ABSTRACT
The Minimal Residual iteration method can be seen in a certain sense as a modification of Steepest Descent able to manage general positive definite matrices. The Conjugate Gradient algorithm is one of the most used well known and effective iterative techniques for solving linear systems with Hermitian and definite matrices. It is remarkable that the iteration can progress without storing a basis for the entire Krylov subspace thanks to the existence of a three-term relation for the Hessenberg matrix, that becomes symmetric and tridiagonal, in the Arnoldi process. The computationally most expensive operations of bi-Lanczos-based algorithms are the two matrix-vector product per each iteration and some vector sums and scalar products. If the matrix has no special properties to enable fast matrix-vector multiplication then Generalized minimal residual method will be fine if it converges in few iterations, as is the case of matrices with clustered spectrum.
