Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear system of 10500 Diophantine equations:

10500

x(i) + sigma x(j) - (10500+1) = 0, for i = 1, 2, 3,..., 10499,

j=1

10500

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]; www.SciRP.org/journal/am.

The starting vectors are shown in line 42, which is 42 A(J44) = -3 + FIX(RND * 7).

0 REM DEFDBL A-Z

2 DEFINT J, X

3 DIM B(10999), N(10999), A(10999), H(10999), L(10999), U(10999), X(10999), D(10999), P(10999), PS(10999), J(10999)

12 FOR JJJJ = -32000 TO 32000

15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 10500

42 A(J44) = -3 + FIX(RND * 7)

43 NEXT J44

128 FOR I = 1 TO 60000

129 FOR KKQQ = 1 TO 10500

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 J = 1 + FIX(RND * 10500)

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 10499

258 SU = SU + X(J44)

266 NEXT J44

311 X(10500) = -X(1) - SU + (10500 + 1)

351 PR = 1

353 FOR J45 = 1 TO 10500

355 PR = PR * X(J45)

359 NEXT J45

422 FOR J41 = 2 TO 10499

439 P(J41) = -ABS(X(J41) + SU + X(10500) - (10500 + 1))

427 NEXT J41

441 P(10500) = -ABS(PR - 1)

451 FOR J77 = 2 TO 10500

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 10500

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 10500

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(10497), A(10498), A(10499), A(10500), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [7]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31989 is shown below:

0 0 0 0 0

0 0 10501 -1 -32000

1 1 1 1 1

1 1 1 0 -31999

1 1 1 1 1

1 1 4 -4 -31998

-1 -1 -1 -1 -1

-1 -1 21000 -2 -31997

0 0 0 0 0

0 0 10501 -1 -31996

1 1 1 1 1

1 1 3 -3 -31995

1 1 1 1 1

1 1 3 -3 -31994

1 1 1 1 1

1 1 4 -4 -31993

-1 -1 -1 -1 -1

-1 -1 21000 -2 -31992

1 1 1 1 1

1 1 3 -3 -31991

1 1 1 1 1

1 1 2 -2 -31990

1 1 1 1 1

1 1 1 0 -31989

Above there is no rounding by hand; it is just straight copying by hand from the screen.

Thus, through JJJJ=-31989, M=0 occurred at JJJJ=-31999 and at JJJJ=-31989. Of the 10500 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 [7], the wall-clock time for obtaining the output through

JJJJ= -31989 was four 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. 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] 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.

[7] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[8] 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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