Dependence of the rate of convergence of the Rayleigh-Ritz method on a nonlinear parameter.

Numerical computation of optimum values for nonlinear parameters in a Rayleigh-Ritz variational trial function is considerably more difBcult than numerical computation of optimum values for linear parameters. Thus, an analytic understanding of the mechanisms that determine these optimum values can be quite useful. Uniform asymptotic expansions can be used to explore these mechanisms for the nonlinear parameter that sets the length scale for a basis set. These uniform asymptotic expansions usually involve two or more difFerent kinds of terms whose relative importance changes as the nonlinear parameter changes, with two difFerent terms being equally important at the point where the nonlinear parameter has its optimum value. Interference effects between these difFerent terms are typical, and tend to become most pronounced near the optimum value. These difFerent kinds of terms arise from singularities of the wave function, from the neighborhood of the classical turning point for the basis functions, and/or from saddle points. Comparisons of theory with (numerical) experiment will be given for Rayleigh-Ritz calculations on three model problems that illustrate the kinds of terms listed above.