Convergence characteristics of the generalized residual cutting method
For researchers solving large sparse linear systems, this work provides a theoretical foundation for GRC and demonstrates its practical advantages over GMRES.
The paper proves that the generalized residual cutting (GRC) method is a Krylov subspace method and is equivalent to the conjugate residual (CR) method for symmetric matrices. Numerically, GRC shows more robust convergence and requires less memory than GMRES for large matrix problems.
The residual cutting (RC) method has been proposed for efficiently solving linear equations obtained from elliptic partial differential equations. Based on the RC, we have introduced the generalized residual cutting (GRC) method, which can be applied to general sparse matrix problems. In this paper, we study the mathematics of the GRC algorithm and and prove it is a Krylov subspace method. Moreover, we show that it is deeply related to the conjugate residual (CR) method and that GRC becomes equivalent to CR for symmetric matrices. Also, in numerical experiments, GRC shows more robust convergence and needs less memory compared to GMRES, for significantly larger matrix sizes.