Stabilization of BiCGSTAB by the generalized residual cutting method

Abstract: The residual cutting (RC) method has been proposed as an outer-inner loop iteration for efficiently solving large and sparse linear systems of equations arising in solving numerically problems of elliptic partial differential equations. Then based on RC the generalized residual cutting (GRC) method has been introduced, which can be applied to more general sparse linear systems problems. In this paper, we show that GRC can stabilize the BiCGSTAB, which is also an iterative algorithm for solving large, sparse, and nonsymmetric linear systems, and widely used in scientific computing and engineering simulations, due to its robustness. BiCGSTAB converges faster and more smoothly than the original BiCG method, by reducing irregular convergence behavior by stabilizing residuals. However, it sometimes fails to converge due to stagnation or breakdown. We attempt to emphasize its robustness by further stabililizing it by GRC, avoiding such failures.