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

Abstract

We have developed a derect 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 m iterations, the number of operations tosolve the LA equations is of the order N 2 m. 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 ( m ⪡ 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 many redundant operations that would arise from treating each column of x independently. We have applied the techniques 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 problems for electron-atom and -molecule collisions. The order of A is given approximately by the product of the number of quadrature points n p and the number of scattering channels nc( N = nc np). Since the method is direct, we need only store the potential, the regular and irregular components of the product Green's function, and the iterates, therefore drastically reducing the central memory requirements. The integrals are constructed from simple recursion relationship involving the stored quantities. In addition, the direct approach results in savings in computational time since the number of operations to generate an iterate scales as n 2c np rather than N 2. We generally start the iteration from the Born solution although better such as the distored wave, may enhance the convergence. As an example, we apply the method to electron collisions with the hydrogen molecular ion at the static level and treat large numbers of coupled channels ( nc ≈ 30).