Explicitly preconditioned conjugate gradient method for the solution of unsymmetric linear systems

For the iterative solution of nonsingular linear system Ay = f of order n the preconditioned system Mx = b is considered, where M = GAF T b = Gf y = F T x, and the matrices G and F are sparse lower triangular. Their nonzero elements are obtained by alternate minimization of β (M T M) with respect to F and G, where β (M T M) = n − 1 tr M T M/(det M T M)1/n . Such preconditioning makes the matrix M closer to an orthogonal one, which improves the efficiency of the standard CGNE-type algorithms. This conclusion is confirmed by the estimate k ≤ ⌈ n log2 β (M T M) + log2(e −1 )⌉ for the iteration number k needed for the e times reduction of the error norm ‖ G(f − Ay k )‖. Numerical examples are given to show the efficiency of the resulting solution algorithms for the indefinite and strongly nonsymmetric linear systems arising from finite difference discretizations of the 2D convection-diffusion equation.

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