Joris Van Deun

Algorithm 882: Near-Best Fixed Pole Rational Interpolation with Applications in Spectral Methods

We present a numerical procedure to compute the nodes and weights in rational Gauss-Chebyshev quadrature formulas. Under certain conditions on the poles, these nodes are near best for rational interpolation with prescribed poles (in the same sense that Chebyshev points are near best for polynomial interpolation). As an illustration, we use these interpolation points to solve a differential equation with an interior boundary layer using a rational spectral method. The algorithm to compute the interpolation points (and, if required, the quadrature weights) is implemented as a Matlab program.

Integrating products of Bessel functions with an additional exponential or rational factor

Abstract We provide two Matlab programs to compute integrals of the form ∫ 0 ∞ e − c x x m ∏ i = 1 k J ν i ( a i x ) d x and ∫ 0 ∞ x m r 2 + x 2 ∏ i = 1 k J ν i ( a i x ) d x with J ν i ( x ) the Bessel function of the first kind and (real) order ν i . The parameter m is a real number such that ∑ i ν i + m > − 1 (to assure integrability near zero), r is real and the numbers c and a i are all strictly positive. The program can deliver accurate error estimates. Program summary Program title: BESSELINTR, BESSELINTC Catalogue identifier: AEAH_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAH_v1_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1601 No. of bytes in distributed program, including test data, etc.: 13 161 Distribution format: tar.gz Programming language: Matlab (version ⩾6.5), Octave (version ⩾ 2.1 . 69 ) Computer: All supporting Matlab or Octave Operating system: All supporting Matlab or Octave RAM: For k Bessel functions our program needs approximately ( 500 + 140 k ) double precision variables Classification: 4.11 Nature of problem: The problem consists in integrating an arbitrary product of Bessel functions with an additional rational or exponential factor over a semi-infinite interval. Difficulties arise from the irregular oscillatory behaviour and the possible slow decay of the integrand, which prevents truncation at a finite point. Solution method: The interval of integration is split into a finite and infinite part. The integral over the finite part is computed using Gauss–Legendre quadrature. The integrand on the infinite part is approximated using asymptotic expansions and this approximation is integrated exactly with the aid of the upper incomplete gamma function. In the case where a rational factor is present, this factor is first expanded in a Taylor series around infinity. Restrictions: Some (and eventually all) numerical accuracy is lost when one or more of the parameters r , c , a i or v i grow very large, or when r becomes small. Running time: Less than 0.02 s for a simple problem (two Bessel functions, small parameters), a few seconds for a more complex problem (more than six Bessel functions, large parameters), in Matlab 7.4 (R2007a) on a 2.4 GHz AMD Opteron Processor 250. References: [1] J. Van Deun, R. Cools, Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions, ACM Trans. Math. Software 32 (4) (2006) 580–596.

Eigenvalue problems to compute almost optimal points for rational interpolation with prescribed poles

Explicit formulas exist for the (n,m) rational function with monic numerator and prescribed poles that has the smallest possible Chebyshev norm. In this paper we derive two different eigenvalue problems to obtain the zeros of this extremal function. The first one is an ordinary tridiagonal eigenvalue problem based on a representation in terms of Chebyshev polynomials. The second is a generalised tridiagonal eigenvalue problem which we derive using a connection with orthogonal rational functions. In the polynomial case (m = 0) both problems reduce to the tridiagonal eigenvalue problem associated with the Chebyshev polynomials of the first kind.

Validated computation of certain hypergeometric functions

We present an efficient algorithm for the validated high-precision computation of real continued fractions, accurate to the last digit. The algorithm proceeds in two stages. In the first stage, computations are done in double precision. A forward error analysis and some heuristics are used to obtain an a priori error estimate. This estimate is used in the second stage to compute the fraction to the requested accuracy in high precision (adaptively incrementing the precision for reasons of efficiency). A running error analysis and techniques from interval arithmetic are used to validate the result. As an application, we use this algorithm to compute Gauss and confluent hypergeometric functions when one of the numerator parameters is a positive integer.

Validated evaluation of special mathematical functions

Because of the importance of special functions, several books and a large collection of papers have been devoted to their use and computation, the most well-known being the Abramowitz and Stegun handbook (Abramowitz and Stegun, 1964) 1] and its successor (Olver et al. 0000) 2]. However, until now no environment offers routines for the provable correct multiprecision and radix-independent evaluation of these special functions. We point out how we make good use of series and limit-periodic continued fraction representations in a package that is being developed at the University of Antwerp. Our scalable precision technique is mainly based on the use of sharpened a priori truncation and round-off error upper bounds for real arguments. The implementation is validated in the sense that it returns a sharp interval enclosure for the requested function evaluation, at the same cost as the evaluation.

Computing near-best fixed pole rational interpolants

We study rational interpolation formulas on the interval [-1,1] for a given set of real or complex conjugate poles outside this interval. Interpolation points which are near-best in a Chebyshev sense were derived in earlier work. The present paper discusses several computation aspects of the interpolation points and the corresponding interpolants. We also study a related set of points (that includes the end points), which is more suitable for applications in rational spectral methods. Some examples are given at the end of this paper.

Validated special functions software

Because of the importance of special functions, several books and a large collection of papers have been devoted to the numerical computation of these functions, the most well-known being the NBS handbook by Abramowitz and Stegun. But up to this date, symbolic and numeric environments offer no routines for the validated evaluation of special functions. We point out how a provable correct function evaluation can be returned efficiently.

Continued Fractions for Special Functions: Handbook and Software

The revived interest in continued fractions stems from the fact that many special functions enjoy easy to handle and rapidly converging continued fraction representations. These can be made to good use in a project that envisages the provably correct (or interval) evaluation of these functions. Of course, first a catalogue of these continued fraction representations needs to be put together. The Handbook of continued fractions for special functions is the result of a systematic study of series and continued fraction representations for several families of mathematical functions used in science and engineering. Only 10% of the listed continued fraction representations can also be found in the famous NBS Handbook edited by Abramowitz and Stegun. More information is given in Sect. 1. The new handbook is brought to life at the website www.cfhblive.ua.ac.be where visitors can recreate tables to their own specifications, and can explore the numerical behaviour of the series and continued fraction representations. An easy web interface supporting these features is discussed in the Sects. 2, 3 and 4.