Nonlinear Integer/Continuous/Discrete Programming Solver

Tuesday, December 6, 2016

Simultaneously Solving in General Integers a Nonlinear System of 33 Simultaneous Equations in 51 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following problem:

X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51) = 1,

X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9,

2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51) = 9,

X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51) = 9,

X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51) = 9,

X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51) = 9,

X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51) = 9,

X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51) = 9,

X(41) ^ 2 + X(42) - 37 = 0,

X(41) - X(42) ^ 2 - 5 = 0,

X(41) + X(42) + X(43) - 3 = 0,

X(39) + X(40) ^ 3 - 5 * X(40) ^ 2 - X(40) = 10,

X(39) + X(40) ^ 3 + X(40) ^ 2 - 14 * X(40) = 29,

- X(35) * (X(35) + 1) + 2 * X(36) = 18,

(X(35) - 1) ^ 2 + (X(36) - 6) ^ 2 = 25

X(37) ^ 2 - 10 * X(37) + X(38) ^ 2 + 8 = 0,

X(37) * X(38) ^ 2 + X(37) - 10 * X(38) + 8 = 0,

EXP(X(32) * X(33) * X(34)) = 1,

3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3 = 0,

3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3 = 6666.6666666,

(X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27) = 714,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
= 2902,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 = 1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 = 659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 = 395,

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) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1 = 348,

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 = 468,

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,

and

-5<= Integers X(i)<= 15 for i=1 through 51 except 0<= Integer X(13)<= 10.

The first eight equations above are based on pages 510 of Burden and Faires [3, p. 510]. The next three equations come from page 649 of Burden, Faires, and Burden [2], and the next two equations are based on page 510 of Burden and Faires [3]. The next four equations come from Burden, Faires, and Burden [2, page 648]. Equation 18 through equation 21 are based on page 100, page 118, 116, and page 111 of Hock and Schittkowski [8], respectively. Equation 22 through equation 26 are based on page 147 of Asaadi [1]. Equation 27 is based on page 122 of Hock and Schittkowski [9]. Equation 30 through equation 32 are based on page 112 of the same book [9]. Equation 33, the last one, is based on page 117 of the same book [9].

One notes line 13, which is 13 A(KNEW) = -5 + FIX(RND * 20.98).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
13 A(KNEW) = -5 + FIX(RND * 20.98)

14 RANDOMIZE JJJJ

16 M = -1D+317

22 REM

91 FOR KK = 1 TO 51

94 A(KK) = A(KNEW)

95 NEXT KK
128 FOR I = 1 TO 16500000

129 FOR K = 1 TO 51

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 51)

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
255 FOR J45 = 1 TO 51

256 IF X(J45) < -5 THEN X(J45) = A(J45)

257 IF X(J45) > 15 THEN X(J45) = A(J45)

259 NEXT J45

264 X(13) = 348 - (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) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

266 IF X(13) < 0 THEN 1670

267 IF X(13) > 10 THEN 1670

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2

278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2

281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2

282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2

283 N22 = -714 + (X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)

284 N23 = -6666.6666666 + 3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3

291 N24 = 0 + 3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3

295 IF (X(32) * X(33) * X(34)) > 80 THEN GOTO 1670

296 N25 = -1 + EXP(X(32) * X(33) * X(34))

298 N26 = -18 - X(35) * (X(35) + 1) + 2 * X(36)

299 N27 = -25 + (X(35) - 1) ^ 2 + (X(36) - 6) ^ 2

301 N28 = X(37) ^ 2 - 10 * X(37) + X(38) ^ 2 + 8

303 N29 = X(37) * X(38) ^ 2 + X(37) - 10 * X(38) + 8

304 X(39) = 10 - X(40) ^ 3 + 5 * X(40) ^ 2 + X(40)

305 N31 = -29 + X(39) + X(40) ^ 3 + X(40) ^ 2 - 14 * X(40)

400 X(41) = X(42) ^ 2 + 5

401 N32 = X(41) ^ 2 + X(42) - 37

402 REM N33 = X(41) - X(42) ^ 2 - 5

403 N34 = X(41) + X(42) + X(43) - 3

499 X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9

500 REM X(44) = -2 * X(45) - X(46) - X(47) + 5

501 N35 = 2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9

503 REM N37 = X(44) +2* X(45) + X(46) + X(47)+x( 48 )+x( 49 )+x( 50 )+x( 51 ) - 9
504 N38 = X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9
505 N39 = X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51) - 9
506 N40 = X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51) - 9
507 N41 = X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51) - 9
508 N42 = X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51) - 9
509 N43 = X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51) - 1

1287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21) - ABS(N22) - ABS(N23) - ABS(N24) - ABS(N25) - ABS(N26) - ABS(N27) - ABS(N28) - ABS(N29) - ABS(N30) - ABS(N31) - ABS(N32) - ABS(N33) - ABS(N34) - ABS(N35) - ABS(N36) - ABS(N37) - ABS(N38) - ABS(N39) - ABS(N40) - ABS(N41) - ABS(N42) - ABS(N43)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 51

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

1670 NEXT I
1888 IF M < -1 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), A(14), A(15), A(16), A(17), A(18), A(19), A(20)

1914 PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), A(30)

1915 PRINT A(31), A(32), A(33), A(34), A(35), A(36), A(37), A(38), A(39), A(40), A(41), A(42), A(43), A(44), A(45), A(46), A(47), A(48), A(49), A(50), A(51), 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=-31512 is shown below:

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 -2 6 0 6
12 9 2 11 2
3 2 1 1 0
0 0 4 9 -2
10 1 1 15 -1
6 1 -4 1 1
1 1 1 1 1
1 -9.020425E-04 -31704

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 -2 6 6
8 15 2 14 0
-1 5 1 1 0
0 13 6 0 -2
10 1 1 15 -1
6 1 -4 1 1
1 1 1 1 1
1 -9.020425E-04 -31648

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 -2 6 0 6
9 4 -3 9 1
5 -3 1 1 0
0 12 0 2 -2
10 1 1 15 -1
6 1 -4 1 1
1 1 1 1 1
1 -9.020425E-04 -31512

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 [16], the wall-clock time through JJJJ=-31512 was eleven hours.

Acknowledgment

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

References
[1] J. Asaadi, A Computational Comparison of Some Nonlinear Programs, Mathematical Programming, Vol. 4 (1973), pp. 144-154.
[2] R. Burden, J. Faires, and A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] R. Burden, and J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] F. Glover, 1986. Future Paths for Integer ng and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[6] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[7] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[8] 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.
[9] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[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] 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.
[13] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[14] 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.
[15] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[16] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[17] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[18] 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.
[19] 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/

Sunday, December 4, 2016

Simultaneously Solving in General Integers a Nonlinear System of 29 Simultaneous Equations in 47 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following problem:

2 * X(44) + X(45) + X(46) + X(47) - 5 = 0,

X(44) + 2 * X(45) + X(46) + X(47) - 5 = 0,

X(44) + X(45) + 2 * X(46) + X(47) - 5 = 0,

X(44) * X(45) * X(46) * X(47) - 1 =0,

X(41) ^ 2 + X(42) - 37 = 0,

X(41) - X(42) ^ 2 - 5 = 0,

X(41) + X(42) + X(43) - 3 = 0,

X(39) + X(40) ^ 3 - 5 * X(40) ^ 2 - X(40) = 10,

X(39) + X(40) ^ 3 + X(40) ^ 2 - 14 * X(40) = 29,

- X(35) * (X(35) + 1) + 2 * X(36) = 18,

(X(35) - 1) ^ 2 + (X(36) - 6) ^ 2 = 25,

X(37) ^ 2 - 10 * X(37) + X(38) ^ 2 + 8 = 0,

X(37) * X(38) ^ 2 + X(37) - 10 * X(38) + 8 = 0,

EXP(X(32) * X(33) * X(34)) = 1,

3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3 = 0,

3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3 = 6666.6666666,

(X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27) = 714,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
= 2902,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 = 1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 = 659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 = 395,

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) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1 = 348,

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 = 468,

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, and

-5<= Integers X(i)<= 15 for i=1 through 47 except 0<= Integer X(13)<= 10.

The first four equations above come from Burden and Faires [3, p. 510]. The next three equations come from page 649 of Burden, Faires, and Burden [2], and the next two equations are based on page 510 of Burden and Faires [3]. The next four equations come from Burden, Faires, and Burden [2, page 648]. Equation 14 through equation 17 are based on page 100, page 118, 116, and page 111 of Hock and Schittkowski [8], respectively. Equation 18 through equation 22 are based on page 147 of Asaadi [1]. Equation 23 is based on page 122 of Hock and Schittkowski [9]. Equation 26 through equation 28 are based on page 112 of the same book [9]. Equation 29, the last one, is based on page 117 of the same book [9].

One notes line 13, which is 13 A(KNEW) = -5 + FIX(RND * 20.98).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
13 A(KNEW) = -5 + FIX(RND * 20.98)

14 RANDOMIZE JJJJ

16 M = -1D+317

22 REM

91 FOR KK = 1 TO 47

94 A(KK) = A(KNEW)

95 NEXT KK
128 FOR I = 1 TO 13000000

129 FOR K = 1 TO 47

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 47)

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
255 FOR J45 = 1 TO 47

256 IF X(J45) < -5 THEN X(J45) = A(J45)

257 IF X(J45) > 15 THEN X(J45) = A(J45)

259 NEXT J45

264 X(13) = 348 - (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) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

266 IF X(13) < 0 THEN 1670

267 IF X(13) > 10 THEN 1670

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2

278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2

281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2

282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2

283 N22 = -714 + (X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)

284 N23 = -6666.6666666 + 3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3

291 N24 = 0 + 3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3

295 IF (X(32) * X(33) * X(34)) > 80 THEN GOTO 1670

296 N25 = -1 + EXP(X(32) * X(33) * X(34))

298 N26 = -18 - X(35) * (X(35) + 1) + 2 * X(36)

299 N27 = -25 + (X(35) - 1) ^ 2 + (X(36) - 6) ^ 2

301 N28 = X(37) ^ 2 - 10 * X(37) + X(38) ^ 2 + 8

303 N29 = X(37) * X(38) ^ 2 + X(37) - 10 * X(38) + 8

304 N30 = -10 + X(39) + X(40) ^ 3 - 5 * X(40) ^ 2 - X(40)

305 N31 = -29 + X(39) + X(40) ^ 3 + X(40) ^ 2 - 14 * X(40)

400 X(41) = X(42) ^ 2 + 5

401 N32 = X(41) ^ 2 + X(42) - 37

402 REM N33 = X(41) - X(42) ^ 2 - 5

403 N34 = X(41) + X(42) + X(43) - 3
500 X(44) = -2 * X(45) - X(46) - X(47) + 5

501 N35 = 2 * X(44) + X(45) + X(46) + X(47) - 5

502 REM N36 = X(44) + 2 * X(45) + X(46) + X(47) - 5

503 N37 = X(44) + X(45) + 2 * X(46) + X(47) - 5
504 N38 = X(44) * X(45) * X(46) * X(47) - 1

1287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21) - ABS(N22) - ABS(N23) - ABS(N24) - ABS(N25) - ABS(N26) - ABS(N27) - ABS(N28) - ABS(N29) - ABS(N30) - ABS(N31) - ABS(N32) - ABS(N33) - ABS(N34) - ABS(N35) - ABS(N36) - ABS(N37) - ABS(N38)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 47

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

1670 NEXT I
1888 IF M < -30 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), A(14), A(15), A(16), A(17), A(18), A(19), A(20)

1914 PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), A(30)

1915 PRINT A(31), A(32), A(33), A(34), A(35), A(36), A(37), A(38), A(39), A(40), A(41), A(42), A(43), A(44), A(45), A(46), A(47), 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=-31862 is shown below:

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
5 12 5 9 1
1 3 1 1 0
0 -2 0 3 -2
10 1 1 15 -1
6 1 -4 1 1
1 1 -9.020425E-04 -31997

6 6 6 6 6
6 6 6 6 6
6 6 6 3 5
6 -4 6 1 6
6 11 5 8 0
2 3 1 1 0
0 0 1 -1 -2
10 1 1 15 -1
6 1 -4 1 1
1 1 -20.0009 -31979

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 0 6
13 9 0 10 0
6 5 1 1 0
0 0 2 1 -2
10 1 1 15 -1
6 1 -4 1 1
1 1 -9.020425E-04 -31862

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 [16], the wall-clock time through JJJJ=-31862 was two hours and fifteen minutes.

Acknowledgment

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

References
[1] J. Asaadi, A Computational Comparison of Some Nonlinear Programs, Mathematical Programming, Vol. 4 (1973), pp. 144-154.
[2] R. Burden, J. Faires, and A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] R. Burden, and J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] F. Glover, 1986. Future Paths for Integer ng and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[6] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[7] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[8] 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.
[9] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[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] 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.
[13] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[14] 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.
[15] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[16] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[17] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[18] 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.
[19] 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/

Saturday, December 3, 2016

Simultaneously Solving in General Integers a Nonlinear System of 18 Simultaneous Nonlinear Equations in 36 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following problem:

- X(35) * (X(35) + 1) + 2 * X(36) = 18,

(X(35) - 1) ^ 2 + (X(36) - 6) ^ 2 = 25

EXP(X(32) * X(33) * X(34)) = 1,

3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3 = 0,

3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3 = 6666.6666666,

(X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27) = 714,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
= 2902,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 = 1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 = 659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 = 395,

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) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1 = 348,

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 = 468,

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,

and

-5<= Integers X(i)<= 15 for i=1 through 36 except 0<= Integer X(13)<= 10.

The first two equations come from Burden, Faires, and Burden [2, page 648]. Equations 3 through 6 are based on page 100, page 118, 116, and page 111 of Hock and Schittkowski [8], respectively. Equations 7 through 11 are based on page 147 of Asaadi [1]. Equation 12 is based on page 122 of Hock and Schittkowski [8], equations 15, and 16, and 17 are based on page 112 of the same book, and equation 18 is based on page 117 of the same book, as well.

One notes line 13, which is 13 A(KNEW) = -5 + FIX(RND * 20.98).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
13 A(KNEW) = -5 + FIX(RND * 20.98)

14 RANDOMIZE JJJJ

16 M = -1D+317

22 REM

91 FOR KK = 1 TO 36

94 A(KK) = A(KNEW)

95 NEXT KK
128 FOR I = 1 TO 3000000

129 FOR K = 1 TO 36

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 36)

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
255 FOR J45 = 1 TO 36

256 IF X(J45) < -5 THEN X(J45) = A(J45)

257 IF X(J45) > 15 THEN X(J45) = A(J45)

259 NEXT J45

264 X(13) = 348 - (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) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

266 IF X(13) < 0 THEN 1670

267 IF X(13) > 10 THEN 1670

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2

278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2

281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2

282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2

283 N22 = -714 + (X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)

284 N23 = -6666.6666666 + 3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3

291 N24 = 0 + 3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3

295 IF (X(32) * X(33) * X(34)) > 80 THEN GOTO 1670

296 N25 = -1 + EXP(X(32) * X(33) * X(34))

298 N26 = -18 - X(35) * (X(35) + 1) + 2 * X(36)

299 N27 = -25 + (X(35) - 1) ^ 2 + (X(36) - 6) ^ 2

1287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21) - ABS(N22) - ABS(N23) - ABS(N24) - ABS(N25) - ABS(N26) - ABS(N27)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 36

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

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)
1912 PRINT A(11), A(12), A(13), A(14), A(15), A(16), A(17), A(18), A(19), A(20)

1914 PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), A(30)

1915 PRINT A(31), A(32), A(33), A(34), A(35), A(36), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [15]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31920 is shown below:

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 0 6
10 10 0 14 1
6 5 1 1 0
0 -4 0 1 -2
10 -9.020425E-04 -31995

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 7 0 15 7
12 9 0 10 0
11 3 1 1 0
0 0 7 -2 -2
10 -22.0009 -31993

6 6 6 6 6
6 6 6 6 7
5 6 6 3 5
6 7 6 0 6
12 4 4 6 1
4 1 1 1 0
0 0 -1 -5 -2
10 -18.0009 -31988

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
11 11 2 8 2
2 -1 1 1 0
0 8 2 0 -2
2 -16.0009 -31982

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 7 -5
11 15 5 11 -1
3 0 1 1 0
0 0 6 7 -2
10 -11.0009 -31937

6 6 6 6 6
6 6 6 6 6
5 6 7 1 4
6 7 6 0 6
6 2 -1 4 -1
4 2 1 1 0
0 -4 3 0 -2
10 -17.0009 -31925

6 6 6 6 6
6 6 6 6 6
6 6 6 3 5
6 7 5 13 6
4 6 -4 4 -1
5 2 1 1 0
0 6 -4 0 -2
10 -18.0009 -31920

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 [15], the wall-clock time through JJJJ=-31920 was 20 minutes.

Acknowledgment

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

References
[1] J. Asaadi, A Computational Comparison of Some Nonlinear Programs, Mathematical Programming, Vol. 4 (1973), pp. 144-154.
[2] R. Burden, J. Faires, and A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[4] F. Glover, 1986. Future Paths for Integer ng and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[5] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[6] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[7] 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.
[8] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[9] 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.
[10] 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.
[11] 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.
[12] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[13] 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.
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[16] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[17] 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.
[18] 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/

Friday, December 2, 2016

Simultaneously Solving in General Integers a Nonlinear System of 14 Simultaneous Nonlinear Equations in 29 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following problem:

3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3 = 6666.6666666,

(X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27) = 714,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
= 2902,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 = 1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 = 659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 = 395,

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) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1 = 348,

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 = 468,

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,

and

0<= Integers X(i)<= 10 for i=1 through 29.

Equation 1 and equation 2 above are based on page 116 and page 111 of Hock and Schittkowski [7], respectively. Equations 3, 4, 5, 6, and 7 are based on page 147 of Asaadi [1]. Equation 8 is based on page 122 of Hock and Schittkowski [7], equations 11, 12, and 13 are based on page 112 of the same book, and equation 14 is based on page 117 of the same book, as well.

One notes line 13, which is 13 A(KNEW) = FIX(RND * 11).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
13 A(KNEW) = FIX(RND * 11)

14 RANDOMIZE JJJJ

16 M = -1D+317

22 REM

91 FOR KK = 1 TO 29

94 A(KK) = A(KNEW)

95 NEXT KK
128 FOR I = 1 TO 1000000

129 FOR K = 1 TO 29

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 29)

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
255 FOR J45 = 1 TO 29

256 IF X(J45) < 0 THEN X(J45) = A(J45)

257 IF X(J45) > 10 THEN X(J45) = A(J45)

259 NEXT J45

264 X(13) = 348 - (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) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

266 IF X(13) < 0 THEN 1670

267 IF X(13) > 10 THEN 1670

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2

278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2

281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2

282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2

283 N22 = -714 + (X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)

284 N23 = -6666.6666666 + 3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3

287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21) - ABS(N22) - ABS(N23)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 29

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), a(28),a(29),M, JJJJ

1670 NEXT I
1888 IF M < -10 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), A(14), A(15), A(16), A(17), A(18), A(19), A(20)

1914 PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [14]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31868 is shown below:

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
7 6 4 4 0
5 3 1 1 -9.020425E-04
-31964

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
10 4 3 6 0
7 1 1 1 -9.020425E-04
-31895

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
8 8 1 10 2
1 1 1 1 -9.020425E-04
-31869

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
5 6 4 7 0
1 4 1 1 -9.020425E-04
-31868

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 [14], the wall-clock time through JJJJ=-31868 was 12 minutes.

Acknowledgment

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

References
[1] J. Asaadi, A Computational Comparison of Some Nonlinear Programs, Mathematical Programming, Vol. 4 (1973), pp. 144-154.
[2] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[3] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[4] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[5] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[6] 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.
[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[8] 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.
[9] 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.
[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] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[12] 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.
[13] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[14] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[15] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[16] 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.
[17] 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/

Simultaneously Solving in General Integers a Nonlinear System of 12 Simultaneous Nonlinear Equations, Improved Edition

Jsun Yui Wong

The computer program listed below seeks one or more integer solutions to the following problem:

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
= 2902,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 = 1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 = 659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 = 395,

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) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1 = 348,

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 = 468,

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,

and

1<= Integers X(i)<= 15 for i=1 through 20.

Equations 1, 2, 3, 4, and 5 above are based on page 147 of Asaadi [1]. Equation 6 is based on page 122 of Hock and Schittkowski [7], equations 9, 10, and 11 are based on page 112 of the same book, and equation 12 is based on page 117 of the same book, as well.

One notes line 13, which is 13 A(KNEW) = 1 + FIX(RND * 14.98).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000

13 A(KNEW) = 1 + FIX(RND * 14.98)

14 RANDOMIZE JJJJ
15 REM

16 M = -1D+17
91 FOR KK = 1 TO 20

93 A(KK) = A(KNEW)

94 REM

95 NEXT KK
128 FOR I = 1 TO 1000000

129 FOR K = 1 TO 20

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 20)

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
255 FOR J45 = 1 TO 20

256 IF X(J45) < 1 THEN X(J45) = A(J45)

257 IF X(J45) > 15 THEN X(J45) = A(J45)

259 NEXT J45

264 X(13) = 348 - (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) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

266 IF X(13) < 1 THEN 1670

267 IF X(13) > 15 THEN 1670

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) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2

278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2

281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2

282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2

287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 20

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

1670 NEXT I
1888 IF M < -5 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), A(14), A(15), A(16), A(17), A(18), A(19), A(20), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [14]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31917 is shown below:
.
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
-9.020425E-04 -31990

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 5 13 6
-9.020425E-04 -31982

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
-9.020425E-04 -31967

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
-9.020425E-04 -31956

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
-9.020425E-04 -31938

6 6 6 6 6
6 6 6 6 6
6 6 6 6 6
6 6 5 13 6
-9.020425E-04 -31917

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 [14], the wall-clock time through JJJJ=-31917 was 8 minutes.

Acknowledgment

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

References
[1] J. Asaadi, A Computational Comparison of Some Nonlinear Programs, Mathematical Programming, Vol. 4 (1973), pp. 144-154.
[2] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[3] F. Glover, 1986. Future Paths for Integer Programming and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[4] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[5] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[6] 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.
[7] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[8] 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.
[9] 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.
[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] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[12] 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.
[13] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[14] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[15] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[16] 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.
[17] Jsun Yui Wong (20C15, 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/