Abstract
The residual polynomial of GPBi-CG method is represented in the product of the Lanczos polynomial R_k(A), the polynomial H_k(A) similar to the Lanczos but different in parameters, and the starting residual vector. We remove R_k(A) from the residual polynomial, namely, we suggest a new method whose residual vector is the product of the starting one and H_k(A) only. The algorithm, which is called MRTR method, is proven to be mathematically equivalent to the conjugate residual(CR)method. Therefore, we can show its convergence as well as its error bound. However, MRTR method is different from CR method in their implementation. MRTR method has less operation cost than CR method per iteration step. Through numerical experiments of nonsingular and singular linear equations we confirm the equivalence on numerical computations. Moreover, we show that MRTR method is more effective than CR method.
