18,002 research outputs found
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
Formal Proofs for Nonlinear Optimization
We present a formally verified global optimization framework. Given a
semialgebraic or transcendental function and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of by suprema of quadratic forms with a well
chosen curvature. Thus, we reduce the initial goal to a hierarchy of
semialgebraic optimization problems, solved by sums of squares relaxations. Our
implementation tool interleaves semialgebraic approximations with sums of
squares witnesses to form certificates. It is interfaced with Coq and thus
benefits from the trusted arithmetic available inside the proof assistant. This
feature is used to produce, from the certificates, both valid underestimators
and lower bounds for each approximated constituent. The application range for
such a tool is widespread; for instance Hales' proof of Kepler's conjecture
yields thousands of multivariate transcendental inequalities. We illustrate the
performance of our formal framework on some of these inequalities as well as on
examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table
Detecting Simultaneous Integer Relations for Several Real Vectors
An algorithm which either finds an nonzero integer vector for
given real -dimensional vectors such
that or proves that no such integer vector with
norm less than a given bound exists is presented in this paper. The cost of the
algorithm is at most exact arithmetic
operations in dimension and the least Euclidean norm of such
integer vectors. It matches the best complexity upper bound known for this
problem. Experimental data show that the algorithm is better than an already
existing algorithm in the literature. In application, the algorithm is used to
get a complete method for finding the minimal polynomial of an unknown complex
algebraic number from its approximation, which runs even faster than the
corresponding \emph{Maple} built-in function.Comment: 10 page
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
Evaluating parametric holonomic sequences using rectangular splitting
We adapt the rectangular splitting technique of Paterson and Stockmeyer to
the problem of evaluating terms in holonomic sequences that depend on a
parameter. This approach allows computing the -th term in a recurrent
sequence of suitable type using "expensive" operations at the cost
of an increased number of "cheap" operations.
Rectangular splitting has little overhead and can perform better than either
naive evaluation or asymptotically faster algorithms for ranges of
encountered in applications. As an example, fast numerical evaluation of the
gamma function is investigated. Our work generalizes two previous algorithms of
Smith.Comment: 8 pages, 2 figure
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