Oblique Projection Implementation for Linear System of Equations

Solving the nonsymmetric linear system of equation Ax = b by some iterative methods have been considered as a popular field of research. Various iterative solvers have been proposed with variety of implementations. ELMRES is on of these iterative algorithms to solve this kind of problems. This method looks for iterates xk among the krylov subspace to minimize ||L(b-Ax)|| by an oblique projection process. To compute the new iterates, upper hessenberg least squares problems should be solved which these approximations in primary implementation are computed by Givens rotations. Here a new ELMRES implementation is proposed to solve these problems without using Givens rotations. This algorithm is fast in point of convergence and needs less computing arithmetic. At the end of this paper, the new implementation is compared with the current version by some well-conditioned and ill-conditioned popular examples, numerically.