16,129 research outputs found
Computing special functions by using quadrature rules
The usual tools for computing special functions are power series, asymptotic expansions, continued fractions, differential equations, recursions, and so on. Rather seldom are methods based on quadrature of integrals. Selecting suitable integral representations of special functions, using principles from asymptotic analysis, we develop reliable algorithms which are valid for large domains of real or complex parameters. Our present investigations include Airy functions, Bessel functions and parabolic cylinder functions. In the case of Airy functions we have improvements in both accuracy and speed for some parts of Amos's code for Bessel functions
On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations
We describe a method for calculating the roots of special functions
satisfying second order linear ordinary differential equations. It exploits the
recent observation that the solutions of a large class of such equations can be
represented via nonoscillatory phase functions, even in the high-frequency
regime. Our algorithm achieves near machine precision accuracy and the time
required to compute one root of a solution is independent of the frequency of
oscillations of that solution. Moreover, despite its great generality, our
approach is competitive with specialized, state-of-the-art methods for the
construction of Gaussian quadrature rules of large orders when it used in such
a capacity. The performance of the scheme is illustrated with several numerical
experiments and a Fortran implementation of our algorithm is available at the
author's website
On the computation of Gaussian quadrature rules for Chebyshev sets of linearly independent functions
We consider the computation of quadrature rules that are exact for a
Chebyshev set of linearly independent functions on an interval . A
general theory of Chebyshev sets guarantees the existence of rules with a
Gaussian property, in the sense that basis functions can be integrated
exactly with just points and weights. Moreover, all weights are positive
and the points lie inside the interval . However, the points are not the
roots of an orthogonal polynomial or any other known special function as in the
case of regular Gaussian quadrature. The rules are characterized by a nonlinear
system of equations, and earlier numerical methods have mostly focused on
finding suitable starting values for a Newton iteration to solve this system.
In this paper we describe an alternative scheme that is robust and generally
applicable for so-called complete Chebyshev sets. These are ordered Chebyshev
sets where the first elements also form a Chebyshev set for each . The
points of the quadrature rule are computed one by one, increasing exactness of
the rule in each step. Each step reduces to finding the unique root of a
univariate and monotonic function. As such, the scheme of this paper is
guaranteed to succeed. The quadrature rules are of interest for integrals with
non-smooth integrands that are not well approximated by polynomials
The Kink Phenomenon in Fejér and Clenshaw-Curtis Quadrature
The Fejér and Clenshaw-Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N, (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like , where is a constant greater than 1. For these values of N the accuracy of both the Fejér and Clenshaw-Curtis rules is almost indistinguishable from that of the more celebrated Gauss-Legendre quadrature rule. For larger N, however, the error decreases at the rate , i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.\ud
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This work was supported by the Royal Society of the UK and the National Research Foundation of South Africa under the South Africa-UK Science Network Scheme. The first author also acknowledges grant FA2005032300018 of the NRF
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