Example 4

In this example (see section 15.1.3) we deal with a complex problem of three equations in three unknowns $\psi,\theta,\phi$. We are looking for a solution in the domain:

$$[4.537856054,4.886921908], [1.570796327,1.745329252], [0.6981317008, 0.8726646262]$$

The system has a solution which is approximately:

$$4.6616603883, 1.70089818026, 0.86938888189$$

This problem is extremely ill conditioned as for the TestDomain the functions intervals are:

$$[-1.45096e+08,1.32527e+08]; [-38293.3,29151.5] ; [-36389.1,27705.7]$$

This program is implemented under the name Test_Solve_General. With espsilonf=0 and epsilon=0.001 and if we stop at the first solution we find with the maximum equation ordering:

$$\psi=[4.664665,4.665347]~~~~ \theta=[1.7034,1.703741]~~~~ \phi=[0.8706193,0.8709602]$$

with 531 boxes. We may also mention the following remarks:

• we get no improvement with the single bisection mode as we need 2435 boxes to find the first solution,
• using the Reverse Storage mode does not lead to any improvement for finding the first root: in this mode we need 5587 boxes to get the first solution,

With the maximum middle-point equation ordering we find:

$$\psi=[4.665347,4.666029]~~~~ \theta=[1.701355,1.701696]~~~~ \phi=[0.8709602,0.8713011]~~~~$$

with 203 boxes. The importance of normalizing the functions appears if we use epsilonf=0.1 and epsilon=0. If we stop at the first solution we find:

$$\begin{eqnarray*} &&\psi=[4.661660388259656,4.661660388340929]\\ &&\theta=[1.70... ...00898180284073]\\ &&\phi=[0.869388881899751,0.869388881940387] \end{eqnarray*}$$

while if we divide the first function by 1000 we find:

$$\begin{eqnarray*} &&\psi=[4.661658091884636,4.661658424779772]\\ &&\theta=[1.70... ...00898570395561]\\ &&\phi=[0.869388105618527,0.869388272066095] \end{eqnarray*}$$

in four time less computation time.