Introduction to KH-integral tools

This site contains programs for illustrating Riemann sums based on tagged divisions. The basis for these programs is introduced in the paper Maple tools for the Kurzweil integral , P. Adams and R. Vyborny, Matematica Bohemica 131. No 4. (2006), 337-346.

There are two Maple10 procedures available: khsum and khbox. The procedure khsum calculates the Riemann sum numerically, and khbox in addition produces a graph. The programs were not designed for high numerical accuracy, so we recommend that you stay within four decimal places.

Both procedures require FOUR input arguments: the integrand, the initial point of the interval, the end point of the interval and the gauge. The integrand and the gauge must be entered as functions, not as expressions. All plot options are also accepted as optional arguments for khbox.

The class of Kurzweil integrable function is wide and in an example involving a complicated function a meaningful gauge can require subintervals too small for the graphical display, in which case khsum can be used. After you run either khsum or khbox the commands "tagdiv;" and "nops(tagdiv);" will provide the the tagged divisions and the number of subintervals, respectively. The existence of a gauge-fine tagged division is equivalent to the completeness of the reals. Obviously, a computer is unable to provide a gauge-fine tagged division for every gauge. If our program cannot find the reqired tagged division it will try to find an approximate gauge-fine division. These are defined in the above paper. However, requiring a high accuracy or using a wild gauge may cause the program to fail.

The filed linked from Examples from the paper ... is called khtools_web.txt. It is a Maple10 text file, rather than a plain text file. After downloading the file (in most browsers by right clicking the link and then choosing save link target as) you should open the file in Maple as Maple text. The command "!!!" on the toolbar then executes the whole file.

Similarly, the file linked to Simple examples is also a Maple text file. The functions involved are Riemann integrable, nevertheles by choosing appropriate gauges the tagged divisions adjust nicely to the behavior of the integrand. The error is controlled by the variable er, and the user can hence select the required level of accuracy. To run these examples, you will first need to download the file khtools.m. Follow the link The programs. Download the file khtools.m and place it in the appropriate directory for Maple to load and save files in Maple's internal format.

If you wish to explore your own examples, you need the file khtools.m in the appropriate directory. Open a worksheet, and type the commands restart: and with(plots):. After invoking read("khtools.m"): you are ready to work out your own examples.

The file linked to Examples with unbounded integrands is also a Maple's text file and as the title says the integrands in the examples are not bounded and consequently not Rieamnn integrable. The error this time is controled by the variable ε. As before the file khtools.m is needed for processing.

With both Simple examples and Examples with unbounded integrands we provide additional pdf files. The purpose is twofold, to allow readers who are unable to use Maple to see and possibly use our work, and also give detailed explanations for the selections of gauges in the examples.

We have recently added some animations to this website. These illustrate the different approximations to the integral by Riemann sums, depending on whether we use Kurzweil theory or Riemann theory. Download the file Anim.mpl and place it in the appropriate directory for Maple to load and save files in Maple's internal format. Then load and open the file AnimWithMPL.mw for the animations.

For our readers who do not wish to use Maple we provide an html file. Unfortunately, the animations in this file run in a loop without the possibility of stopping and restarting.