A direct iterative-variational method for solving large sets of linear algebraic equations

Conference · Fri Jan 01 00:00:00 EST 1988

DOI:https://doi.org/10.1016/0010-4655(89)90174-4· OSTI ID:6794257

Schneider, B I; Collins, L A

We have developed a direct iterative-variational technique for solving large systems of linear equations: Ax = b, where N is the order of the matrix A and the length of the vectors x and b. The method, which has analogues in the conjugate gradient and Lanczos schemes as well as the direct configuration interaction procedures of quantum chemistry, involves the construction of an orthonormal basis from successive applications of the general linear algebraic (LA) matrix, A, to an initial guess for the solution vector. The solution vector is expanded in this basis, and the coefficients are determined from a variational prescription. For n iterations, the number of operations to solve the LA equations is of the order N/sup 2/n. Since the basis is orthonormal, the procedure is guaranteed to converge within N iterations, provided that the basis vectors remain linearly independent. In practice, the convergence is much more rapid (n << N). Another advantage of the method is that the whole matrix A need not be stored. In the more general case of multiple right-hand-sides (x and b, matrices), the method can be applied simultaneously to all of the solutions, thus saving many redundant operations that would arise from treating each column of x independently. We have applied the technique to the solution of LA systems that arise from converting radial coupled integrodifferential equations to an integral representation on a discrete quadrature, in particular, to a variety of problems for electron-atom and -molecule collisions. 21 refs.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• DOE Contract Number: W-7405-ENG-36
• OSTI ID: 6794257
• Report Number(s): LA-UR-88-2823; CONF-8810119-3; ON: DE88016149
• Resource Relation: Journal Volume: 53; Journal Issue: 1-3; Conference: Workshop on practical iterative methods for large scale computations, Minneapolis, MN, USA, 23 Oct 1988; Other Information: Portions of this document are illegible in microfiche products
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
DIFFERENTIAL EQUATIONS
NUMERICAL SOLUTION
INTEGRAL EQUATIONS
GREEN FUNCTION
ITERATIVE METHODS
MATRICES
SCATTERING
VARIATIONAL METHODS
WAVE FUNCTIONS
EQUATIONS
FUNCTIONS
657000* - Theoretical &amp; Mathematical Physics
990230 - Mathematics &amp; Mathematical Models- (1987-1989)