• Range Enclosure with graphical representation
  • Functions of one real variable
  • Functions of two real variables

Range Enclosure with graphical representation

written by  Grimmer, Markus, Department of Mathematics, University of Wuppertal, Germany, http://www.math.uni-wuppertal.de/wrswt
<? 1999-2002 Scientific Computing/Software Engineering Research Group, University of Wuppertal, Germany>

NOTE: This worksheet demonstrates the use of the Maple package intpakX v1.0  for interval arithmetic.

This document is not the package.  It only shows how to work with the functions and types provided by intpakX v1.0 .  You must create the package in an empty directory before loading the package ( i.e., /usr/maple/intpakX/lib)  Once created, load the package as follows:

Warning, the name changecoords has been redefined

Warning, the assigned name midpoint now has a global binding

Warning, the protected names ilog10, max and min have been redefined and unprotected

Range Enclosure with graphical representation

With intpakX v1.0 , you can enclose the ranges of two- or three-dimensional functions.

Functions of one real variable

The range enclosure is computed using a combination of the following methods each of which can be found as a separate procedure.
(For details, see the intpakX v1.0  worksheet.)

Range enclosure using interval evaluation; Range enclosure using mean value form;
Range enclosure using monotony properties; Range enclosure using Taylor forms.

The range enclosure of a function can be computed as follows:

>	X:=[0.5,2.];

>	f3:=x->exp(-x^2)*sin(Pi*x^3);

>	compute_range(f3,X,6);

`Start range enclosure =  `, [-.7788007834, .7788007834]

Range_enclosure after step 6 =  [-.2834388814,.5563221619]

As parameters, you have to give the function, the range in which to evaluate the function, and the number of iteration steps to be done.

The numerical results are stored in the list_of_ranges variable while the graphical output is directly displayed on the screen.

>	list_of_ranges;

Optional parameters:
"linear" or "quadratic": Influences the method used for evaluation and yields linear or quadratic convergence.
"adaptive": Uses adaptive instead of uniform subdivision when subdividing intervals.
"Nx=n" with an integer number n: Subdivides the start interval into n parts before computation starts.

Functions of two real variables

In the three dimensional case, interval evaluation is always used for computations.
The procedure call is similar to the 2D call.

>	T:=[evalf(Pi/8),evalf(Pi/2)]; S:=[evalf(Pi/8),evalf(Pi/2)];

>	g:=(x,y)->exp(-x*y)*sin(Pi*x^2*y^2);

>	compute_range3d(g,T,S,4);

`Start range enclosure = `, [-.8570898115, .8570898115]

Range enclosure after step 1 =  [-.8570898115,.8570898115]

Range enclosure after step 2 =  [-.6800891261,.8570898115]

Range enclosure after step 3 =  [-.6800891261,.8486122905]

Range enclosure after step 4 =  [-.5093193828,.7559256232]

In addition, you can use the options of the plot3d maple command.

Disclaimer:  While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.