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
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http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
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[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/
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