Solving a nonlinear system of 1000 nonlinear equations involving 1000 general integer variables
Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program listed below seeks to solve simultaneously the following system of 1000 nonlinear equations involving 1000 general integer variables; this problem is based on Problem 5 in Sharma and Gupta [80, p. 7, Problem 5]:
X(i) ^2 * X(i+1) -1 = 0, (i = 1, 2,..., 999)
X(1000) ^ 2 * X(1) -1 = 0.
One notes the following line 1905 and line 1988, which are 1905 PRINT A(1), A(2), A(3), A(4), A(5) and 1988 PRINT A(996), A(997), A(998), A(999), A(1000), M, JJJJ.
0 DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
116 REM
118 A(1) = INT(0 + RND * 10)
128 FOR i = 1 TO 50000
129 FOR KKQQ = 1 TO 1000
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + 1000)
140 B = 1 + FIX(RND * 1000)
144 IF RND < .5 THEN 160 ELSE GOTO 167
160 r = (1 - RND * 2) * A(B)
164 X(B) = A(B) + (RND ^ (RND * 15)) * r
165 GOTO 168
167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)
168 NEXT IPP
177 FOR J44 = 1 TO 1000
178 X(J44) = INT(X(J44))
179 NEXT J44
231 FOR J44 = 2 TO 1000
234 X(J44) = 1 / (X(J44 - 1)) ^ 2
239 NEXT J44
333 REM
334 FOR J44 = 1 TO 1000
335 IF X(J44) < 0 THEN 1670
337 NEXT J44
461 REM
463 PD1 = -ABS(X(1000) ^ 2 * X(1) - 1)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 1000
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT i
1905 PRINT A(1), A(2), A(3), A(4), A(5)
1988 PRINT A(996), A(997), A(998), A(999), A(1000), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31994 is shown below:
1 1 1 1 1
1 1 1 1 1
0 -32000
0 1.#INF 0 1.#INF 0
1.#INF 0 1.#INF 0 1.#INF
-1.#IND -31999
0 1.#INF 0 1.#INF 0
1.#INF 0 1.#INF 0 1.#INF
-1.#IND -31998
0 1.#INF 0 1.#INF 0
1.#INF 0 1.#INF 0 1.#INF
-1.#IND -31997
1 1 1 1 1
1 1 1 1 1
0 -31996
1 1 1 1 1
1 1 1 1 1
0 -31995
1 1 1 1 1
1 1 1 1 1
0 -31994
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31994 was 6 minutes, counting from "Starting program...". One can see the computational results above and the discussion in Sharma and Gupta [80, Problem 5].
The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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