1,939 research outputs found
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
Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix
This paper studies how to sketch element-wise functions of low-rank matrices.
Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we
aim for finding an approximated low-rank representation of the (possibly
high-rank) matrix [f(Aij)]. To this end, we propose an efficient
sketching-based algorithm whose complexity is significantly lower than the
number of entries of A, i.e., it runs without accessing all entries of [f(Aij)]
explicitly. The main idea underlying our method is to combine a polynomial
approximation of f with the existing tensor sketch scheme for approximating
monomials of entries of A. To balance the errors of the two approximation
components in an optimal manner, we propose a novel regression formula to find
polynomial coefficients given A and f. In particular, we utilize a
coreset-based regression with a rigorous approximation guarantee. Finally, we
demonstrate the applicability and superiority of the proposed scheme under
various machine learning tasks
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
A numerical method for oscillatory integrals with coalescing saddle points
The value of a highly oscillatory integral is typically determined
asymptotically by the behaviour of the integrand near a small number of
critical points. These include the endpoints of the integration domain and the
so-called stationary points or saddle points -- roots of the derivative of the
phase of the integrand -- where the integrand is locally non-oscillatory.
Modern methods for highly oscillatory quadrature exhibit numerical issues when
two such saddle points coalesce. On the other hand, integrals with coalescing
saddle points are a classical topic in asymptotic analysis, where they give
rise to uniform asymptotic expansions in terms of the Airy function. In this
paper we construct Gaussian quadrature rules that remain uniformly accurate
when two saddle points coalesce. These rules are based on orthogonal
polynomials in the complex plane. We analyze these polynomials, prove their
existence for even degrees, and describe an accurate and efficient numerical
scheme for the evaluation of oscillatory integrals with coalescing saddle
points
Fast Computing on Vehicle Dynamics Using Chebyshev Series Expansions.
This article focusses on faster computation techniques
to integrate mechanical models in electronic advanced active
safety applications. It shows the different techniques of
approximation in series of functions and differential equations
applied to vehicle dynamics. This allows the achievement of
approximate polynomial and rational solutions with a very fast
and efficient computation. Firstly, the whole theoretical basic
principles related to the techniques used are presented:
orthogonality of functions, function expansion in Chebyshev and
Jacobi series, approximation through rational functions, the
Minimax-Remez algorithm, orthogonal rational functions
(ORF’s) and the perturbation of dynamic systems theory, that
reduces the degree of the expansion polynomials used.
As an application, it is shown the obtaining of approximate
solutions to the longitudinal dynamics, vertical dynamics, steering
geometry and a tyre model, all obtained through development in
series of orthogonal functions with a computation much faster
than those of its equivalents in the classic vehicle theory. These
polynomial partially symbolic solutions present very low errors
and very favourable analytical properties due to their simplicity,
becoming ideal for real time computation as those required for
the simulation of evasive manoeuvres prior a crash. This set of
techniques had never been applied to vehicle dynamics before.pre-print748 K
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