Richard Mikael Slevinsky

A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in

New formulae for higher order derivatives and applications

{\cal O}(m^2n)

The double exponential sinc collocation method for singular Sturm-Liouville problems

operations using an adaptive QR factorization, where

On The Use of Conformal Maps for the Acceleration of Convergence of the Trapezoidal Rule and Sinc Numerical Methods

m

Computation of Tail Probabilities via Extrapolation Methods and Connection with Rational and Padé Approximants

is the bandwidth and

A comparative study of numerical steepest descent, extrapolation, and sequence transformation methods in computing semi-infinite integrals

n

One- and two-center ETF-integrals of first order in relativistic calculation of NMR parameters

is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to

Useful properties of the coefficients of the Slevinsky-Safouhi formula for differentiation

{\cal O}(m n)

A spectral method for nonlocal diffusion operators on the sphere

operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface.

NEW TECHNIQUES IN NUMERICAL INTEGRATION: THE COMPUTATION OF MOLECULAR INTEGRALS OVER EXPONENTIAL-TYPE FUNCTIONS

We present new formulae (the Slevinsky-Safouhi formulae I and II) for the analytical development of higher order derivatives. These formulae, which are analytic and exact, represent the kth derivative as a discrete sum of only k+1 terms. Involved in the expression for the kth derivative are coefficients of the terms in the summation. These coefficients can be computed recursively and they are not subject to any computational instability. As examples of applications, we develop higher order derivatives of Legendre functions, Chebyshev polynomials of the first kind, Hermite functions and Bessel functions. We also show the general classes of functions to which our new formula is applicable and show how our formula can be applied to certain classes of differential equations. We also presented an application of the formulae of higher order derivatives combined with extrapolation methods in the numerical integration of spherical Bessel integral functions.