Abstract In this article, different types of Gaussian quadrature methods have been presented to find the numerical integration a neutrosophic valued function. A new definition distance between two number has defined and it proved that set all form complete metric space. Also, continuity on closed-bounded interval in sense $$(\alpha ,\beta ,\gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/...

Galerkin discretizations of integral operators in R d require the evaluation of integrals R S (1) R S (2) f (x, y) dydx where S (1) , S (2) are d-dimensional simplices and f has a singularity at x = y. In [3] we constructed a family of hp-quadrature rules Q N with N function evaluations for a class of integrands f allowing for algebraic singularities at x = y, possibly non-integrable with respe...

Recently Laurie presented a fast algorithm for the computation of (2n + 1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. We describe modifications of this algorithm that allow the computation of Gauss-Kronrod quadrature rules with complex conjugate nodes and weights or with real nodes and positive and negative weights.

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is proved to be equivalent to the linearized Bhatnagar-Gross-Krook (BGK) equation. Therefore, when the same Gauss-Hermite quadrature is used, LB method closely ass...

Kronrod extensions to two classes of Gauss and Lobatto integration rules for the evaluation of Cauchy principal value integrals are derived. Since in one frequently occurring case, the Kronrod extension involves evaluating the derivative of the integrand, a new extension is introduced using n + 2 points which requires only values of the integrand. However, this new rule does not exist for all n...

We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O’Hara and Smith in the 1960s, the phenomenon is explained as ...

This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and val...

A new numerical method for solving the kinetic collection equation (KCE) is proposed, and its accuracy and convergence are investigated. The method, herein referred to as the bin integral method with Gauss quadrature (BIMGQ), makes use of two binwise moments, namely, the number and mass concentration in each bin. These two degrees of freedom define an extended linear representation of the numbe...

In their paper published in 1952, Hestenes and Stiefel considered the conjugate gradient (CG) method an iterative method which terminates in at most n steps if no rounding errors are encountered [24, p. 410]. They also proved identities for the A-norm and the Euclidean norm of the error which could justify the stopping criteria [24, Theorems 6:1 and 6:3, p. 416]. The idea of estimating errors i...