Abstract
The restarted CMRH method (changing minimal residual method based on the Hessenberg process) using fewer operations and storage is an alternative method to the restarted generalized minimal residual method (GMRES) method for linear systems. However, the traditional restarted CMRH method, which completely ignores the history information in the previous cycles, presents a slow speed of convergence. In this paper, we propose a heavy ball restarted CMRH method to remedy the slow convergence by bringing the previous approximation into the current search subspace. Numerical examples illustrate the effectiveness of the heavy ball restarted CMRH method.
Highlights
We are concerned with the CMRH method introduced in [1,2] for the solution of n × n linear system Ax = b
Motivated by the locally optimal and heavy ball CMRES method [14] which is proposed by including the approximation before the last to bring in more information in the previous cycles, we naturally develop a heavy ball restarted CMRH method denoted by HBCMRH(m) to make up the loss of previous search spaces
We proposed a heavy ball restarted CMRH method (HBCMRH(m) for short; i.e., Algorithm 2) for a linear system Ax = b
Summary
We are concerned with the CMRH method (changing minimal residual method based on the Hessenberg process) introduced in [1,2] for the solution of n × n linear system Ax = b. One of the reasons is because the basis vectors generated by the Hessenberg process are not orthonormal This leads to the CMRH residual vector being not orthogonal to the subspace Km(A, r0) or A × Km(A, r0). In the heavy ball restarted CMRH method proposed in this paper, for salvaging the loss of the previous search space, we take the previous approximation into the current search to bring sufficient history information of the previous Krylov subspace. Notations · 1, · 2 and · ∞ are the 1-norm, 2-norm, and ∞-norm of a vector or matrix, respectively
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