On the approximation of an integral by a sum of random variables

We approximate the integral of a smooth function on [0,1], where values are only known at n random points (i.e., a random sample from the uniform-(0,1) distribution), and at 0 and 1. Our approximations are based on the trapezoidal rule and Simpson's rule (generalized to the non-equidistant case), respectively. In the first case, we obtain an n2-rate of convergence with a degenerate limiting distribution; in the second case, the rate of con-vergence is as fast as n3½, whereas the limiting distribution is Gaussian then

On the new trapezoid rule with the same precision as that of the Simpson's rule

In this study, we first try to establish the generalized trapezoid rule in the theory of numerical integrals by the inverse calculation from the order of error estimations. Next, we shall prove that there exists one and only one trapezoid rule that has the same precision as that of the Simpson's rule. Finally, we shall give the concrete form of the above new trapezoid rule

Analysis and design of integration formulas for a random integrand

Analysis of integration formulas and procedure for designing optimal integration formul

Numerical residual perturbation solution applied to an earth satellite including luni-solar effects

Mathematical model and computer program for numerical solution of earth orbit differential equations of motio

Flux Jacobian matrices and generaled Roe average for an equilibrium real gas

Inviscid flux Jacobian matrices and their properties used in numerical solutions of conservation laws are extended to general, equilibrium gas laws. Exact and approximate generalizations of the Roe average are presented. Results are given for one-dimensional flow, and then extended to three-dimensional flow with time-varying grids

Vibronic dephasing of anharmonic molecules. II. Impurity molecules isolated in low-temperature matrices

The quantum‐mechanical theory of vibronic dephasing presented in the first paper of this series is applied to the case of a diatomic impurity dissolved in a solid rare‐gas host. An explicit expression for the pure dephasing rate T_2′^(−1) is derived in terms of microscopic properties of the impurity and host, and the effects of variations in the parameters characterizing these properties are investigated. The expression for T_2′^(−1) is applied specifically to the system Cl_2/Ar in order to relate the results to those of previous classical‐trajectory calculations and of experimental measurements. The significance of anharmonicity in the intramolecular potential curve (of the impurity) is demonstrated

Deriving the Composite Simpson Rule by Using Bernstein Polynomials for Solving Volterra Integral Equations

In this paper we use Bernstein polynomials for deriving the modified Simpson's 3/8 , and the composite modified Simpson's 3/8 to solve one dimensional linear Volterra integral equations of the second kind , and we find that the solution computed by this procedure is very close to exact solution