How To Solve Nonlinear Systems of Equations: An Illustration with 3 Simultaneous Nonlinear Equations with 3 Unknowns, Third Edition

Jsun Yui Wong

The computer program listed below tries to solve simultaneously the following system of 3 nonlinear equations from Farid [32]:

311 * COS(0.5236 - X(3)) + X(1) * COS(X(2) + X(3)) - 311=0,

311 * SIN(.5236 - X(3)) - (X(1) * SIN(X(2) + X(3)))=0,

(X(1) ^ 2) * (TAN(.5236) * COS(X(2)) * COS(X(2) - .7526) - (1 / TAN(.5236)) * (SIN(X(2)) * SIN(X(2) - .7526))) - 7036.18=0.

One notes line 501 and line 1061, which are 501 X(1) = (-311 * COS(0.5236 - X(3)) + 311) / (COS(X(2) + X(3))) and 1061 PD1 = - ABS(LHS2) - ABS(LHS3), respectively, and jointly engage all of these three equations.

While line 112 of the preceding edition is 112 A(J44) = FIX(RND * 6), here line 112 is 112 A(J44) = FIX(RND * 16).

0 DEFDBL A-Z

1 REM DEFINT I, J, K, A, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99)

5 DIM AA(99), HR(32), HHR(32), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(99)

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -3D+30

110 A(1) = FIX(RND * 500)

111 FOR J44 = 2 TO 3

112 A(J44) = FIX(RND * 16)

117 NEXT J44

128 FOR I = 1 TO 12500

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 3)

148 J = 1 + FIX(RND * 3)

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 166

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

166 NEXT IPP

223 IF X(1) < 0 THEN 1670

501 X(1) = (-311 * COS(0.5236 - X(3)) + 311) / ((COS(X(2) + X(3))))

515 LHS2 = 311 * SIN(.5236 - X(3)) - (X(1) * SIN(X(2) + X(3)))

519 LHS3 = (X(1) ^ 2) * (TAN(.5236) * COS(X(2)) * COS(X(2) - .7526) - (1 / TAN(.5236)) * (SIN(X(2)) * SIN(X(2) - .7526))) - 7036.18

1061 PD1 = -ABS(LHS2) - ABS(LHS3)

1111 IF PD1 <= M THEN 1670

1452 M = PD1

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1457 NEXT KLX

1458 LLHS2 = LHS2: LLHS3 = LHS3

1557 GOTO 128

1670 NEXT I

1677 IF M < -.0000001 THEN 1999 ELSE GOTO 1919

1919 PRINT A(1), A(2), A(3), M, JJJJ

1920 REM PRINT LLHS2, LLHS3

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [121]. Distinct solutions of one run through JJJJ= -30155 are shown below:

592.2261740862392 9.211837669959447 3.043873235209662

-2.660999198766945D-11 -31820

592.2261740862394 2.928652362779861 15.61024384956883

-1.036518093577854D-12 -31767

592.2261740862394 2.928652362779861 3.04387323520966

-8.916617444398867D-13 -31741

146.6871682461996 3.950714907905677 .9997481449580253

-1.67152403030002D-12 -31078

592.2261740862391 9.211837669959447 28.176614463928

-2.497790703176356D-11 -30927

592.2261740862391 9.211837669959447 9.327058542389233

-2.472305693501653D-11 -30627

146.6871682461998 10.23390021508526 .9997481449580591

-3.329202397324934D-11 -30470

146.6871682461997 3.950714907905677 7.282933452137614

-5.24304211158011D-12 -30386

592.2261740862394 2.928652362779861 9.327058542389244

-9.37347421903212D-13 -30155

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 [121]. The **wall-clock time** (not CPU time) for obtaining the output shown above was 27 seconds, counting from “Starting program…”.

**Acknowledgement**

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

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[120] 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.)

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

[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

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[124] Xin-She Yang, Test problems in optimization, arXiv, 3 August 2010, in Engineering Optimization: An Introduction with Metaheuristic Applications (Eds Xin-She Yang), John Wiley and Sons, (2010).