Abstract
In this paper, a new iterative solution method is proposed for solving multiple linear systems A(i)x(i)=b(i), for 1≤ i ≤ s, where the coefficient matrices A(i) and the right-hand sides b(i) are arbitrary in general. The proposed method is based on the global least squares (GL-LSQR) method. A linear operator is defined to connect all the linear systems together. To approximate all numerical solutions of the multiple linear systems simultaneously, the GL-LSQR method is applied for the operator and the approximate solutions are obtained recursively. The presented method is compared with the well-known LSQR method. Finally, numerical experiments on test matrices are presented to show the efficiency of the new method.
Highlights
IntroductionWe want to solve, using global least squares (GL-LSQR) method, the following linear systems: A(i)x(i) = b(i), 1 £ i £ s (1)
We want to solve, using global least squares (GL-LSQR) method, the following linear systems: A(i)x(i) = b(i), 1 £ i £ s (1)where A(i) are arbitrary matrices of order n, and in general A(i) 1 A( j) and b(i) 1 b( j) for i 1 j
We do not need to store the basis vectors, we do not need to predetermine a subspace dimension and the approximate solutions and residuals are cheaply computed at every stage of the algorithm because they are updated with short-term recurrence
Summary
We want to solve, using global least squares (GL-LSQR) method, the following linear systems: A(i)x(i) = b(i), 1 £ i £ s (1). We recall some fundamental properties of GL-LSQR algorithm [13], which is an iterative method for solving the multiple linear systems (2). For solving the matrix Equation (6), the GL-LSQR method uses a procedure, namely Global-Bidiag procedure, to reduce A to the global lower bidiagonal form. For the Global-Bidiag procedure, we have the following propositions. By using the Global-Bidiag procedure, the GL-LSQR algorithm constructs an approximate solution of the form X k = Vk * yk , where yk = éêë yk(1), , yk(k) ùúûT Î k , which solves the least-squares problem, min X. More details about the GL-LSQR algorithm can be found [13]
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