ARM Club 1/ARM_CLUB_CD.iso/contents/apps/maths/progs/solver/!Solver/!Help/Readme

Text File | 1991-05-03 | 11.6 KB | 335 lines

***************************** Copyright Notice ****************************
*                                                                         *
*                          (C)1991 Duco Fennema                           *
*                                                                         *
*    Read the 'Terms of Use' chapter at the end of this file to know      *
*             what you may and may not do with this program.              *
*                                                                         *
***************************************************************************


                    ReadMe file for !Solver v1.10


                equation solver (like HP calculators)





        Contents___________________________________________________

        This file contains 11 chapters in this order.
          -Why !Solver
          -Using !Solver
          -Using !Solver as a Evaluator
          -Using the library
          -Extending the library
          -How it works
          -Alternative solving methods
          -Acknowledgements
          -Terms of use
          -Release information
          -Correspondence



        Why !Solver ?______________________________________________

        Do you know those handy Hewlett Packard calculators with a
        Solver program?
        Well I do, my brother has one, and it is very handy.
        I found it indispensable when calculating the maximum gain
        from given a fase margin for a second order process with
        delay time.
        Thus an idea came to my mind, to give my Archimedes the same
        capability.



        Using !Solver______________________________________________

        The Solver main window has three writable icons.

        In the top icon 'Equation in X' you may enter an equation
        in X (it has to be a capital X).
        You may enter a function in X on both sides of the "=", if
        you enter a function in X only, Solver tries to find a
        value for X which gives f(X)=0.

        In the second icon 'Lower Limit' you have to enter the
        lower limit for X.
        In the third icon 'Upper Limit' you have to enter the upper
        limit for X.

        If you press return in the third icon or click on 'Perform'
        !Solver will try to find a value for X between the upper
        and lower limit for which the equation is true.

        If there is more than one value of X for which the equation
        is true !Solver will only find one of them.
        To make sure there are no more solutions you could fiddle
        the limits a bit but some mathematical insight would come
        in handy here.
        In most cases though if you move the limit nearest to the
        answer towards to the other limit in such a way that the
        found answer now lies just outside the limits then chances
        are Solver will find another answer.

        After you have started the Solver, the 'Perform' icon will
        change to 'Interrupt'.
        If you click on 'Interrupt' Solver will stop and you may
        then change the limits or the equation.

        During the Solving process you can monitor it's progres in
        the three icons in the second box.
        The first icon 'Evaluating' gives the value under test,
        while the second icon 'Deviation' gives the deviation from
        the correct answer.
        The third icon 'Messages' reports any errors that might
        occur during the process and also tells you when it is
        finished.

        When Solver has found an answer the 'Messages' icon will
        say "Found after ... iterations" and the answer will be in
        the 'Evaluating' icon.

        If Solver can not find an answer, that means  no answer
        with a zero deviation, it will stop after 160 iterations
        and the 'Messages' icon will report this to you.
        After this the 'Evaluating' icon will display an answer but
        is has a deviation as shown in the 'Deviation' icon.

        If you click on 'Interrupt' during the solving process, the
        'Messages' icon will say "Halted after ... iterations" and
        the 'Evaluating' and 'Deviation' icon display the progress
        so far.
        If you click on 'Perform' after this the solving process
        will start all over again.
        It is not a resume but a restart.



        Using !Solver as a Evaluator_______________________________

        You may also use !Solver as a straight Evaluator.

        To do so click on the 'Solve' icon, it will change to
        'Evaluate' and the 'Messages' icon will say "Evaluate
        mode".

        After this you may enter a function in X in the 'Equation
        in X' icon.
        The 'Lower Limit' and 'Upper Limit' icons have a slightly
        different use now.
        In the 'Lower Limit' icon you may enter an initial value for
        X while the 'Upper Limit' icon must contain a step value.

        Now each time you click on the 'Perform' icon, the function
        will be evaluated for X='Lower Limit' and 'Upper Limit'
        is added to 'Lower Limit' afterwards.
        The 'Evaluating' icon gives the evaluated value of X as
        always, but the 'Deviation' icon now gives the result.

        Use the interactive Help application from the applications
        disc 1 if you want any help when using !Solver.



        Using the library__________________________________________

        The library functions may be used in exactly the same way
        as Basic functions.
        To find out how many functions there are and what they do try
        the folowing.
        Put !Solver in the Evaluation mode and enter into the three
        writable icons.

            Equation in X > FNFunctionInfo(X)
            Lower Limit   > -2
            Upper Limit   > 1

        Clicking on the 'Perform' icon causes the 'Messages' icon to
        display information about the function library.
        Keep doing this until no further messages are displayed.



        Extending the library______________________________________

        You may add your own routines to the function library, but
        some rules must be obeyed.
        These rules can be found in the library header.
        Also if you make any worthwhile additions, I would like to
        receive a copy from you to be incorporated into later
        releases of !Solver in the true spirit of shareware.



        How it works_______________________________________________

        !Solver works with the relatively simple methode of
        successive halving

        It takes the right side of the equation, brackets it,
        subtracts it from the left side of the equation and tries
        to find a value for X for which the resulting function
        becomes zero.

           g(X)=f(X) ==> g(X)-{f(X)}=0

        After that a test is made to see if one of the limits has a
        negative result and if so than that limit is taken as the
        lower limit.

        Now it just takes the mean from the upper and lower limit
        and tests it.
        If the result is negative we found a new lower limit
        otherwise we found a new upper limit.
        This process just goes on until we reach a zero result.

        The reason that Solver always stops after 160 iterations is
        that every possible answer should have been found by then.
        This fact can be proven by the following.

        Solver uses Basic floating point numbers to evaluate the
        given equation and these have the folowing range and
        accuracy.
            range    -1.7x10^38 to 1.7x10^38 (-1.7E38 to 1.7E38)
            accuracy  1  x10^-9              (1E-9)

        This means there are

            ([most positive]-[most negative])/[accuracy]

        definite points in this range.

        To find how many times we have to halve this range to end
        at a definite point, we calculate which power of two gives
        the same value as (pos-neg)/acc.

            2^i=(pos-neg)/acc

        To solve this we need some trigonometry.

            log(2^i) = log((pos-neg)/acc)

            i*log(2) = log(pos-neg)-log(acc)

                   i = {log(pos-neg)-log(acc)}/log(2)

        So, to find i from Basic we need to evaluate

            i=(LOG(3.4E38)-LOG(1E-9))/LOG(2)

        Sadly this is too much for Basic and it gives a 'Number too
        big' error.
        Time for another trick, we divide both the range and the
        accuracy by two.
        This gives the same number of definite points but the log
        arguments stay within range.
        Now we evaluate:

            i=(LOG(1.7E38)-LOG(2E-9))/LOG(2)

        If you enter the right part of this in the Solver and
        Evaluate it the answer will be 155.89.... which is almost
        accurate because it should be 157.89....

        And I think that should conclude our maths for this moment.



        Alternative solving methods________________________________

        There are a number of other methods to find a zero crossing
        for a given function, most notably Newton Raphson and
        Regula Falsi.
        They both converge a lot faster than successive halving but
        both have drawbacks as well.


        Newton Raphson works by iteration of the folowing lines:

            y=x-f(x)/f'(x)
            x=y

        As you can see it needs the first derivative to be
        determined and has to evaluate two functions in every
        iteration.


        Regula Falsi works by iteration of the folowing lines:

            y=x1-((x1-x0)/f(x1)-f(x0))*f(x1)
            x0=x1
            x1=y

        As you can see a lot more calculations are needed, but with
        some clever coding only one function evaluation is
        neccessary for every iteration.

        Personally I did not feel the need to implement one of these
        methods into !Solver.
        If you have written a working version suitable for
        incorporation in !Solver I would like to hear from you.



        Acknowledgements___________________________________________

        Thanks to:
          -Joris Roling for his !Help application (you are probably
           staring at that right now).
          -Freddy Huttner for his excellent !NewModes module.
           (the 56Hz 896x360x16 mode is great to work with).
          -Acorn for developing this tremendous machine.
          -My girlfriend for putting up with so much lack of
           attention.
          -And all others I forgot to mention.



        Terms of Use_______________________________________________

        You may freely copy as long as:
          -all files in this application directory are included.
          -this is not for commercial purposes
          -you do not charge for copying.
          -you do not put this program or parts of it on a disc for
           which you are charging (other than the costs of the disc
           itself).
          -the application is not tampered with.
          -this Readme file is always included without any change
           made to its contents.
          -you do not upload this on a bulletin board which charges
           for its downloads.

        The author reserves the right to amend these terms in cases
        where he deems misuse.



        Release information________________________________________

        v1.00 (26-12-90) First release (for selected persons only)
        v1.10 (04-01-91) Library added.



        Correspondence_____________________________________________

        Send remarks, bugs and praise to:

          Duco Fennema
          Eursingerweg 34
          9411 BB  BEILEN
          HOLLAND

          Telephone:
          05930-22277

          From outside Holland:
          <International entry code> 031 5930 22277