2007/07/03 by Dean Darnell, Ronald B. Morgan, Walter Wilcox · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Matrix Theory and Algorithms #Numerical methods for differential equations #hep-lat #math-ph #math.MP
paper · pdf · doi:10.1016/j.laa.2008.04.019
published as LinearAlgebraAppl.429:2415-2434,2008 · 19 pages, 9 figures
arxiv created 2007/07/03 · openalex publication_date 2008/06/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple right-hand sides. First, for a single right-hand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including lattice quantum chromodynamics where the matrices are complex and non-Hermitian. Some Krylov iterative methods such as GMRES and BiCGStab have been used to solve multiply shifted systems for about the cost of solving just one system. Restarted GMRES can be improved by deflating eigenvalues for matrices that have a few small eigenvalues. We show that a particular deflated method, GMRES-DR, can be applied to multiply shifted systems. In quantum chromodynamics, it is common to have multiple right-hand sides with multiple shifts for each right-hand side. We develop a method that efficiently solves the multiple right-hand sides by using a deflated version of GMRES and yet keeps costs for all of the multiply shifted systems close to those for one shift. An example is given showing this can be extremely effective with a quantum chromodynamics matrix.