The computer program listed below seeks one or more integer solutions to the following given system of two nonlinear equations:
(X(1) - 1) ^ 2 + (X(1) - X(2)) ^ 2 + (X(3) - 1) ^ 2 + (X(4) - 1) ^ 4 + (X(5) - 1) ^ 6 = 4
(X(1) - 1) ^ 2 + (X(1) - X(2)) ^ 2 + (X(2) - X(3)) ^ 2 + (X(3) - X(4)) ^ 4 + (X(4) - X(5)) ^ 4 = 1.
The two equation are based on page 97 and page 99 of Hock and Schittkowski [6], respectively.
One notes the starting vectors of line 94, which is 94 A(KK) = -300000 + FIX(RND * 600001).
0 REM DEFDBL A-Z
3 DEFINT X
4 DIM X(100042), A(100042), K(100033)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+57
91 FOR KK = 1 TO 5
94 A(KK) = -300000 + FIX(RND * 600001)
95 NEXT KK
128 FOR I = 1 TO 1000
129 FOR K = 1 TO 5
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 5)
182 IF RND < .5 THEN 183 ELSE GOTO 189
183 R = (1 - RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * R
188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1
190 REM IF A(B) = 0 THEN X(B) = 1 ELSE X(B) = 0
191 NEXT IPP
224 N1 = (X(1) - 1) ^ 2 + (X(1) - X(2)) ^ 2 + (X(3) - 1) ^ 2 + (X(4) - 1) ^ 4 + (X(5) - 1) ^ 6 - 4
276 N2 = (X(1) - 1) ^ 2 + (X(1) - X(2)) ^ 2 + (X(2) - X(3)) ^ 2 + (X(3) - X(4)) ^ 4 + (X(4) - X(5)) ^ 4 - 1
277 P = -ABS(N1) - ABS(N2)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 5
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1888 IF M < -10 THEN 1999
1910 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [13]. Copied by hand from the screen, the computer program’s output through JJJJ=-27005 is summarized below:
.
.
.
0 0 0 0 1
-2 -27070
0 0 0 0 1
-2 -27058
0 0 0 0 0
0 -27046
0 0 0 0 0
0 -27037
2 2 2 2 2
0 -27036
1 0 0 0 0
0 -27033
0 0 0 0 0
0 -27019
0 -1 0 0 1
-3 -27016
1 2 2 2 2
0 -27013
0 0 0 0 0
0 -27006
1 0 0 0 0
0 -27005
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 [13], the wall-clock time through JJJJ=-27005 was 20 seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[2] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[3] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[4] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[5] 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.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[7] 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.
[8] 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.
[9] 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.
[10] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[11] 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.
[12] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[13] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[14] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[15] 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.
[16] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/
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