## The block least squares method for solving nonsymmetric linear systems with multiple right-hand sides.(English)Zbl 1096.65040

Summary: We present the block least squares method for solving nonsymmetric linear systems with multiple right-hand sides. This method is based on the block bidiagonalization. We first derive two algorithms by using two different convergence criteria. The first one is based on independently minimizing the 2-norm of each column of the residual matrix and the second approach is based on minimizing the Frobenius norm of residual matrix. We then give some properties of these new algorithms. Finally, some numerical experiments on test matrices from Harwell-Boeing collection are presented to show the efficiency of the new method.

### MSC:

 65F20 Numerical solutions to overdetermined systems, pseudoinverses 65F10 Iterative numerical methods for linear systems

LSQR
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### References:

 [1] Chan, T.; Wang, W., Analysis of projection methods for solving linear systems with multiple right-hand sides, SIAM J. sci. comput., 18, 1698-1721, (1997) · Zbl 0888.65033 [2] A. EL Guennouni, K. Jbilou, H. Sadok, The block Lanczos method for linear systems with multiple right-hand sides, ano.univ-lille 1.fr/pub/1999/ano 396.html-2k. [3] EL Guennouni, A.; Jgilon, K.; Sadok, H., A block bicgstab algorithm for multiple linear systems, Electron. trans. numer. anal., 16, 129-142, (2003) · Zbl 1065.65052 [4] Freund, R.; Malhotra, M., A block-QMR algorithm for non-Hermitian linear systems with multiple right-hand sides, Linear algebra appl., 254, 119-157, (1997) · Zbl 0873.65021 [5] Golub, G.H.; Kahan, W., Algorithm LSQR is based on the Lanczos process and bidiagonalization procedure, SIAM J. numer. anal., 2, 205-224, (1965) · Zbl 0194.18201 [6] Golub, G.H.; Van Loan, C.F., Matrix computations, (1983), Johns Hopkins University Press Baltimore, MD · Zbl 0559.65011 [7] Jbilou, K.; Messaoudi, A.; Sadok, H., Global FOM and GMRES algorithms for matrix equations, Appl. numer. math., 31, 1, 49-63, (1999) · Zbl 0935.65024 [8] K. Jbilou, H. Sadok, Global Lanczos-based methods with applications, Technical Report LMA 42, Univérsité du Littora, Calais, France, 1997. [9] P. Joly, Résolution de Systéms Linéaires Avec Plusieurs Second Members par la Méthode du Gradient Conjugué, Tech. Rep. R-91012, Publications du Laboratire d’Analyse Numérique, Univérsité Pierre et Marie Curie, Paris, 1991. [10] O’Leary, D., The block conjugate gradient algorithm and related methods, Linear algebra appl., 29, 293-322, (1980) · Zbl 0426.65011 [11] Paige, C.C.; Saunders, M.A., LSQR: an algorithm for sparse linear equations and sparse least squares, ACM trans. math. software, 8, 1, 43-71, (1982) · Zbl 0478.65016 [12] Saad, Y., On the Lanczos method for solving symmetric linear systems with several right-hand sides, Math. comput., 48, 651-662, (1987) · Zbl 0615.65038 [13] Simoncini, V.; Gallopolous, E., An iterative method for nonsymmetric systems with multiple right-hand sides, SIAM J. sci. comput., 16, 917-933, (1995) · Zbl 0831.65037 [14] Van Der Vorst, H., An iterative solution method for solving f(A)=b, using Krylov subspace information obtained for the symmetric positive definite matrix A, J. comput. appl., 18, 249-263, (1987) · Zbl 0621.65022 [15] B. Vital, Etude de Quelques Méthodes de Résolution de Problèmes Linéares de Grande Taille sur Multiprocesseur, Ph.D. thesis, Univérsité de Rennes, Rennes, France, 1990.
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