143 research outputs found
Numerically satisfactory solutions of Kummer recurrence relations
Pairs of numerically satisfactory solutions as for
the three-term recurrence relations satisfied by the families of
functions _1\mbox{F}_1(a+\epsilon_1 n; b +\epsilon_2 n;z),
, are given.
It is proved that minimal
solutions always exist, except
when and is in the positive or negative real axis, and that
_1\mbox{F}_1 (a+\epsilon_1 n; b +\epsilon_2 n;z)
is minimal as whenever
. The minimal solution is identified for any
recurrence direction, that is, for any integer values of and
.
When the
confluent limit
\lim_{b\rightarrow \infty}{}_1\mbox{F}_1(\gamma b;b;z)=e^{\gamma z},
with fixed,
is the main tool for identifying minimal solutions
together with a connection formula; for ,
\lim_{a\rightarrow +\infty} {}_1\mbox{F}_1(a;b;z)
/{}_0\mbox{F}_1(;b;az)=e^{z/2} is the main tool to be
considered
Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions
Conical functions appear in a large number of applications in physics and
engineering. In this paper we describe an extension of our module CONICAL for
the computation of conical functions. Specifically, the module includes now a
routine for computing the function , a
real-valued numerically satisfactory companion of the function for . In this way, a natural basis for solving
Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication
Identifying minimal and dominant solutions for Kummer recursions
We identify minimal and dominant solutions of three-term recurrence relations for the confluent hypergeometric functions and , where (not both equal to 0). The results are obtained by applying Perron's theorem, together with uniform asymptotic estimates derived by T.M. Dunster for Whittaker functions with large parameter values. The approximations are valid for complex values of , and , with
Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders
We consider the problem of computing satisfactory pairs of solutions of the
differential equation for Legendre functions of non-negative integer order
and degree , where is a non-negative real
parameter. Solutions of this equation are the conical functions
and ,
. An algorithm for computing a numerically satisfactory pair of solutions
is already available when (see \cite{gil:2009:con},
\cite{gil:2012:cpc}).In this paper, we present a stable computational scheme
for a real valued numerically satisfactory companion of the function
for , the function
. The
proposed algorithm allows the computation of the function on a large parameter
domain without requiring the use of extended precision arithmetic.Comment: To be published in Numerical Algoritm
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
Computing the Kummer function for small values of the arguments
We describe methods for computing the Kummer function for small values of ,
with special attention to small values of . For these values of the connection formula
that represents as a linear combination of two -functions needs a limiting
procedure. We use the power series of the -functions and consider the terms for which this
limiting procedure is needed. We give recursion relations for higher terms in the expansion, and we
consider the derivative as well.
We also discuss the performance for small of an asymptotic approximation
of the Kummer function in terms of modified Bessel functions
Basic Methods for Computing Special Functions
This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are
frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website
On the computation and inversion of the cumulative noncentral beta distribution function
The computation and inversion of the noncentral beta distribution Bp,q(x, y) (or the noncentral F-distribution, a particular case of Bp,q(x, y)) play an important role in different applications. In this paper we study the stability of recursions satisfied by Bp,q(x, y) and its complementary function and describe asymptotic expansions useful for computing the function when the parameters are large. We also consider the inversion problem of finding x or y when a value of Bp,q(x, y) is given. We provide approximations to x and y which can be used as starting values of methods for solving nonlinear equations (such as Newton) if higher accuracy is needed
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