Substance

An easiest method for solving systems of nonlinear equations: another illustration

An easiest method for solving systems of nonlinear equations: another illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following system of nonlinear equations from Pkrzysiek Pavel [70]:

(X(2) + X(3) * X(2) - 5)=0
(X(1) + X(3) * X(1) - 4)=0
(X(1) * X(2) - 720)=0.

The key line of the following computer program is 575 PD1 = -ABS(X(2) + X(3) * X(2) - 5) - ABS(X(1) + X(3) * X(1) - 4) - ABS(X(1) * X(2) - 720).

 

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
87 RANDOMIZE JJJJ
88 M = -3D+30

117 FOR J44 = 1 TO 3

121 A(J44) = 0 + RND * 20

126 NEXT J44

127 FOR i = 1 TO 30000

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)

156 IF RND < .5 THEN 160 ELSE GOTO 167

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

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

166 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
211 FOR J44 = 1 TO 3

213 REM X(J44) = INT(X(J44))

219 NEXT J44
422 FOR J44 = 1 TO 3
255 REM

455 NEXT J44
575 PD1 = -ABS(X(2) + X(2) * X(3) - 5) - ABS(X(1) + X(1) * X(3) - 4) - ABS(X(1) * X(2) - 720)
669 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
1778 IF M < -.01 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its output of one run through
JJJJ= -26727 is summarized below. GW-BASIC also can handle this computer program.

.
.
.
23.99856456315997       30.00179440337307       -.8333433016447183
-4.784646378603544D-04       -26727

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -26727 was 5 minutes, counting from “Starting program…”. One can compare the computational results presented above with the results in Pkrzysiek Pavel [70].
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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An easiest method for solving nonlinear systems of equations: an illustration

An easiest method for solving nonlinear systems of equations: an illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below is based on the computer program in Wong [106] and aims to solve directly the following system of four nonlinear equations from Sharma, Guba, and Gupta [82, p. 705]:

X(3) * X(2) - X(3) - X(4) - X(1) - X(2) - 1 = 0
X(3) * X(1) - 2 * X(3) - X(4) - X(1) - X(2) - 1 = 0
X(3) * X(4) - 4 * X(3) - 2 * X(4) - 2 * X(1) - 2 * X(2) - 2 = 0
X(3) ^ 2 - 8 * X(3) - 4 * X(4) - 4 * X(1) - 4 * X(2) - 4 = 0

where the four variables are integers in the present paper.

One notes that the main line below is 699, which is 699 PD1 = -ABS(X(3) * X(2) - X(3) - X(4) - X(1) - X(2) - 1) - ABS(X(3) * X(1) - 2 * X(3) - X(4) - X(1) - X(2) - 1) - ABS(X(3) * X(4) - 4 * X(3) - 2 * X(4) - 2 * X(1) - 2 * X(2) - 2) - ABS(X(3) ^ 2 - 8 * X(3) - 4 * X(4) - 4 * X(1) - 4 * X(2) - 4)

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
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 4
121 A(J44) = 0 + RND * 100
124 NEXT J44
127 FOR i = 1 TO 1000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 4)
140 B = 1 + FIX(RND * 4)
156 IF RND < .5 THEN 160 ELSE GOTO 169
160 R = (1 - RND * 2) * A(B)
164 X(B) = A(B) + (RND ^ (RND * 15)) * R
166 GOTO 188
169 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

188 NEXT IPP
211 FOR J44 = 1 TO 4

213 X(J44) = INT(X(J44))

219 NEXT J44
685 FOR J44 = 1 TO 4
686 IF X(J44) < 0 THEN 1670
687 IF X(J44) > 100 THEN 1670
688 NEXT J44
699 PD1 = -ABS(X(3) * X(2) - X(3) - X(4) - X(1) - X(2) - 1) - ABS(X(3) * X(1) - 2 * X(3) - X(4) - X(1) - X(2) - 1) - ABS(X(3) * X(4) - 4 * X(3) - 2 * X(4) - 2 * X(1) - 2 * X(2) - 2) - ABS(X(3) ^ 2 - 8 * X(3) - 4 * X(4) - 4 * X(1) - 4 * X(2) - 4)

777 p = PD1
1111 IF p <= M THEN 1670
1452 M = p
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT i
1778 IF M < -1000 THEN 1999
1920 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output of one run through JJJJ= -31996 is show below. GW-BASIC also can handle this computer program.

0 0 0 0 -8
-32000
3       2       12       6       0
-31999
0 0 0 0 -8
-31998
3 2 12 6 0
-31997
3 2 12 6 0
-31996

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the total wall-clock time (not CPU time) for obtaining the output through JJJJ=-31996 was 1 second, counting from "Starting program...". Please see Sharma, Guba, and Gupta [82, p. 705].

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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Erratum in the preceding paper ( An easiest method for solving nonlinear systems of equations )

Jsun Yui Wong

The title of the preceding paper is An easiest method for solving nonlinear systems of equations.

 

An easiest method for solving nonlinear systems of equations

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below is based on the computer program in Wong [105] and aims to solve directly the following nonlinear system of equations from Maranas and Floudas [58, 59]:

2 * X(1) + X(2) + X(3) + X(4) + X(5) - 6 = 0
X(1) + 2 * X(2) + X(3) + X(4) + X(5) - 6 = 0
X(1) + X(2) + 2 * X(3) + X(4) + X(5) - 6 = 0
X(1) + X(2) + X(3) + 2 * X(4) + X(5) - 6 = 0
X(1) * X(2) * X(3) * X(4) * X(5) - 1 = 0
-2 <= X(i) <= 2, i=1, ... , 5.

One notes the main line below is line 567, which is 567 PD1 = -ABS(2 * X(1) + X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + 2 * X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + 2 * X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + X(3) + 2 * X(4) + X(5) - 6) - ABS(X(1) * X(2) * X(3) * X(4) * X(5) - 1) .

 

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
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 5
121 A(J44) = -2 + RND * 4
124 NEXT J44
127 FOR i = 1 TO 200000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 5)

140 B = 1 + FIX(RND * 5)
156 IF RND < .5 THEN 160 ELSE GOTO 169

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

164 X(B) = A(B) + (RND ^ (RND * 15)) * r
166 GOTO 188

169 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

188 NEXT IPP
211 FOR J44 = 1 TO 5
213 REM X(J44) = INT(X(J44))
219 NEXT J44
421 FOR J44 = 1 TO 5
433 IF X(J44) < -2 THEN 1670
435 IF X(J44) > 2 THEN 1670
455 NEXT J44
567 PD1 = -ABS(2 * X(1) + X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + 2 * X(2) + X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + 2 * X(3) + X(4) + X(5) - 6) - ABS(X(1) + X(2) + X(3) + 2 * X(4) + X(5) - 6) - ABS(X(1) * X(2) * X(3) * X(4) * X(5) - 1)

679 p = PD1
1111 IF p <= M THEN 1670
1452 M = p
1454 FOR KLX = 1 TO 5

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1670 NEXT i

1778 IF M < -.00009 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31980 is shown below. GW-BASIC also can handle this computer program.

.9999186728934321       .9999308699505136       .9999308735024195
.9999309655166014       1.000347743896858       -7.135564008152223D-05
-31980

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31980 was 10 seconds, counting from “Starting program…”. One can compare the computational results presented above with the computational results in Maranas and Floudas [58, 59].

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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Computer program for solving mixed-integer nonlinear programming (MINLP) problems, including systems of nonlinear equations: applied to a system of 4 integer nonlinear equations

Computer program for solving mixed-integer nonlinear programming (MINLP) problems, including systems of nonlinear equations: applied to a system of 4 integer nonlinear equations

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following system of four nonlinear equations from Sharma, Guba, and Gupta [82, p. 705]:

X(3) * X(2) - X(3) - X(4) - X(1) - X(2) - 1 = 0
X(3) * X(1) - 2 * X(3) - X(4) - X(1) - X(2) - 1 = 0
X(3) * X(4) - 4 * X(3) - 2 * X(4) - 2 * X(1) - 2 * X(2) - 2 = 0
X(3) ^ 2 - 8 * X(3) - 4 * X(4) - 4 * X(1) - 4 * X(2) - 4 = 0
where the four variables are integers in the present paper.

One notes line 213, which is 213 X(J44) = INT(X(J44)).

 

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
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 4
121 A(J44) = 0 + RND * 100
124 NEXT J44
127 FOR i = 1 TO 1000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 4)
140 B = 1 + FIX(RND * 4)
156 IF RND < .5 THEN 160 ELSE GOTO 169
160 R = (1 - RND * 2) * A(B)
164 X(B) = A(B) + (RND ^ (RND * 15)) * R
166 GOTO 188
169 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

188 NEXT IPP
211 FOR J44 = 1 TO 4

213 X(J44) = INT(X(J44))

219 NEXT J44

335 X(1) = X(3) * X(2) - X(3) - X(4) - X(2) - 1
685 FOR J44 = 1 TO 4
686 IF X(J44) < 0 THEN 1670
687 IF X(J44) > 100 THEN 1670
688 NEXT J44
699 PD1 = -ABS(X(3) * X(1) - 2 * X(3) - X(4) - X(1) - X(2) - 1) - ABS(X(3) * X(4) - 4 * X(3) - 2 * X(4) - 2 * X(1) - 2 * X(2) - 2) - ABS(X(3) ^ 2 - 8 * X(3) - 4 * X(4) - 4 * X(1) - 4 * X(2) - 4)
777 p = PD1
1111 IF p <= M THEN 1670
1452 M = p
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT i
1778 IF M < -1000 THEN 1999
1920 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output of one run through JJJJ= -31989 is show below. GW-BASIC also can handle this computer program.

 

0 7 4 16 -144
-31999
0 5 2 2 -76
-31996
3       2       12       6       0
-31995
4 3 8 8 -64
-31994
0 11 2 8 -148
-31993
3 2 12 6 0
-31989

 

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the total wall-clock time (not CPU time) for obtaining the output through JJJJ=-31989 was 1 second, counting from "Starting program...". Please see Sharma, Guba, and Gupta [82, p. 705].

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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Computer program for solving mixed-integer nonlinear programming (MINLP) problems, including systems of nonlinear equations: applied to a boundary value problem with fifteen nonlinear equations, third edition

Computer program for solving mixed-integer nonlinear programming (MINLP) problems, including systems of nonlinear equations: applied to a boundary value problem with fifteen nonlinear equations, third edition

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following system of 15 nonlinear equations from Sharma and Gupta [81, Problem 6, p. 599]:

y( k-1 ) -2 y( k ) + y( k+1 ) + h^2 y( k ) ^3 = 0, k = 1, 2, 3, ... , m-1

The present case has m=16, h= 1/m, n=15 and other conditions shown in Sharma and Gupta [81, Problem 6, p. 599].

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

87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 15
121 A(J44) = .05 + RND * .5
124 NEXT J44
127 FOR i = 1 TO 1000000

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)

156 IF RND < .5 THEN 160 ELSE GOTO 169

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

164 X(B) = A(B) + (RND ^ (RND * 15)) * R

166 GOTO 188
169 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

188 NEXT IPP
211 FOR J44 = 1 TO 15

213 REM X(J44) = INT(X(J44))

219 NEXT J44
333 X(2) = -(0 - 2 * X(1) + 0 * X(2) + .00390625 * X(1) ^ 3) - 0 * ABS(X(14) - 2 * X(15) + 1 + .00390625 * X(15) ^ 3)

337 X(14) = -0 * ABS(0 - 2 * X(1) + X(2) + .00390625 * X(1) ^ 3) - (0 * X(14) - 2 * X(15) + 1 + .00390625 * X(15) ^ 3)
339 REM GOTO 666
341 FOR J44 = 3 TO 15

343 X(J44) = -(X(J44 - 2) - 2 * X(J44 - 1) + 0 * X(J44) + .00390625 * X(J44 - 1) ^ 3)
344 NEXT J44

680 SUM = 0

682 FOR J44 = 3 TO 15

683 SUM = SUM - (X(J44 - 2) - 2 * X(J44 - 1) + X(J44) + .00390625 * X(J44 - 1) ^ 3)
684 NEXT J44

685 FOR J44 = 1 TO 15

686 IF X(J44) < 0 THEN 1670
687 IF X(J44) > 1 THEN 1670

688 NEXT J44

698 PD1 = - ABS(0 - 2 * X(1) + X(2) + .00390625 * X(1) ^ 3) - ABS(X(14) - 2 * X(15) + 1 + .00390625 * X(15) ^ 3)

700 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
1778 IF M < -9999999 THEN 1999
1920 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 [103]. Its complete output of one run through JJJJ= -31998 is summarized below. GW-BASIC also can handle this computer program.

6.599763320036463D-02    .1319941434902927    .1979816207259937
.2639388845380347    .3298242742545747    .395569509201723
.4610729241597304    .5261935245263726    .5907449789924144
.6544911044873543    .7171421345765487    .7783524329539742
.837720734425025    .8947925814807637    .9490659166292828
-4.04533654642181D-17       -32000

6.599763320036463D-02 .1319941434902927 .1979816207259937
.2639388845380347 .3298242742545747 .395569509201723
.4610729241597304 .5261935245263726 .5907449789924144
.6544911044873543 .7171421345765487 .7783524329539742
.837720734425025 .8947925814807637 .9490659166292828
-4.04533654642181D-17 -31999

6.599763320036463D-02 .1319941434902927 .1979816207259937
.2639388845380347 .3298242742545747 .395569509201723
.4610729241597304 .5261935245263726 .5907449789924144
.6544911044873543 .7171421345765487 .7783524329539742
.837720734425025 .8947925814807637 .9490659166292828
-4.04533654642181D-17 -31998

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 25 seconds, counting from “Starting program…”. Please see Sharma and Gupta [81, Problem 6, p. 599].

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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Computer program for solving mixed-integer nonlinear programming (MINLP) problems, including systems of nonlinear equations

Computer program for solving mixed-integer nonlinear programming (MINLP) problems, including systems of nonlinear equations

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following system of 15 nonlinear equations from Sharma and Gupta [79, Problem 6, p. 9, open access]:

                                         15
X(i) - COS(2 * X(i) - sigma X(J44) ) = 0, 1 <= i <=15
                                         J44=1

where X(1) through X(15) are continuous variables.

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
87 RANDOMIZE JJJJ
88 M = -3D+30
117 FOR J44 = 1 TO 15

121 A(J44) = .981 + RND * .001

124 NEXT J44
127 FOR i = 1 TO 400000

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)

156 IF RND < .5 THEN 160 ELSE GOTO 169

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

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

166 GOTO 188
169 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

188 NEXT IPP
211 FOR J44 = 1 TO 15

213 REM X(J44) = INT(X(J44))

219 NEXT J44

421 FOR J44 = 1 TO 15

433 IF X(J44) < 0 THEN 1670
435 IF X(J44) > 2 THEN 1670

455 NEXT J44

458 SU = 0

461 FOR J44 = 1 TO 15
465 SU = SU + X(J44)

469 NEXT J44

511 SUM = 0
514 FOR J44 = 1 TO 15
515 SUM = SUM - ABS(X(J44) - COS(2 * X(J44) - SU))
522 NEXT J44
671 PD1 = SUM
679 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
1778 IF M < -4.5D-05 THEN 1999
1915 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 [103]. Its output of one run through
JJJJ= -31410 is summarized below. GW-BASIC also can handle this computer program.

.
.
.
.9814705053091656    .9814725691355511    .9814725776001751
.9814726164108965    .9814725740186614    .981472617155851
.9814726244209022    .9814726004879182  .9814768457650682
.9814915879201344    .981472609757574      .9814730166517885
.9814725813543811    .981472604189234     .981472538568285
-1.602300759429021D-05 -31410

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31410 was 28 minutes, counting from “Starting program…”. Please see Sharma and Gupta [79, Problem 6, p. 9, open access].

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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