Preconditioning the Lanczos Algorithm for Sparse Symmetric Eigenvalue Problems

A method for computing a few eigenpairs of sparse symmetric matrices is presented and analyzed that combines the power of preconditioning techniques with the efficiency of the Lanczos algorithm. The method is related to Davidson’s method and its generalizations, but can be less expensive for matrices that are fairly sparse. A double iteration is used. An effective termination criterion is given for the inner iteration. Quadratic convergence with respect to the outer loop is shown.

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

[2]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[3]  O. Axelsson.  On the efficiency of a class of a-stable methods , 1974 .

[4]  E. Davidson.  The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices , 1975 .

[5]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[6]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[7]  J. G. Lewis.  Algorithms for sparse matrix eigenvalue problems , 1977 .

[8]  A. K. Cline.  Several Observations on the Use of Conjugate Gradient Methods , 1978 .

[9]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[10]  Axel Ruhe,et al.  The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems , 1980 .

[11]  D. Scott.  Solving Sparse Symmetric Generalized Eigenvalue Problems without Factorization , 1981 .

[12]  D. Scott.  The Advantages of Inverted Operators in Rayleigh–Ritz Approximations , 1982 .

[13]  Iain S. Duff,et al.  Sparse matrix test problems , 1982 .

[14]  G. Golub,et al.  Block Preconditioning for the Conjugate Gradient Method , 1985 .

[15]  R. Morgan,et al.  Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices , 1986 .

[16]  D. Szyld.  Criteria for Combining Inverse and Rayleigh Quotient Iteration , 1988 .

[17]  R. Morgan.  Davidson's method and preconditioning for generalized eigenvalue problems , 1990 .

[18]  R. Morgan.  Computing Interior Eigenvalues of Large Matrices , 1991 .