The fast Fourier transform method and ill-conditioned matrices

摘要

We study the problem of finding numerical solutions of the linear algebraic equation, a*x=b, where a denotes an N × N ill-conditioned coefficient matrix. It is well-known that Gaussian elimination methods coupled with pivoting strategies are ineffective in this setting due to round-off error. We propose a new and simple application of the fast Fourier transform (FFT) method. Other viable methods, such as the QR method (QRM) or the singular value decomposition method (SVDM), have been proposed in the literature. The goal of this paper is to investigate the performance of the proposed method and compare it to other popular methods. The comparison is illustrated by computer simulation results using MATLAB.