Restarting the Nonsymmetric Lanczos Algorithm for Eigenvalues and Linear Equations Including Multiple Right-Hand Sides

Abstract

A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalues and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding near-breakdown and discuss maintaining biorthogonality. A system of linear equations can be solved simultaneously with the eigenvalue computations. Deflation from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. The right and left eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right-hand sides. Introduction. The nonsymmetric Lanczos algorithm [20]is a way to find eigenvalues of large sparse matrices. It is also the basis of many of the leading methods for solving large nonsymmetric systems of linear equations. Nonsymmetric Lanczos is very efficient because of three-term recurrences for generating biorthogonal bases for its Krylov subspaces. However, it is not popular for computing eigenvalues and eigenvectors because of roundoff error concerns. Also, since it is a nonrestarted method, a large amount of storage is needed for the eigenvector computation. So the implicitly restarted Arnoldi method (IRAM) [46] is generally used for finding eigenvalues and eigenvectors of a nonsymmetric matrix. However, the orthogonalization expense in IRAM can be significant as a proportion of total computations, especially for fairly sparse matrices.We give an approach for restarting nonsymmetric Lanczos, and we use it to compute both right and left eigenvectors. Right and left approximate eigenvectors are saved at the restart and used for the next cycle. Linear equations can be solved simultaneously, and the presence of approximate eigenvectors keeps the convergence from being slowed as much by the restarting. This approach is called NLan-DR, for nonsymmetric Lanczos with deflated restarting. It gives an alternative to IRAM when both right and left eigenvectors are desired. The restarting limits the storage, but it also allows for new approaches for dealing with roundoff error. We consider both the roundoff effects of the occurrence of near-breakdown and the loss of biorthogonality.Our application here is the solution of linear equations with multiple right-hand sides. Right and left eigenvectors are used to deflate eigenvalues. The eigenvectors are

Recycling BiCGSTAB with an Application to Parametric Model Order Reduction

“…Whenever the matrix changes in the sequence, we call RBiCG to perform the linear solve. This helps to approximate both left and right invariant subspaces, which are not easily available from the RBiCGSTAB iterations (sometimes a left invariant subspace is available from a right invariant subspace [2,24]). The primary system right-hand side comes from the PDE.…”

“…This corresponds to linear systems where only the right-hand sides change. This is an effective strategy because it has been shown that the recycle space can be useful for multiple consecutive systems [25,21,24,1]. Moreover, using RBiCG for all systems will be expensive since an unnecessary dual system would be solved at each step in the sequence.…”

“…Third, we also give numerical experiments demonstrating the usefulness of our approach, while [20] discusses only a theoretical framework. Also, [2,24,1] focus on variants of deflated BiCGSTAB for multiple right hand sides and do not discuss changing matrices.…”

Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple right-hand sides

“…However, without the locally optimal restarting technique, its spectral approximations are not accurate eigenvectors and therefore have been used mainly in applications where the matrix changes between right‐hand sides. On the other hand, the deflated nonsymmetric Lanczos in is a thick restarted eigensolver. For deflation, other methods project the obtained eigenvectors at every step ( GMRes , Recycled BiCG ) or at every restart ( GMRes‐Proj ).…”

Improved seed methods for symmetric positive definite linear equations with multiple right‐hand sides

“…Block methods are a popular way to solve systems with multiple right-hand sides (see for example [1][2][3][4][5]). Another approach is deflating eigenvalues (e.g., [6][7][8][9][10][11]). However, we will concentrate on a third approach, the seed conjugate gradient (CG) method [12][13][14][15][16][17][18][19][20][21][22].…”