Substance

A Computer Program for the Air Transport Model of Markowitz and Manne, Third Edition

Jsun Yui Wong

The computer program listed below seeks to solve the air transport model of Markowitz and Manne [1, pages 95-100].

While line 128 of thepreceding paper is 128 FOR I = 1 TO 1000, here line 128 is 128 FOR I = 1 TO 2000. 

0 REM DEFDBL A-Z
2 DEFINT I, J, K
3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000

15 RANDOMIZE JJJJ

16 M = -1D+37
41 FOR J44 = 1 TO 48
42 A(J44) = FIX(RND * 3)
43 NEXT J44
51 FOR J44 = 37 TO 48
52 A(J44) = FIX(RND * 3)
53 NEXT J44
126 IMAR = 10 + FIX(RND * 5000)
128 FOR I = 1 TO 2000

129 FOR KKQQ = 1 TO 48
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 47))

181 J = 1 + FIX(RND * 48)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP
201 FOR J88 = 37 TO 48
202 IF X(J88) > 100 THEN X(J88) = A(J88)

203 X(J88) = CINT(X(J88))

204 NEXT J88
211 REM X(43) = 5 - X(44) - X(45)

212 X(48) = X(43) + X(44) + X(45) - X(38) - X(41)
266 X(46) = X(37) + X(38) + X(39) - X(40) - X(43)
267 X(47) = X(40) + X(41) + X(42) - X(37) - X(44)

301 X(1) = X(20) + X(29) - X(4) - X(7) + 6
302 X(5) = X(11) + X(30) - X(8) + 6 - X(2)
303 X(9) = X(12) + X(21) - X(6) + 6 - X(3)
304 X(10) = X(22) + X(31) - X(13) - X(16) + 5
305 X(14) = X(2) - X(11) - X(17) + 20 + X(33)
306 X(18) = X(24) + X(3) - X(12) - X(15) + 31

307 X(19) = X(34) + X(13) - X(22) - X(25) + 2

308 X(23) = X(4) + X(35) - X(20) - X(26) + 24
309 X(27) = X(6) + X(15) - X(21) - X(24) + 11
310 X(28) = X(16) + X(25) - X(31) - X(34) + 14
311 X(32) = X(7) + X(26) - X(29) - X(35) + 36

312 X(36) = X(8) + X(17) - X(30) - X(33) + 7

371 FOR J44 = 1 TO 48
372 IF X(J44) < 0 THEN X(J44) = A(J44)

373 NEXT J44

431 PS(21) = -7.5 * X(37) + X(1) + X(2) + X(3)
432 PS(22) = -7.2 * X(38) + X(4) + X(5) + X(6)
433 PS(23) = -7.5 * X(39) + X(7) + X(8) + X(9)
434 PS(24) = -7.5 * X(40) + X(10) + X(11) + X(12)
435 PS(25) = -7.5 * X(41) + X(13) + X(14) + X(15)
436 PS(26) = -7.5 * X(42) + X(16) + X(17) + X(18)
437 PS(27) = -7.2 * X(43) + X(19) + X(20) + X(21)
438 PS(28) = -7.5 * X(44) + X(22) + X(23) + X(24)
439 PS(29) = -5.6 * X(45) + X(25) + X(26) + X(27)
440 PS(30) = -7.5 * X(46) + X(28) + X(29) + X(30)
441 PS(31) = -7.5 * X(47) + X(31) + X(32) + X(33)
442 PS(32) = -5.6 * X(48) + X(34) + X(35) + X(36)

454 FOR J44 = 21 TO 32
455 IF (PS(J44)) > 0 THEN PS(J44) = PS(J44) ELSE PS(J44) = 0

456 NEXT J44
459 POB1 = -4.5 * X(37) - 8.3 * X(38) - 2.9 * X(39) - 4.5 * X(40) - 4.2 * X(41) - 6.9 * X(42) - 8.3 * X(43) - 4.2 * X(44) - 10.9 * X(45) - 2.9 * X(46) - 6.9 * X(47) - 10.9 * X(48)
461 POB3 = -999999999# * (1000 * PS(21) ^ 4 + 1000 * PS(22) ^ 4 + PS(23) ^ 4 + PS(24) ^ 4 + PS(25) ^ 4 + PS(26) ^ 4 + PS(27) ^ 4 + PS(28) ^ 4 + PS(29) ^ 4 + PS(30) ^ 4 + PS(31) ^ 4 + PS(32) ^ 4)

463 P1NEWMAY = POB1 + POB3
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 48
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -155 THEN 1999

1900 GOTO 1945
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1912 PRINT A(6), A(7), A(8), A(9), A(10)
1913 PRINT A(11), A(12), A(13), A(14), A(15)
1914 PRINT A(16), A(17), A(18), A(19), A(20)
1915 PRINT A(21), A(22), A(23), A(24), A(25)
1916 PRINT A(26), A(27), A(28), A(29), A(30)
1917 PRINT A(31), A(32), A(33), A(34), A(35)
1945 PRINT A(36), A(37), A(38), A(39), A(40)
1946 PRINT A(41), A(42), A(43), A(44), A(45)
1947 PRINT A(46), A(47), A(48), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [3]. The complete output through JJJJ=32000 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

5.594838       2       0       1       1
4       7       0       5       0
2       5       1       -153.7021       9562

4.819988       1       1       1       1
3       7       0       5       0
2       5       1       -153.301       19259

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [3], the wall-clock time for obtaining the output through JJJJ=32000 was 70minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Harry M. Markowitz and Alan S. Mann. On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[2] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[3] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[4] Jsun Yui Wong (2015, August 27). The Domino Method of General Integer Nonlinear Programming Applied to the Air Transport Model of Markowitz and Manne, Third Edition Extended. https://computerprogramsandresults.wordpress.com/2015/08/27/the-domino-method-of-general-integer-nonlinear-programming-applied-to-the-air-transport-model-of-markowitz-and-manne-third-edition-extended/

A Computer Program for the Air Transport Model of Markowitz and Manne, Second Edition

Jsun Yui Wong

Based on an earlier computer program in Wong [4], the computer program listed below seeks to solve the air transport model of Markowitz and Manne [1, pages 95-100].

While the older paper [4] has 211 X(43) = 5 - X(44) - X(45), here line 211 is 211 REM X(43) = 5 - X(44) - X(45). That means X(43) = 5 - X(44) - X(45) from the extra analysis of Markowitz and Manne [1, p. 99] is not used in the present paper.

While the older paper [4] has 461 POB3 = -999999999# * ( PS(21) ^ 4 + PS(22) ^ 4 + PS(23) ^ 4 + PS(24) ^ 4 + PS(25) ^ 4 + PS(26) ^ 4 + PS(27) ^ 4 + PS(28) ^ 4 + PS(29) ^ 4 + PS(30) ^ 4 + PS(31) ^ 4 + PS(32) ^ 4), here line 461 is
461 POB3 = -999999999# * (1000 * PS(21) ^ 4 + 1000 * PS(22) ^ 4 + PS(23) ^ 4 + PS(24) ^ 4 + PS(25) ^ 4 + PS(26) ^ 4 + PS(27) ^ 4 + PS(28) ^ 4 + PS(29) ^ 4 + PS(30) ^ 4 + PS(31) ^ 4 + PS(32) ^ 4); the intent of this change is to get additional domino effect.

One also notes line 202, which is 202 IF X(J88) > 100 THEN X(J88) = A(J88).

The following computer program uses qb64v1000-win [3], which was also used for the older paper [4].

0 REM DEFDBL A-Z
2 DEFINT I, J, K
3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000

15 RANDOMIZE JJJJ

16 M = -1D+37
41 FOR J44 = 1 TO 48
42 A(J44) = FIX(RND * 3)
43 NEXT J44
51 FOR J44 = 37 TO 48
52 A(J44) = FIX(RND * 3)
53 NEXT J44
126 IMAR = 10 + FIX(RND * 5000)
128 FOR I = 1 TO 1000

129 FOR KKQQ = 1 TO 48
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 47))

181 J = 1 + FIX(RND * 48)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ 3) * R
192 NEXT IPP
201 FOR J88 = 37 TO 48
202 IF X(J88) > 100 THEN X(J88) = A(J88)

203 X(J88) = CINT(X(J88))

204 NEXT J88
211 REM X(43) = 5 - X(44) - X(45)

212 X(48) = X(43) + X(44) + X(45) - X(38) - X(41)
266 X(46) = X(37) + X(38) + X(39) - X(40) - X(43)
267 X(47) = X(40) + X(41) + X(42) - X(37) - X(44)

301 X(1) = X(20) + X(29) - X(4) - X(7) + 6
302 X(5) = X(11) + X(30) - X(8) + 6 - X(2)
303 X(9) = X(12) + X(21) - X(6) + 6 - X(3)
304 X(10) = X(22) + X(31) - X(13) - X(16) + 5
305 X(14) = X(2) - X(11) - X(17) + 20 + X(33)
306 X(18) = X(24) + X(3) - X(12) - X(15) + 31

307 X(19) = X(34) + X(13) - X(22) - X(25) + 2

308 X(23) = X(4) + X(35) - X(20) - X(26) + 24
309 X(27) = X(6) + X(15) - X(21) - X(24) + 11
310 X(28) = X(16) + X(25) - X(31) - X(34) + 14
311 X(32) = X(7) + X(26) - X(29) - X(35) + 36

312 X(36) = X(8) + X(17) - X(30) - X(33) + 7

371 FOR J44 = 1 TO 48
372 IF X(J44) < 0 THEN X(J44) = A(J44)

373 NEXT J44

431 PS(21) = -7.5 * X(37) + X(1) + X(2) + X(3)
432 PS(22) = -7.2 * X(38) + X(4) + X(5) + X(6)
433 PS(23) = -7.5 * X(39) + X(7) + X(8) + X(9)
434 PS(24) = -7.5 * X(40) + X(10) + X(11) + X(12)
435 PS(25) = -7.5 * X(41) + X(13) + X(14) + X(15)
436 PS(26) = -7.5 * X(42) + X(16) + X(17) + X(18)
437 PS(27) = -7.2 * X(43) + X(19) + X(20) + X(21)
438 PS(28) = -7.5 * X(44) + X(22) + X(23) + X(24)
439 PS(29) = -5.6 * X(45) + X(25) + X(26) + X(27)
440 PS(30) = -7.5 * X(46) + X(28) + X(29) + X(30)
441 PS(31) = -7.5 * X(47) + X(31) + X(32) + X(33)
442 PS(32) = -5.6 * X(48) + X(34) + X(35) + X(36)

454 FOR J44 = 21 TO 32
455 IF (PS(J44)) > 0 THEN PS(J44) = PS(J44) ELSE PS(J44) = 0

456 NEXT J44
459 POB1 = -4.5 * X(37) - 8.3 * X(38) - 2.9 * X(39) - 4.5 * X(40) - 4.2 * X(41) - 6.9 * X(42) - 8.3 * X(43) - 4.2 * X(44) - 10.9 * X(45) - 2.9 * X(46) - 6.9 * X(47) - 10.9 * X(48)
461 POB3 = -999999999# * (1000 * PS(21) ^ 4 + 1000 * PS(22) ^ 4 + PS(23) ^ 4 + PS(24) ^ 4 + PS(25) ^ 4 + PS(26) ^ 4 + PS(27) ^ 4 + PS(28) ^ 4 + PS(29) ^ 4 + PS(30) ^ 4 + PS(31) ^ 4 + PS(32) ^ 4)

463 P1NEWMAY = POB1 + POB3
466 P = P1NEWMAY
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 48
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -155 THEN 1999

1900 GOTO 1945
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1912 PRINT A(6), A(7), A(8), A(9), A(10)
1913 PRINT A(11), A(12), A(13), A(14), A(15)
1914 PRINT A(16), A(17), A(18), A(19), A(20)
1915 PRINT A(21), A(22), A(23), A(24), A(25)
1916 PRINT A(26), A(27), A(28), A(29), A(30)
1917 PRINT A(31), A(32), A(33), A(34), A(35)
1945 PRINT A(36), A(37), A(38), A(39), A(40)
1946 PRINT A(41), A(42), A(43), A(44), A(45)
1947 PRINT A(46), A(47), A(48), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [3]. The complete output through JJJJ=32000 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

5.512149       1       1       1       1
3       7       0       5       0
2       5       1       -153.3       3510

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

At JJJJ=3510, M=-153.3 is optimal. See Markowitz and Manne [1].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [3], the wall-clock time for obtaining the output through JJJJ=32000 was 50 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Harry M. Markowitz and Alan S. Mann. On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[2] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[3] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[4] Jsun Yui Wong (2015, August 27). The Domino Method of General Integer Nonlinear Programming Applied to the Air Transport Model of Markowitz and Manne, Third Edition Extended. https://computerprogramsandresults.wordpress.com/2015/08/27/the-domino-method-of-general-integer-nonlinear-programming-applied-to-the-air-transport-model-of-markowitz-and-manne-third-edition-extended/

 

A Computer Program with Additional Domino Effect Solving a Nonlinear Diophantine System of 10 Integer Unknowns and 9 Equations, Second Edition

Jsun Yui Wong

Based on the computer program in Wong [3], the following computer program seeks to solve the nonlinear system of nine equations on page 745 of Perez, Amaya, and Correa [1]. Only integer solutions are of interest.

While line 329 of the earlier edition is 329 PD1 = -100 * ABS(N(11)) - 100 * ABS(N(12)) - ABS(N(13)) - ABS(N(14)) - ABS(N(15)) - ABS(N(16)), here line 329 is
329 PD1 = -500 * ABS(N(11)) - 500 * ABS(N(12)) - ABS(N(13)) - ABS(N(14)) - ABS(N(15)) - ABS(N(16)).

0 DEFDBL A-Z
1 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)
88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 10
112 A(J44) = CINT(RND * 20)

113 NEXT J44

128 FOR I = 1 TO 1000
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 10)
144 GOTO 167

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

167 IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP
177 X(1) = -3 + X(3) - 2 * X(5) + X(7) + X(9)

179 X(10) = (-24 - X(1) ^ 2 + 2 * (X(2) + X(4)) ^ 3 - X(5) + 3 * X(6) + X(7) - 4 * X(9)) / 15

183 X(8) = -8 + X(2) + (2 * X(4)) ^ 2 - 6 * X(6) + 2 * X(10)

192 FOR J44 = 1 TO 10
193 IF X(J44) < 0 THEN GOTO 1670

194 NEXT J44

196 N(11) = 31 - 2 * X(1) - (X(2) + 3 * X(4)) ^ 3 - (5 * X(7)) ^ 2 + 6 * X(8) - X(9) + 9 * X(10)

197 N(12) = -16 - X(1) + 3 * X(2) - 4 * X(4) - X(6) + 6 * X(7) - X(8) + 2 * X(9)

199 N(13) = 27 - (2 * X(1) + X(2)) ^ 2 - 3 * X(3) + 10 * X(5) + (X(6) + 3 * X(7)) ^ 3 + X(8) + 6 * X(9)

201 N(14) = 23 - 5 * X(1) - 2 * X(2) + 8 * X(4) + 3 * X(5) - 4 * X(6) - X(7) + X(9)

203 N(15) = -9 - 3 * X(1) - 2 * X(2) + 5 * X(3) + X(4) ^ 4 + 2 * X(5) - X(6) - 4 * X(7) + 10 * X(8) - 8 * X(9)

205 N(16) = -25 - 3 * X(1) + (2 * X(2)) ^ 2 - 10 * X(3) + 9 * X(4) - 3 * X(5) - X(6) + 2 * X(7) + 8 * X(8) - 12 * X(9) + 5 * X(10)

329 PD1 = -500 * ABS(N(11)) - 500 * ABS(N(12)) - ABS(N(13)) - ABS(N(14)) - ABS(N(15)) - ABS(N(16))

466 P = PD1

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1551 NN(11) = N(11)
1552 NN(12) = N(12)
1553 NN(13) = N(13)
1554 NN(14) = N(14)
1555 NN(15) = N(15)
1556 NN(16) = N(16)
1557 REM GOTO 128

1670 NEXT I
1889 IF M < 0 THEN 1999

1904 PRINT A(1), A(2), A(3)
1906 PRINT A(4), A(5), A(6)
1907 PRINT A(7), A(8), A(9), A(10)
1917 PRINT M, JJJJ
1919 PRINT NN(11), NN(12), NN(13), NN(14)
1920 PRINT NN(15), NN(16)
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [2]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-22889 is shown below:

3       4       5
0       1       1
1       2       2       6
0       -31040
0       0       0       0
0       0

3 4 5
0 1 1
1 2 2 6
0       -30758
0 0 0 0
0 0

3 4 5
0 1 1
1 2 2 6
0       -29121
0 0 0 0
0 0

3 4 5
0 1 1
1 2 2 6
0       -29092
0 0 0 0
0 0

3 4 5
0 1 1
1 2 2 6
0       -27691
0 0 0 0
0 0

3 4 5
0 1 1
1 2 2 6
0       -24901
0 0 0 0
0 0

3 4 5
0 1 1
1 2 2 6
0       -24418
0 0 0 0
0 0

3 4 5
0 1 1
1 2 2 6
0       -24001
0 0 0 0
0 0

3 4 5
0 1 1
1 2 2 6
0       -23404
0 0 0 0
0 0

3 4 5
0 1 1
1 2 2 6
0       -22889
0 0 0 0
0 0

Above there is no rounding by hand; it is just straight copying by hand from the screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [2], the wall-clock time for obtaining the output through JJJJ= -22889 was 25 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[2] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[3] Jsun Yui Wong (2013, November 11). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html

Solving the Diophantine Equation X(1) ^ 5 = X(2) ^ 25 + X(2) ^ 24 + ... + X(2) + 7

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear Diophantine equation from Tengely [2, p. 8]. Only integer solution/s are of interest.

X(1) ^ 5 = X(2) ^ 25 + X(2) ^ 24 + ... + X(2) + 7.

One notes line 112, which is 112 A(J44) = -5000 + FIX(RND * 10000). These many possibilities for A(J44) are used on purpose to test the present method; line 112 gives cold starts.

0 DEFDBL A-Z
1 DEFINT 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), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 2

112 A(J44) = -5000 + FIX(RND * 10000)

115 NEXT J44

128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * .3)
140 B = 1 + FIX(RND * 2)

144 GOTO 166

150 R = (1 - RND * 2) * A(B)
160 X(B) = (A(B) + RND ^ 3 * R)
166 IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP
201 SSUM = 0
204 FOR J44 = 1 TO 25
206 SSUM = SSUM + X(2) ^ J44

209 NEXT J44
210 IF RND < .5 THEN 211 ELSE GOTO 214

211 X(1) = ABS((SSUM + 7)) ^ (1 / 5)

214 N(7) = -ABS(-X(1) ^ 5 + SSUM + 7)

322 PD1 = N(7)

1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1557 REM GOTO 128
1670 NEXT I

1889 IF M < -25 THEN 1999

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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [3]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31585 is shown below:

1.430969081105256       -1       -1.196959198423997D-15
-31932

2       1       0       -31886

1.430969081105256       -1       -1.196959198423997D-15
-31671

2       1       0       -31668

2       1       0       -31622

1.430969081105256       -1       -1.196959198423997D-15
-31621

2       1       0       -31585

Above there is no rounding by hand; it is just straight copying by hand from the screen. The solution (2 1) with M=0 shown above is acceptable.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [3], the wall-clock time for obtaining the output through JJJJ= -31585 was 17 seconds, not including "Creating .EXE file..." time; the total time was 23 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[2] Szabolcs Tengely (1976) Effective Methods for Diophantine Equations. https://www.math.leidenuniv.nl/scripties/Tengely.pdf

[3] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[4] Jsun Yui Wong (2013, November 12). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html

A Computer Program Solving a Nonlinear Diophantine System of 3 Integer Unknowns and 2 Equations

Jsun Yui Wong

Based on the computer program in Wong [3], the following computer program seeks to solve the following nonlinear system of Diophantine equations from Perez, Amaya, and Correa [1, p. 741]. Only integer solutions are of interest.

X(1) + X(2) ^ 2 - X(3)  = 115

4 * X(1) + 8 * X(2) + 7 * X(3) ^ 2  = 335.

0 DEFDBL A-Z
1 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)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 3
112 A(J44) = CINT(RND * 20)

113 NEXT J44
128 FOR I = 1 TO 1000
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 GOTO 167

160 R = (1 - RND * 2) * A(B)
166 GOTO 168
167 IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP
181 X(1) = 115 - X(2) ^ 2 + X(3)
192 REM FOR J44 = 1 TO 3

193 REM IF X(J44) < 0 THEN 1670

194 REM NEXT J44

195 N(4) = 335 - 4 * X(1) - 8 * X(2) - 7 * X(3) ^ 2

322 PD1 = -ABS(N(4))

335 PDU = PD1
466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1551 NN(4) = N(4)
1557 REM GOTO 128

1670 NEXT I
1889 IF M < -99999! THEN 1999
1904 PRINT A(1), A(2), A(3)
1917 PRINT M, JJJJ
1919 PRINT NN(4)
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [2]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31994 is shown below:

-163       17       11
-4       -32000
4

20       10       5
0       -31999
0

-163 17 11
-4 -31998
4

20 10 5
0 -31997
0

-163       17       11
-4       -31996
4

20       10       5
0       -31995
0

-163       17       11
-4       -31994
4

Above there is no rounding by hand; it is just straight copying by hand from the screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [2], the wall-clock time for obtaining the output through JJJJ= -31994 was 2 seconds, not including "Creating .EXE file..." time; the total time was 10 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[2] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[3] Jsun Yui Wong (2013, November 12). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html

 

A Computer Program Solving a Nonlinear Diophantine System of 6 Integer Unknowns and 6 Equations

Jsun Yui Wong

Based on the computer program in Wong [3], the following computer program seeks to solve the nonlinear system of six simultaneous equations on page 741 of Perez, Amaya, and Correa [1]. Only integer solutions are of interest.

0 DEFDBL A-Z
1 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)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 6
112 A(J44) = CINT(RND * 20)

113 NEXT J44

128 FOR I = 1 TO 1000
129 FOR KKQQ = 1 TO 6
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 6)
144 GOTO 167

160 R = (1 - RND * 2) * A(B)
163 IF RND < .5 THEN 165 ELSE GOTO 167
165 IF RND < .5 THEN X(B) = CINT(A(B) + RND ^ 3 * R) ELSE X(B) = A(B) + RND ^ 3 * R
166 GOTO 168
167 IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP
181 X(2) = -4 + 5 * X(3) - 3 * X(5) + X(6)
183 X(1) = (153 - 18 * X(3) + 5 * X(5) - 17 * X(6)) / 3
187 X(4) = (-277 - X(1) + 5 * X(2) + 8 * X(3) + 15 * X(5) + 10 * X(6)) / 6
192 FOR J44 = 1 TO 6
193 IF X(J44) < 0 THEN 1670
194 NEXT J44
195 N(7) = 1772 - 6 * X(1) + 99 * X(2) - X(3) - (15 * X(6)) ^ 2
197 N(8) = 1772 - 5 * X(1) - 10 * X(2) + 5 * X(3) - X(5) ^ 3 - 8 * X(6)
199 N(9) = 150 - (X(1) + X(2)) ^ 2 + 7 * X(3) - 5 * X(4) - 12 * X(5) + 8 * X(6)
322 PD1 = -ABS(N(7)) - ABS(N(8)) - ABS(N(9))

466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1551 NN(7) = N(7)
1552 NN(8) = N(8)
1553 NN(9) = N(9)
1557 REM GOTO 128

1670 NEXT I
1889 IF M < -99999000 THEN 1999

1904 PRINT A(1), A(2), A(3)
1906 PRINT A(4), A(5), A(6)
1917 PRINT M, JJJJ
1919 PRINT NN(7), NN(8), NN(9)
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [2]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31687 is shown below:

6       3       8
1       12       3
0       -31915
0       0       0

6       3       8
1       12       3
0       -31852
0       0       0

6       3       8
1       12       3
0       -31797
0       0       0

6       3       8
1       12       3
0       -31770
0       0       0

6       3       8
1       12       3
0       -31767
0       0       0

6       3       8
1       12       3
0       -31751
0       0       0

.6666666666666666       1       10
12.8888888888889       16       3
-2494.555555555555       -31687
-168       -2311.333333333333       15.22222222222221

Above there is no rounding by hand; it is just straight copying by hand from the screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [2], the wall-clock time for obtaining the output through JJJJ= -31687 was 2 seconds, not including "Creating .EXE file..." time; the total time was 10 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[2] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[3] Jsun Yui Wong (2013, November 12). Solving Nonlinear Systems of Equations with the Domino Method, Second Edition. http://myblogsubstance.typepad.com/substance/2013/11/solving-nonlinear-systems-of-equations-with-the-domino-method-second-edition.html