Regarding Integration of Functions...

Many functions are defined by an integral.

Integrals are useful in probability theory to compute the probability of obtaining a value over a given interval.

Integrals come up in the computation of surfaces and of many physical quantities related to mechanics, energy, power and others and finally numerical integrals are accurate procedures that formally consist of an infinite sum of infinitesimal quantities.

We have already introduced the use of (see menu under)  "Simpson Method " in a real world application.

We want to show how different methods offer strategies that deal with number of iterations, estimated precision of the results and absolute values of the difference between the result of the integration and the true result. This time we implemented under the Smalltalk environment.

We chose three methods of integration approximation: Trapeze, Simpson and Romberg to illustrate progress of function evaluation in each situation. The function to integrate:

is specified as a block closure to be implemented in Smalltalk:

In general the trapeze integration algorithm is not used in practice. The interest here is to define a general framework for numerical integration for comparison purposes.

The method takes its origin in the series expansion of an integral as expressed by the Euler-McLaurin formulism.

An estimation of the integral is obtained by summing the areas of the trapezes corresponding to each partition:

The corresponding estimation for the integral is:

With the initial condition

Simpson integration algorithm consists in replacing the function to integrate by a second-order Lagrange interpolation polynomial. For a given interval of integration, [a,b], the function is evaluated at the extremities of the interval and at its middle point c = (a + b) /2.

The second-order Lagrange interpolation is given by,

As in the case of the trapeze algorithm, the precision of the algorithm is estimated by looking at the differences between the estimation obtained previously and the current estimation.

Romberg's algorithm introduces the parameter k, where k - 1 is the degree of the interpolation's polynomial. The result of an integral is estimated by extrapolating the series I^(n-k),...,I⁽ⁿ⁾ at the value e⁽ⁿ⁾ = 0. Since,

the rule calls for interpolation over successive powers of 1/4, starting with 1:1, 1/4, 1/16, 1/256 and so forth. The Romberg's algorithm requires at least k iterations. A polynomial of fourth degree (that is, k=5) is generally suficient. This algorithm converges very quickly and, in general, only a few iterations are needed after the five initial ones.

An example comparing the results of the three algorithms is shown in the next tab under RESULTS. As the reader may recall the function to integrate is the inverse function [f(x) = 1 / x] . The integration interval is from 1 to 2 so that the value of the result is known, namely, ln 2.

• Integration carried for various values of the desired precision the reader can compare the attained precision (predicted and real).

• Number of iterations required for each algorithm recall that the number of required function evaluations grows exponentially with the number of iterations.