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**1 - 3**of**3**### Iranian Journal of Numerical Analysis and Optimization

"... The block LSMR algorithm for solving linear systems with multiple right-hand sides F. Toutounian and M. Mojarrab LSMR (Least Squares Minimal Residual) is an iterative method for the solution of the linear system of equations and least-squares problems. This paper presents a block version of the LSM ..."

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The block LSMR algorithm for solving linear systems with multiple right-hand sides F. Toutounian and M. Mojarrab LSMR (Least Squares Minimal Residual) is an iterative method for the solution of the linear system of equations and least-squares problems. This paper presents a block version of the LSMR algorithm for solving linear sys-tems with multiple right-hand sides. The new algorithm is based on the block bidiagonalization and derived by minimizing the Frobenius norm of the resid-ual matrix of normal equations. In addition, the convergence of the proposed algorithm is discussed. In practice, it is also observed that the Frobenius norm of the residual matrix decreases monotonically. Finally, numerical ex-periments from real applications are employed to verify the eectiveness of the presented method.

### POLYNOMIAL PRECONDITIONED GMRES AND GMRES-DR

"... We look at solving large nonsymmetric systems of linear equations using polynomial preconditioned Krylov methods. We give a simple way to find the polynomial. It is shown that polynomial preconditioning can significantly improve restarted GMRES for difficult problems, and the reasons for this are ex ..."

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We look at solving large nonsymmetric systems of linear equations using polynomial preconditioned Krylov methods. We give a simple way to find the polynomial. It is shown that polynomial preconditioning can significantly improve restarted GMRES for difficult problems, and the reasons for this are examined. Stability is discussed and algorithms are given for increased stability. Next we apply polynomial preconditioning to GMRES with deflated restarting. It is shown that this is worthwhile for sparse matrices and for problems with many small eigenvalues. Multiple right-hand sides are also considered.