Methods of Numerical Integration

Value of Standard Value of Peak all values are known short of the value of the peak, which thus can be determined. This technique is remarkably precise and only assumes that the thickness (density) of the paper is consistent throughout the sheet. 

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This improved procedure is an example of the Runge-Kutta method of numerical integration. Because the derivative was evaluated at two points in the interval, this is called a second-order Runge-Kutta process. We chose to evaluate the mean derivative at points Pq and Pi, but because there is an infinite number of points in the interval, an infinite number of choices for the two points could have been made. In calculating the average for such choices appropriate weights must be assigned. 

When all else fails, recourse to numerical methods is indicated. Some of the classic methods of numerical integration are described in Chapter 13. However, it should be emphasized that numerical methods are to be used as a last resort Not only are they subject to errors (often not easily evaluated), but they do not yield analytical results that can be employed in fbrther derivations (see p. 43). 

The two well-known methods of numerical integration described in the previous sections can be generalized. Represent the sum on the right-hand side of Eq. (59) as Sb( ). This function converges but very slowly towards... 

More accurate methods of numerical integration are described in the references (5). 

The convolution integral in (1.19) and (1.20) can be solved by the method of numerical integration proposed by Nicholson and Olmstead [47], The time t is divided into m time increments t = md. It is assumed that within each time increment the function 1 can be replaced by the average value Ij ... 

The simplest method of numerical integration is the trapezoidal method, where the abscissa is divided into intervals and each resulting area is estimated as the abscissa interval times the average of the initial and final ordinate in the interval ... 

A more accurate method of numerical integration is by the use of Simpson s rule. This method, however, requires that the integration range be divided into an even number of intervals of equal width h. This requires an odd number of points on the abscissa, which are numbered from 0 to n. Simpson s rule gives... 

We have used the Numerov-Cooley method of numerical integration to solve the Schrodinger equation... 

A method of numerical integration (or quadrature, as it is also called) is required to evaluate I in any of these cases. The specific techniques we will present are algebraic, but the general approach to the problem is best visualized graphically. For the moment, we will suppose that all we have relating x and y is a table of data points, which we may graph on a plot of y versus x. 

By the method of numerical integration of differential kinetic equations of the Scheme 3 it has been found that this scheme quantitatively describes experimental data on kinetics of accumulation of acetic acid and radicals at CA photolysis (Figure 2.1. - 2.4) at the following set of kinetic parameters ... 

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

Davis PJ, Rabinowitz P (1984) Methods of numerical integration. Academic, New York... 

There is another promising approach that is inspired by the methods of numerical integration. 

The underlying method of numerical integration is often called the GAUSS integration or Gauss quadrature. 

Runge-Kutta Merson method of numerical integration was used for the solution of this equation. 

The Monte Carlo method is a method of numerical integration that is becoming more and more popular as computers become faster. To find the area under a curve using the Monte Carlo method, the area in question is circumscribed by a rectangular area of known value, as shown in Fig. 11-3. Then it is determined whether points (x, y), randomly chosen within this area, fall above or below the curve and widiin the limits of the integration a and b. The probability that a point falls below the curve is equal to the number of points that actually fall below the curve, divided by the total number of points in the area Y AX. The area under the curve, then, is... 

This method works well provided a large number (several thousand) of points are tried. Because of the large number of calculations that must be performed, the speed of the computer becomes an important part of tiie decision whether to use this method of numerical integration. 

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