Monday, May 31, 2021

Answering "What if ..." Can Sometimes Help Solve a Nonlinear System of Equations

  

Jsun Yui Wong

 

Here the nonlinear system of interest is:

 

         3 * X(1) ^ 3 + 2 * X(2) - 5 + SIN(X(1) - X(2)) * SIN(X(1) + X(2))        =0,

 

         -X(J44 - 1) * EXP(X(J44 - 1) - X(J44)) + X(J44) * (4 + 3 * X(J44) ^ 2) + 2 * X(J44 + 1) + SIN(X(J44) - X(J44 + 1)) * SIN(X(J44) + X(J44 + 1)) - 8      =0,  J44 = 2 TO 14999,    

 

         -X(14999) * EXP(X(14999) - X(15000)) + 4 * X(15000) - 3        =0.

 

This nonlinear system of interest is based on the nonlinear system 12 in La Cruz and Raydan [58, p. 598, Appendix:Test Functions]:

 

The computer program below tries to answer what if there is a solution vector with all integers.  Then one can have line 19 and line 224, which are 19 A(J44) = INT(-3 + RND * 6) and 224 X(J44) = INT(X(J44)), respectively.

 

0 DEFDBL A-Z

 

2 REM    DEFINT K

 

3 DIM B(99), N(99), A(200111), H(99), L(200111), U(99), X(200111), D(111), P(511), PS(33), J(200111), J44(200111), KKQQ(200111), KLX(200111), W(200111), LSUM(200111)

 

12 FOR JJJJ = -32000 TO 32000 STEP .01

 

    13 RANDOMIZE JJJJ

 

    16 M = -1D+37

 

    18 FOR J44 = 1 TO 15000

 

        19 A(J44) = INT(-3 + RND * 6)

 

    25 NEXT J44

    128 FOR I = 1 TO 50000

        129 FOR KKQQ = 1 TO 15000

 

            130 X(KKQQ) = A(KKQQ)

 

        131 NEXT KKQQ

 

        135 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

            148 J = 1 + FIX(RND * 15100)

 

            154 IF RND < .5 THEN GOTO 157 ELSE GOTO 164

 

            157 R = (1 - RND * 2) * (A(J))

 

            160 X(J) = A(J) + (RND ^ (RND * 30)) * R

 

            161 GOTO 174

 

            164 IF RND < .5 THEN X(J) = A(J) - FIX(1 + RND * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)

 

        174 NEXT IPP

 

 

        222 FOR J44 = 1 TO 15000

 

 

            224 X(J44) = INT(X(J44))

 

        226 NEXT J44

 

 

        1012 L(1) = 3 * X(1) ^ 3 + 2 * X(2) - 5 + SIN(X(1) - X(2)) * SIN(X(1) + X(2))

 

        1020 FOR J44 = 2 TO 14999

            1026 L(J44) = -X(J44 - 1) * EXP(X(J44 - 1) - X(J44)) + X(J44) * (4 + 3 * X(J44) ^ 2) + 2 * X(J44 + 1) + SIN(X(J44) - X(J44 + 1)) * SIN(X(J44) + X(J44 + 1)) - 8

 

        1027 NEXT J44

 

        1046 L(15000) = -X(14999) * EXP(X(14999) - X(15000)) + 4 * X(15000) - 3

 

        1048 SUMM = 0

 

        1051 FOR J44 = 1 TO 15000

            1055 SUMM = SUMM + ABS(L(J44))

 

        1057 NEXT J44

 

 

        1113 P = -ABS(SUMM)

 

 

        1119 IF P <= M THEN 1670

 

        1452 M = P

 

        1454 FOR KLX = 1 TO 15000

 

            1459 A(KLX) = X(KLX)

 

        1460 NEXT KLX

 

        1557 GOTO 128

 

    1670 NEXT I

 

    1913 PRINT A(1), A(2), A(3), A(4), A(5)

 

    1914 PRINT A(6), A(7), A(8), A(14993), A(14994)

 

    1915 PRINT A(14995), A(14996), A(14997), A(14998), A(14999)

 

    1916 PRINT A(15000), M, JJJJ

1999 NEXT JJJJ

 

 

This BASIC computer program was run with qb64v1000-win [120].  The complete output of one run through JJJJ=-31999.99 is shown below:

 

1  1  1  1  1

1  1  1  1  1

1  1  1  1  1

1   -11.71828182845905   -32000  

 

1  1  1  1  1

1  1  1  1  1

1  1  1  1  1

1      0      -31999.99 

 

One notes that only 16 values of the 15000 values of A(1) through A(15000) are shown above, in accordance with line 1913 through line 1916.

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

The system properties of the computer system used include Intel Core 2 Duo CPU E8400 @ 3.00GHz  3.00GHz, 4.00GB of RAM, and qb64v1000-win [120].  The wall-clock time (not CPU time) for obtaining the output shown above was 3 hours and 44 minutes, counting from “Starting program…”.     

 

Acknowledgement

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

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[113] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

 

[114] Eric W. Weisstein, “Diophantine Equation–8th Powers.” https://mathworld.wolfram.com/DiophantineEquation8thPowers.html.

 

[115] Eric W. Weisstein, “Euler’s Sum of 0wers Conjecture.” https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

 

[116] Eric W. Weisstein, “Diophantine Equation–5th Powers.” https://mathworld.wolfram.com/DiophantineEquation5thPowers.html.

 

[117] Eric W. Weisstein, “Diophantine Equation–10th Powers.” https://mathworld.wolfram.com/DiophantineEquation10thPowers.html.

 

[118] Eric W. Weisstein, “Diophantine Equation–9th Powers.” https://mathworld.wolfram.com/DiophantineEquation9thPowers.html.

 

[119] Rick Wicklin (2018), Solving a system of nonlinear equations with SAS.  blogs.sas.com>iml>2018/02/28.  (One can directly read this on Google.)

 

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

 

[121] Jsun Yui Wong (2014, March 13). The Domino Method Applied to a System of Four Simultaneous Nonlinear Equations. Retrieved from http://myblogsubstance.typepad.com/substance/2014/03.

                    

[122] Jsun Yui Wong (2021, May 3).  Solving Another Big Instance (n, m, Dimensions=1200) of the Keane Benchmark Test Problem. https://nonlinearintegerprogrammingsolver.blogspot.com/2021/05/solving-another-big-instance-n-m.html

 

[123] Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990). Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.

 

 

 

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