The computer program listed below seeks to solve the following nonlinear Diophantine equation from Oliveira [18, p. 55; www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf]; also see Abraham, Sanyal, and Sanglikar [1; https://arxiv.org/ftp/arxiv/papers/1003/1003.2724.pdf]:
X(1)^2+ X(2)^2+X(3)^2+...+X(500)^2=500000.
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
91 FOR KK = 1 TO 500
94 A(KK) = FIX(1 + RND * 20)
95 NEXT KK
128 FOR I = 1 TO 20000 STEP 1
129 FOR K = 1 TO 500
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 500)
183 REM R = (1 - RND * 2) * A(B)
188 REM IF RND < .2 THEN X(B) = FIX(A(B) + RND * R) ELSE IF RND < .25 THEN X(B) = FIX(A(B) + RND ^ 3 * R) ELSE IF RND < .333 THEN X(B) = FIX(A(B) + RND ^ 5 * R) ELSE IF RND < .5 THEN X(B) = FIX(A(B) + RND ^ 7 * R) ELSE X(B) = FIX(A(B) + RND ^ 9 * R)
191 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(A(B) + 1)
199 NEXT IPP
201 FOR J43 = 1 TO 500
203 IF X(J43) < 1 THEN 1670
207 NEXT J43
211 SUML = 0
221 FOR J44 = 1 TO 500
231 SUML = SUML + X(J44) ^ 2
251 NEXT J44
261 PZ = -ABS(SUML - 500000)
1111 P = PZ
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 500
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -99999 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1913 PRINT A(6), A(7), A(8), A(9), A(10)
1975 PRINT A(491), A(492), A(493), A(494), A(495)
1977 PRINT A(496), A(497), A(498), A(499), A(500)
1989 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [21]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31998 is shown below.
21 30 35 20 15
33 54 44 29 45
26 41 60 25 48
37 34 17 9 5
0 -32000
38 20 41 22 30
39 41 32 17 24
29 22 26 29 10
32 25 35 39 41
0 -31999
27 41 24 38 24
12 26 46 31 34
45 30 48 41 22
22 11 39 45 7
0 -31998
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 500 unknowns, only the 20 A's of line 1911 through line 1977 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [21], the wall-clock time for obtaining the output through JJJJ= -31998 was seven seconds, not including "Creating .EXE file..." time--the total time was twenty seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[17] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[18] Hime Aguiar E. Oliveira, Junior, Diophantine Equations and Fuzzy Adapttive Simulated Annealing. WSEAS Transactions on Mathematics, Volume 13, 2014. www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf.
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[21] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[22] 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-19×19-syste.html.
[23] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.
[23] 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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