The following computer program seeks to solve the system of nonlinear equations of the discrete boundary value problem on pages 402-405 of Gilat and Subramanian [6, Example 9.4]. Also see Gilat and Subramanian [7].
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 6
94 A(KK) = 293 + RND * 200
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 6
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 6)
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(6) = 293
605 FOR J49 = 6 TO 3 STEP -1
610 P(J49) = X(J49 - 2) - (2 + .02 ^ 2 * betaa) * X(J49 - 1) - .02 ^ 2 * betab * X(J49 - 1) ^ 4 + X(J49) + .02 ^ 2 * (betaa * 293 + betab * 293 ^ 4)
612 NEXT J49
660 PS = 0
661 FOR J33 = 6 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 6
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)
1919 PRINT A(6), 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= -31999 is shown below.
473 423.3441267021215 383.3134062263356
349.8410248260544 320.4456637140555
293 -3.890045567611633D-07 -32000
473 423.3441269631637 383.313406594297
349.8410252159446 320.4456639230223
293 -1.725536257490834D-07 -31999
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= -31999 was five seconds, not including "Creating .EXEC file..." time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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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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