Jsun Yui Wong
The computer program listed below seeks to solve the following nonlinear system of 20500 Diophantine equations:
20500
x(i) + sigma x(j) – (20500+1) = 0, for i = 1, 2, 3,…, 20499,
j=1
20500
pi x(j) -1 = 0.
j=1
This present system is based on the Brown almost linear function in La Cruz, Marinez, and Raydan [3, p. 25]; http://www.ime.unicamp.br/~martinez/lmrreport.pdf. See also Han and Han [2, p. 227, Example 3]; http://www.SciRP.org/journal/am.
The starting vectors are shown in line 42, which is 42 A(J44) = -1 + FIX(RND * 3).
0 REM DEFDBL A-Z
2 DEFINT J, X
3 DIM B(20999), N(20999), A(20999), H(20999), L(20999), U(20999), X(20999), D(20999), P(20999), PS(20999), J(20999)
12 FOR JJJJ = -32000 TO 32000
15 RANDOMIZE JJJJ
16 M = -1D+37
41 FOR J44 = 1 TO 20500
42 A(J44) = -1 + FIX(RND * 3)
43 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 20500
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 J = 1 + FIX(RND * 20503)
183 REM R = (1 – RND * 2) * A(J)
187 IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
189 REM X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP
251 SU = 0
254 FOR J44 = 1 TO 20499
258 SU = SU + X(J44)
266 NEXT J44
311 X(20500) = -X(1) – SU + (20500 + 1)
351 PR = 1
353 FOR J45 = 1 TO 20500
355 PR = PR * X(J45)
359 NEXT J45
422 FOR J41 = 2 TO 20499
439 P(J41) = -ABS(X(J41) + SU + X(20500) – (20500 + 1))
427 NEXT J41
441 P(20500) = -ABS(PR – 1)
451 FOR J77 = 2 TO 20500
452 IF P(J77) < 0 THEN P(J77) = P(J77) ELSE P(J77) = 0
454 NEXT J77
577 SP = 0
578 FOR J99 = 2 TO 20500
579 SP = SP + P(J99)
580 NEXT J99
595 P = SP
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 20500
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M < -99 THEN 1999
1947 PRINT A(1), A(2), A(3), A(4), A(20497), A(20498), A(20499), A(20500), 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=-31999 is shown below:
0 0 0 0 0
0 0 20501 -1 -32000
1 1 1 1 1
1 1 1 0 -31999
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Thus, through JJJJ=-31999, M=0 was obtained at JJJJ=-31999. Of the 20500 A’s, only the 8 A’s of line 1947 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 [8], the wall-clock time for obtaining the output through JJJJ= -31999 was two hours.
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. http://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] 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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