Confluent prony approximation

AbstractProny approximation for f(x) fits an exponential sum Sn(x) ≏∑i=1n Aieαix to m values f(xj), m ≥ 2n, j = 0(1)m − 1, for xj's equally-spaced at intervals of h. The αi's are obtained from eαih, the roots of an nth degree algebraic equation whose coefficients are found from the f(xj)'s. For nodes xj spaced unequally, there is neither a Prony approximation, nor, for interpolation, an exponential analogue of divided differences of tabulated functions. But when all the nodes xj coalesce to x0, there is a confluent Prony approximation (CPA), where Sn(j)(x0) ⋟ tf(j)(x0), j = 0(1)m − 1. Here the αi's themselves are the roots of an equation whose coefficients are found similarly, but from the f(j)(x0)'s. CPA obtains exponential terms in Sn(x) that would be undetectable at nodes xj away from χ0. For m = 2n, CPA becomes interpolation that is equivalent to Gaussian quadrature for I(x) ≡ ∫abw(t)extdt, where w(t), a and b depend upon f(x), the moments being f(j)(x0), j = 0(1)2n − 1. In tests on three different f(x)′s, CPA converged better than the Taylor series at x0. Numerical integration of differential equations involving functions of an exponential nature may employ CPA for predictors in conjunction with Obreschkoff formulas for correctors