Jsun Yui Wong
"Solving systems of
nonlinear equations is perhaps the most difficult problem in all of numerical
computations," Rice [77, 1993, p.355].
The computer program listed
below seeks to solve simultaneously the following system of nonlinear/linear
equations:
(X(1) ^ 2 + X(3) ^ 2) ^ .5 - 1 =0,
(X(2) ^ 2 + X(4) ^ 2) ^ .5 - 1 =0,
X(1) * X(2) + X(3) * X(4) =0,
-.58 * X(2) - .19 * X(3) =0.
These four equations are
based on the system in Stackoverflow [85].
One notes line 1125, which
is 1125 P = -ABS((X(1) ^ 2 + X(3) ^ 2) ^ .5## - 1) - ABS((X(2) ^ 2 + X(4) ^ 2)
^ .5## - 1) - ABS(X(1) * X(2) + X(3) * X(4)) - ABS(-.58 * X(2) - .19 * X(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 4
121 A(J44) = FIX(RND * 6)
122 REM A(J44) = (RND * 5)
123 NEXT J44
128 FOR I = 0 TO FIX(RND * 100000)
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 + RND * 2.3)
143 j = 1 + FIX(RND * 4)
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
1125 P = -ABS((X(1) ^ 2 + X(3) ^ 2) ^
.5## - 1) - ABS((X(2) ^ 2 + X(4) ^ 2) ^ .5## - 1) - ABS(X(1) * X(2) + X(3) *
X(4)) - ABS(-.58 * X(2) - .19 * X(3))
1221 IF P <= M THEN 1670
1420 M = P
1444 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1894 IF M < -.00001 THEN 1999
1923 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program
was run with QB64v1000-win [104].
Selected candidate solutions of a single run through JJJJ= -31914 are
shown below:
-.9999999999798488 -4.061979906966781D-06 4.06197128225776D-06
-.9999999999895248 -1.584196553836798D-06 -31987
.9999999999981254 5.407536964059212D-07 -5.407536964611527D-07
.9999999999903195 -2.109208042962987D-07 -31984
.
.
.
1 0
0 1 0
-31972
.
.
.
-1 0
0 -1 0
-31954
.
.
.
-1 0
0 1 0
-31949
.
.
.
-1 0
0 -1 0
-31933
.
.
.
-1 0
0 1 0
-31915
1 0
0 -1
-31914
One notes the four distinct solutions shown above; see Stackoverflow [85].
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
[104], the wall-clock time (not CPU time) for obtaining the output through JJJJ =
-31914 took 11 seconds, counting from "Starting program...". One can compare the computational results
above with those in Stackoverflow [85].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom
Clark.
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