Solving in General Integers a Nonlinear System of Five Simultaneous Nonlinear Equations

Jsun Yui Wong

The computer program listed below seeks one integer solution to the following given system of five nonlinear equations:

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 3 + X(11) ^ 3 + X(12) ^ 3 + X(13) ^ 3 = 1188

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2205.868

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2206.889

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2208.886

-.5 * (X(1) * X(4) – X(2) * X(3) + X(3) * X(9) – X(5) * X(9) + X(5) * X(8) – X(6) * X(7)) = 0.

Equations 2, 3, and 4 above are based on page 112 of Hock and Schittkowski [6]. The last is based on page 117 of Hock and Schittkowski [6].

One notes line 94, which is 94 A(KK) = 1 + FIX(RND * 10). .

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 13
94 A(KK) = 1 + FIX(RND * 10)

95 NEXT KK
128 FOR I = 1 TO 10000000

129 FOR K = 1 TO 13
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 13)
182 IF RND < .1 THEN 183 ELSE GOTO 189
183 R = (1 – RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * R
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 X(B) = CINT(A(B))
188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) – FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
191 NEXT IPP
265 FOR J45 = 1 TO 13
266 IF X(J45) < 1 THEN X(J45) = A(J45)
267 NEXT J45
268 N11 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) – 2205.868
269 N12 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) – 2206.889
270 N13 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) – 2208.886
273 N14 = -.5 * (X(1) * X(4) – X(2) * X(3) + X(3) * X(9) – X(5) * X(9) + X(5) * X(8) – X(6) * X(7)) – 0

275 N15 = X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 3 + X(11) ^ 3 + X(12) ^ 3 + X(13) ^ 3 – 1188

277 P = -ABS(N11) – ABS(N12) – ABS(N13) – ABS(N14) – ABS(N15)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 13
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1888 IF M < -3 THEN 1999

1910 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10)
1912 PRINT A(11), A(12), A(13), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [13]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31976 is shown below:

6       6       3       9       4
7       5       6       26       5
4       2       3       -1.964422       -31993

6       6       6       6       6
6       6       6       6       8
7       2       1       -9.020425E-04
-31976

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 [13], the wall-clock time through JJJJ=-31976 was 28 minutes.

Acknowledgment

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

References

[1] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[2] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[3] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[4] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[5] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. http://www.SciRP.org/journal/am.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[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] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[11] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.
[12] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[13] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[14] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[15] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html.
[16] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/

ERRATUM: The General Mixed Integer Nonlinear Programming (MINLP) Computer Program/Solver Solving Simultaneously a Nonlinear System of Two Simultaneous Nonlinear Equations Involving 150,000 General Integer Variables

Jsun Yui Wong

The last paragraph should read as follows:

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [12], the wall-clock time for reaching M=0 was eight hours and a half.

The General Mixed Integer Nonlinear Programming (MINLP) Computer Program/Solver Solving Simultaneously a Nonlinear System of Two Simultaneous Nonlinear Equations Involving 150,000 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks one integer solution to the following problem:

X(1) * X(2) * X(3) * X(4) * …* X(150000) – 1 = 0,

149999
sigma [ 100 * (X(k + 1) – X(k) ^ 2) ^ 2 + (1 – X(k)) ^ 2 ] = 0,
k=1

and each unknown = 0, 1, or 2.

The first equation above is a part of the Brown almost linear function [1, p. 660]. The second is based on the Rosenbrock function in Schitkowski [11, pp. 118-123].

One notes line 88, line 94, line 189, line 195, and line 196.

0 REM DEFDBL A-Z
3 DEFINT X
4 DIM X(150042), A(150042), L(150033), K(150033)

5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+57

88 A(1) = 2

91 FOR KK = 2 TO 150000

94 A(KK) = 1

95 NEXT KK
128 FOR I = 1 TO 30000000

129 FOR K = 1 TO 150000

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
171 B = 1 + FIX(RND * 150005)

175 GOTO 189

181 B = 1 + FIX(RND * 100003)
182 IF RND < .5 THEN 183 ELSE GOTO 189

183 R = (1 – RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * R

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 X(B) = CINT(A(B))
188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) – 1 ELSE X(B) = A(B) + 1
190 REM X(B) = FIX(RND * 2.98)

191 NEXT IPP
193 REM GOTO 201
194 FOR J49 = 1 TO 150000
195 IF X(J49) < 0 THEN X(J49) = FIX(RND * 2.98)

196 IF X(J49) > 2 THEN X(J49) = FIX(RND * 2.98)

197 NEXT J49

201 PRODBROWN = 1
203 FOR J33 = 1 TO 150000
206 PRODBROWN = PRODBROWN * X(J33)

209 NEXT J33

222 N1 = PRODBROWN – 1
223 GOTO 257
231 SUMSCHI = 0
236 FOR J22 = 1 TO 150000
239 SUMSCHI = SUMSCHI + X(J22) ^ 2

241 NEXT J22
244 N9 = -SUMSCHI + 150000
246 REM N5 = -(X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2) * (X(1) ^ 4 + X(2) ^ 4 + X(3) ^ 4 + X(4) ^ 4 + X(5) ^ 4 + X(6) ^ 4 + X(7) ^ 4 + X(8) ^ 4) + (X(1) ^ 3 + X(2) ^ 3 + X(3) ^ 3 + X(4) ^ 3 + X(5) ^ 3 + X(6) ^ 3 + X(7) ^ 3 + X(8) ^ 3) ^ 2
257 sumrose = 0
259 FOR j44 = 1 TO 149999
261 sumrose = sumrose + 100 * (X(j44 + 1) – X(j44) ^ 2) ^ 2 + (1 – X(j44)) ^ 2
263 NEXT j44
265 N6 = sumrose – 0
277 P = -ABS(N6) – ABS(N1)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 150000
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(49999), A(50000), M, JJJJ

1667 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(149992), A(149993), A(149994), A(149995), A(149996), A(149997), A(149998), A(149999), A(150000), M, JJJJ
1670 NEXT I
1888 IF M < -100 THEN 1999
1910 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(29996), A(29997), A(49998), A(49999), A(50000)
1912 PRINT A(1111), A(1112), A(1113), A(1114), A(1115), A(3336), A(3337), A(3338), A(3339), A(3340), A(5996), A(5997), A(25998), A(25999), A(50000), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [12]. Copied by hand from the screen, the computer program’s output through M=0 is shown below:

2 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-2910 -32000

2 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1905 -32000

1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
-1002 -32000

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

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

Of the 150,000 A’s, only the 15 A’s of line 1667 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 [12], the wall-clock time for reaching M=0 was 12 hours.

Acknowledgment

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

References

[1] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[2] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[3] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[4] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[5] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. http://www.SciRP.org/journal/am.
[6] 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.
[7] 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.
[8] 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.
[9] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[10] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.
[11] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[12] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[13] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[14] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html.
[15] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/

Solving in General Integers a Nonlinear System of Four Simultaneous Nonlinear Equations

Jsun Yui Wong

The computer program listed below seeks one or more integer solutions to the following given system of four nonlinear equations:

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2205.868
10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2206.889

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2208.886

-.5 * (X(1) * X(4) – X(2) * X(3) + X(3) * X(9) – X(5) * X(9) + X(5) * X(8) – X(6) * X(7)) = 0.

The first three equations are based on page 112 of Hock and Schittkowski [6]. The last is based on page 117 of Hock and Schittkowski [6].

One notes line 94, line 182, and line 189, which are
94 A(KK) = 1 + FIX(RND * 15), 182 IF RND < .1 THEN 183 ELSE GOTO 189, and 189 IF RND < .5 THEN X(B) = A(B) – FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99), respectively.

0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(342), A(342), L(333), K(333)
5 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 9

94 A(KK) = 1 + FIX(RND * 15)
95 NEXT KK

128 FOR I = 1 TO 4000000
129 FOR K = 1 TO 9
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 9)
182 IF RND < .1 THEN 183 ELSE GOTO 189
183 R = (1 – RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * R
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 X(B) = CINT(A(B))

188 GOTO 191

189 IF RND < .5 THEN X(B) = A(B) – FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
191 NEXT IPP

265 FOR J45 = 1 TO 9

266 IF X(J45) < 1 THEN X(J45) = A(J45)
267 NEXT J45

268 N11 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) – 2205.868
269 N12 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) – 2206.889

270 N13 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) – 2208.886

273 N14 = -.5 * (X(1) * X(4) – X(2) * X(3) + X(3) * X(9) – X(5) * X(9) + X(5) * X(8) – X(6) * X(7)) – 0
277 P = -ABS(N11) – ABS(N12) – ABS(N13) – ABS(N14)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 9

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1888 IF M < -.3 THEN 1999

1910 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [13]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31026 is shown below:

2       3      4       1       6
2       16       9       6       -.2988313
-31898

18       2       6       2       6
6       6       2       8       -.2470762
-31858

7       6       6       10       9
10       10       14       20       -.2746716
-31847

6       6       6       6       6
6       6       6       2       -9.020425E-04
-31828

.
.
.
6       6       6       6       6
6       6       6       5       -9.020425E-04
-31026

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 [13], the wall-clock time through JJJJ=-31026 was five hours and a half.

Acknowledgment

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

References
[1] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[2] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[3] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[4] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[5] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. http://www.SciRP.org/journal/am.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[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] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[11] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.
[12] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[13] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[14] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[15] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html.
[16] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/

Solving in General Integers a Nonlinear System of Five Simultaneous Nonlinear Equations with Cold Starts A(KK) = -300000 + FIX(RND * 600001)

Jsun Yui Wong

The computer program listed below seeks one or more integer solutions to the following given system of five nonlinear equations:

(X(1) – 1) ^ 2 + (X(1) – X(2)) ^ 2 + (X(3) – 1) ^ 2 + (X(4) – 1) ^ 4 + (X(5) – 1) ^ 6 = 4

(X(1) – 1) ^ 2 + (X(1) – X(2)) ^ 2 + (X(2) – X(3)) ^ 2 + (X(3) – X(4)) ^ 4 + (X(4) – X(5)) ^ 4 = 1

(4 * X(1) – X(2)) ^ 2 + (X(2) + X(3) – 2) ^ 2 + (X(4) – 1) ^ 2 + (X(5) – 1) ^ 2 = 42

(X(1) – X(2)) ^ 2 + (X(2) + X(3) – 2) ^ 2 + (X(4) – 1) ^ 2 + (X(5) – 1) ^ 2 = 6

(X(1) – 1) ^ 2 + (X(1) – X(2)) ^ 2 + (X(2) – X(3)) ^ 2 = 1.

The first two equations are based on page 97 and page 99 of Hock and Schittkowski [6], respectively. The third and fourth are based on page 75 and page 76 of Hock and Schittkowski [6], respectively. The fifth equation comes from Schittkowski [12, p. 165].

One notes the starting vectors of line 94, which is 94 A(KK) = -300000 + FIX(RND * 600001).

0 REM DEFDBL A-Z

3 DEFINT X

4 DIM X(100042), A(100042), K(100033)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+57

91 FOR KK = 1 TO 5
94 A(KK) = -300000 + FIX(RND * 600001)
95 NEXT KK

128 FOR I = 1 TO 1000
129 FOR K = 1 TO 5
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 5)

182 IF RND < .5 THEN 183 ELSE GOTO 189
183 R = (1 – RND * 2) * A(B)

186 X(B) = A(B) + (RND ^ 3) * R

188 GOTO 191

189 IF RND < .5 THEN X(B) = A(B) – 1 ELSE X(B) = A(B) + 1

190 REM IF A(B) = 0 THEN X(B) = 1 ELSE X(B) = 0

191 NEXT IPP
211 N1 = (X(1) – 1) ^ 2 + (X(1) – X(2)) ^ 2 + (X(3) – 1) ^ 2 + (X(4) – 1) ^ 4 + (X(5) – 1) ^ 6 – 4

213 N2 = (X(1) – 1) ^ 2 + (X(1) – X(2)) ^ 2 + (X(2) – X(3)) ^ 2 + (X(3) – X(4)) ^ 4 + (X(4) – X(5)) ^ 4 – 1
224 N3 = (4 * X(1) – X(2)) ^ 2 + (X(2) + X(3) – 2) ^ 2 + (X(4) – 1) ^ 2 + (X(5) – 1) ^ 2 – 42
276 N4 = (X(1) – X(2)) ^ 2 + (X(2) + X(3) – 2) ^ 2 + (X(4) – 1) ^ 2 + (X(5) – 1) ^ 2 – 6

277 N5 = (X(1) – 1) ^ 2 + (X(1) – X(2)) ^ 2 + (X(2) – X(3)) ^ 2 – 1
278 P = -ABS(N1) – ABS(N2) – ABS(N3) – ABS(N4) – ABS(N5)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 5

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

1670 NEXT I
1888 IF M < -10 THEN 1999
1910 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ
This computer program was run with qb64v1000-win [13]. Copied by hand from the screen, the computer program’s output through JJJJ=-31911 is summarized below:
.
.
.
2       2       2       1       0
-5       -31996
2       2       2       2       2
0       -31980
2       2       2       1       0
-5       -31969
2       2       2       2       1
-4       -31947
2       2       2       2       2
0       -31940
2       2       2       1       0
-5       -31917
2       2       2       2       2
0       -31911

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 [13], the wall-clock time through JJJJ=-31911 was 10 seconds.

Acknowledgment

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

References
[1] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[2] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[3] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[4] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[5] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. http://www.SciRP.org/journal/am.
[6] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[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] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[11] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.
[12] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[13] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[14] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[15] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html.
[16] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/