Simpson’s rule integration procedure

Figure 2.2 The Simpson s rule integration procedure, on an EXCEL spreadsheet, to determine the normalization constant, N, for the Is radial function in hydrogen defined in Table 1.1. The constant N in cell F 7 is the inverse of the square root of the value of the integral in cell F 6.

Figure 2.3 Convergence tests on the Simpson s rule integration procedure to evaluate equation 2.4 on an EXCEL spreadsheet.

Most of the design details should be familiar. Provide for the Simpson s rule integration procedure in the usual manner in columns A to C from cell A 12 to cell C 3013. 

Parabolic Rule (Simpson s Rule) This procedure consists of subdividing the interval a [Pg.298]

To provide for the alternating multiplication factors of 2 and 4 in equation 2.2, the spreadsheet design needs to include a conditional multiplier for the construction of the Simpson s rule integration as a sum over the projected values on the mesh points. MOD() is one of the functions available in the library of the function wizard in EXCEL, which is activated by the toolbar icon f. MOD() returns the remainder in a division. The presence or absence of such a remainder, then, permits a logical operator such as the IF() operator to be triggered as TRUE or FALSE. For example, for TRUE the cell entry can be set to 2, for FALSE the entry can be set to 4 and this procedure enables the various multiplications in equation 2.2. Thus, enter the formula... 

For the case where t8 = rp — tp, Equation 13 can be integrated directly to give ITto1al, the total amount of tertiary ion formed. For the other two cases, integration cannot be performed directly, and values of ITtotal were evaluated numerically on a KDF 9 computer, using a procedure for Simpson s rule. (Numerical evaluation of the directly integrable case provided a check on this procedure.) Ip and I8 are then given by... 

Simpson s Rules. There are better procedures for approximating the integral of a function that make use of quadratic and cubic forms rather than linear segments. The simplest of these is Simpson s one-third rule ... 

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. 

We will now integrate Equation (E8-6.8) using Simpson s rule after feinting a table (E8-6.1) to calculate as a function of X. This procedure is similar to that... 

When the integration involves an equal number of equally spaced data points, a special procedure must be devised to estimate the area associated with the last segment in the data set. Suppose the last three points are designated n — 2, n— 1, and n, where n is an even number. Simpson s rule may be applied in the normal way up to the (n — l)th point. In order to estimate the area associated with the last segment located between the (n — l)th and th point, a quadratic equation is fitted to the last three points. Using the above definitions of P and y, this yields the following values ... 

A Simpson s rule procedure modified to handle odd numbers of intervals is used to calculate the integrals. The program calculates and prints the time... 

In the Simpson s rule procedure, a similar division of the area, defined between the bounds of the integration and the curve of the function is made. However, for Simpson s rule, it is necessary to choose an even number of intervals. Then, in the same notation as for the Trapezium rule procedure. 

Construct the standard Simpson s rule numerical integration procedure in cells A 34 to C 3034. 

Fig4-l.xls provides for the various numerical integrations for the calculation of the Is orbital energy given in equation 4.9. As usual, a part of the spreadsheet is shown in Figure 4.1 and you can see that the design follows directly from Chapter 2 and the Simpson s rule procedure for integration of exponential functions. 

The worksheet IDzcalc contains the Simpson s rule procedure for carrying out the various integrations to return the terms in the Fock matrix, for the double-zeta basis defined in cells D 6 and E 6 [Table 1.3] and normalized as in equation 5.49, cells D 7 and E 7. Note now the Y(r) factors are handled in columns G to I, with the sums of the exponents for integration following equation 5.20 entered as appropriate sums in cells G 17 to 1 17. 

The alternative integration procedure using Simpson s rule gives... 

In spite of different procedures and design tools, we sometimes come across some typical design requirements that are not covered by any known procedure. Some imderstanding of basic mathematics and fundamental process engineering can solve a large number of problems without much difficulty. For example, a volume calculation of a horizontal vessel requires simple integration, volume calculation of an inclined vessel using Simpson s rule, etc. 

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