Inverting Symmetric Positive Definite Matrices using Divide and Conquer Mathematical Technique with LU Factorization

Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman, Shady Seed El Okuer and Musa Adam Abdullah
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Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman: Professor, Department of Mathematics, Faculty of Education, Omdurman Islamic University, Sudan
Shady Seed El Okuer: PhD, Gaza, Palestine
Musa Adam Abdullah: Department of Mathematics, College of Computer Science and Information Technology, University of Holy Quran and Tasssel of Science, Sudan

European Journal of Mathematics and Statistics, 2025, vol. 6, issue 1, 8-14

Abstract: We looked at the solution of a system of equations Ax=b with symmetric positive definite coefficient matrix A that has singular and nearly singular values. Our technique, which is based on the Divide and Conquer strategy, combines the LU Factorization algorithm with the Divide-and-Conquer technique (D&C algorithm). The matrix was transformed into a product of the type LU using the LU Factorization, where L is a lower triangular matrix and U is an upper triangular matrix. MATLAB was used to implement the algorithm and simulate it as a user-subroutine. In order to reduce the round-off error, particularly for sensitive systems, the user-subroutine takes into account MATLAB characteristics. A non-singular matrix and an ill-conditioned matrix were both numerically demonstrated. Analysis was done on the impact of round-off error. We contrasted the results with those from earlier studies that employed LU factorization.

Keywords: Divide and conquer algorithm; LU algorithm; mathematical software; nearly singular matrix (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:epw:ejmath:v:6:y:2025:i:1:id:14387

DOI: 10.24018/ejmath.2025.6.1.387

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