The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

Semi-classical scalar products in the generalised SU(2) model

In these notes we review the field-theoretical approach to the computation of the scalar product of multi-magnon states in the Sutherland limit where the magnon rapidities condense into one or several macroscopic arrays. We formulate a systematic procedure for computing the 1/M expansion of the on-shell/off-shell scalar product of M-magnon states in the generalised integrable model with SU(2)-invariant rational R-matrix. The coefficients of the expansion are obtained as multiple contour integrals in the rapidity plane.Comment: 13 pages, 3 figures. Based on a talk delivered at the X. International Workshop "Lie Theory and Its Applications in Physics", (LT-10), Varna, Bulgaria, 17-23 June 201

Computing the Gamma function using contour integrals and rational approximations

Some of the best methods for computing the gamma function are based on numerical evaluation of Hankel's contour integral. For example, Temme evaluates this integral based on steepest-decent contours by the trapezoid rule. Here we investigate a different approach to the integral: the application of the trapezoid rule on Talbot-type contours using optimal parameters recently derived by Weideman for computing inverse Laplace transforms. Relatedly, we also investigate quadrature formulas derived from best approximations to exp(z) on the negative real axis, following Cody, Meinardus and Varga. The two methods are closely related and both converge geometrically. We find that the new methods are competitive with existing ones, even though they are based on generic tools rather than on specific analysis of the gamma function

Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals

Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of $f(A)$, where $A$ is a negative definite matrix and $f$ is the exponential function or one of the related ``$\varphi$ functions'' such as $\varphi_1(z) = (e^z-1)/z$. Building on previous work by Trefethen and Gutknecht, Gonchar and Rakhmanov, and Lu, we propose two methods for the fast evaluation of $f(A)$ that are especially useful when shifted systems $(A+zI)x=b$ can be solved efficiently, e.g. by a sparse direct solver. The first method method is based on best rational approximations to $f$ on the negative real axis computed via the Carathéodory-Fejér procedure, and we conjecture that the accuracy scales as $(9.28903\dots)^{-2n}$, where $n$ is the number of complex matrix solves. In particular, three matrix solves suffice to evaluate $f(A)$ to approximately six digits of accuracy. The second method is an application of the trapezoid rule on a Talbot-type contour

Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves

We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves. Since it is easy to find such exact meromorphic one-forms from algebraic geometry, this idea provides a new highly efficient algorithm for integral reduction. We demonstrate the power of this method via several complicated two-loop diagrams with internal massive legs. No explicit elliptic or hyperelliptic integral computation is needed for our method.Comment: minor changes: more references adde

Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights

We design convergent multipoint Pade interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on our earlier work for the choice of the interpolation points, and dwell on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a segment. We also elaborate on the $\bar\partial$-extension of the Riemann-Hilbert technique, initiated by McLaughlin and Miller on the line to relax analyticity assumptions. This yields strong asymptotics for the denominator polynomials of the multipoint Pade interpolants, from which convergence follows.Comment: 42 pages, 3 figure

Fast and stable contour integration for high order divided differences via elliptic functions

In this paper, we will present a new method for evaluating high order divided differences for certain classes of analytic, possibly, operator valued functions. This is a classical problem in numerical mathematics but also arises in new applications such as, e.g., the use of generalized convolution quadrature to solve retarded potential integral equations. The functions which we will consider are allowed to grow exponentially to the left complex half plane, polynomially to the right half plane and have an oscillatory behaviour with increasing imaginary part. The interpolation points are scattered in a large real interval. Our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule