For solving non-Hermitian linear systems, a famous method is Bi-Conjugate Gradient method (Bi-CG), but its performance is not usually satisfactory because of the unstable convergent behavior and the expensive computational cost. Recently a family of methods named Product-type methods were developed to enhance the Bi-CG method by redefining the residual vector as r_n :=H_n (A)r^B_n, where r^B_n is the residual vector of the Bi-CG method and H_n(A) is an n-degree polynomialnamed accelerating polynomial. In this research, we present the computational varieties based on the deriving processes of the Product-type methods, and discuss the construction of various algorithms depending on these varieties. Finally, among these algorithms we try to find some variations that are expected to be more numerically stable and have higher convergent speed than the present Product-type methods.
Demo code in Matlab