Sunday, January 31, 2016

Testing the Domino Method of Nonlinear Integer/Continuous/Discrete Programming with a Nonlinear System of Equations from a Discrete Boundary Value Problem

Jsun Yui Wong

The following computer program seeks a solution to the system of nonlinear equations of a discrete boundary value problem on page 29 of La Cruz, Martinez, and Raydan [7, p. 29, Test function 41, Discrete boundary value problem]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf.

One notes line 93, which comes from La Cruz, Martinez, and Raydan [7, p. 29] and is 93 A(KK) = (h * (KK * h - 1)).

0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), P(32768), K(32768)

5 FOR JJJJ = -32000 TO 32000
    14 RANDOMIZE JJJJ
    16 M = -1D+50

    22 h = 1 / (119 + 1)


    91 FOR KK = 1 TO 119

        93 A(KK) = (h * (KK * h - 1))

    95 NEXT KK



    128 FOR I = 1 TO 500000 STEP 1


        129 FOR K = 1 TO 119

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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 122)

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


            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 IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1
            188 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE X(B) = A(B) + RND ^ 7 * R

        199 NEXT IPP


        566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3


        605 FOR J49 = 2 TO 118
            611 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * J49) ^ 3 - X(J49 - 1) + X(J49 + 1)

        613 NEXT J49
        615 P0 = 0

        617 FOR j11 = 2 TO 118
            619 P0 = P0 - ABS(P(j11))
        629 NEXT j11


        622 FOR j33 = 1 TO 119

            633 IF ABS(X(j33)) > 3 THEN 1670


        655 NEXT j33


        677 P119 = 2 * X(119) + .5 * h ^ 2 * (X(119) + h * 119) ^ 3 - X(118)




        999 P = -ABS(P119) + P0



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 119

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

    1670 NEXT I
    1890 IF M < -9 THEN 1999


    1912 PRINT A(1), A(2), A(3)

    1917 PRINT A(117), A(118), A(119)

    1939 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [16]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31999 is shown below.

-2.68801128311508D-11            -3.36663472559238D-11        -1.202984329827381D-10
-1.688823992425807D-05          -1.318892470245795D-05       -2.352394767238498D-05
-8.339331381840622D-12          -32000

-2.687959843339338D-11           -3.366531846040526D-11       -1.202995204456031D-10
-1.688824009365936D-05           -1.318892476496033D-05       -2.35239477694988D-05
-8.327981091731493D-12          -31999

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

Of the 119 unknowns, only the six A's of line 1912 and line 1917 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 [16], the wall-clock time for obtaining the output through JJJJ= -31999 was one minute.

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] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.     .

[10] 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.

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

[12] 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.

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

[14] 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.

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

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

[17] 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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