Quadrature formulas

Using the Gaussian quadrature formulas gives the following values for various ... 

Some other typical Gaussian quadrature formulas are (a,b) W(x) Gauss-... 

Weights and zeros for the above formulas (and for other Gaussian formulas) may he found in references such as Stroud Gaussian Quadrature Formulas, Prentice-Hall, 1966). 

The approximation of the integral on the right-hand side of (5) can be done using various quadrature formulae, for instance, by the formula of trapezoids... 

Using the Gaussian quadrature formulas gives the following values for various values ofm. Clearly, three internal points, requiring evaluation of the integrand at only three points, gives excellent results. 

Note that each environment in the micromixing model will have its own set of concentrations can and moments mkn, reflecting the fact that the PSD is coupled to the chemistry and will thus be different at every SGS point in the flow. The PD algorithm is applied separately in each environment to compute the weights (wmn) and abscissa (lmn) from the quadrature formula as follows ... 

Day s method essentially replaces the integral in Eq. (B.6) by a different quadrature formula at each of the times tut2, and t3. Each of these quadrature formulas approximates its appropriate integral with an error which is 0(At5). Therefore, this method gives very accurate starting values for Ky(h), Ky(t2), and Ky(t3). 

For an isotropic medium, by substituting the Gaussian quadrature formula for the integral in Eq. (4.78), the integral-differential equation may be reduced into a system of ordinary linear differential equations. Specifically, the integral can be treated as [Chandrasekhar, 1960]... 

Unlike Xi, which in principle cannot be evaluated analytically at arbitrary a [90] for Xjnt an exact solution is possible for arbitrary values of the anisotropy parameter. Two ways were proposed to obtain quadrature formulas for Tjnt. One method [91] implies a direct integration of the Fokker-Planck equation. Another method [58] involves solving three-term recurrence relations for the statistical moments of W. The emerging solution for Tjnt can be expressed in a finite form in terms of hypergeometric (Kummer s) functions. Equivalence of both approaches was proved in Ref. 92. 

Some quadrature formula, such as Simpson s rule, gives the solution faster than does numerical integration of the differential equation by a general method. 

Stroud, A.H. and Secrest, D. (1966). Gatissian Quadrature Formulas (Prentice Hall, Englewood Cliffs, N.J.). 

A better solution is to use one of the many existing quadrature formulas—algebraic expressions that provide estimates of the integrals of tabulated data. Several such formulas are presented and illustrated in Appendix A.3 the use of one of them, Simpson s rule, is required for the integration of heat capacity data in several problems at the end of this chapter. 

The many existing quadrature formulas differ only in the choice of functions to fit to the data points. Two of the simplest approximations are to fit straight lines between successive points and sum the area under the lines, and to fit parabolas to successive triplets of points and sum the areas under the parabolas. These approximations lead to the quadrature formulas known respectively as the trapezoidal rule and Simpson s rule. We will discuss each in turn. 

The trapezoidal rule is an approximation, as are all quadrature formulas. Figure A.3-1 illustrates the nature of the error introduced by its use. The integral to be evaluated... 

A second and more accurate quadrature formula is the one most often used. It is applicable only to an odd number of equally spaced data points and is based on fitting parabolic functions to successive groups of three points. 

When you have an analytical expression for an integrand y(x) but you cannot perform the integration from jr = a to -c = b analytically, the procedure is to evaluate y at a series of values of X from a to b—that is, to generate a data table—and then to use a quadrature formula such as Simpson s rule to estimate the integral. Now, however, you have the choice of the number of evaluations of y(x) to make. 

Eq (13) can be discretized using any quadrature formula and written in algebraic form as follows ... 

Figure A-4 Simpson s four-point rule illustratioa 4. Five-point quadrature formula ...

V. 1. Lebedev (1977) Spherical quadrature formulas exact to order-25-order-29. Siberian Mathematical Journal 18(1), pp. 99-107... 

The Gaussian quadrature formula was used to perform the integration. The above procedure was repeated for each isomer in the set finally, the mean thermodynamic quantities (5)-(10) were calculated. 

Abstract An improvement is made to an automatic quadrature due to Ninomiya (J. Inf. Process. 3 162-170, 1980) of adaptive type based on the Newton-Cotes rule by incorporating a doubly-adaptive algorithm due to Favati, Lotti and Romani (ACM Trans. Math. Softw. 17 207-217,1991 ACM Trans. Math. Softw. 17 218-232, 1991). We compare the present method in performance with some others by using various test problems including Kahaner s ones (Computation of numerical quadrature formulas. In Rice, J.R. (ed.) Mathematical Software, 229-259. Academic, Orlando, FL, 1971). 

Eavati, P., Lotti, G., Romani, E. Theoretical and practical efficiency measures for symmetric interpolatory quadrature formulas. BIT 34,546-557 (1994)... 

Kahaner, D.K. Computation of numerical quadrature formulas. In Rice, J.R. (ed.) Mathematical Software, 229-259. Academic, Orlando, FL (1971)... 

Many quadrature formulas approximate the integral by a weighted sum of the values of the integrand at particular points on the interval of integration, that is, by [149, 50, 134, 93] ... 

One of the easiest ways to obtain useful quadrature formulas for our purpose is to use Lagrangian interpolation on an equally spaced mesh and integrate the result. This class of quadrature formulas is called the Newton-Cotes... 

An alternative closed Newton-Cotes quadrature formula of second order can be obtained by a polynomial of degree 1 which passes through the end points. This quadrature formula is called the trapezoid rule. In 2D this surface integral approximation requires the integrand values at the GCV corners. 

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