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

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.