THE ORTHOGONAL FACTORIZATION OF A LARGE SPARSE MATRIX

Publisher Summary Methods based upon Gaussian elimination have been used to solve large sparse systems of equations more extensively than methods based upon orthogonalization. Although orthogonalization techniques give superior numerical stability, this is offset by the increase in storage requirements caused by fill-in. However, the use of orthogonalization can be beneficial in some applications, for example, in linear least-squares and linear programming. In the first case, the orthogonal factorization can give a significant amount of statistical information. In the linear programming case, a factorization is repeatedly modified and an algorithm with a more stable method of modifying the factors may require fewer reinversions. This chapter describes the application of the orthogonal factorization to both the large sparse linear least-squares problem and the large-scale linear programming problem. It is possible to take advantage of sparsity of a large matrix by arranging the orthogonal factorization of a matrix to be in product form.