• Journal of applied mathematics & informatics
• /
• v.26 no.1_2
• /
• pp.213-222
• /
• 2008

Iterative methods are often suitable for solving least squares problems min$||Ax-b||_2$, where A $\epsilon\;\mathbb{R}^{m{\times}n}$ is large and sparse. The well known LSQR algorithm is among the iterative methods for solving these problems. A good preconditioner is often needed to speedup the LSQR convergence. In this paper we present the numerical experiments of applying a well known preconditioner for the LSQR algorithm. The preconditioner is based on the $A^T$ A-orthogonalization process which furnishes an incomplete upper-lower factorization of the inverse of the normal matrix $A^T$ A. The main advantage of this preconditioner is that we apply only one of the factors as a right preconditioner for the LSQR algorithm applied to the least squares problem min$||Ax-b||_2$. The preconditioner needs only the sparse matrix-vector product operations and significantly reduces the solution time compared to the unpreconditioned iteration. Finally, some numerical experiments on test matrices from Harwell-Boeing collection are presented to show the robustness and efficiency of this preconditioner.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.4
• /
• pp.895-903
• /
• 2011

A two dimensional version of LSQR iterative algorithm which takes advantages of working solely with the 2-dimensional arrays is developed and applied to the image deblurring problem. The efficiency of the method comparing to the Fourier-based LSQR method and the 2-D version CGLS algorithm methods proposed by Hanson ([4]) is analyzed.

• Journal of the Chungcheong Mathematical Society
• /
• v.22 no.2
• /
• pp.177-186
• /
• 2009

The global least squares method (Gl-LSQR) is a generalization of LSQR method for solving linear system with multiple right hand sides. In this paper, we present how to apply this algorithm for solving the image restoration problem and illustrate the usefulness and effectiveness of this method from numerical experiments.

• Journal of applied mathematics & informatics
• /
• v.27 no.1_2
• /
• pp.1-12
• /
• 2009

Iterative algorithms are proposed for the least-squares symmetric solution of AXB = E with a submatrix constraint. We characterize the linear mappings from their independent element space to the constrained solution sets, study their properties and use these properties to propose two matrix iterative algorithms that can find the minimum and quasi-minimum norm solution based on the classical LSQR algorithm for solving the unconstrained LS problem. Numerical results are provided that show the efficiency of the proposed methods.

• Journal of applied mathematics & informatics
• /
• v.34 no.1_2
• /
• pp.95-106
• /
• 2016

In this paper, according to the classical LSQR algorithm forsolving least squares (LS) problem, an iterative method is proposed for finding the minimum-norm pure imaginary solution of the quaternionic least squares (QLS) problem. By means of real representation of quaternion matrix, the QLS's correspongding vector algorithm is rewrited back to the matrix-form algorthm without Kronecker product and long vectors. Finally, numerical examples are reported that show the favorable numerical properties of the method.