Transpose-Free Gl-BCG Algorithm for Linear Systems with Multiple Right-Hand Sides

In the present paper, we present a new transpose-free global method for solving large nonsymmetric linear systems with multiple right-hand sides. We first give the scalar polynomial interpretation of the classical global biconjugate gradient algorithm using formal orthogonal polynomials. The global conjugate gradient squared algorithm can be derived by using this. Although related to the transpose of a matrix, the global conjugate gradient squared algorithm does not need multiplication by the transpose of a matrix. We also show to apply the method for solving the Lyapunov matrix equation. Finally, some numerical examples are given to illustrate the proposed method.