Thursday, September 15, 2016

Simultaneously Solving in Integers a System of Four Simultaneous Nonlinear Equations Involving 10000 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks one integer solution or more to the following given system of four nonlinear equations:

X(1) * X(2) * X(3) * X(4) ...* X(10000)    - 1 = 0

-(X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2) * (X(1) ^ 4 + X(2) ^ 4 + X(3) ^ 4 + X(4) ^ 4 + X(5) ^ 4 + X(6) ^ 4 + X(7) ^ 4 + X(8) ^ 4) + (X(1) ^ 3 + X(2) ^ 3 + X(3) ^ 3 + X(4) ^ 3 + X(5) ^ 3 + X(6) ^ 3 + X(7) ^ 3 + X(8) ^ 3) ^ 2 = 0

9999
sigma    [ 100 * (X(k + 1) - X(k) ^ 2) ^ 2 + (1 - X(k)) ^ 2 ]  = 0
k=1
   
   10000
 - sigma   X(i}^2    + 10000   = 0.
     i=1

The first equation above is a part of the Brown almost linear function [1, p. 660]. The second equation above comes from Schittkowski [8, p. 187].  The third is based on the Rosenbrock function in Schitkowski [8, pp. 118-123].  The last comes from Schittkowski [8, p.194].

One notes the starting vector 94 A(KK) = -3 + FIX(RND * 7).


0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(10042), A(10042), L(10033), K(10033)

5 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ
    16 M = -1D+17
    91 FOR KK = 1 TO 10000


        94 A(KK) = -3 + FIX(RND * 7)



    95 NEXT KK

    128 FOR I = 1 TO 1000000 STEP 1



        129 FOR K = 1 TO 10000

            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)

            181 B = 1 + FIX(RND * 10000)
            182 IF RND < .5 THEN 183 ELSE GOTO 189


            183 R = (1 - RND * 2) * A(B)



            186 X(B) = A(B) + (RND ^ 3) * R


            187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
            188 GOTO 191

            189 IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

        191 NEXT IPP


        201 PRODBROWN = 1
        203 FOR J33 = 1 TO 10000
            206 PRODBROWN = PRODBROWN * X(J33)


        209 NEXT J33


        222 N1 = PRODBROWN - 1


        231 SUMSCHI = 0
        236 FOR J22 = 1 TO 10000

            239 SUMSCHI = SUMSCHI + X(J22) ^ 2


        241 NEXT J22
        244 N9 = -SUMSCHI + 10000


        246 N5 = -(X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2) * (X(1) ^ 4 + X(2) ^ 4 + X(3) ^ 4 + X(4) ^ 4 + X(5) ^ 4 + X(6) ^ 4 + X(7) ^ 4 + X(8) ^ 4) + (X(1) ^ 3 + X(2) ^ 3 + X(3) ^ 3 + X(4) ^ 3 + X(5) ^ 3 + X(6) ^ 3 + X(7) ^ 3 + X(8) ^ 3) ^ 2


        257 sumrose = 0

        259 FOR j44 = 1 TO 9999

            261 sumrose = sumrose + 100 * (X(j44 + 1) - X(j44) ^ 2) ^ 2 + (1 - X(j44)) ^ 2
        263 NEXT j44
        265 N6 = sumrose - 0

        277 P = -ABS(N1) - ABS(N5) - ABS(N6) - ABS(N9)


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 10000

            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
    1670 NEXT I
    1888 IF M < -5000 THEN 1999

    1912 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(9996), A(9997), A(9998), A(9999), A(10000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [9]. Copied by hand from the screen, the computer program’s complete output through
JJJJ=-31999 is shown below:

1      1      1      1      1
1      1      1      1      1
1      1      1      2      4
-920      -32000

1      1      1      1      1
1      1      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.

Of the 10000 A's, only the 15 A's of line 1912 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 [9], the wall-clock time for obtaining the output through JJJJ= -31999 was 100 minutes.

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] 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] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.

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

[10] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.

[11] 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.

[12] 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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