The computer program listed below seeks to solve in integers the following system of nonlinear equations:
3 z =x ^ 2 - 2 x y
2 t = x ^ 3 + 96 * z ^ 2 - 1
(x - 2y) ^ 2 - 3x = 18.
The system above is based on a system in [9, of the section on "Systems with More Than One Equation"].
0 REM DEFDBL A-Z
2 DEFINT J
3 DIM N(32999), A(32999), H(32999), L(32999), U(32999), X(32999), D(32999), P(32999), PS(32999), J(32999)
12 FOR JJJJ = -32000 TO 32000
15 RANDOMIZE JJJJ
16 M = -1D+37
41 FOR J44 = 1 TO 4
42 A(J44) = -50 + FIX(RND * 111)
43 NEXT J44
128 FOR I = 1 TO 300
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 J = 1 + FIX(RND * 4)
182 IF RND < .5 THEN 183 ELSE GOTO 191
183 R = (1 - RND * 2) * A(J)
189 X(J) = A(J) + FIX(RND * R)
190 GOTO 192
191 IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1
192 NEXT IPP
441 P(12500) = -ABS(3 * X(3) - X(1) ^ 2 + 2 * X(1) * X(2))
443 PNEW(4) = -ABS(2 * X(4) - X(1) ^ 3 - 96 * X(3) ^ 2 + 1)
448 PNEWzz(4) = -ABS((X(1) - 2 * X(2)) ^ 2 - 3 * X(1) - 18)
452 IF PNEW(4) < 0 THEN PNEW(4) = PNEW(4) ELSE PNEW(4) = 0
455 IF PNEWzz(4) < 0 THEN PNEWzz(4) = PNEWzz(4) ELSE PNEWzz(4) = 0
459 IF P(12500) < 0 THEN P(12500) = P(12500) ELSE P(12500) = 0
597 P = P(12500) + (PNEW(4)) + PNEWzz(4)
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 0 THEN 1999
1947 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-15554 is shown below:
-3 0 3 418 0
-30527
-3 0 3 418 0
-27402
-3 0 3 418 0
-26082
-3 0 3 418 0
-19939
-3 -3 -3 418 0
-19761
-3 0 3 418 0
-16701
-3 -3 -3 418 0
-15554
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 [8], the wall-clock time for obtaining the output through
JJJJ= -15554 was forty seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Jean-Marie de Konnick, Armel Mercier, 1001 Problems in Classical Number Theory. American Mathematical Society, Providence, Rhode Island, 2007.
[2] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. www.SciRP.org/journal/am.
[3] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
[4] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[5] 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.
[6] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[7] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.
[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[9] WOLFRAM, Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[10] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html.
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