A nested Krylov subspace method to compute the sign function of large complex matrices

Abstract

We present an acceleration of the well-established Krylov-Ritz methods to compute the sign function of large complex matrices, as needed in lattice QCD simulations involving the overlap Dirac operator at both zero and nonzero baryon density. Krylov-Ritz methods approximate the sign function using a projection on a Krylov subspace. To achieve a high accuracy this subspace must be taken quite large, which makes the method too costly. The new idea is to make a further projection on an even smaller, nested Krylov subspace. If additionally an intermediate preconditioning step is applied, this projection can be performed without affecting the accuracy of the approximation, and a substantial gain in efficiency is achieved for both Hermitian and non-Hermitian matrices. The numerical efficiency of the method is demonstrated on lattice configurations of sizes ranging from 4^4 to 10^4, and the new results are compared with those obtained with rational approximation methods.

17 pages, 12 figures, minor corrections, extended analysis of the preconditioning step

References in corpus (5)

• Overlap Dirac operator at nonzero chemical potential and random matrix theory
• An iterative method to compute the overlap Dirac operator at nonzero chemical potential
• Domain-wall and overlap fermions at nonzero quark chemical potential
• Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential
• Comparing iterative methods to compute the overlap Dirac operator at nonzero chemical potential

Cited by in corpus (9)

• Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential
• Level spacings for weakly asymmetric real random matrices and application to two-color QCD with chemical potential
• A numerical method to compute derivatives of functions of large complex matrices and its application to the overlap Dirac operator at finite chemical potential
• From Sylvester's determinant identity to Cramer's rule
• A Method to Calculate Conserved Currents and Fermionic Force for the Lanczos Approximation to the Overlap Dirac Operator
• A method to compute derivatives of functions of large complex matrices
• Double-pass variants for multi-shift BiCGstab(ell)
• A new deflated block GCROT($m,k$) method for the solution of linear systems with multiple right-hand sides
• A SQMRCGstab Algorithm for Families of Shifted Linear Systems