Banded Doolittle’s Method for the Solutions of Large Systems of Equations

Abstract

Abstract Doolittle’s method is a much simplified form of Cholesky’s method for solving systems of linear simultaneous equations without inverting the coefficient matrix. In Cholesky’s method the coefficient matrix must be symmetric, the two conjugate triangular matrices must be transposes of each other, and the solution process involves square roots of sums. In Doolittle’s method the coefficient matrix need not be symmetric and there is no constrained relationship between the two conjugate matrices. In this article, Doolittle’s method is used to formulate the banded matrix solution of linear sets of equations of form [A]{X} = {B}, where [A] is symmetric. Such systems of equations occur for many engineering problems, especially for structural finite element formulations where [A] corresponds to the stiffness matrix. In the banded form of [A] the non-zero diagonal rays of one-half of [A] is formed in column form such that [A] is replaced with two conjugate matrices in column forms. The system of equations can then be solved for {X} without inverting [A]. In this paper generalized expressions for the elements of the conjugate matrices, an intermediate unknown parameter matrix, and the {X} matrix are provided for use in programming the method. A curved beam stress analysis is presented to illustrate that the computation time required by the banded Doolittle method is many times less than that of methods such as the regular Doolittle method, the matrix inversion method (using Gauss-Jordan row transformations), Gauss elimination methods, and the banded Cholesky method. Hence, an efficient and less costly process for solving large size engineering systems is offered. A tested FORTRAN program using the banded Doolittle’s method in a subroutine is given in the Appendix.