Monday, May 23, 2016

Solving a Boundary Value Problem, Part 2

Jsun Yui Wong

The following computer program seeks to solve the system of nonlinear equations of the discrete boundary value problem on pp. 402-405 of Gilat and Subramanian [6, Example 9.4].  Also see Gilat and Subramanian [7].  While the preceding paper has five sub-intervals, the present paper has twenty sub-intervals.  

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

    33 betaa = (40 * .016) / (240 * 1.6D-05)

    35 betab = (.4 * 5.67D-08 * .016) / (240 * 1.6D-05)


    91 FOR KK = 1 TO 21


        94 A(KK) = 293 + RND * 200


    95 NEXT KK

    128 FOR I = 1 TO 20000000 STEP 1


        129 FOR K = 1 TO 21


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 22)


            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
        566 X(1) = 473

        577 X(21) = 293


        605 FOR J49 = 21 TO 3 STEP -1

            610 P(J49) = X(J49 - 2) - (2 + .005 ^ 2 * betaa) * X(J49 - 1) - .005 ^ 2 * betab * X(J49 - 1) ^ 4 + X(J49) + .005 ^ 2 * (betaa * 293 + betab * 293 ^ 4)

        612 NEXT J49

        660 PS = 0

        661 FOR J33 = 21 TO 3 STEP -1


            663 PS = PS + ABS(P(J33))


        668 NEXT J33


        1111 P = -PS



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 21


            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)

    1912 PRINT A(6), A(7), A(8), A(9), A(10)

    1913 PRINT A(11), A(12), A(13), A(14), A(15)

    1914 PRINT A(16), A(17), A(18), A(19), A(20)

    1919 PRINT A(21), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [18]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -32000 is shown below.

473                                    459.455895068255           446.6932249241943
434.6475924043841         423.2590645622598
412.4716933656588         402.2330921869252         392.4940590698486
383.2082393784908         374.331821732474
365.8232621978106         357.6430325441322         349.7533890177581
342.1181586515432         334.7025405843482
327.4729202056817         320.3966942320405         313.442105080995
306.5780831026808         299.7740953874252
293                                   -2.877462553516411D-08         -32000

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 [18], the wall-clock time for obtaining the output through JJJJ= -32000 was four minutes.

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.

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

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

[19] 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.

[20] 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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