Solving simultaneously with absolute values a system of 15 nonlinear equations
Jsun Yui Wong
Similar to the computer programs of the preceding papers, the computer program listed below seeks to solve the following system of of 15 nonlinear equations from Sharma and Gupta [79, p. 9, Problem 6]:
X(i) - COS(2* X(i) -( X(1) +...+ X(15))=0, 1<= i <= 15.
One notes line 221 and line 224, which are 221 J45 = 1 + FIX(RND * 15) and 224 IF RND < .5 THEN X(J45) = A(J45) - RND * .001 * A(J45) ELSE X(J45) = A(J45) + RND * .001 * A(J45).
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
114 REM A(1) = -100 + RND * 200
115 FOR J44 = 1 TO 15
116 A(J44) = RND * 2
122 NEXT J44
123 REM
128 FOR i = 1 TO 800000
129 FOR KKQQ = 1 TO 15
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 15)
140 B = 1 + FIX(RND * 15)
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
199 REM
221 J45 = 1 + FIX(RND * 15)
222 REM
224 IF RND < .5 THEN X(J45) = A(J45) - RND * .001 * A(J45) ELSE X(J45) = A(J45) + RND * .001 * A(J45)
333 FOR J44 = 1 TO 20
335 IF X(J44) < 0 THEN 1670
337 NEXT J44
339 SUMOF15 = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)
343 FOR J44 = 1 TO 15
345 LHS(J44) = X(J44) - COS(2 * X(J44) - SUMOF15)
347 NEXT J44
464 PD1 = -ABS(LHS(1) - 0) - ABS(LHS(2) - 0) - ABS(LHS(3) - 0) - ABS(LHS(4) - 0) - ABS(LHS(5) - 0) - ABS(LHS(6) - 0) - ABS(LHS(7) - 0) - ABS(LHS(8) - 0) - ABS(LHS(9) - 0) - ABS(LHS(10) - 0) - ABS(LHS(11) - 0) - ABS(LHS(12) - 0) - ABS(LHS(13) - 0) - ABS(LHS(14) - 0) - ABS(LHS(15) - 0)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 20
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT i
1777 IF M < -.000022 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(11), A(12), A(13), A(14), A(15), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [102]. Its output of one run through JJJJ= -31930 is summarized below:
.9814752930912436 .9814744062820444 .9814694999619514
.981473149751699 .9814744943561806 .981472067410644
.9814737255296876 .9814747157693836 .9814751151087455
.9814724950905058 .9814756955382911 .9814767097465355
.9814717440167047 .9814732118406873 .9814747203085347
-1.3377424339045D-05 -32000
.
.
.
.981474116329137 .981473777530832 .9814720832938884
.9814727362039528 .981475856713149 .9814752533700367
.9814725212349293 .9814749820740133 .9814734406963844
.981473081242802 .9814743270270298 .9814734088339313
.9814700351700231 .981477797892055 .9814741423984115
-1.198692079995567D-05 -31930
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 = -31930 was 6 minutes, counting from "Starting program...".
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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