Sunday, December 13, 2015

Seeking an Integer Solution to a System of 20155 Simultaneous Equations from the Literature

Jsun Yui Wong

The following computer program seeks to find an integer solution to Problem 7 in Cao [2, page 9, Problem 7 (penalty I function)]--http://dx.doi.org/10.1155/2014/251587.  See also La Cruz et al. [5, page 25]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf.  The present paper considers the case of 20155 equations with 20155 variables.  One notes the initial guess, 94 A(KK) = FIX(RND * 1.9).

0 DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(32768), A(32768), P(32768), K(32768)

5 FOR JJJJ = -32000 TO -32000

    14 RANDOMIZE JJJJ
    16 M = -1D+50


    91 FOR KK = 1 TO 20155

        94 A(KK) = FIX(RND * 1.9)


    95 NEXT KK

    128 FOR I = 1 TO 12000000 STEP 1


        129 FOR K = 1 TO 20155


            131 X(K) = A(K)
        132 NEXT K

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 20158)


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

            187 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 IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1


        191 NEXT IPP


        666 sumssqq = 0
        669 FOR J44 = 1 TO 20155

            677 sumssqq = sumssqq + X(J44) ^ 2


        688 NEXT J44


        770 REM


        772 P(20155) = (1 / (4 * 20155)) * (sumssqq) - 1 / 4

        773 FOR J44 = 1 TO 20154


            775 P(J44) = -ABS(((1 / 10 ^ 5) ^ .5 * (X(J44) - 1)))

        777 NEXT J44
        822 P = 0


        833 FOR J44 = 1 TO 20154

            837 P = P + P(J44)



        855 NEXT J44


        999 P = P - ABS(P(20155))


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 20155


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P

        1666 PRINT A(20155), M, JJJJ

        1668 IF M > -.00001 THEN 1912


    1670 NEXT I        
    1890 REM  IF M < -.00001 THEN 1999
    1912 PRINT A(1), A(2), A(3)
    1913 PRINT A(4), A(5), A(6)

    1914 PRINT A(7), A(8), A(9)


    1915 PRINT A(3152), A(3153), A(3154)


    1916 PRINT A(4152), A(4153), A(4154)


    1917 PRINT A(5152), A(5153), A(5154)


    1918 PRINT A(6149), A(6150), A(6151)

    1919 PRINT A(20152), A(20153), A(20154)

    1939 PRINT A(20155), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.


.
.
.

1          -33.8039589923291         -32000
1          -33.8007843108295         -32000
.
.
.
0        -3.18708536961386D-03           -32000
0        -1.240387000744233D-05         -32000
1         0       -32000
1      1      1
1      1      1
1      1      1
1      1      1
1      1      1
1      1      1
1      1      1
1      1      1
1          0       -32000

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

Of the 20155 unknowns, only the 25 A's of line 1912 through line 1939 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 [11], the wall-clock time for obtaining the output through JJJJ= -32000 was two hours and ten minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.

[2] Huiping Cao, Global Convergence of Schubert's Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages.  Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587

[3] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.

[4] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229.
http://www.SciRP.org/journal/am

[5]  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

[6]  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.    

[7] 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.

[8] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf

[9] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.

[10] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.

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

[12] M. Ziani, F. Guyomarc'h, An Autoadaptive Limited Memory Broyden's Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211.  web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf

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