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

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 44 to 104 , and the new results are compared with those obtained with rational approximation methods.

• Type: Journal
• Year: 2011
• Where: CPHYSICS
• Authors: Jacques Bloch, Simon Heybrock
• Keywords: CPHYSICS 2011, Education, Krylov Subspace, Lattice Qcd Simulations, Nonzero Baryon Density