Title: A new form of LSMR for solving linear systems and least-squares problems

Authors: Maryam Mojarrab; Afsaneh Hasanpour; Somayyeh Ghadamyari

Addresses: Department of Mathematics, University of Sistan and Baluchestan, Zahedan, 45845-98167, Iran ' Department of Mathematics, University of Sistan and Baluchestan, Zahedan, 45845-98167, Iran ' Department of Mathematics, University of Sistan and Baluchestan, Zahedan, 45845-98167, Iran

Abstract: The least squares minimal residual (LSMR) method of Fong and Saunders (2011) is an algorithm for solving linear systems Ax = b and least-squares problems min∥Ax - b∥₂ that is analytically equivalent to the MINRES method applied to a normal equation A^TAx = A^T b so that the quantities ∥A^Tr_k∥₂ are minimised (where r_k = b - Ax_k is the residual for current iterate x_k). This method is based on the Golub-Kahan bidiagonalisation 1 process, which generates orthonormal Krylov basis vectors. Here, the Golub-Kahan bidiagonalisation 2 process is implemented in the LSMR algorithm. This substitution makes the algorithm simpler than the standard algorithm. Also, numerical results show the new form to be competitive.

Keywords: bidiagonalisation process; linear system; least-squares problem; Krylov subspace method; LSMR; least squares minimal residual.

DOI: 10.1504/IJCSM.2023.134561

International Journal of Computing Science and Mathematics, 2023 Vol.18 No.3, pp.266 - 275

Accepted: 08 Mar 2023
Published online: 27 Oct 2023 *

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