A Polynomial Preconditioner for the CMRH Algorithm

Abstract

Many large and sparse linear systems can be solved efficiently by restarted GMRES and CMRH methods Sadok 1999. The CMRH(m) method is less expensive and requires slightly less storage than GMRES(m). But like GMRES, the restarted CMRH method may not converge. In order to remedy this defect, this paper presents a polynomial preconditioner for CMRH-based algorithm. Numerical experiments are given to show that the polynomial preconditioner is quite simple and easily constructed and the preconditioned CMRH(m) with the polynomial preconditioner has better performance than CMRH(m).

Preconditioning Complex Symmetric Linear Systems

“…Moreover z = 0 is never a root of (34). Thus the roots are bounded in the interval (−1, 1) for n ≥ 0 and λ ∈ (0, 1].…”

“…In fact, polynomial preconditioners based on Chebyshev polynomials need accurate estimate of minimum and maximum eigenvalue, while least squares polynomials were not computed using stable recurrences, limiting the degree of available stable polynomials [3,4,31,38,22]. However, in the last years, polynomial preconditioner went back to the top after the works of Lu-Lai-Xu [34]. Notwithstanding anything contained above, here we propose as a preconditioner the use of a polynomial approximation of a modified HSS step.…”

Heavy Ball Restarted CMRH Methods for Linear Systems

“…The corresponding convergence analysis of the CMRH method and its relation to the GMRES method can be found in [5,6]. There are a great deal of past and recent works and interests in developing the Hessenberg process and CMRH method for linear systems [7][8][9][10][11][12][13]. Specifically, in [7,8], Duminil presents an implementation for parallel architectures and an implementation of the left preconditioned CMRH method.…”

“…Specifically, in [7,8], Duminil presents an implementation for parallel architectures and an implementation of the left preconditioned CMRH method. A polynomial preconditioner and flexible right preconditioner for CMRH methods are considered in [9] and [10], respectively. In [12,13], the variants of the Hessenberg and CMRH methods are introduced for solving multi-shifted non-Hermitian linear systems.…”

MINRES Seed Projection Methods for Solving Symmetric Linear Systems with Multiple Right-Hand Sides

“…The global methods [19,20] are also a class of important methods. Following the work [20], many other global methods have been developed, including the global BiCG and global BiCGSTAB methods [21], the global Hessenberg and global CMRH methods [22], and the polynomial preconditioned global CMRH method [23,24]. Generally, the global methods are more appropriate for large and sparse systems.…”