Jsun Yui Wong
"Solving systems of nonlinear equations is perhaps the most difficult
problem in all of numerical computations," Rice [76, 1993, p. 354].
The computer program listed below seeks to solve simultaneously the
following system of nonlinear equations:
3 * X(1) - COS(X(2) * X(3)) -
1 / 2 =0
X(1) ^ 2 - 81 * (X(2) + .1) ^
2 + SIN(X(3)) + 1.06 =0
2.718281828 ^ (-X(1) * X(2))
+ 20 * X(3) + (10 * 3.141592654 - 3) / (3)) =0
where X(1) through X(3) are continuous variables, given, for example,
-1<=X(i)<=1 for i=1, 2, 3.
The three equations above are from Burden, Faires, and Burden [16, p.
654].
One notes line 1125, which is 1125 P = -ABS(3 * X(1) - COS(X(2) * X(3)) - 1
/ 2) - ABS(X(1) ^ 2 - 81 * (X(2) + .1) ^ 2 + SIN(X(3)) + 1.06) -
ABS(2.718281828 ^ (-X(1) * X(2)) + 20 * X(3) + (10 * 3.141592654 - 3) / (3)).
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111),
PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22),
UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22),
C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
83 RANDOMIZE JJJJ
87 M = -4E+299
120 FOR J44 = 1 TO 3
121 A(J44) = -1 + FIX(RND *
3)
122 REM A(J44) = -1 + (RND * 2)
123 NEXT J44
128 FOR I = 0 TO FIX(RND * 50000)
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 +
RND * 3.3)
143 j = 1 + FIX(RND * 3)
154 IF RND < .5 THEN
GOTO 156 ELSE GOTO 162
156 REM
157 R = (1 - RND * 2) *
(A(j))
160 X(j) = A(j) + (RND ^
(RND * 15)) * R
161 GOTO 169
162 IF RND < .5 THEN
X(j) = A(j) - FIX(1 + RND * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)
169 NEXT IPP
171 FOR J44 = 1 TO 3
172 IF X(J44) < -1
THEN 1670
174 IF X(J44) > 1 THEN
1670
175 NEXT J44
1125 P = -ABS(3 * X(1) -
COS(X(2) * X(3)) - 1 / 2) - ABS(X(1) ^ 2 - 81 * (X(2) + .1) ^ 2 + SIN(X(3)) +
1.06) - ABS(2.718281828 ^ (-X(1) * X(2)) + 20 * X(3) + (10 * 3.141592654 - 3) /
(3))
1221 IF P <= M THEN 1670
1420 M = P
1444 FOR KLX = 1 TO 3
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1894 IF M < -.00009 THEN 1999
1923 PRINT A(1), A(2), A(3), M,
JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [102]. The complete output of a single run through
JJJJ= -31997 is shown below:
.500000000000001 -3.659687308162791D-12 -.5235987756667582
-3.14960731107039D-15 -31999
.5 0 -.523598775666633 -5.98519378589707D-11
-31998
.5000000000000001 0
-.5235987756666667
-5.920867463850288D-11 -31997
Above there is no rounding by hand; it is just straight copying by hand
from the monitor screen. On a personal computer with Processor Intel (R) Core
(TM) 2 Duo CPU E8400 @ 3.0 GHz 3.0 GHz,
4.00 GB of RAM (3.9 GB usable), 64-bit
Operating System, and QB64v1000-win [102], the wall-clock time (not CPU time)
for obtaining the output through
JJJJ = -31997 took 4 seconds,
counting from "Starting program...". One can compare the computational results
above with those in Burden et al. [16,
p. 655, Table 10.3].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom
Clark.
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