Jsun Yui Wong
The computer program listed below seeks to solve the immediately following integer nonlinear system of equations:
2 * X(1) + X(2) + X(3) + X(4) +X(5) = 6
X(1) + 2 * X(2) + X(3) + X(4) + X(5) = 6
X(1) + X(2) + 2 * X(3) + X(4) + X(5) = 6
X(1) + X(2) + X(3) + 2 * X(4) + X(5) = 6
X(1) * X(2) * X(3) * X(4) * X(5) =1
-32<=X(i)<=32, i=1, 2, 3. 4, 5,
X(1) through X(5) are integer variables.
The problem above is based on Floudas et al.’s Test Problem 5 on p. 329 [7, Brown’s almost linear system], which is as follows
2 * X(1) + X(2) + X(3) + X(4) +X(5) = 6
X(1) + 2 * X(2) + X(3) + X(4) + X(5) = 6
X(1) + X(2) + 2 * X(3) + X(4) + X(5) = 6
X(1) + X(2) + X(3) + 2 * X(4) + X(5) = 6
X(1) * X(2) * X(3) * X(4) * X(5) =1
-2<=X(i)<=2, i=1, 2, 3. 4, 5,
X(1) through X(5) are continuous variables.
0 DEFDBL A-Z
2 DEFINT K
3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 31996. STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
33 FOR J44 = 1 TO 5
34 A(J44) = -32 + RND * 64
37 NEXT J44
128 FOR I = 1 TO 10000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 j = 1 + FIX(RND * 5)
189 r = (1 – RND * 2) * A(j)
190 X(j) = A(j) + (RND ^ (RND * 10)) * r
195 NEXT IPP
211 FOR J44 = 1 TO 5
214 X(J44) = INT(X(J44))
218 NEXT J44
225 FOR J44 = 1 TO 5
226 IF X(J44) < -32## THEN 1670
227 IF X(J44) > 32## THEN 1670
228 NEXT J44
229 GOTO 243
243 X(5) = -2 * X(1) – X(2) – X(3) – X(4) + 6
244 IF X(5) < -32## THEN 1670
245 IF X(5) > 32## THEN 1670
246 LHS2 = X(1) + 2 * X(2) + X(3) + X(4) + X(5) – 6
247 LHS3 = X(1) + X(2) + 2 * X(3) + X(4) + X(5) – 6
248 LHS4 = X(1) + X(2) + X(3) + 2 * X(4) + X(5) – 6
249 LHS5 = X(1) * X(2) * X(3) * X(4) * X(5) – 1
455 POBA = -ABS(LHS2) – ABS(LHS3) – ABS(LHS4) – ABS(LHS5)
466 P = POBA
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 5
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 0 THEN 1999
1907 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [55]. The complete output through JJJJ = -31836.4000000262 is shown below:
1 1 1 1 1
0 -31959.75000000644
1 1 1 1 1
0 -31881.09000001903
1 1 1 1 1
0 -31836.4000000262
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [55], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31836.4000000262 was 6 minutes, total, including the time for “Creating .EXE file.”
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.
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