Substance

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

Jsun Yui Wong

Based on the computer program in [5], the computer program listed below seeks to solve the air transport model of Markowitz and Manne [1, pages 95-100].

0 REM DEFDBL A-Z
2 DEFINT I, 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 STEP .1

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 REM 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
1511 PPOB1 = POB1

1515 PPOB3 = POB3

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
1949 PRINT PPOB1, PPOB3

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [3]. The complete output through JJJJ=-18573.55 is shown below.

5.346795          1          1          1          1
3          7          0          5          0
2          5          1          -153.3          -31190.08
-153.3          -4.087735E-05

5.506616          1          1          1          1
3          7          0          5          0
2          5          1          -153.9542          -29453.19
-153.3          -.6541663

5.460964          1          1          1          1
3          7          0          5          0
2          5          1          -153.3006          -28532.3
-153.3          -5.850007E-04

5.583451          1          1          1          1
3          7          0           5          0
2          5          1          -153.3          -18573.55
-153.3          -6.001241E-06

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

The best solution shown above is at JJJJ=-18573.55

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=-18573.55 was two hours and a half.

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/

[5] Jsun Yui Wong (2016, July 6). A Computer Program for the Air Transport Model of Markowitz and Manne, Third Edition. https://myblogsubstance.typepad.com/substance/2016/07/a-computer-program-for-the-air-transport-model-of-markowitz-and-manne-third-edition/

Solving for Integer Solutions of Nonlinear Systems of Equations, Including Exponential Diophantine Equations

Jsun Yui Wong

The computer program below seeks to solve simultaneously the following system of 24 equations:

X(11) + X(12) ^ 2 - X(3)=115,
4 * X(11) + 8 * X(12) + 7 * X(3) ^ 2 =335,
1 + 3 ^ X(8) - 2 ^ X(5) * 5 * X(6)=0,
5 ^ X(4) + 5 ^ X(5) + 3 ^ X(6) – 3 ^ X(8)=0,
3 ^ X(5) + 5 ^ X(4) + 7 ^ X(6) – 11 ^ X(7)=0,
13 ^ X(5) + 7 ^ X(4) – 3 ^ X(8) – 5 ^ X(6)=0,
17 ^ X(4) + 7 ^ X(5) – 3 ^ X(6) – 5 ^ X(7)=0,
5 ^ X(5) + 5 ^ X(6) – 3 ^ X(8) – 7 ^ X(4)=0,
2 ^ X(5) + 7 ^ X(6) – 3 ^ X(8) – 5 ^ X(4)=-1,
-X(1) -3 + X(3) – 2 * X(5) + X(7) + X(9)=0,
(-24 – X(1) ^ 2 + 2 * (X(2) + X(4)) ^ 3 – X(5) + 3 * X(6) + X(7) – 4 * X(9)) -15*X(10)=0,
-X(8) -8 + X(2) + (2 * X(4)) ^ 2 – 6 * X(6) + 2 * X(10)=0,
– 2 * X(1) – (X(2) + 3 * X(4)) ^ 3 – (5 * X(7)) ^ 2 + 6 * X(8) – X(9) + 9 * X(10)=-31,
– X(1) + 3 * X(2) – 4 * X(4) – X(6) + 6 * X(7) – X(8) + 2 * X(9)=16,
– (2 * X(1) + X(2)) ^ 2 – 3 * X(3) + 10 * X(5) + (X(6) + 3 * X(7)) ^ 3 + X(8) + 6 * X(9)=-27,
– 5 * X(1) – 2 * X(2) + 8 * X(4) + 3 * X(5) – 4 * X(6) – X(7) + X(9)=-23,
– 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)=9,
– 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)=25,
5 * X(1) + 10 * X(2) – 5 * X(3) + X(5) ^ 3 + 8 * X(6)=39,
6 * X(1) + X(3) – 99 * X(2) + (15 * X(6)) ^ 2=-148,
(X(1) + X(2)) ^ 2 – 7 * X(3) + 5 * X(4) + 12 * X(5) – 8 * X(6)=18,
3 * X(1) + 18 * X(3) – 5 * X(5) + 17 * X(6) = 111,
-X(1) + 5 * X(2) + 8 * X(3) – 6 * X(4) + 15 * X(5) + 10 * X(6) – 82=82,
X(2) – 5 * X(3) + 3 * X(5) – X(6) + 19=-19.

These equations, including the six exponential diophantine equations above, are based on Table 1, the 9×10 system, the 6×6 system, the 2x3 system, and 1 + 3 ^ X(1) = 2 ^ X(2) * 5 * X(3)=0 of Perez, Amaya, and Correa [4].

0 REM 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 32111

89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 12
113 A(J44) = FIX(RND * 40)

115 NEXT J44

128 FOR I = 1 TO 1000
129 FOR KKQQ = 1 TO 12
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 12)
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) = CINT(A(B) + RND ^ 3 * R)

166 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(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 12
193 IF X(J44) < 0 THEN 1670
194 NEXT J44
195 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)

209 N(17) = -39 + 5 * X(1) + 10 * X(2) - 5 * X(3) + X(5) ^ 3 + 8 * X(6)
210 N(18) = 148 + 6 * X(1) + X(3) - 99 * X(2) + (15 * X(6)) ^ 2

212 N(19) = -18 + (X(1) + X(2)) ^ 2 - 7 * X(3) + 5 * X(4) + 12 * X(5) - 8 * X(6)
213 N(20) = 5 ^ X(5) + 5 ^ X(6) - 3 ^ X(8) - 7 ^ X(4)

214 N(21) = 1 + 2 ^ X(5) + 7 ^ X(6) - 3 ^ X(8) - 5 ^ X(4)

333 REM

338 N(22) = 3 * X(1) + 18 * X(3) - 5 * X(5) + 17 * X(6) - 111

344 REM

348 N(23) = -X(1) + 5 * X(2) + 8 * X(3) - 6 * X(4) + 15 * X(5) + 10 * X(6) - 82

355 REM

359 N(24) = X(2) - 5 * X(3) + 3 * X(5) - X(6) + 19

366 N(25) = 17 ^ X(4) + 7 ^ X(5) - 3 ^ X(6) - 5 ^ X(7)

368 N(26) = 13 ^ X(5) + 7 ^ X(4) - 3 ^ X(8) - 5 ^ X(6)
369 N(27) = 3 ^ X(5) + 5 ^ X(4) + 7 ^ X(6) - 11 ^ X(7)

371 N(28) = 5 ^ X(4) + 5 ^ X(5) + 3 ^ X(6) - 3 ^ X(8)

372 N(29) = X(11) + X(12) ^ 2 - X(3) - 115

373 N(30) = 4 * X(11) + 8 * X(12) + 7 * X(3) ^ 2 - 335

374 N(31) = 1 + 3 ^ X(8) - 2 ^ X(5) * 5 * X(6)

555 P = -ABS(N(11)) - ABS(N(12)) - ABS(N(13)) - ABS(N(14)) - ABS(N(15)) - ABS(N(16)) - ABS(N(17)) - ABS(N(18)) - ABS(N(19)) - ABS(N(20)) - ABS(N(21)) - ABS(N(22)) - ABS(N(23)) - ABS(N(24)) - ABS(N(25)) - ABS(N(26)) - ABS(N(27)) - ABS(N(28)) - ABS(N(29)) - ABS(N(30)) - ABS(N(31))

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 12
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -1 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), A(11), A(12)

1917 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [6]. Copied by hand from the screen, the computer program’s complete output through JJJJ=32000 is shown below:

3       4       5
0       1       1
1       2       2       6       20
10
0       -22660

3       4       5
0       1       1
1       2       2       6       20
10
0       -20533

3       4       5
0       1       1
1       2       2       6       20
10
0       -10305

3       4       5
0       1       1
1       2       2       6       20
10
0       5600

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 [6], the wall-clock time through JJJJ=32000 was eight minutes.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), Pages 11376-11387.

[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] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.

[4] 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.

[5] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.

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

 

Searching for Integer Solutions of Nonlinear Systems of Equations, Including Exponential Diophantine Equations

Jsun Yui Wong

The computer program below seeks to solve simultaneously the following system of 20 equations:

3 ^ X(5) + 5 ^ X(4) + 7 ^ X(6) - 11 ^ X(7)=0,
13 ^ X(5) + 7 ^ X(4) - 3 ^ X(8) - 5 ^ X(6)=0,
17 ^ X(4) + 7 ^ X(5) - 3 ^ X(6) - 5 ^ X(7)=0,
5 ^ X(5) + 5 ^ X(6) - 3 ^ X(8) - 7 ^ X(4)=0,
2 ^ X(5) + 7 ^ X(6) - 3 ^ X(8) - 5 ^ X(4)=-1,
-X(1) -3 + X(3) - 2 * X(5) + X(7) + X(9)=0,
(-24 - X(1) ^ 2 + 2 * (X(2) + X(4)) ^ 3 - X(5) + 3 * X(6) + X(7) - 4 * X(9)) -15*X(10)=0,
-X(8) -8 + X(2) + (2 * X(4)) ^ 2 - 6 * X(6) + 2 * X(10)=0,
- 2 * X(1) - (X(2) + 3 * X(4)) ^ 3 - (5 * X(7)) ^ 2 + 6 * X(8) - X(9) + 9 * X(10)=-31,
- X(1) + 3 * X(2) - 4 * X(4) - X(6) + 6 * X(7) - X(8) + 2 * X(9)=16,
- (2 * X(1) + X(2)) ^ 2 - 3 * X(3) + 10 * X(5) + (X(6) + 3 * X(7)) ^ 3 + X(8) + 6 * X(9)=-27,
- 5 * X(1) - 2 * X(2) + 8 * X(4) + 3 * X(5) - 4 * X(6) - X(7) + X(9)=-23,
- 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)=9,
- 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)=25,
5 * X(1) + 10 * X(2) - 5 * X(3) + X(5) ^ 3 + 8 * X(6)=39,
6 * X(1) + X(3) - 99 * X(2) + (15 * X(6)) ^ 2=-148,
(X(1) + X(2)) ^ 2 - 7 * X(3) + 5 * X(4) + 12 * X(5) - 8 * X(6)=18,
3 * X(1) + 18 * X(3) - 5 * X(5) + 17 * X(6) = 111,
-X(1) + 5 * X(2) + 8 * X(3) - 6 * X(4) + 15 * X(5) + 10 * X(6) - 82=82,
X(2) - 5 * X(3) + 3 * X(5) - X(6) + 19=-19.

These equations, including the five exponential diophantine equations above, are based on Table 1, the 9x10 system, and the 6x6 system of Perez, Amaya, and Correa [4].

One notes line 113, which is 113 A(J44) = FIX(RND * 50).

0 REM 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
113 A(J44) = FIX(RND * 50)

115 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)
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) = CINT(A(B) + RND ^ 3 * R)

166 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(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 1670
194 NEXT J44
195 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)

209 N(17) = -39 + 5 * X(1) + 10 * X(2) - 5 * X(3) + X(5) ^ 3 + 8 * X(6)
210 N(18) = 148 + 6 * X(1) + X(3) - 99 * X(2) + (15 * X(6)) ^ 2

212 N(19) = -18 + (X(1) + X(2)) ^ 2 - 7 * X(3) + 5 * X(4) + 12 * X(5) - 8 * X(6)
213 N(20) = 5 ^ X(5) + 5 ^ X(6) - 3 ^ X(8) - 7 ^ X(4)

214 N(21) = 1 + 2 ^ X(5) + 7 ^ X(6) - 3 ^ X(8) - 5 ^ X(4)

333 REM

338 N(22) = 3 * X(1) + 18 * X(3) - 5 * X(5) + 17 * X(6) - 111

344 REM

348 N(23) = -X(1) + 5 * X(2) + 8 * X(3) - 6 * X(4) + 15 * X(5) + 10 * X(6) - 82

355 REM

359 N(24) = X(2) - 5 * X(3) + 3 * X(5) - X(6) + 19

366 N(25) = 17 ^ X(4) + 7 ^ X(5) - 3 ^ X(6) - 5 ^ X(7)

368 N(26) = 13 ^ X(5) + 7 ^ X(4) - 3 ^ X(8) - 5 ^ X(6)
369 N(27) = 3 ^ X(5) + 5 ^ X(4) + 7 ^ X(6) - 11 ^ X(7)

555 P = -ABS(N(11)) - ABS(N(12)) - ABS(N(13)) - ABS(N(14)) - ABS(N(15)) - ABS(N(16)) - ABS(N(17)) - ABS(N(18)) - ABS(N(19)) - ABS(N(20)) - ABS(N(21)) - ABS(N(22)) - ABS(N(23)) - ABS(N(24)) - ABS(N(25)) - ABS(N(26)) - ABS(N(27))

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -1 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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [6]. Copied by hand from the screen, the computer program’s complete output through JJJJ=32000 is shown below:

3        4        5
0       1       1
1       2       2        6
0       4608

3       4       5
0       1       1
1       2       2       6
0       10639

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 [6], the wall-clock time through JJJJ=32000 was six minutes.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), Pages 11376-11387.

[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] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.

[4] 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.

[5] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.

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

 

Solving for Integer Solutions of Nonlinear Systems of Equations, Including Exponential Diophantine Equations

Jsun Yui Wong

The computer program below seeks to solve simultaneously the following system of 19 equations:

13 ^ X(5) + 7 ^ X(4) - 3 ^ X(8) - 5 ^ X(6)=0,
17 ^ X(4) + 7 ^ X(5) - 3 ^ X(6) - 5 ^ X(7)=0,
5 ^ X(5) + 5 ^ X(6) - 3 ^ X(8) - 7 ^ X(4)=0,
2 ^ X(5) + 7 ^ X(6) - 3 ^ X(8) - 5 ^ X(4)=-1,
-X(1) -3 + X(3) - 2 * X(5) + X(7) + X(9)=0,
(-24 - X(1) ^ 2 + 2 * (X(2) + X(4)) ^ 3 - X(5) + 3 * X(6) + X(7) - 4 * X(9)) -15*X(10)=0,
-X(8) -8 + X(2) + (2 * X(4)) ^ 2 - 6 * X(6) + 2 * X(10)=0,
- 2 * X(1) - (X(2) + 3 * X(4)) ^ 3 - (5 * X(7)) ^ 2 + 6 * X(8) - X(9) + 9 * X(10)=-31,
- X(1) + 3 * X(2) - 4 * X(4) - X(6) + 6 * X(7) - X(8) + 2 * X(9)=16,
- (2 * X(1) + X(2)) ^ 2 - 3 * X(3) + 10 * X(5) + (X(6) + 3 * X(7)) ^ 3 + X(8) + 6 * X(9)=-27,
- 5 * X(1) - 2 * X(2) + 8 * X(4) + 3 * X(5) - 4 * X(6) - X(7) + X(9)=-23,
- 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)=9,
- 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)=25,
5 * X(1) + 10 * X(2) - 5 * X(3) + X(5) ^ 3 + 8 * X(6)=39,
6 * X(1) + X(3) - 99 * X(2) + (15 * X(6)) ^ 2=-148,
(X(1) + X(2)) ^ 2 - 7 * X(3) + 5 * X(4) + 12 * X(5) - 8 * X(6)=18,
3 * X(1) + 18 * X(3) - 5 * X(5) + 17 * X(6) = 111,
-X(1) + 5 * X(2) + 8 * X(3) - 6 * X(4) + 15 * X(5) + 10 * X(6) - 82=82,
X(2) - 5 * X(3) + 3 * X(5) - X(6) + 19=-19.

These equations, including the four exponential diophantine equations above, are based on Table 1, the 9x10 system, and the 6x6 system of Perez, Amaya, and Correa [4].

One notes line 113, which is 113 A(J44) = FIX(RND * 40).

0 REM 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
113 A(J44) = FIX(RND * 40)

115 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)
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) = CINT(A(B) + RND ^ 3 * R)

166 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(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 1670
194 NEXT J44
195 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)

209 N(17) = -39 + 5 * X(1) + 10 * X(2) - 5 * X(3) + X(5) ^ 3 + 8 * X(6)
210 N(18) = 148 + 6 * X(1) + X(3) - 99 * X(2) + (15 * X(6)) ^ 2

212 N(19) = -18 + (X(1) + X(2)) ^ 2 - 7 * X(3) + 5 * X(4) + 12 * X(5) - 8 * X(6)
213 N(20) = 5 ^ X(5) + 5 ^ X(6) - 3 ^ X(8) - 7 ^ X(4)

214 N(21) = 1 + 2 ^ X(5) + 7 ^ X(6) - 3 ^ X(8) - 5 ^ X(4)

333 REM

338 N(22) = 3 * X(1) + 18 * X(3) - 5 * X(5) + 17 * X(6) - 111

344 REM

348 N(23) = -X(1) + 5 * X(2) + 8 * X(3) - 6 * X(4) + 15 * X(5) + 10 * X(6) - 82

355 REM

359 N(24) = X(2) - 5 * X(3) + 3 * X(5) - X(6) + 19

366 N(25) = 17 ^ X(4) + 7 ^ X(5) - 3 ^ X(6) - 5 ^ X(7)

368 N(26) = 13 ^ X(5) + 7 ^ X(4) - 3 ^ X(8) - 5 ^ X(6)

555 P = -ABS(N(11)) - ABS(N(12)) - ABS(N(13)) - ABS(N(14)) - ABS(N(15)) - ABS(N(16)) - ABS(N(17)) - ABS(N(18)) - ABS(N(19)) - ABS(N(20)) - ABS(N(21)) - ABS(N(22)) - ABS(N(23)) - ABS(N(24)) - ABS(N(25)) - ABS(N(26))

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -1 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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [5]. Copied by hand from the screen, the computer program’s complete output through JJJJ=32000 is shown below:

3       4       5
0       1       1
1       2       2       6
0       -8228

3       4       5
0       1       1
1       2       2       6
0       1698

3       4       5
0       1       1
1       2       2       6
0       2829

3       4       5
0       1       1
1       2       2       6
0       16645

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 [5], the wall-clock time through JJJJ=32000 was six minutes.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), Pages 11376-11387.

[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] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.

[4] 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.

[5] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.

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

 

Finding Integer Solutions of Nonlinear Systems of Equations, Including Exponential Diophantine Equations

Jsun Yui Wong

The computer program below seeks to solve simultaneously the following system of 18 equations:

17 ^ X(4) + 7 ^ X(5) - 3 ^ X(6) - 5 ^ X(7)=0,
5 ^ X(5) + 5 ^ X(6) - 3 ^ X(8) - 7 ^ X(4)=0,
2 ^ X(5) + 7 ^ X(6) - 3 ^ X(8) - 5 ^ X(4)=-1,
-X(1) -3 + X(3) - 2 * X(5) + X(7) + X(9)=0,
(-24 - X(1) ^ 2 + 2 * (X(2) + X(4)) ^ 3 - X(5) + 3 * X(6) + X(7) - 4 * X(9)) -15*X(10)=0,
-X(8) -8 + X(2) + (2 * X(4)) ^ 2 - 6 * X(6) + 2 * X(10)=0,
- 2 * X(1) - (X(2) + 3 * X(4)) ^ 3 - (5 * X(7)) ^ 2 + 6 * X(8) - X(9) + 9 * X(10)=-31,
- X(1) + 3 * X(2) - 4 * X(4) - X(6) + 6 * X(7) - X(8) + 2 * X(9)=16,
- (2 * X(1) + X(2)) ^ 2 - 3 * X(3) + 10 * X(5) + (X(6) + 3 * X(7)) ^ 3 + X(8) + 6 * X(9)=-27,
- 5 * X(1) - 2 * X(2) + 8 * X(4) + 3 * X(5) - 4 * X(6) - X(7) + X(9)=-23,
- 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)=9,
- 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)=25,
5 * X(1) + 10 * X(2) - 5 * X(3) + X(5) ^ 3 + 8 * X(6)=39,
6 * X(1) + X(3) - 99 * X(2) + (15 * X(6)) ^ 2=-148,
(X(1) + X(2)) ^ 2 - 7 * X(3) + 5 * X(4) + 12 * X(5) - 8 * X(6)=18,
3 * X(1) + 18 * X(3) - 5 * X(5) + 17 * X(6) = 111,
-X(1) + 5 * X(2) + 8 * X(3) - 6 * X(4) + 15 * X(5) + 10 * X(6) - 82=82,
X(2) - 5 * X(3) + 3 * X(5) - X(6) + 19=-19.

These equations, including the three exponential diophantine equations above, are based on Table 1, the 9x10 system, and the 6x6 system of Perez, Amaya, and Correa [3].

0 REM 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 32111

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

115 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)
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) = CINT(A(B) + RND ^ 3 * R)

166 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(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 1670
194 NEXT J44
195 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)

209 N(17) = -39 + 5 * X(1) + 10 * X(2) - 5 * X(3) + X(5) ^ 3 + 8 * X(6)
210 N(18) = 148 + 6 * X(1) + X(3) - 99 * X(2) + (15 * X(6)) ^ 2

212 N(19) = -18 + (X(1) + X(2)) ^ 2 - 7 * X(3) + 5 * X(4) + 12 * X(5) - 8 * X(6)
213 N(20) = 5 ^ X(5) + 5 ^ X(6) - 3 ^ X(8) - 7 ^ X(4)

214 N(21) = 1 + 2 ^ X(5) + 7 ^ X(6) - 3 ^ X(8) - 5 ^ X(4)

333 REM

338 N(22) = 3 * X(1) + 18 * X(3) - 5 * X(5) + 17 * X(6) - 111

344 REM

348 N(23) = -X(1) + 5 * X(2) + 8 * X(3) - 6 * X(4) + 15 * X(5) + 10 * X(6) - 82

355 REM

359 N(24) = X(2) - 5 * X(3) + 3 * X(5) - X(6) + 19

366 N(25) = 17 ^ X(4) + 7 ^ X(5) - 3 ^ X(6) - 5 ^ X(7)

555 P = -ABS(N(11)) - ABS(N(12)) - ABS(N(13)) - ABS(N(14)) - ABS(N(15)) - ABS(N(16)) - ABS(N(17)) - ABS(N(18)) - ABS(N(19)) - ABS(N(20)) - ABS(N(21)) - ABS(N(22)) - ABS(N(23)) - ABS(N(24)) - ABS(N(25))

1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -1 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

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [5]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-525 is shown below:

3       4       5
0       1       1
1       2       2       6
0       -16298

3 4 5
0 1 1
1 2 2 6
0 -15882

3 4 5
0 1 1
1 2 2 6
0 -13257

3 4 5
0 1 1
1 2 2 6
0 -3182

3 4 5
0 1 1
1 2 2 6
0 -525

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 [5], the wall-clock time through JJJJ=-525 was four minutes.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), Pages 11376-11387.

[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] 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.

[4] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.

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

 

Solutions in Integers of Nonlinear Systems of Equations

Jsun Yui Wong

The computer program below seeks to solve simultaneously the following system of six equations from page 741 of Perez, Amaya, and Correa [3].

5 * X(1) + 10 * X(2) - 5 * X(3) + X(5) ^ 3 + 8 * X(6)=1772,

3 * X(1) + 18 * X(3) - 5 * X(5) + 17 * X(6)=153,

6 * X(1) + X(3) - 99 * X(2) + (15 * X(6)) ^ 2=1772,

- X(1) + 5 * X(2) + 8 * X(3) - 6 * X(4) + 15 * X(5) + 10 * X(6)=277,

(X(1) + X(2)) ^ 2 - 7 * X(3) + 5 * X(4) + 12 * X(5) - 8 * X(6)=150,

X(2) - 5 * X(3) + 3 * X(5) - X(6)=-4.

One notes line 112, which is 112 A(J44) = -1000 + RND * 2000.

0 REM DEFDBL A-Z
1 DEFINT I, J, K, X
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99)
5 DIM AA(99), HR(32), HHR(32), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(99)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 6
112 A(J44) = -1000 + RND * 2000

113 NEXT J44
128 FOR I = 1 TO 500
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)
150 R = (1 - RND * 2) * A(B)
155 REM IF RND < .5 THEN 160 ELSE GOTO 167
160 X(B) = A(B) + RND ^ 3 * R

165 GOTO 168
167 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(A(B) + 1)
168 NEXT IPP
407 N(1) = -1772 + 5 * X(1) + 10 * X(2) - 5 * X(3) + X(5) ^ 3 + 8 * X(6)

411 N(2) = -153 + 3 * X(1) + 18 * X(3) - 5 * X(5) + 17 * X(6)

415 N(3) = -1772 + 6 * X(1) + X(3) - 99 * X(2) + (15 * X(6)) ^ 2

419 N(4) = -277 - X(1) + 5 * X(2) + 8 * X(3) - 6 * X(4) + 15 * X(5) + 10 * X(6)

421 N(5) = -150 + (X(1) + X(2)) ^ 2 - 7 * X(3) + 5 * X(4) + 12 * X(5) - 8 * X(6)

443 N(6) = 4 + X(2) - 5 * X(3) + 3 * X(5) - X(6)

922 PD1 = -ABS(N(1)) - ABS(N(2)) - ABS(N(3)) - ABS(N(4)) - ABS(N(5)) - ABS(N(6))

1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1555 FOR J44 = 1 TO 6

1557 NN(J44) = N(J44)

1559 NEXT J44

1561 GOTO 128
1670 NEXT I
1889 IF M < -1 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5)
1905 PRINT A(6), M, JJJJ
1911 PRINT NN(1), NN(2), NN(3)
1912 PRINT NN(4), NN(5), NN(6)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [5]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-30840 is shown below:

6       3       8       1       12
3       0       -31979
0       0       0
0       0       0

6       3       8       1       12
3       0       -31773
0       0       0
0       0       0

6       3       8       1       12
3       0       -31407
0       0       0
0       0       0

6       3       8       1       12
3       0       -31289
0       0       0
0       0       0

6       3       8       1       12
3       0       -31169
0       0       0
0       0       0

6       3       8       1       12
3       0       -30840
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 [5], the wall-clock time through JJJJ=-30840 was 15 seconds.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), Pages 11376-11387.

[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] 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.

[4] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.

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

Finding Integer Solutions of Nonlinear Systems of Equations

Jsun Yui Wong

Based on the computer program in Wong [54], the computer program below seeks to solve the following nonlinear system of simultaneous equations from Bellido [4]. This system represents a problem of robotics [4, p. 347] and is as follows:

X(1)^2 +X(2)^2 +X(3)^2 -12*X(1) -68 =0
X(4)^2 +X(5)^2 +X(6)^2 -12*X(5) -68 =0
X(7)^2 +X(8)^2 +X(9) ^2 -24*X(8) -12*X(9) +100 =0
X(1) *X(4) +X(2) *X(5)+X(3)*X(6)-6 * X(1)-6*X(5) -52 =0
X(1) *X(7) +X(2) *X(8)+X(3)*X(9)-6 * X(1)-12*X(8)-6*X(9) +64 =0
X(4) *X(7) +X(5) *X(8)+X(6)*X(9)-6 * X(5)-12*X(8)-6*X(9) +32 =0
2*X(2)+2*X(3) -X(4)-X(5)-2*X(6) -X(7) -X(9) +18 =0
X(1) +X(2) +2*X(3) +2*X(4)+2*X(6) -2*X(7) +X(8) -X(9) -38 =0
X(1) +X(3) -2*X(4)+X(5)-X(6)+2*X(7)-2*X(8) + 8 =0.

0 REM DEFDBL A-Z
1 DEFINT I, J, K, X
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)
4 REM DIM PE(35,35)
5 REM DIM SD(35,35)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 9
112 A(J44) = RND * 200

113 NEXT J44
126 REM IMAR=10+FIX(RND*4000)
128 FOR I = 1 TO 500
129 FOR KKQQ = 1 TO 9
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 10)
160 R = (1 - RND * 2) * A(B)
161 IF RND < .5 THEN 164 ELSE GOTO 167
164 X(B) = (A(B) + (RND ^ 3 * R))
165 GOTO 168
167 IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1
168 NEXT IPP
177 N(1) = (X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 - 12 * X(1)) - 68
179 N(2) = (X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 - 12 * X(5)) - 68
183 N(3) = (X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 - 24 * X(8) - 12 * X(9)) + 100
185 N(4) = (X(1) * X(4) + X(2) * X(5) + X(3) * X(6) - 6 * X(1) - 6 * X(5)) - 52
187 N(5) = (X(1) * X(7) + X(2) * X(8) + X(3) * X(9) - 6 * X(1) - 12 * X(8) - 6 * X(9)) + 64
189 N(6) = (X(4) * X(7) + X(5) * X(8) + X(6) * X(9) - 6 * X(5) - 12 * X(8) - 6 * X(9)) + 32
195 N(7) = (2 * X(2) + 2 * X(3) - X(4) - X(5) - 2 * X(6) - X(7) - X(9)) + 18
196 N(8) = (X(1) + X(2) + 2 * X(3) + 2 * X(4) + 2 * X(6) - 2 * X(7) + X(8) - X(9)) - 38
197 N(9) = (X(1) + X(3) - 2 * X(4) + X(5) - X(6) + 2 * X(7) - 2 * X(8)) + 8
334 PD1 = -ABS(N(1)) - ABS(N(2)) - ABS(N(3)) - ABS(N(4)) - ABS(N(5)) - ABS(N(6)) - ABS(N(7)) - ABS(N(8)) - ABS(N(9))
335 PDU = PD1

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 9
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1521 NN(1) = N(1): NN(2) = N(2): NN(3) = N(3): NN(4) = N(4): NN(5) = N(5)
1522 NN(6) = N(6): NN(7) = N(7): NN(8) = N(8): NN(9) = N(9)
1557 GOTO 128
1670 NEXT I
1889 IF M < -3 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5)
1906 PRINT A(6), A(7), A(8), A(9)
1907 PRINT M, JJJJ
1910 PRINT NN(1), NN(2), NN(3), NN(4), NN(5)
1911 PRINT NN(6), NN(7), NN(8), NN(9)
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [52]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-29711 is shown below:

12 8 2 8 12
2 8 16 6
0 -31329
0 0 0 0 0
0 0 0 0

12       8       2       8       12
2       8       16       6
0       -31294
0       0       0       0       0
0       0       0       0

12       8       2       8       12
2       8       16       6
0       -31100
0       0       0       0       0
0       0       0       0

12       8       2       8       12
2       8       16       6
0       -30981
0       0       0       0       0
0       0       0       0

4       0       10       0       4
10       0       8       14
0       -29711
0       0       0       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 [52], the wall-clock time through JJJJ=-29711 was 20 seconds.

Acknowledgment

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

References

[1] Egon Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, 517-546 (1965).

[2] Egon Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] M. C. Bartholomew-Biggs, A Numerical Comparison between Two Approaches to the Nonlinear Programming Problem, in Towards Global Optimisation 2, L. C. W. Dixon, G. P. Szego, editors, North-Holland Publishing Company (1978).

[4] A.-M. Bellido, Construction of Iteration Functions for the Simultaneous Computation of the Solutions of Equations and Algebraic Systems. Numerical Algorithms, 6 (1994) 317-351.

[5] J. Bracken, G. P. McCormick, Selected Applications of Nonlinear Programming. New York: John Wiley and Sons, 1968.

[6] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition. Brooks/Cole 2011.

[7] M. F. Cardoso, R. L. Salcedo, S. Feyo de Azevedo, D. Barbosa (1997), A Simulated Annealing Approach to the Solution of MINLP Problems. Computers and Chemical Engineering, 21 (12) pp.1349-1364.

[8] S. H. Chew, Q. Zheng, Integral Global Optimization. Volume 298 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, 1988.

[9] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., 1980.

[10] William Conley, Optimization: A Simplified Approach. New York/Princeton: Petrocelli Books, Inc., 1981.

[11] William Conley (1982), An Economic Order Quantity Problem. International Journal of Mathematical Education in Science and Technology, Vol. 13, No. 3, pp. 265-268.

[12] William Conley, Computer Optimization Techniques, Revised Edition. New York/Princeton: Petrocelli Books, Inc., 1984.

[13] William Conley (1994), Multi-stage Monte Carlo and non-linear test problems. International Journal of Systems Science, Vol. 25, No. 1, pp. 155-171.

[14] William Conley (2007), Simulation Optimization and Correlation with Multi Stage Monte Carlo Optimization. International Journal of Systems Science, Vol. 38, No. 12, pp. 1013-1019.

[15] William Conley (2008), Ecological Optimization of Pollution Control Equipment and Planning from a Simulation Perspective. International Journal of Systems Science, Vol. 39, No. 1, pp. 1-7.

[16] G. Dantzig, R. Fulkerson, S. Johnson, Solution of a Large-Scale Traveling Salesman Problem. Operations Research, Vol. 2, No. 4 (Nov., 1954), pp. 393-410.

[17] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[18] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming Theory and Applications. John Wiley and Sons, 1967.

[19] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[20] Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization. Oxford University Press, 1995.

[21] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.

[22] R. S. Garfinkel, G. L. Nemhauser, Integer Programming. John Wiley, Inc. 1972.

[23] Amos Gilat and Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. John Wiley and Sons, Inc. 2008.

[24] Donald Greenspan, Vincenzo Casulli, Numerical Analysis for Applied Mathematics, Science, and Enginerring. Addison-Wesley Publishing Company, 1988.

[25] Peter L. Hammer, Sergiu Rudeanu, Boolean Methods in Operations Research and Related Areas. New York: Springer-Verlag, 1968.

[26] Peter L. Hammer, Sergiu Rudeanu, Psudo-Boolean Programming. Operations Research, March/April 1969 Vol. 17 No. 2, pp. 233-261.

[27] D. M. Himmelblau (1972), Applied Nonlinear Programming. New York: McGraw-Hill.

[28] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag 1981.

[29] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[30] Ali Husseinzadeh Kashan (2011), An Efficient Algorithm for Constrained Global Optimization and Application to Mechanical Engineering Design: League Championship Algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[31] G. R. Kocis, I. E. Grossmann, Global Optimization of Nonconvex Mixed-Integer Nonlinear Programming Problems in Process Synthesis. Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988, pages 1407-1421.

[32] G. R. Kocis, I. E. Grossmann, A Modelling amd Decomposition Strategy for MINLP Optimization. Computers and Chemical Engineering, Vol. 13, No. 7, July 1989, pages 797-819.

[33] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[34] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[35] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer, 2006.

[36] D. Q. Mayne, N. Maratos (1979), A First Order, Exact Penalty Function Algorithm for Equality Constrained Optimization Problems. Mathematical Programming 16 (1979) 303-324.

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

[38] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems. ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.

[39] 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.

[40] C. Mohan, H. T. Nguyen (1999), A Controlled Random Search Technique Incorporating the Simulated Annealing Concept for Solving Integer and Mixed Integer Global Optimization Problems.
Computational Optimization and Applications, July 1999, Vol. 14 Issue 1, pp.103-132.

[41] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis. Cambridge University Press, 2009.

[42] Chi-Kong Ng, Lian-Sheng Zhang, Duan Li, Wei-Wen Tian (2005), Discrete Filled Function Method for Discrete Optimization. Computational Optimization and Applications, 31, pp. 87-115, 2005.

[43] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition. Cambridge University Press, 2000.

[44] D. A. Paviani (1969), A New Method for the Solution of the General Nonlinear Programming Problem, Ph.D Dissertation, University of Texas, Austin, Texas.

[45] D. R. Plane, C. McMillan, Discrete Optimization. Englewood Cliffs: Prentice-Hall, 1971.

[46] John R. Rice, Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.

[47 H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[48] Terry Shoup, A Practical Guide to Computer Methods for Engineers. Englewood Cliffs: Prentice-Hall, 1979.

[49] M. Syslo, N. Deo, J. Kowalik, Discrete Optimization Algorithms with Pascal Programs. Englewood Cliffs: Prentice-Hall, 1983.

[50] C. A. Trauth, R. E. Woolsey (1969), Integer Linear Programming: A Study in Computational Efficiency. Management Science, Volume 15, 1969, 481-493.

[51] J.-F. Tsai, M.-H. Lin (2007), Finding All Solutions of Systems of Nonlinear Equations with Free Variables. Engineering Optimization, 39:6, 649-659.

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

[53] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07

[54] Jsun Yui Wong (2013, October 13). A Computer Program for Solving Nonlinear Systems of Simultaneous Diophantine Equations, Part 3. http://myblogsubstance.typepad.com/substance/2013/10/a-computer-program- for-solving-nonlinear-systems-of-simultaneous-diophantine-equations-part-3.html

Solving for Integer Solutions of Nonlinear Systems of Equations

Jsun Yui Wong

Based on the computer program in Wong [65], the following computer program seeks to solve simultaneously the nonlinear system of nine equations on page 745 of Perez, Amaya, and Correa [54].

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)
4 REM DIM PE(35,35)
5 REM DIM SD(35,35)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 10
112 A(J44) = CINT(RND * 15)
113 NEXT J44
126 REM IMAR=10+FIX(RND*4000)
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)
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) = CINT(A(B) - 1) ELSE X(B) = CINT(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 1670
194 NEXT J44
195 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)
214 FOR J44 = 11 TO 16
216 IF ABS(N(J44)) > 0# THEN PN(J44) = -1000000# * ABS(N(J44)) ELSE PN(J44) = 0#
218 NEXT J44
322 PD1 = PN(11) + PN(12) + PN(13) + PN(14) + PN(15) + PN(16)
335 PDU = PD1
466 P = PDU
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 GOTO 128
1670 NEXT I
1889 IF M < -99999! 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 [63]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31624 is shown below:

3       4       5
0       1       1
1       2       2       6
0       -31694
0       0       0       0
0       0

3       4       5
0       1       1
1       2       2       6
0       -31624
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 [63], the wall-clock time through JJJJ=-31624 was 30 seconds.

Acknowledgement

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

References

[1] Egon Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, 517-546 (1965).

[2] Egon Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] M. C. Bartholomew-Biggs, A Numerical Comparison between Two Approaches to the Nonlinear Programming Problem, in Towards Global Optimisation 2, L. C. W. Dixon, G. P. Szego, editors, North-Holland Publishing Company (1978).

[4] A.-M. Bellido, Construction of Iteration Functions for the Simultaneous Computation of the Solutions of Equations and Algebraic Systems. Numerical Algorithms, 6 (1994) 317-351.

[5] J. T. Betts (1977), An Accelerated Multiplier Method for Nonlinear Programming. Journal of Optimization Theory and Applications, Volume 21, pp. 137-174.

[6] J. Bracken, G. P. McCormick, Selected Applications of Nonlinear Programming. New York: John Wiley and Sons, 1968.

[7] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition. Brooks/Cole 2011.

[8] M. F. Cardoso, R. L. Salcedo, S. Feyo de Azevedo, D. Barbosa (1997), A Simulated Annealing Approach to the Solution of MINLP Problems. Computers and Chemical Engineering, 21 (12) pp.1349-1364.

[9] S. H. Chew, Q. Zheng, Integral Global Optimization. Volume 298 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, 1988.

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Computational Optimization and Applications, July 1999, Vol. 14 Issue 1, pp.103-132.

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[54] 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.

[55] D. R. Plane, C. McMillan, Discrete Optimization. Englewood Cliffs: Prentice-Hall, 1971.

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

[64] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07

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

The Domino Method Applied to Martin Gardiner’s System of Six Simultaneous Diophantine Equations, Second Edition

Jsun Yui Wong

Based on the computer program in Wong [18], the computer program listed below seeks to solve simultaneously the following system of six Diophantine equations taken from page 4 of Gardiner [6]:
X(7) = 5*X(1) +1
4*X(1) = 5*X(2) +1
4*X(2) = 5*X(3) +1
4*X(3) = 5*X(4) +1
4*X(4) = 5*X(5) +1
4*X(5) = 5*X(6) +1

0 REM DEFDBL A-Z
1 DEFINT I, J, K, X
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 7
112 A(J44) = -10000 + RND * 20000

113 NEXT J44
128 FOR I = 1 TO 500
129 FOR KKQQ = 1 TO 7
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 7)
150 R = (1 - RND * 2) * A(B)

157 X(B) = (A(B) + RND ^ 3 * R)
168 NEXT IPP
231 X(1) = (X(7) - 1) / 5
232 X(2) = (4 * X(1) - 1) / 5
233 X(3) = (4 * X(2) - 1) / 5
234 X(4) = (4 * X(3) - 1) / 5
235 X(5) = (4 * X(4) - 1) / 5
236 X(6) = (4 * X(5) - 1) / 5

251 N(1) = (X(7) - 5 * X(1) - 1)
252 N(2) = (4 * X(1) - 5 * X(2) - 1)
253 N(3) = (4 * X(2) - 5 * X(3) - 1)
254 N(4) = (4 * X(3) - 5 * X(4) - 1)
255 N(5) = (4 * X(4) - 5 * X(5) - 1)
256 N(6) = (4 * X(5) - 5 * X(6) - 1)

322 PD1 = -ABS(N(1)) - ABS(N(2)) - ABS(N(3)) - ABS(N(4)) - ABS(N(5)) - ABS(N(6))
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1453 FOR KLX = 1 TO 7
1454 NN(KLX) = N(KLX)
1455 A(KLX) = X(KLX)
1556 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 0 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5)
1905 PRINT A(6), A(7), M, JJJJ
1908 PRINT NN(1), NN(2), NN(3), NN(4), NN(5), NN(6)
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [16]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-29814 is shown below:

-1       -1       -1       -1       -1
-1       -4       0       -31834
0       0       0       0       0
0

-3126       -2501       -2001       -1601       -1281
-1025       -15629       0       -31717
0      0      0      0      0
0

3124       2499       1999       1599       1279
1023      15621      0      -31704
0      0      0      0      0
0

-1      -1      -1      -1      -1
-1      -4       0       -31448
0      0      0      0      0
0

-3126       -2501       -2001       -1601       -1281
-1025       -15629       0       -29910
0      0      0      0      0
0

-1      -1      -1      -1      -1
-1      -4       0       -29889
0      0      0      0      0
0

-1      -1      -1      -1      -1
-1      -4       0       -29886
0      0      0      0      0
0

-1       -1       -1       -1       -1
-1       -4       0       -29814
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 [16], the wall-clock time through JJJJ=-29814 was 15 seconds.

Acknowledgement

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[4] Martin Gardner (1979), Mathematical Circus. Knopf (1979).

[5] Martin Gardner (1983), Diophantine Analysis and Fermat’s Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). http://www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler’s Conjecture on Sums of Like Powers. Bulletin of the American Mathematical Society, Vol. 72, 1966, page 1079.

[8] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler’s Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[9] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[10] 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.

[11] 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.

[12] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[13] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[14] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

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

[17] Jsun Yui Wong (2013, November 26), A Computer Program for Solving Systems of Diophantine Nonlinear Equations, Part 2. Retrieved from http://myblogsubstance.typepad.com/substance/2013/11/index.html.

[18] Jsun Yui Wong (2014, February 14), The Domino Method Applied to Martin Gardiner’s System of Six Simultaneous Diophantine Equations. Retrieved from https://computerprogramsandresults.wordpress.com/2014/02/page/3/

Solving in Integers Nonlinear Systems of Equations by Computer--More Computational Results

Jsun Yui Wong

The computer program listed below is about the following quotation from http://math.stackexchange.com [8]:

"Do there exist integers x, y, z, w that satisfy

(x+1)^2+y^2=(x+2)^2+z^2

(x+2)^2+z^2=(x+3)^2+w^2?"

One notes line 94, which is 94 A(KK) = -2500 + FIX(RND * 5001).

0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(342), A(342), L(333), K(333), N(111), NN(111)

12 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ

16 M = -1D+317

91 FOR KK = 1 TO 4

94 A(KK) = -2500 + FIX(RND * 5001)

95 NEXT KK

128 FOR I = 1 TO 500000

129 FOR K = 1 TO 4

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 4)

182 REM IF RND < .5 THEN 183 ELSE GOTO 189

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

186 X(B) = A(B) + (RND ^ 3) * r

188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)

191 NEXT IPP

603 N(91) = (X(1) + 1) ^ 2 + X(2) ^ 2 - (X(1) + 2) ^ 2 - X(3) ^ 2

606 N(92) = (X(1) + 2) ^ 2 + X(3) ^ 2 - (X(1) + 3) ^ 2 - X(4) ^ 2

1433 P = -ABS(N(91)) - ABS(N(92))

1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 4

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P

1670 NEXT I
1888 IF M < -3 THEN 1999

1914 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [10]. Copied by hand from the screen, the computer program’s only solution through JJJJ=-31688 is shown below:

5322       -264       -243       -220       0
-31688

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 [10], the wall-clock time through JJJJ=-31688 was six minutes.

One notes that A(1)=5322 is not within the starting interval, 94 A(KK) = -2500 + FIX(RND * 5001), and that

(5322       264       243       220)  is a solution also.

Acknowledgment

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

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[5] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[6] 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.
[7] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[8] Puzzled417 (http://math.stackexchange.com/users/297795/puzzled417), Integer solutions to nonlinear system of equations $(x+1)^2+y^2 = (x+2)^2+z^2$ and $(x+2)^2+z^2 = (x+3)^2+w^2$, URL (version: 2017-01-06): http://math.stackexchange.com/q/2085253
[9] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.
[10] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.