Friday, June 3, 2016

The Domino Method Applied to Solving a Partial Differential Equation

Jsun Yui Wong

The following computer program seeks to solve the elliptical equation of Example 10.1 of Yang [21, pp.118-119].

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
    18 H = .25


    91 FOR KK = 0 TO 4


        94 IF RND < .5 THEN A(KK) = 1 - RND ELSE A(KK) = 1 + RND

    95 NEXT KK

    128 FOR I = 1 TO 10000 STEP 1




        129 FOR K = 0 TO 4
            131 X(K) = A(K)
        132 NEXT K

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 4)

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

            188 IF RND < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
        199 NEXT IPP
        566 X(0) = 1
        577 X(4) = 0


        586 FOR J44 = 2 TO 3



            611 X(J44) = .0625 - X(J44 - 2) + 2 * X(J44 - 1)


        613 NEXT J44


        714 PNEW = X(2) - 2 * X(3) - .0625


        999 P = -ABS(PNEW)
        1451 IF P <= M THEN 1670

        1657 FOR KEW = 0 TO 4


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
    1670 NEXT I
    1890 REM IF M < -.1 THEN 1999

    1911 PRINT A(0), A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [19]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31997 is shown below.

1           .6562499999994446                   .3749999999988891
.1562499999983337              0               -2.221778316879863D-12
-32000

1           .6562500000000195                   .3750000000000391
.1562500000000586              0               -7.815970093361102D-14
-31999

1           .6562500000000126                   .3750000000000251
.1562500000000376              0               -5.018208071305708D-14
-31998

1           .6562500000000527                   .3750000000001055
.1562500000001582              0               -2.109423746787797D-13
-31997

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

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [19], the wall-clock time for obtaining the output through JJJJ= -31997 was three seconds, not including "Creating .EXE file..." time.

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] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.

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

[6] 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.

[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.

[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.

[9] 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.

[10] 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.

[11] 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.

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

[13] 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.

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

[15] 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.

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

[17] 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.

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

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

[20] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.

[21]  Xin-She Yang. Introduction to Computational Mathematics.  World Scientific Publishing Co. Pte. Ltd., 2008.

[22] 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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