Nonorthogonal bases in variational calculations and the loss of numerical accuracy

The most common method for calculating accurate numerical solutions for complicated linear differential equations - for example, finding eigenvalues and eigenfunctions of the Schrodinger equation for many-electron atoms - is the variational method with some convenient basis of functions. This leads to a finite matrix representation of the operators involved; and standard numerical operations - such as Gaussian elimination - may be employed. When the basis functions are not orthogonal, one expects substantial loss of numerical accuracy in those matrix manipulations; and so multiple-precision arithmetic is often required for useful results. In this paper, for the first time, we offer a way to estimate the rate at which numerical cancellations will grow in severity as one increases the basis size. For the familiar case of using simple power series, x^n, n<N as the basis instead of orthogonal polynomials, we predict a loss of about 2N bits or 4N bits of numerical accuracy.