4,393 research outputs found
Chebyshev Expansions for Solutions of Linear Differential Equations
A Chebyshev expansion is a series in the basis of Chebyshev polynomials of
the first kind. When such a series solves a linear differential equation, its
coefficients satisfy a linear recurrence equation. We interpret this equation
as the numerator of a fraction of linear recurrence operators. This
interpretation lets us give a simple view of previous algorithms, analyze their
complexity, and design a faster one for large orders
Rational chebyshev spectral methods for unbounded solutions on an infinite interval using polynomial-growth special basis functions
AbstractIn the method of matched asymptotic expansions, one is often forced to compute solutions which grow as a polynomial in y as |y| → ∞. Similarly, the integral or repeated integral of a bounded function f(y) is generally unbounded also. The kth integral of a function f(y) solves . We describe a two-part algorithm for solving linear differential equations on y ϵ [−∞, ∞] where u(y) grows as a polynomial as |y| → ∞. First, perform an explicit, analytic transformation to a new unknown v so that v is bounded. Second, expand v as a rational Chebyshev series and apply a pseudospectral or Galerkin discretization. (For our examples, it is convenient to perform a preliminary step of splitting the problem into uncoupled equations for the parts of u which are symmetric and antisymmetric with respect to y = 0, but although this is very helpful when applicable, it is not necessary.) For the integral and interated integrals and for constant coefficient differential equations in general, the Galerkin matrices are banded with very low bandwidth. We derive an improvement on the “last coefficient error estimate” of the author's book which applies to series with a subgeometric rate of convergence, as is normally true of rational Chebyshev expansions
An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments
We describe a method for the rapid numerical evaluation of the Bessel
functions of the first and second kinds of nonnegative real orders and positive
arguments. Our algorithm makes use of the well-known observation that although
the Bessel functions themselves are expensive to represent via piecewise
polynomial expansions, the logarithms of certain solutions of Bessel's equation
are not. We exploit this observation by numerically precomputing the logarithms
of carefully chosen Bessel functions and representing them with piecewise
bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions
of orders between and 1\sep,000\sep,000\sep,000 at essentially any
positive real argument. In that regime, it is competitive with existing methods
for the rapid evaluation of Bessel functions and has several advantages over
them. First, our approach is quite general and can be readily applied to many
other special functions which satisfy second order ordinary differential
equations. Second, by calculating the logarithms of the Bessel functions rather
than the Bessel functions themselves, we avoid many issues which arise from
numerical overflow and underflow. Third, in the oscillatory regime, our
algorithm calculates the values of a nonoscillatory phase function for Bessel's
differential equation and its derivative. These quantities are useful for
computing the zeros of Bessel functions, as well as for rapidly applying the
Fourier-Bessel transform. The results of extensive numerical experiments
demonstrating the efficacy of our algorithm are presented. A Fortran package
which includes our code for evaluating the Bessel functions as well as our code
for all of the numerical experiments described here is publically available
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
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