TABLE OF CONTENTS The basic methods for solving systems of linear algebraic equations were discussed in this chapter. Since these methods use matrices and determinants, the basic properties of matrices and determinants were presented.
Several direct solution methods were also discussed. Among them were Cramer s rule, Gaussian elimination and its variants, the Gauss-Jordan method, and the LU decomposition method. Cramer s rule is impractical for solving systems with more than three or four equations. Gaussian elimination is the best choice for solving linear systems. For systems of equations having a constant coefficients matrix but many right-hand side vectors, LU decomposition is the method of choice. The LU decomposition method has been used for the solution of the tridiagonal system. Direct methods are generally used when the number of equations is small, or most of the coefficients of the equations are nonzero, or the system of equations is not diagonally dominant, or the system of equations is ill-conditioned. Several iterative methods were discussed. Among them were the Jacobi method, the Gauss-Seidel method, and the SOR method. All methods converge if the coefficient matrix is strictly diagonally dominant. The SOR is the best method of choice. Although the determination of the optimum value of the relaxation factor ? is difficult, it is generally worthwhile if the system of equations is to be solved many times for right-hand side vectors. The need for estimating parameters is removed in the conjugate gradient method, which, although more complicated to code, can rival the SOR method...
