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Divide-and-conquer strategy with Cholesky’s factorization for inverting symmetric positive definite matrices. (English) Zbl 1413.65039

Summary: In this paper we study the solution of a system of equations \(\mathbf{Ax}=\mathbf{b}\) with singular and nearly singular, symmetric positive definite coefficient matrix \(\mathbf{A}\). Our algorithm based on, the Divide and Conquer strategy leading to the Divide-and-Conquer Algorithm (D&C algorithm) with, Cholesky’s factorization algorithm. The Cholesky’s factorization will be used to convert the matrix into a product of the form \(\mathbf{LL}^T\), where \(\mathbf{L}\) is a lower triangular matrix. The algorithm will be implemented on MATLAB and simulated as a user-subroutine. The user-subroutine is considering MATLAB features for reducing the round-off error especially for sensitive systems. Numerical examples will be given of a non-singular matrix and another for ill-conditioned matrix. The effect of round-off error will be analyzed. Results will be compared with previous ones, where LU factorization is used.

MSC:

65F05 Direct numerical methods for linear systems and matrix inversion
15A23 Factorization of matrices

Software:

Matlab
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