Nonorthogonal bases in variational calculations and the loss of numerical accuracy

TL;DR

This paper investigates how using nonorthogonal bases in variational calculations leads to significant numerical accuracy loss, providing estimates of the growth rate of cancellations as basis size increases.

Contribution

It introduces a method to estimate the rate of numerical accuracy loss in variational calculations with nonorthogonal bases, specifically for power series.

Findings

Loss of about 2N bits of accuracy with power series basis

Loss of about 4N bits of accuracy with power series basis

Provides a predictive estimate for numerical cancellations

Abstract

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.