Finding Integer Solutions of Nonlinear Systems of Equations

Jsun Yui Wong

The computer program below seeks to solve simultaneously the following system of 14 equations:  

 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.

These equations, including the two exponential diophantine equations above, are based on the equations in 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 32000
    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)


        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))


        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=-10032 is shown below:

3      4      5
0      1      1
1      2      2      6
0      -30380

3      4      5
0      1      1
1      2      2      6
0      -22163

3      4      5
0      1      1
1      2      2      6
0      -15830

3      4      5
0      1      1
1      2      2      6
0      -10884

3      4      5
0      1      1
1      2      2      6
0      -10032

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=-10032 was three 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.