A note on the pin-pointing solution of ill-conditioned linear system of equations

Abstract

Volokh and Vilnay [Appl. Math. Lett. 13 (2000), pp. 119-124] proposed a method for computing an accurate solution of nearly singular linear systems of equations. The method is processed in two stages. Their method uses the truncated singular value decomposition of the initial coefficient matrix at the first stage and the Gaussian elimination procedure for well-conditioned reduced system of linear equations at the second stage. In this note, we propose a multistage approach based on the Volokh and Vilnay's method. Some numerical results are given to show the efficiency of the proposed method.

A new iterative refinement of the solution of ill-conditioned linear system of equations

“…For the implementation of the proposed method (hereafter denoted by TSM), x 1 is computed by the WUM and then x m , m ≥ 2 are computed via Equation (4). Hence, the new method can be summarized as follows.…”

“…One approach is to use the scaling strategy, but scaling of the equations and unknowns must proceed on a problem-by-problem basis. Hence, the general scaling strategy is unreliable [3,4,10]. Another approach, which we focus our attention on it, is to use an iterative refinement of the solution obtained by a direct solver.…”

Settlement monitoring data fusion approach for high-speed railways based on GNSS and InSAR

“…However, truncation parameter selection is difficult, and it is not effective for solving the ill-conditioned matrix with large condition numbers. 15,16 The spectrum correction iterative method avoids the parameter selection problem of Tikhonov regular method and truncated singular value method, and the estimated value has the unbiased property. 17 When the condition number of the ill-conditioned matrix is too large, the convergence speed of the spectrum correction iterative method slows down.…”

“…The truncated singular value method is based on the singular value decomposition technology, which cuts off the small singular value that seriously expands the variance. However, truncation parameter selection is difficult, and it is not effective for solving the ill-conditioned matrix with large condition numbers 15 , 16 . The spectrum correction iterative method avoids the parameter selection problem of Tikhonov regular method and truncated singular value method, and the estimated value has the unbiased property 17 .…”