Liu, Chein-Shan; Hong, Hong-Ki; Atluri, Satya N. Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems. (English) Zbl 1231.65071 CMES, Comput. Model. Eng. Sci. 60, No. 3, 279-308 (2010). Summary: We propose novel algorithms to calculate the inverses of ill-conditioned matrices, which have broad engineering applications. The vector-form of the conjugate gradient method (CGM) is recast into a matrix-form, which is named as the matrix conjugate gradient method (MCGM). The MCGM is better than the CGM for finding the inverses of matrices. To treat the problems of inverting ill-conditioned matrices, we add a vector equation into the given matrix equation for obtaining the left-inversion of matrix (and a similar vector equation for the right-inversion) and thus we obtain an over-determined system. The resulting two modifications of the MCGM, namely the MCGM1 and MCGM2, are found to be much better for finding the inverses of ill-conditioned matrices, such as the Vandermonde matrix and the Hilbert matrix. We propose a natural regularization method for solving an ill-posed linear system, which is theoretically and numerically proven in this paper, to be better than the well-known Tikhonov regularization. The presently proposed natural regularization is shown to be equivalent to using a new preconditioner, with better conditioning. The robustness of the presently proposed method provides a significant improvement in the solution of ill-posed linear problems, and its convergence is as fast as the CGM for the well-posed linear problems. Cited in 1 ReviewCited in 20 Documents MSC: 65F22 Ill-posedness and regularization problems in numerical linear algebra 65F10 Iterative numerical methods for linear systems Keywords:ill-posed linear system; inversion of ill-conditioned matrix; left-inversion; right-inversion; regularization vector; Vandermonde matrix; Hilbert matrix; Tikhonov regularization Software:NewtonLib PDFBibTeX XMLCite \textit{C.-S. Liu} et al., CMES, Comput. Model. Eng. Sci. 60, No. 3, 279--308 (2010; Zbl 1231.65071) Full Text: DOI