Related papers: A general correction for numerical integration rul…

In this paper, we propose a numerical method of computing an integral whose integrand is a slowly decaying oscillatory function. In the proposed method, we consider a complex analytic function in the upper-half complex plane, which is…

We study the numerical anisotropy existent in compact difference schemes as applied to hyperbolic partial differential equations, and propose an approach to reduce this error and to improve the stability restrictions based on a previous…

Numerical evaluations of Feynman integrals often proceed via a deformation of the integration contour into the complex plane. While valid contours are easy to construct, the numerical precision for a multi-loop integral can depend…

We present a method of concatenated quantum error correction in which improved classical processing is used with existing quantum codes and fault-tolerant circuits to more reliably correct errors. Rather than correcting each level of a…

An important question in the theory of approximate integration is to study the conditions on the nodes $x_{k,n}$ and weights $w_{k,n}$ that allow an estimate of the form $$ \sup_{f\in \mathcal{B}_\gamma}|\sum_k…

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes…

Partial differential equations sometimes have critical points where the solution or some of its derivatives are discontinuous. The simplest example is a discontinuity in the initial condition. It is well known that those decrease the…

In this paper, we propose a new and simple approach to the approximation algorithms that are modified and improved from our published results. The computational and graphical examples are presented with the aid of Maple procedures.

We establish an integration by parts formula in an abstract framework in order to study the regularity of the law for processes solution of stochastic differential equations with jumps, including equations with discontinuous coefficients…

We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces…

Corrected trapezoidal rules are proved for $\int_a^b f(x)\,dx$ under the assumption that $f"\in L^p([a,b])$ for some $1\leq p\leq\infty$. Such quadrature rules involve the trapezoidal rule modified by the addition of a term…

We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We…

Using typical solution strategies to compute the solution curve of challenging problems often leads to the break down of the algorithm. To improve the solution process, numerical continuation methods have proved to be a very efficient tool.…

The elementary resolution of singularities algorithm of the author's earlier paper (math.CA/0609217) is developed further, replacing the quasibump functions in the blown up coordinates with the characteristic function of a rectangle times a…

Quasi-Monte Carlo (QMC) methods are being adopted in statistical applications due to the increasingly challenging nature of numerical integrals that are now routinely encountered. For integrands with $d$-dimensions and derivatives of order…

This paper offers what seems at first to be a minor technical correction to the current practice of computing indefinite integrals, and introduces the idea of a "Kahanian constant of integration". However, the total impact of this minor…

The two-dimensional n-body problem of classical mechanics is a non-integrable Hamiltonian system for n > 2. Traditional numerical integration algorithms, which are polynomials in the time step, typically lead to systematic drifts in the…

Closed formulae for all Gaussian or optimal, 1-parameter quadrature rules in a compact interval [a, b] with non uniform, asymmetric subintervals, arbitrary number of nodes per subinterval for the spline classes $S_{2N, 0}$ and $S_{2N+1,…

We introduce quasi-Monte Carlo rules for the numerical integration of functions $f$ defined on $[0,1]^s$, $s \ge 1$, which satisfy the following properties: the Fourier-, Fourier cosine- or Walsh coefficients of $f$ are absolutely summable…

We present a new approach to termination analysis of numerical computations in logic programs. Traditional approaches fail to analyse them due to non well-foundedness of the integers. We present a technique that allows overcoming these…