Deflated and Restarted Symmetric Lanczos Methods for Eigenvalues and Linear Equations with Multiple Right-Hand Sides

Abstract

Abstract.A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. Some reorthogonalization is necessary to control roundoff error, and several approaches are discussed. The eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right-hand sides. Experiments are given with large matrices from quantum chromodynamics that have many right-hand sides.Key words. linear equations, deflation, eigenvalues, Lanczos, conjugate gradient, QCD, multiple right-hand sides, symmetric, Hermitian AMS subject classifications. 65F10, 65F15, 15A06, 15A181. Introduction. We consider a large matrix A that is either real symmetric or complex Hermitian. We are interested in solving the system of equations Ax = b, possibly with multiple right-hand sides, and in solving the associated eigenvalue problem. Both eigenvalues and eigenvectors are desired. Symmetric and Hermitian problems can take advantage of fast algorithms such as the conjugate gradient method (CG) [22,52] for linear equations and the related Lanczos algorithm [25,46] for eigenvalues. However, regular CG can be improved upon for the case of multiple right-hand sides, and Lanczos may have storage and accuracy issues while computing eigenvectors. We give new methods for these problems.An approach is presented called Lanczos with deflated restarting or Lan-DR. It simultaneously solves the linear equations and computes the eigenvalues and eigenvectors. A restarted Krylov subspace approach is used for the linear equations, but it also saves approximate eigenvectors at the restart and uses them in the subspace for the next cycle. The restarting of the Lanczos algorithm does not slow down the convergence as it normally would, because once the approximate eigenvectors are accurate enough, they essentially remove or deflate the associated eigenvalues from the problem. The eigenvalue portion of Lan-DR has already been presented in [68], but as mentioned, we add on the solution of linear equations. Also, some reorthogonalization is necessary to control roundoff error. We give some new approaches for this reorthogonalization. We also give a Minres/harmonic version of the algorithm.

Eigenvalue Computations Based on IDR

“…The first similar transition from a linear equation solver to an eigenvalue solver, namely, from the Hestenes and Stiefel variant of CG, which is the OrthoMin-variant of CG, to the Lanczos method tailored to symmetric matrices was given in the book of Householder [20]. The concurrent solution of linear systems and the corresponding eigenvalue problems has recently attracted interest in connection with using projection based singular preconditioners for deflating unwanted eigenvectors; see, e.g., [1].…”

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

“…In [1], the restarted Lanczos method Lan-DR is proposed for both eigenvalues and linear equations. The eigenvalue portion is the same as TRLAN, but several different reorthogonalizations are discussed, including a combination of selective reorthogonalization and periodic reorthogonalization.…”

“…Periodic can be done against all previous Lanczos vectors or only against the Ritz vectors (then it is a combination of periodic and selective rebiorthogonalization). See [1] for further discussion and examples in the symmetric case. Example 1.…”

Deflated Hermitian Lanczos Methods for Multiple Right-Hand Sides

“…These methods simultaneously solve the linear equations and compute eigenvalues and eigenvectors. Recently, we have also developed similar methods for hermitian systems [3], and we will partially describe that work here. See also Ref.…”