Wednesday, January 20, 2016

The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking an Integer Solution to a Five-Diagonal System of Nonlinear Equations from the Literature

Jsun Yui Wong

The following computer program seeks an integer solution to the five-diagonal system of nonlinear equations on page 27 of La Cruz, Martinez, and Raydan [7, p. 27, Test function 35, Five-diagonal system]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf.  The present case has 32035 nonlinear equations and 32035 unknowns.

0 DEFDBL A-Z
3 DEFINT J, K, X

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

5 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ

    16 M = -1D+50

    91 FOR KK = 1 TO 32035


        94 A(KK) = RND



    95 NEXT KK

    128 FOR I = 1 TO 1000000 STEP 1



        129 FOR K = 1 TO 32035


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 32038)

            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


        555 X(1) = (4 * X(2) ^ 2 - X(2) + X(3) ^ 2) / 4
        557 IF (8 * X(2) * (X(2) ^ 2 - X(1)) - 2 * (1 - X(2)) + 4 * (X(2) - X(3) ^ 2) + X(3) ^ 2) < 0 THEN 1670


        559 X(4) = (8 * X(2) * (X(2) ^ 2 - X(1)) - 2 * (1 - X(2)) + 4 * (X(2) - X(3) ^ 2) + X(3) ^ 2) ^ .5



        605 FOR J44 = 3 TO 32033


            607 Xnew = 8 * X(J44) * (X(J44) ^ 2 - X(J44 - 1)) - 2 * (1 - X(J44)) + 4 * (X(J44) - X(J44 + 1) ^ 2) + X(J44 - 1) ^ 2 - X(J44 - 2) + X(J44 + 1)

            618 IF Xnew < 0 THEN 1670

            621 X(J44 + 2) = Xnew ^ .5



        641 NEXT J44

        651 FOR j47 = 1 TO 32035


            666 IF ABS(X(j47)) > 30 THEN 1670


        688 NEXT j47


        699 P1 = 8 * X(32035) * (X(32035) ^ 2 - X(32034)) - 2 * (1 - X(32035)) + X(32034) ^ 2 - X(32033)


        711 P2 = 8 * X(32034) * (X(32034) ^ 2 - X(32033)) - 2 * (1 - X(32034)) + 4 * (X(32034) - X(32035) ^ 2) + X(32033) ^ 2 - X(32032)


        999 P = -ABS(P1) - ABS(P2)



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 32035


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1666 REM PRINT A(1), A(2), A(32035), M, JJJJ

        1668 IF M > -.000001 THEN 1912


    1670 NEXT I
    1890 IF M < -1 THEN 1999


    1912 PRINT A(1), A(2), A(3)


    1917 PRINT A(4), A(5), A(6)
    1943 PRINT A(32030), A(32031), A(32032)


    1946 PRINT A(32033), A(32034), A(32035)

    1947 PRINT M, JJJJ


1999 NEXT JJJJ

This computer program was run with qb64v1000-win [15]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31996 is shown below.

1      1      1
1      1      1
1      1      1
1      1      1
0      -32000

1      1      1
1      1      1
1      1      1
1      1      1
0      -31998

1      1      1
1      1      1
1      1      1
1      1      1
0      -31997

1      1      1
1      1      1
1      1      1
1      1      1
0      -31996

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

Of the 32035 unknowns, only the 12 A's of line 1912 through line 1946 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 [15], the wall-clock time for obtaining the output through JJJJ= -31996 was 16 minutes.

Acknowledgment

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

References

[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.

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

[3] 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

[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
 
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292.  Published Online July 2013.  http://www.scirp.org/journal/ojapps

[6] 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

[7] 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

[8] 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.

[9] 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.

[10]  J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.

[11] 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

[12]  NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf

[13] 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.

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

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

[16] 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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