SECTION E. TUTORIAL AND EXAMPLES

The following examples are currently available:

1. Integration of a trivariate normal density, using the algorithms TRINT, GAULG3, NEWCT3, CLARK3, and DUTT3. The program first compares all five of these algorithms when used to integrate a rectangular region. It then compares DUTT3 and CLARK3 when used to integrate over several infinite wedges with increasingly smaller mass.

2. Solution of a nonlinear five-equation system using NEWRAP. The equations are

x_{1}[e^{x}^{2}log(x_{3})+x_{4}]^{x5}-1=0 x_{1}+x_{2}x_{3}+x_{4}x_{5}-3=0 x_{1}-x_{2}+2x_{3}^{2}-x_{4}-x_{5}=0 x_{1}^{x}^{2}+x_{2}^{x}^{1}-x_{4}^{x}^{5}-x_{3}^{x}^{4}=0 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}-5=0

The iterations start from a point at
which each of the x's is equal to 1.5. It is easy to verify that
a solution is provided by x_{1} = x_{2} = x_{3}
= x_{4} = x_{5} = 1.

3. Linear programming, using both the steepest ascent method for pivot choice and the method of greatest total improvement at each step. The example uses five inequality contraints.

4. Finding the roots of a polynomial equation. The equation is

x^{6}-7x^{5}+7x^{4}-3x^{3}+4x^{2}-5x+4=0,

which has two real roots and two pairs of complex conjugate roots.

5.Maximizing and plotting a probit likelihood. The program generates some random data and maximizes the likelihood function, first using DFP and then using GRADX. After that is done, the third parameter, X(3), is held at its optimized value, and the contours of the likelihood are plotted, with X(1) on the horizontal and X(2) on the vertical axis. A "+" marks the occurrence of the maximum; in the contour plot this occurs approximately at X(1)=0.04 and X(2)=0.108; agreeing with the results of the optimization runs. The DFP run is set up as if we were going to do a suboptimization with respect to a smaller set of paramaters than the full set, but in fact the optimization is with respect to all three parameters.

6. Solving a nonlinear equation by REGFAL. We shall use REGFAL to solve the the following equation for x:

It is obvious that the solution is x = 0.0. The algorithm is given the search range (-3.0, 3.0) and finds the solution with excellent accuracy.

7. Calculating a density plot by DENSPLOT. We generate for the x-variable 1000 uniformly distributed numbers between 0 and 10 and choose as our y-variable y_{i}=x_{i}^{1/2}+0.1u_{i}, where u_{i} is uniformly distributed on (0-1).

8. Calculating a global minimum by GENALG. This calculates the (approximate) location of the global maximum between 0 and 10 along both the X and Y axes of the function

f(x,y)=x sin(4x) + 1.1 sin(2y)