HIGH-ORDER NEWTON - COTES INTEGRATION METHODS IN SCATTERING-THEORY
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We have implemented high-order Newton-Cotes integration formulas through 16th order for use in the integral equation formulation of scattering theory. We have found that the high-order integration rules give high accuracy results with fewer grid points than low-order rules. Use of Newton-Cotes formulas beyond 16th order was prohibited by numerical instabilities. These formulas were applied to both a model integral and to the calculation of the photoionization of N2 leading to the (2u)-1B2u+ and (3g)-1 X2g+ states of N2+ at a photon energy of 34 eV. High-order rules were found to be unstable when applied to the integration of functions which rise as r21 + 2 at high 1 in the partial wave expansions. An empirical formula is presented which predicts when high-order rules diverge as a function of l in a given integration region. Using a combination of high and low-order rules eliminated the divergence and still yielded accurate results in the N2 photoionization calculations. Results for the photoionization of N2 showed that Simpson's rule is the most efficient Newton-Cotes formula for calculations in which the desired error was larger than 0.14%. For smaller errors, high-order integration rules are more efficient. 1988.