620,555 research outputs found
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
Numerical Calculation of Bessel Functions
A new computational procedure is offered to provide simple, accurate and
flexible methods for using modern computers to give numerical evaluations of
the various Bessel functions. The Trapezoidal Rule, applied to suitable
integral representations, may become the method of choice for evaluation of the
many Special Functions of mathematical physics.Comment: 10 page
Stochastic B-series analysis of iterated Taylor methods
For stochastic implicit Taylor methods that use an iterative scheme to
compute their numerical solution, stochastic B--series and corresponding growth
functions are constructed. From these, convergence results based on the order
of the underlying Taylor method, the choice of the iteration method, the
predictor and the number of iterations, for It\^o and Stratonovich SDEs, and
for weak as well as strong convergence are derived. As special case, also the
application of Taylor methods to ODEs is considered. The theory is supported by
numerical experiments
Geometric numerical integration of nonholonomic systems and optimal control problems
A geometric derivation of numerical integrators for nonholonomic systems and
optimal control problems is obtained. It is based in the classical technique of
generating functions adapted to the special features of nonholonomic systems
and optimal control problems.Comment: 6 pages, 1 figure. Submitted to IFAC Workshop on Lagrangian and
Hamiltonian Methods for Nonlinear Control, Sevilla 200
Numerical calculation of Bessel, Hankel and Airy functions
The numerical evaluation of an individual Bessel or Hankel function of large
order and large argument is a notoriously problematic issue in physics.
Recurrence relations are inefficient when an individual function of high order
and argument is to be evaluated. The coefficients in the well-known uniform
asymptotic expansions have a complex mathematical structure which involves Airy
functions. For Bessel and Hankel functions, we present an adapted algorithm
which relies on a combination of three methods: (i) numerical evaluation of
Debye polynomials, (ii) calculation of Airy functions with special emphasis on
their Stokes lines, and (iii) resummation of the entire uniform asymptotic
expansion of the Bessel and Hankel functions by nonlinear sequence
transformations.
In general, for an evaluation of a special function, we advocate the use of
nonlinear sequence transformations in order to bridge the gap between the
asymptotic expansion for large argument and the Taylor expansion for small
argument ("principle of asymptotic overlap"). This general principle needs to
be strongly adapted to the current case, taking into account the complex phase
of the argument. Combining the indicated techniques, we observe that it
possible to extend the range of applicability of existing algorithms. Numerical
examples and reference values are given.Comment: 18 pages; 7 figures; RevTe
Numerical aspects of special functions
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions)
Numerical aspects of special functions
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions)
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
Particle algorithms for optimization on binary spaces
We discuss a unified approach to stochastic optimization of pseudo-Boolean
objective functions based on particle methods, including the cross-entropy
method and simulated annealing as special cases. We point out the need for
auxiliary sampling distributions, that is parametric families on binary spaces,
which are able to reproduce complex dependency structures, and illustrate their
usefulness in our numerical experiments. We provide numerical evidence that
particle-driven optimization algorithms based on parametric families yield
superior results on strongly multi-modal optimization problems while local
search heuristics outperform them on easier problems
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