Friday, December 11, 2015

Seeking an Integer Solution with Hot Starts of Another System of 5765 Nonlinear Equations

Jsun Yui Wong

Using qb64v1000-win [9], the following computer program seeks to find an integer solution to Problem 6 in Cao [2, page 9, Problem 6 (exponential problem 2)]--http://dx.doi.org/10.1155/2014/251587.  See also La Cruz et al. [5, page 21]--www.ime.unicamp.br/~martinez/lmrreport.pdf.  The present paper considers the case of 5765 nonlinear equations with 5765 variables.  One notes the hot starts, 94 A(KK) = FIX(RND * 1.9).

0 REM  DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(52768), A(52768), K(52768), P(52222)

5 FOR JJJJ = -32000 TO -32000

    14 RANDOMIZE JJJJ

    16 M = -1D+50

    91 FOR KK = 1 TO 5765
        94 A(KK) = FIX(RND * 1.9)

    95 NEXT KK

    128 FOR I = 1 TO 12000000 STEP 1

        129 FOR K = 1 TO 5765
            131 X(K) = A(K)
        132 NEXT K

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 5765)

            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
        222 FOR J44 = 1 TO 5765

            227 IF X(J44) > 80 THEN 1670
        229 NEXT J44
        770 X(1) = 0
        771 FOR J44 = 2 TO 5765
            774 P(J44) = -ABS(.1 * J44 * (EXP(X(J44)) + X(J44 - 1) - 1))

        777 NEXT J44
        800 P = 0

        801 FOR J44 = 2 TO 5765
            822 P = P + P(J44)

        888 NEXT J44

        1111 P = P

        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 5765

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

        1666 PRINT A(5765), M, JJJJ

        1668 IF M > -.0001 THEN 1912
    1670 NEXT I
    1890 REM IF M < -500 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(557), A(558), A(559)

    1917 PRINT A(4777), A(4778), A(4779)
    1928 PRINT A(4877), A(4878), A(4879)
    1947 PRINT A(5762), A(5763), A(5764)

    1948 PRINT A(5762), A(5763), A(5764)

    1949 PRINT A(5765), 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        -2088833        -32000
1        -2088358        -32000
.
.
.
0     -2964.686    -32000
0     -1854.712     -32000
0     -970.8984     -32000
0     -378.7567     -32000
0       0      -32000
0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0       0      -32000

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

Of the 5765 unknowns, only the 25 A's of line 1912 through line 1949 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 13 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.    
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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