We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is

An efficient algorithm for the accurate computation of Gauss–Legendre and Gauss– Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton’s root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The n-point quadrature rule is computed in O(n) operations to an accuracy of essentially double precision for any n ≥ 100.

This paper discusses about the solution of fuzzy Volterra integral equation of first-kind (F-VIE1) using spectral method. The parametric form of fuzzy driving term is applied for F-VIE1, then three classifications for (F-VIE1) are searched to solve them. These classifications are considered based on the interval sign of the kernel. The Gauss-Legendre points and Legendre weights for arithmetics ...

This paper presents a Gaussian quadrature method for the evaluation of the triple integral ( , , ) T I f x y z d xd yd z = ∫∫∫ , where ) , , ( z y x f is an analytic function in , , x y z and T refers to the standard tetrahedral region:{( , , ) 0 , , 1, 1} x y z x y z x y z ≤ ≤ + + ≤ in three space( , , ). x y z Mathematical transformation from ( , , ) x y z space to ( , , ) u v w space maps th...

Let I[f ] = ∫ 1 −1 f(x) dx, where f ∈ C ∞(−1, 1), and let Gn[f ] = ∑n i=1 wnif(xni) be the n-point Gauss–Legendre quadrature approximation to I[f ]. In this paper, we derive an asymptotic expansion as n → ∞ for the error En[f ] = I[f ]−Gn[f ] when f(x) has general algebraic-logarithmic singularities at one or both endpoints. We assume that f(x) has asymptotic expansions of the forms f(x) ∼ ∞ ∑ ...

in this paper, an effective technique is proposed to determine thenumerical solution of nonlinear volterra-fredholm integralequations (vfies) which is based on interpolation by the hybrid ofradial basis functions (rbfs) including both inverse multiquadrics(imqs), hyperbolic secant (sechs) and strictly positive definitefunctions. zeros of the shifted legendre polynomial are used asthe collocatio...

In the context of numerical methods for conservation laws, not only the preservation of the primary conserved quantities can be of interest, but also the balance of secondary ones such kinetic energy in case of the Euler equations of gas dynamics. In this work, we construct a kinetic energy preserving discontinuous Galerkin method on Gauss-Legendre nodes based on the framework of summation-by-p...

— We construct and analyze efficient fully discrete Galerkin type methods, that are of high order of accuracy and conservative in the L sense,for approximating the solution ofaform of the linear Schrödinger équation with a time-dependent coefficient, found e.g. in underwater acoustics. The time stepping procedures are based on the class of implicit Runge-Kutta methods known as the q-stage Gauss...