LU Methods for Solving Linear Systems

This chapter presents the principle of LU factorization to solve a linear system. To calculate the LU factorization of an invertible matrix A, we need is to determine L, a lower triangular matrix and U, an upper triangular matrix, such that A = LU. The chapter then details Gauss factorization and, subsequently, Gauss-Jordan factorization for matrices of any kind. The notion of the pivoting of rows is also discussed in detail. Finally, the special case of symmetric matrices is considered and illustrated by Crout factorization, and later, by Cholesky factorization. The main cost of a direct method resides in the factorization phase. The Gauss factorization algorithm is based on the recurrent application of the method of partial factorization. The chapter shows that the Gauss-Jordan factorization is obtained simply by starting with the Gauss factorization and then by factorizing the diagonal term into each row of U.