Preconditioned conjugate residual methods for the solution of spectral equations (Q1094814)

The authors consider an iterative procedure for solving a system of linear equations \(A\phi =f\), where A is an ill-conditioned full real matrix with positive definite symmetric part. Such systems arise when the spectral method is applied for discretizing a selfadjoint elliptic partial differential equation. The considered method is based on the generalized conjugate direction method proposed by \textit{O. Axelsson} [Lect. Notes Math. 773, 1-11 (1980; Zbl 0421.65023)]. To accelerate the rate of convergence a preconditioning technique is applied. The construction of two preconditioning matrices H is presented. Both matrices H are certain perturbations of a finite difference operator connected with the grid points the same as those for the spectral method. Results of numerical experiments are included and comparison with other procedures is given.