An adaptive conjugate gradient normal residuals algorithm for solving large linear systems

In this paper, an adaptive algorithm based on the normal equations for solving large nonsymmetric linear systems is presented. The new algorithm is a hybrid method combining polynomial preconditioning with the CGNR method. Residual polynomial is used in the preconditioning to estimate the eigenvalues of the s.p.d. matrix A^TA, and the residual polynomial is generated from several steps of CGNR by recurrence. The algorithm is adaptive during its implementation. The robustness is maintained, and the iteration convergence is speeded up. A numerical test result is also reported.