Related papers: End-point corrections for the midpoint rule
In this work, we construct a new general two-point quadratre rules for the Riemann--Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$, where the integrand $f$ is assumed to be satisfied with the H\"{o}lder condition on…
Many high dimensional integrals can be reduced to the problem of finding the relative measures of two sets. Often one set will be exponentially larger than the other, making it difficult to compare the sizes. A standard method of dealing…
Inaccurate circuits make possible the conservation of limited resources, such as energy. But effective design of such circuits requires an understanding of resulting tradeoffs between accuracy and design parameters, such as voltages and…
We present a novel numerical method for solving ODEs while preserving polynomial first integrals. The method is based on introducing multiple quadratic auxiliary variables to reformulate the ODE as an equivalent but higher-dimensional ODE…
In this paper, we present a study of an a posteriori estimator for the discretization error of a non-standard finite difference scheme applied to boundary value problems defined on an infinite interval. In particular, we show how…
Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear…
In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and…
Currently, mass matrices are computed by means of sufficiently accurate numerical integration schemes. Two-point and nine-point (Gauss) quadrature remain frequently used. We derive an exact, easy to implement integration rule for six-node…
This research investigates using a mixed-precision iterative refinement method using posit numbers instead of the standard IEEE floating-point format. The method is applied to solve a general linear system represented by the equation $Ax =…
The traditional approach to fault tolerant computing involves replicating computation units and applying a majority vote operation on individual result bits. This approach, however, has several limitations; the most severe is the resource…
Interval arithmetic is a simple way to compute a mathematical expression to an arbitrary accuracy, widely used for verifying floating-point computations. Yet this simplicity belies challenges. Some inputs violate preconditions or cause…
Point processes are finding growing applications in numerous fields, such as neuroscience, high frequency finance and social media. So classic problems of classification and clustering are of increasing interest. However, analytic study of…
It is well known that, in theory, the general secure multi-party computation problem is solvable using circuit evaluation protocols. However, the communication complexity of the resulting protocols depend on the size of the circuit that…
A new numerical method for solving a scalar ordinary differential equation with a given initial condition is introduced. The method is using a numerical integration procedure for an equivalent integral equation and is called in this paper…
The textbook Newton's iteration is practically inapplicable on solutions of nonlinear systems with singular Jacobians. By a simple modification, a novel extension of Newton's iteration regains its local quadratic convergence toward…
For continuous boundary data, including data of polynomial growth, modified Poisson integrals are used to write solutions to the half space Dirichlet and Neumann problems in $\mathbb{R}^{n}$. Pointwise growth estimates for these integrals…
Neural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may…
We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields…
An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…
It is well-known that many famous pooling designs are constructed from mathematical structures by the "containment matrix" method. In this paper, we propose another method and obtain a family of pooling designs with surprisingly high degree…