Substance

A very general computer program for solving simultaneously nonlinear/linear systems of equations: another illustration

A very general computer program for solving simultaneously nonlinear/linear systems of equations: another illustration

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 nonlinear equations from Sharma and Gupta [79, p. 8, Problem 1]:
X(1) + e ^ X(2) - COS(X(2))
3 * X(1) - X(2) - SIN(X(2)).

 

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
115 FOR J44 = 1 TO 2
116 A(J44) = -1 + RND * 2

122 NEXT J44
128 FOR i = 1 TO 70000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

140 B = 1 + FIX(RND * 2)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 13)) * 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

221 REM J45 = 1 + FIX(RND * 15)

224 REM IF RND < .5 THEN X(J45) = A(J45) - RND * .01 * A(J45) ELSE X(J45) = A(J45) + RND * .01 * A(J45)

333 FOR J44 = 1 TO 2

334 REM IF X(J44) < -10 THEN 1670

335 IF X(J44) < 0 THEN 1670
336 IF X(J44) > 1 THEN 1670

338 NEXT J44

340 REM SUMOF13 = 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)

343 REM FOR J44 = 1 TO 13

351 LHS(1) = X(1) + 2.718281828 ^ X(2) - COS(X(2))

352 LHS(2) = 3 * X(1) - X(2) - SIN(X(2))

353 REM LHS(3) = 12 * X(3) + SIN(X(3)) - 1
377 REM NEXT J44

464 PD1 = -ABS(LHS(1) - 0) - ABS(LHS(2) - 0)

466 P = PD1
1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i
1777 IF M < -.000001 THEN 1999
1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31999 is shown below:

1.785791033462545D-13 2.678685098552974D-13 -4.463652405382785D-13
-32000
2.829662332959287D-17   7.244493481874815D-17   -4.824699668432263D-17
-31999

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By unot sing the following computer system and QB64v1000-win [102], the wall-clock time (CPU time) for obtaining the output through JJJJ = -31999 was 1 second, 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.
References
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A very general computer program for solving simultaneously nonlinear/linear systems of equations

A very general computer program for solving simultaneously nonlinear/linear systems of 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 nonlinear equations from Sharma and Gupta [79, p. 9, Problem 3]:
10 * X(1) + SIN(X(1) + X(2)) - 1=0
8 * X(2) - COS(X(3) - X(2)) ^ 2 - 1=0
12 * X(3) + SIN(X(3)) - 1=0.

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
115 FOR J44 = 1 TO 3
116 A(J44) = -0 + RND * 2

122 NEXT J44
128 FOR i = 1 TO 50000

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 3)

140 B = 1 + FIX(RND * 3)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 13)) * 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

221 REM J45 = 1 + FIX(RND * 15)

224 REM IF RND < .5 THEN X(J45) = A(J45) - RND * .01 * A(J45) ELSE X(J45) = A(J45) + RND * .01 * A(J45)

333 FOR J44 = 1 TO 3

334 REM IF X(J44) < -10 THEN 1670

335 IF X(J44) < 0 THEN 1670
336 IF X(J44) > 1 THEN 1670

338 NEXT J44

340 REM SUMOF13 = 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)

343 REM FOR J44 = 1 TO 13

351 LHS(1) = 10 * X(1) + SIN(X(1) + X(2)) - 1

352 LHS(2) = 8 * X(2) - COS(X(3) - X(2)) ^ 2 - 1

353 LHS(3) = 12 * X(3) + SIN(X(3)) - 1
377 REM NEXT J44

464 PD1 = -ABS(LHS(1) - 0) - ABS(LHS(2) - 0) - ABS(LHS(3) - 0)

466 P = PD1
1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i
1777 IF M < -2 * .000001 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31998 is shown below:

6.897834917266645D-02 .2464424186091825 7.692891198753697D-02
-5.127083653466702D-15 -32000
6.897834917266855D-02 .2464424186091602 .076928911987537
-1.903428044520983D-13 -31999
6.897834917266661D-02    .246442418609183    7.692891198753694D-02
-8.5586919496000572D-16    -31998

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 = -31998 was 1 second, 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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Solving simultaneously with absolute values a system of 13 nonlinear equations

Solving simultaneously with absolute values a system of 13 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 13 nonlinear equations from Sharma and Gupta [79, p. 9, Problem 5]:

13
SIGMA x j -e ^ -x i = 0, where j and i are subscripts, 1 <= i <= 13.
J=1, j not= i

Please see Sharma and Gupta [79, p. 9, Problem 5].

 

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

115 FOR J44 = 1 TO 13
116 A(J44) = RND * .2
122 NEXT J44
128 FOR i = 1 TO 3000000
129 FOR KKQQ = 1 TO 13
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 13)

140 B = 1 + FIX(RND * 13)

144 IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 13)) * 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

221 J45 = 1 + FIX(RND * 15)

224 IF RND < .5 THEN X(J45) = A(J45) - RND * .01 * A(J45) ELSE X(J45) = A(J45) + RND * .01 * A(J45)

333 FOR J44 = 1 TO 13
335 IF X(J44) < 0 THEN 1670

337 NEXT J44

340 SUMOF13 = 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)

343 FOR J44 = 1 TO 13

348 LHS(J44) = -X(J44) + SUMOF13 - 2.718281828 ^ (-X(J44))

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)

466 P = PD1
1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 13

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i
1777 IF M < -2 * .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), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31989 is shown below:

7.716825256090046D-02    .07716507895442    7.7143556383307906D-02
7.716778537070106D-02   7.721214909202685D-02   7.7043556383307906D-02
7.712151417075941D-02   .0773250018266174   7.706336852546762D-02
7.7115421094670047D-02   .0771271157026914   7.709124802496208D-02
7.711984811915304D-02     -4.66345718129757D-05     -31992

7.742064054012318D-02 .0770510964834874 7.712501771192205D-02
.0771215615068219 7.7135825787199067D-02 7.7067929445802159D-02
7.710201075738224D-02 7.713566574533654D-02 7.72105511715949405D-02
.0770698123999177 7.713593576793588D-02 .0769623456715555
7.713830086087917D-02 -7.61027844666945D-05 -31989

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 = -31989 was 3.5 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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Solving simultaneously with absolute values a system of 15 nonlinear equations, Improved Edition

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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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 equatio

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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Solving the following system of 8 nonlinear equations

Solving the following system of 8 nonlinear equations

Jsun Yui Wong

Similar to the computer programs of the preceding papers, the computer program listed below aims to solve simultaneously the following system of 8 nonlinear equations from Sharma and Gupta [79, p. 9, Problem 4]:

x i2 * ( X(i) + x i+1 ) -1.5 = 0, i=1, 2, 3,..., 7 ; (The first i an underscript, and the first 2 is a superscript. )
X 8 2 * ( x 1+X 8 ) - 1.5 = 0. (The first 8 is an underscript, and the 2 is a superscript.)

The two line above involve superscripts and underscritps; so one constrasts them with line 207 and line 463, which are 207 X(8) = (1.5 - X(7) ^ 3) / X(7) ^ 2 and 463 PD1 = -ABS((X(8) ^ 2 * (X(1) + X(8)) - 1.5)), respectively. It is even better if one can see Sharma and Gupta [79, p. 9, Problem 4], which is open access for free.

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 A(1) = RND * 2

128 FOR i = 1 TO 100000

129 FOR KKQQ = 1 TO 8

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 8)

140 B = 1 + FIX(RND * 8)
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

201 X(2) = (1.5 - X(1) ^ 3) / X(1) ^ 2

202 X(3) = (1.5 - X(2) ^ 3) / X(2) ^ 2

203 X(4) = (1.5 - X(3) ^ 3) / X(3) ^ 2

204 X(5) = (1.5 - X(4) ^ 3) / X(4) ^ 2

205 X(6) = (1.5 - X(5) ^ 3) / X(5) ^ 2

206 X(7) = (1.5 - X(6) ^ 3) / X(6) ^ 2

207 X(8) = (1.5 - X(7) ^ 3) / X(7) ^ 2

333 FOR J44 = 1 TO 8

335 IF X(J44) < 0 THEN 1670

337 NEXT J44
463 PD1 = -ABS((X(8) ^ 2 * (X(1) + X(8)) - 1.5))

466 P = PD1
1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 8

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i
1777 IF M < -.0000000001 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31982 is shown below:

.90856029641607 .9085602964160693 .9085602964160726
.908560296416056 .9085602964161392 .908560296415723
.9085602964178046 .9085602964073962 -3.579944923403494D-11
-32000
.9085602964160698    .9085602964160698    .9085602964160698
.9085602964160698    .9085602964160698    .9085602964160698
.9085602964160698    .9085602964160698    -2.612927235690066D-17
-31999
.9085602964160698 .9085602964160698 .9085602964160698
.9085602964160698 .9085602964160698 .9085602964160698
.9085602964160698 .9085602964160698 -2.612927235690066D-17
-31997
.9085602964160698 .9085602964160698 .9085602964160698
.9085602964160698 .9085602964160698 .9085602964160698
.9085602964160698 .9085602964160698 -2.612927235690066D-17
-31982

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 = -31982 was 20 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.

References

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Using the preceding computer program as a template to solve nonlinear systems of equations: another illustration

Using the preceding computer program as a template to solve nonlinear systems of equations:
another illustration

Jsun Yui Wong

The computer program listed below aims to solve the following system of nonlinear equations from Google [34]:
X(1) ^ 2 + X(2) ^ 2 = 31
X(1) ^ 3 + X(2) ^ 3 = 154.

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 A(1) = -100 + RND * 200

116 A(2) = -100 + RND * 200

128 FOR i = 1 TO 20000

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 3)
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 IF (31 - X(2) ^ 2) < 0 THEN 1670

222 X(1) = (31 - X(2) ^ 2) ^ .5

333 FOR J44 = 1 TO 3

335 REM X(J44) = INT(X(J44))

337 NEXT J44

460 PD1 = -ABS(X(1) ^ 3 + X(2) ^ 3 - 154) - ABS(X(1) ^ 2 + X(2) ^ 2 - 31)

466 P = PD1
1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i
1777 IF M < -1 THEN 1999

1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31996 is shown below:

5.391164991562635      -1.3911649915634      -4.47055586993983D-14
-32000
5.391164991562635      -1.3911649915634      -4.336114800551627D-14
-31999
5.302775637731995      1.697224362268005      -3.620367894363596D-15
-31998
5.391164991562635      -1.3911649915634      -6.242575900650138D-14
-31997
1.697224362268005      5.302775637731995      -8.762088277158853D-15
-31996

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 = -31996 was 3 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.

References
[1] Siby Abraham, Sugata Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and Mgmt., Vol. III, Issue 1, January-June 2013.
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https://math.stackexchange.com/questions/2186612/easy-and-rel...
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[5] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[6] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[7] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[8] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.
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[12] Hirak Basumatary (1 January 2019). Solve system of equations and inequalities with multiple solutions?
https://www.mathworks.com/matlabcentral/answers/437815-solve-system-of-equations-and-inequalities-with-multiple-solutions?_tid=prof_contriblnk
[13 ] Ahmad Bazzi (January 20, 2022). Multidimensional Newton--Approximate nonlinear equations by sequence of linear equations--lecture 6. (Youtube is where I saw this work.)
http://bazziahmad.com/
[14] Madhulima Bhandari (24 February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?
https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab
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http://purkh.com/index.php/mathlab

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[17] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.

[18] Matthew Chan, Yillian Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with absolute values.
https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute_values#Numerical_ Example

[19] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.

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Using the preceding computer program as a template to solve simultaneously 2*x+y=5-2z^2, y^3+4*z=4, and x*y + z= e^z, second edition

Using the preceding computer program as a template to solve simultaneously 2*x+y=5-2z^2, y^3+4*z=4, and x*y + z= e^z, second edition

Jsun Yui Wong

The computer program listed below aims to solve the following system of nonlinear equations from Google search [34]:
2*x+y=5-2*z^2
y^3+4*z=4
x*y + z= e^z.

 

0 REM 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) = RND * 20
116 A(2) = RND * 20
119 REM A(3) = RND * 2
128 FOR i = 1 TO 10000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)

140 B = 1 + FIX(RND * 3)
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

222 X(3) = (4 - X(2) ^ 3) / 4
224 X(1) = (5 - 2 * X(3) ^ 2 - X(2)) / 2

333 FOR J44 = 1 TO 3

335 IF X(J44) < 0 THEN 1670

337 NEXT J44
459 REM REM PD1 = -ABS(X(1) ^ 5 + X(2) ^ 5 - 464)
460 PD1 = -ABS(X(1) * X(2) + X(3) - 2.718281828 ^ X(3))
466 P = PD1
1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i
1777 IF M < -1 THEN 1999

1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

 

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31998 is shown below:

1.422469       .9753885       .7680081       -8.071313E-09
-32000
1.422469       .9753885       .7680081       -8.071313E-09
-31999
1.422469       .9753885       .7680081       -8.071313E-09
-31998

 

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 = -31998 was 3 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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[34] Google. This paper's system of nonlinear equations was certainly obtained from Google search: https://support.ptc.com/help/mathcad/r9.0/en/index.html#page/PTC_Mathcad_Help/example_solv_nonlin_syst_of_eq.html

 

[35] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[36] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8...

[37] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw Hill, Boston, 2010.

[38] Willi Hock, Klaus Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[39] Xue-Ping Hou, Pei-Ping Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial geometric programming problems, Abstract and Applied Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263
[40] C. H. Huang, H. Y. Kao. (2009). An effective linear approximation method for geometric programming problems, IEEE Publications, 1743-1746.
[41] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mathematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

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https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

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[60] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
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[63] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

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Using the preceding computer program as a template to solve simultaneously 2*x+y=5-2z^2, y^3+4*z=4, and x*y + z= e^z

Using the preceding computer program as a template to solve simultaneously 2*x+y=5-2z^2, y^3+4*z=4, and x*y + z= e^z

Jsun Yui Wong

The computer program listed below aims to solve the following system of nonlinear equations from Google search [34]:

2*x+y=5-2z^2
y^3+4*z=4
x*y + z= e^z.

 

0 REM 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) = RND * 20
116 A(2) = RND * 20
119 REM A(3) = RND * 2
128 FOR i = 1 TO 10000
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)

140 B = 1 + FIX(RND * 3)
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

222 X(3) = (4 - X(2) ^ 3) / 4
224 X(1) = (5 - 2 * X(3) ^ 2 - X(2)) / 2

333 FOR J44 = 1 TO 3

335 IF X(J44) < 0 THEN 1670

337 NEXT J44
459 REM REM PD1 = -ABS(X(1) ^ 5 + X(2) ^ 5 - 464)
460 PD1 = -ABS(X(1) * X(2) + X(3) - 2.718281828 ^ X(3))
466 P = PD1
1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i
1777 IF M < -1 THEN 1999

1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

 

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31998 is shown below:

1.422469       .9753885       .7680081       -8.071313E-09
-32000
1.422469       .9753885       .7680081       -8.071313E-09
-31999
1.422469       .9753885       .7680081       -8.071313E-09
-31998

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 = -31998 was 3 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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[102] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

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Using the preceding computer program as a template to solve simultaneously X(1) ^ 2 = X(2) + 241 and X(2) ^ 2 = X(1) +241 from Communolizer [20]

Using the preceding computer program as a template to solve simultaneously
X(1) ^ 2 = X(2) + 241 and X(2) ^ 2 = X(1) +241 from Communolizer [20]

Jsun Yui Wong

The computer program listed below aims to solve the following system of nonlinear equations from Communolizer [20]:
X(1) ^ 2 = X(2) + 241
X(2) ^ 2 = X(1) +241.

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 A(1) = -100 + RND * 200

128 FOR i = 1 TO 20000

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 3)
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
220 REM IF (13 - X(2) ^ 2) < 0 THEN 1670

222 X(2) = X(1) ^ 2 - 241

333 FOR J44 = 1 TO 3

335 REM X(J44) = INT(X(J44))

337 NEXT J44
459 PD1 = -ABS(X(2) ^ 2 - X(1) - 241)
466 P = PD1
1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1670 NEXT i
1777 IF M < -9999999 THEN 1999
1904 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31996 is shown below:

-16        15       0       -32000
-15.03222456700907       -15.03222456700908       -3.60822483003176D-13
-31999
-16       15       0       -31998
16.03222456700906       16.03222456700894       -4.110434215200948D-12
-31997
15       -16       0       -31996

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 = -31996 was 3 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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