Sunday, May 29, 2016

Solving the Boundary Value Problem of Gilat and Subramaniam's Example 9.5

Jsun Yui Wong

The following computer program seeks to solve the discrete boundary value problem of Gilat and Subramaniam's Example 9.5  [7]. Also see Gilat and Subramaniam [8, Example 11.5].

0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ
    16 M = -1D+50

    55 H = 1 / 8


    91 FOR KK = 1 TO 9


        94 IF RND < .5 THEN A(KK) = 1 - RND ELSE A(KK) = 1 + RND


    95 NEXT KK

    128 FOR I = 1 TO 100000 STEP 1


        129 FOR K = 1 TO 9


            131 X(K) = A(K)
        132 NEXT K

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 9)



            183 R = (1 - RND * 2) * A(B)
            188 IF RND < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R


        199 NEXT IPP

        571 X(1) = 1


        580 X(3) = ((4 + H ^ 2) * X(2) - 2 - H ^ 2 * EXP(-.2 * H)) / 2




        581 FOR J44 = 4 TO 9 STEP 1
            582 X(J44) = (-2 * X(J44 - 2) + (4 + H ^ 2) * X(J44 - 1) - H ^ 2 * EXP(-.2 * (J44 - 2) * H)) / 2


        584 NEXT J44


        610 PNEW = ABS(2 * H * X(9) + X(7) - 4 * X(8) + 3 * X(9))


        1111 P = -PNEW

        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 9


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
    1670 NEXT I
    1890 REM IF M < -.1 THEN 1999
    1911 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [19]. Copied by hand from the screen, the computer program’s output through JJJJ= -31997 is shown below.

 1                                          .943153486079235             .8860557500807424
 .828448844750844                .7700662000336197            .7106306551757421
 .649852405260003                .5874268461945267            .5230323013901858
-1.110223024625157D-16             -32000

 1                                          .943153486079235             .8860557500807424
 .828448844750844                .7700662000336197            .7106306551757421
 .649852405260003                .5874268461945267            .5230323013901858
-1.110223024625157D-16             -31999

 1                                          .943153486079235             .8860557500807424
 .828448844750844                .7700662000336197            .7106306551757421
 .649852405260003                .5874268461945267            .5230323013901858
-1.110223024625157D-16             -31998

 1                                          .943153486079235             .8860557500807424
 .828448844750844                .7700662000336197            .7106306551757421
 .649852405260003                .5874268461945267            .5230323013901858
-1.110223024625157D-16             -31997

Above there is no rounding by hand; it is just straight copying by hand from the screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [19], the wall-clock time for obtaining the output through JJJJ= -31997 was fifteen seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

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[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.

[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.

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[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.

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[13] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[14 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.

[15] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf.

[16] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
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[17] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.

[18] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[20] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19x19-syste.html.

[21] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.

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