Solving simultaneously with absolute values a system of 15 nonlinear equations, Improved Edition
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.
Here line 123 is 128 FOR i = 1 TO 100000, whereas line 128 of the earlier edition is 128 FOR i = 1 TO 8000000.
One notes new line 201, which is 201 X(1) = COS(X(1) - (0 + 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)))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 100000
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
201 X(1) = COS(X(1) - (0 + 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)))
202 X(2) = COS(X(2) - (X(1) + 0 + 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)))
203 X(3) = COS(X(3) - (X(1) + X(2) + 0 + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
204 X(4) = COS(X(4) - (X(1) + X(2) + X(3) + 0 + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
205 X(5) = COS(X(5) - (X(1) + X(2) + X(3) + X(4) + 0 + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
206 X(6) = COS(X(6) - (X(1) + X(2) + X(3) + X(4) + X(5) + 0 + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
207 X(7) = COS(X(7) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + 0 + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
208 X(8) = COS(X(8) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + 0 + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
209 X(9) = COS(X(9) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + 0 + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
210 X(10) = COS(X(10) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + 0 + X(11) + X(12) + X(13) + X(14) + X(15)))
211 X(11) = COS(X(11) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + 0 + X(12) + X(13) + X(14) + X(15)))
212 X(12) = COS(X(12) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + 0 + X(13) + X(14) + X(15)))
213 X(13) = COS(X(13) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + 0 + X(14) + X(15)))
214 X(14) = COS(X(14) - (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) + 0 + X(15)))
215 X(15) = COS(X(15) - (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) + 0))
220 FOR J44 = 1 TO 1
221 REM J45 = 1 + FIX(RND * 20)
222 REM
223 REM J46 = 1 + FIX(RND * 15)
224 REM IF RND < .5 THEN X(J45) = A(J45) - RND * .001 * A(J45) ELSE X(J45) = A(J45) + RND * .001 * A(J45)
225 REM IF RND < .5 THEN X(J46) = A(J46) - RND * .001 * A(J46) ELSE X(J46) = A(J46) + RND * .001 * A(J46)
226 NEXT J44
333 FOR J44 = 1 TO 15
335 IF X(J44) < 0 THEN 1670
337 NEXT J44
339 REM 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)
341 LHS(1) = X(1) - COS(X(1) - (0 + 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)))
342 LHS(2) = X(2) - COS(X(2) - (X(1) + 0 + 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 LHS(3) = X(3) - COS(X(3) - (X(1) + X(2) + 0 + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
344 LHS(4) = X(4) - COS(X(4) - (X(1) + X(2) + X(3) + 0 + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
345 LHS(5) = X(5) - COS(X(5) - (X(1) + X(2) + X(3) + X(4) + 0 + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
346 LHS(6) = X(7) - COS(X(6) - (X(1) + X(2) + X(3) + X(4) + X(5) + 0 + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
347 LHS(7) = X(8) - COS(X(7) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + 0 + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
348 LHS(8) = X(9) - COS(X(8) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + 0 + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
349 LHS(9) = X(10) - COS(X(9) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + 0 + X(10) + X(11) + X(12) + X(13) + X(14) + X(15)))
350 LHS(10) = X(1) - COS(X(10) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + 0 + X(11) + X(12) + X(13) + X(14) + X(15)))
351 LHS(11) = X(11) - COS(X(11) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + 0 + X(12) + X(13) + X(14) + X(15)))
352 LHS(12) = X(12) - COS(X(12) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + 0 + X(13) + X(14) + X(15)))
353 LHS(13) = X(13) - COS(X(13) - (X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + 0 + X(14) + X(15)))
354 LHS(14) = X(14) - COS(X(14) - (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) + 0 + X(15)))
355 LHS(15) = X(15) - COS(X(15) - (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) + 0))
443 FOR J44 = 1 TO 15
445 REM LHS(J44) = X(J44) - COS(2 * X(J44) - SUMOF15)
447 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 15
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT i
1777 IF M < -.001 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 complete output of one run through JJJJ= -31905 is shown below:
.981473779988513 .981473779988513 .981473779988513
.981473779988513 .981473779988513 .981473779988513
.981473779988513 .981473779988513 .981473779988513
.981473779988513 .981473779988513 .9814737799885133
.9814737799885133 .9814737799885133 .9814737799885133
0 -31968
.9814737799885133 .9814737799885133 .9814737799885133
.9814737799885133 .9814737799885133 .981473779988513
.981473779988513 .981473779988513 .981473779988513
.981473779988513 .981473779988513 .981473779988513
.9814737799885133 .9814737799885133 .9814737799885133
-3.33066907387547D-16 -31905
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 = -31905 was 60 seconds, 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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