A Case for a Biorthogonal Jacobi-Davidson Method: Restarting and Correction Equation

We propose a biorthogonal Jacobi--Davidson method (biJD), which can be viewed as an explicitly biorthogonalized, restarted Lanczos method, that uses the approximate solution of a correction equation to expand its basis. Through an elegant formulation, the algorithm allows for all the functionalities and features of the Jacobi--Davidson method (JD), but it also includes some of the advantages of nonsymmetric Lanczos. The main motivation for this work stems from a correction equation and a restarting scheme that are possible with biJD but not with JD. Specifically, a correction equation using the left approximate eigenvectors available in biJD yields cubic asymptotic convergence, as opposed to quadratic with the JD correction equation. In addition, a successful restarting scheme for symmetric JD depends on the Lanczos three-term recurrence and thus can only apply to biJD. Finally, methods that require a multiplication with the adjoint of the matrix need to be reconsidered on today's computers with memory hierarchies, as this multiplication can be performed with minimal additional cost. We describe the algorithm, its features, and the possible functionalities. In addition, we develop an appropriate correction equation framework and analyze the effects of the new restarting scheme. Our numerical experiments confirm that biJD is a highly competitive method for a difficult problem.

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

[2]  Henk A. van der Vorst,et al.  Approximate solutions and eigenvalue bounds from Krylov subspaces , 1995, Numer. Linear Algebra Appl..

[3]  Gerard L. G. Sleijpen,et al.  Effective preconditioning techniques for eigenvalue problems , 1999 .

[4]  Jack Dongarra,et al.  A Test Matrix Collection for Non-Hermitian Eigenvalue Problems , 1997 .

[5]  A. Bultheel,et al.  Implicitly restarting Lanczos , 1998 .

[6]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[7]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

[8]  M. Eiermann,et al.  Geometric aspects of the theory of Krylov subspace methods , 2001, Acta Numerica.

[9]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[10]  Ronald B. Morgan,et al.  On restarting the Arnoldi method for large nonsymmetric eigenvalue problems , 1996, Math. Comput..

[11]  E. Davidson,et al.  Improved Algorithms for the Lowest Few Eigenvalues and Associated Eigenvectors of Large Matrices , 1992 .

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

[13]  Bernard Philippe,et al.  The Davidson Method , 1994, SIAM J. Sci. Comput..

[14]  J. H. van Lenthe,et al.  A space‐saving modification of Davidson's eigenvector algorithm , 1990 .

[15]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[16]  E. Sturler,et al.  Truncation Strategies for Optimal Krylov Subspace Methods , 1999 .

[17]  Yousef Saad,et al.  Deflated and Augmented Krylov Subspace Techniques , 1997, Numer. Linear Algebra Appl..

[18]  R. Freund.  Quasi-kernel polynomials and their use in non-Hermitian matrix iterations , 1992 .

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

[20]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[21]  Y. Saad,et al.  Restarting techniques for the (Jacobi-)Davidson symmetric eigenvalue methods , 1998 .

[22]  H. V. D. Vorst,et al.  Jacobi-Davidson style QR and QZ algorithms for the partial reduction of matrix pencils , 1996 .

[23]  H. V. D. Vorst,et al.  Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems , 1995 .

[24]  Qiang Ye,et al.  ABLE: An Adaptive Block Lanczos Method for Non-Hermitian Eigenvalue Problems , 1999, SIAM J. Matrix Anal. Appl..

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

[26]  Ronald B. Morgan,et al.  A Restarted GMRES Method Augmented with Eigenvectors , 1995, SIAM J. Matrix Anal. Appl..

[27]  Kesheng Wu,et al.  Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods , 1998, SIAM J. Sci. Comput..

[28]  Gerard L. G. Sleijpen,et al.  Alternative correction equations in the Jacobi-Davidson method , 1999 .

[29]  Zhishun A. Liu,et al.  A Look Ahead Lanczos Algorithm for Unsymmetric Matrices , 1985 .

[30]  Gene H. Golub,et al.  Adaptively Preconditioned GMRES Algorithms , 1998, SIAM J. Sci. Comput..

[31]  A. Knyazev.  A Preconditioned Conjugate Gradient Method for Eigenvalue Problems and its Implementation in a Subspace , 1991 .

[32]  Gerard L. G. Sleijpen,et al.  A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems , 1996, SIAM J. Matrix Anal. Appl..