On an application of the QMR_SYM method to complex symmetric shifted linear systems

Abstract

We consider the solution of complex symmetric shifted linear systems. Such systems arise in large scale electronic structure theory and there is a strong need for the fast solution of the systems. In this paper, we describe an algorithm for solving the systems, which is based on the QMR SYM method for solving complex symmetric linear systems.

Solution of generalized shifted linear systems with complex symmetric matrices

“…The propositions are generalizations of Propositions 3.1 and 3.2 in[31], which can be easily proved in a similar manner. Similar to Proposition 1, there is an efficient way to evaluate the residual 2-norms:Proposition 3.…”

“…Two kinds of the weight and the corresponding algorithms were described; one weight was a standard QMR approach, and the other was given for solving a large number of generalized shifted linear systems. Resulting Algorithms 3 and 4 reduce to Algorithms 1 and 3 in [31] when matrix B is the identity matrix. These algorithms can, therefore, be regarded as extensions of Algorithms 1 and 3 in [31] to generalized shifted linear systems with complex symmetric matrices.…”

“…Resulting Algorithms 3 and 4 reduce to Algorithms 1 and 3 in [31] when matrix B is the identity matrix. These algorithms can, therefore, be regarded as extensions of Algorithms 1 and 3 in [31] to generalized shifted linear systems with complex symmetric matrices.…”

“…The Weighted Quasi-Minimal Residual (WQMR) approach for solving standard shifted linear systems (3) was proposed in [31]. In this section, we develop a WQMR approach for solving generalized shifted linear systems (1).…”

Extremely Scalable Algorithm for 108-atom Quantum Material Simulation on the Full System of the K Computer

“…Krylov-subspace methods with (generalized) shifted linear equations have been investigated in particular from 2000's, partially because the strategy is suitable to parallelism. Since the solver algorithms are mathematical, they are applicable to many scientific areas, such as, QCD [15], large-scale electronic state calculation [1], [16], [17], [18], [19], quantum many-body electron problem [20], nuclear shell model problem [21], first-principle electronic excitation problem [22], and first-principle transport calculation [23]. In the present paper, the multiple Arnoldi solver [1] is used, in which Eq.…”