmetric matrix case one of the most elegant algorithms theoretically is the method of minimized iterations developed by Lanczos in 1950. This method reduces the original matrix to tri-diagonal form from which the eigensystem can easily be found. The method can be used iteratively, and here the convergence properties and different possible eigenvalue intervals are first considered assuming infinite precision computation. Next rounding error analyses are given for the method both with and without re-orthogonalization. It is shown that the method has been unjustly neglected, in fact a particular computation algorithm for the method without reorthogonalization is shown to have remarkably good error properties. As well as this the algorithm is very fast and can be programmed to require very little store compared with other comparable methods, and this suggests that this variant of the Lanczos process is likely to become an extremely useful algorithm for finding several extreme eigenvalues, and their eigenvectors if needed, of very large sparse symmetric matrices.

[1]  W. Arnoldi.  The principle of minimized iterations in the solution of the matrix eigenvalue problem , 1951 .

[2]  R. C. Thompson,et al.  Principal submatrices II: the upper and lower quadratic inequalities☆ , 1968 .

[3]  G. Stewart.  Accelerating the orthogonal iteration for the eigenvectors of a Hermitian matrix , 1969 .

[4]  Tosio Kato.  On the Upper and Lower Bounds of Eigenvalues , 1949 .

[5]  C. Fox.  Variational Methods for Eigenvalue Problems. By S. H. Gould. Pp. xiv, 179. 48s. 1957. (University of Toronto Press and Oxford University Press) , 1959, The Mathematical Gazette.

[6]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[7]  A. Jennings.  A direct iteration method of obtaining latent roots and vectors of a symmetric matrix , 1967, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  W. Givens.  Numerical Computation of the Characteristic Values of a Real Symmetric Matrix , 1954 .

[9]  S. Kaniel.  Estimates for Some Computational Techniques - in Linear Algebra , 1966 .

[10]  Friedrich L. Bauer,et al.  On certain methods for expanding the characteristic polynomial , 1959, Numerische Mathematik.

[11]  J. H. Wilkinson,et al.  Solution of symmetric and unsymmetric band equations and the calculation of eigenvectors of band matrices , 1967 .

[12]  Maurice Clint,et al.  The Evaluation of Eigenvalues and Eigenvectors of Real Symmetric Matrices by Simultaneous Iteration , 1970, Comput. J..

[13]  J. Westlake.  Handbook of Numerical Matrix Inversion and Solution of Linear Equations , 1968 .

[14]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[15]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[16]  H. Rutishauser,et al.  TheLR transformation method for symmetric matrices , 1963 .

[17]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[18]  R. Tewarson.  On the transformation of symmetric sparse matrices to the triple diagonal form , 1968 .

[19]  J. H. Wilkinson.  Elementary proof of the Wielandt-Hoffman Theorem and of its generalization , 1970 .

[20]  N. Lehmann.  Zur Verwendung optimaler Eigenwerteingrenzungen bei der Lösung symmetrischer Matrizenaufgaben , 1966 .

[21]  B. Tabarrok,et al.  ON KRON'S EIGENVALUE PROCEDURE AND RELATED METHODS OF FREQUENCY ANALYSIS , 1968 .

[22]  H. Schwarz.  Tridiagonalization of a symetric band matrix , 1968 .

[23]  G. Temple.  The Theory of Rayleigh's Principle as Applied to Continuous Systems , 1928 .

[24]  T. Sebe,et al.  Variational buildup of nuclear shell model bases , 1969 .