Updating the symmetric indefinite factorization with applications in a modified Newton's method. [SYMUPD; also PIVIXI and PIV2X2 for pivoting

Abstract

In recent years the use of quasi-Newton methods in optimization algorithms has inspired much of the research in an area of numerical linear algebra called updating matrix factorizations. Previous research in this area has been concerned with updating the factorization of a symmetric positive definite matrix. Here, a numerical algorithm is presented for updating the Symmetric Indefinite Factorization of Bunch and Parlett. The algorithm requires only O(n/sup 2/) arithmetic operations to update the factorization of an n x n symmetric matrix when modified by a rank-one matrix. An error analysis of this algorithm is given. Computational results are presented that investigate the timing and accuracy of this algorithm. Another algorithm is presented for the unconstrained minimization of a nonlinear functional. The algorithm is a modification of Newton's method. At points where the Hessian is indefinite the search for the next iterate is conducted along a quadratic curve in the plane spanned by a direction of negative curvature and a gradient-related descent direction. The stopping criteria for this search take into account the second-order derivative information. The result is that the iterates are shown to converge globally to a critical point at which the Hessian is positive semidefinite. Computational results are presented which indicate that the method is promising. 6 figures, 9 tables.