Updating the Symmetric Indefinite Factorization with applications in a modified Newton's method. [SYMUPD, for IBM 370/195]

Technical Report · Sat Jan 01 04:00:00 EST 1977

OSTI ID:7320710

Sorensen, D C

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 a 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 takes 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 positively semidefinite. Computational results are presented which indicate that the method is promising. 4 figures, 9 tables.

• Research Organization: Argonne National Lab., IL (USA)
• DOE Contract Number: W-31109-ENG-38
• OSTI ID: 7320710
• Report Number(s): ANL-77-XX-73
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ACCURACY
ALGEBRA
ALGORITHMS
COMPUTER CODES
COMPUTERS
ERRORS
FACTORIZATION
FORTRAN
FUNCTIONALS
FUNCTIONS
IBM COMPUTERS
ITERATIVE METHODS
MATHEMATICAL LOGIC
MATHEMATICS
MATRICES
NEWTON METHOD
NONLINEAR PROBLEMS
NUMERICAL ANALYSIS
OPTIMIZATION
PROGRAMMING LANGUAGES
S CODES