Simultaneously Solving in General Integers a Nonlinear System of 41 Simultaneous Equations, Including Transcendental Equations, in 41 General Integer Variables

Jsun Yui Wong

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

   X(84) = -2 + X(82) ^ 2 + X(83) ^ 2,

   -1 + LOG(X(82)) - X(83) - X(84) =0,

   X(82) + EXP(X(83)) + X(84) - 1 =0,

  X(78) = -X(79) - X(80) - X(81),
     
  LOG(X(78)) + X(79) + LOG(X(80)) + X(81) + 2 =0,

  X(78) * X(79) * X(80) - X(81) =0,

  X(78) - EXP(X(79)) + X(80) + EXP(X(81)) - 2 =0,

 X(74) = -X(75) ^ 3 + 5 * X(75) ^ 2 + X(75) + 10,
     
 -29 + X(74) + X(75) ^ 3 + X(75) ^ 2 - 14 * X(75) =0,

 X(76) =  -X(77) ^ 3 + 5 * X(77) ^ 2 + 2 * X(77) + 11,

  -29 + X(76) + X(77) ^ 3 + X(77) ^ 2 - 14 * X(77)=0,

 -5 + 10 * X(71) - 2 * X(72) ^ 2 + X(72) - 2 * X(73) ^ 2 = 0,

 -12 + 8 * X(72) ^ 2 + 4 * X(73) ^ 2 = 0,

  8 + 8 * X(72) * X(73) = 0,

  X(67) = -10 + X(65) ^ 2 + 10 * X(66),

  -12 + 15 * X(65) + X(66) - 4 * X(67) = 0,

   24 + X(66) - 25 * X(67) = 0,

   -7 + 10 * X(68) - 2 * X(69) ^ 2 + X(69) - 2 * X(70) ^ 2  = 0,

  -12 + 8 * X(69) ^ 2 + 4 * X(70) ^ 2 = 0,

  8 + 8 * X(69) * X(70) = 0,

 X(54) = -X(52) ^ 2,

 X(52) * COS(X(53)) - X(54) = 0,

 (EXP(X(52) + X(54))) * SIN(X(53) / 2) + X(54) = 0,

 X(55) ^ 5 + X(56) ^ 3 * X(57) ^ 4 + 1 = 0,

 X(55) * X(56) * X(57) = 0,

 X(57) ^ 4 - 1 = 0,

 X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51)  =  1,

 X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9,

  2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51)   =  9,
       
   X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51)   =  9,
 
   X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51)   =  9,

   X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51)   =  9,

   X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51)   =  9,

   X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51)   =  9,
   
   -18 - X(61) * (X(61) + 1) + 2 * X(62) =0,

   -25 + (X(61) - 1) ^ 2 + (X(62) - 6) ^ 2=0,

   X(63) ^ 2 - 10 * X(63) + X(64) ^ 2 + 8=0,

  X(63) * X(64) ^ 2 + X(63) - 10 * X(64) + 8=0,
   
 X(41) ^ 2 + X(42) - 37  = 0,                                        

 X(41) - X(42) ^ 2 - 5 = 0,

 X(41) + X(42) + X(43) - 3 = 0,

and

-10<= Integers X(i)<= 20 for i.= 44 through 84.

The first seven equations above are based on page 47 of Greenspan and Casulli [4].   The next four equations are based on page 510 of  Burden and Faires [3], and the next nine are based on page 604 of Burden and Faires [2].  The next six equations come from Johnson and Riess [5]-- the first three from p. 194 and the other three from p. 198.  The next eight equations are based on p. 510 of Burden and Faires [3, p. 510].  The last seven equations come from p. 648 and p.649 of Burden, Faires, and Burden [1].  

One notes line 13, which is 13 A(KNEW) = -10 + FIX(RND * 30.98).


0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
    13 A(KNEW) = -10 + FIX(RND * 30.98)


    14 RANDOMIZE JJJJ

    16 M = -1D+317


    91 FOR KK = 44 TO 84


        94 A(KK) = A(KNEW)


    95 NEXT KK
    128 FOR I = 1 TO 20000



        129 FOR K = 44 TO 84


            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 44 + FIX(RND * 41)



            182 IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R
            187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
        191 NEXT IPP
        255 FOR J45 = 44 TO 84




            256 IF X(J45) < -10 THEN X(J45) = A(J45)



            257 IF X(J45) > 20 THEN X(J45) = A(J45)



        259 NEXT J45


        451 X(54) = -X(52) ^ 2


        453 N44 = X(52) * COS(X(53)) - X(54)
        455 N45 = (EXP(X(52) + X(54))) * SIN(X(53) / 2) + X(54)

        461 N46 = X(55) ^ 5 + X(56) ^ 3 * X(57) ^ 4 + 1



        463 N47 = X(55) * X(56) * X(57)



        465 N48 = X(57) ^ 4 - 1



        499 X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9



        501 N35 = 2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9

        504 N38 = X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9
        505 N39 = X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51) - 9
        506 N40 = X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51) - 9
        507 N41 = X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51) - 9
        508 N42 = X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51) - 9
        509 N43 = X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51) - 1


        601 X(58) = X(59) ^ 2 + 5



        602 N49 = X(58) ^ 2 + X(59) - 37


        604 N50 = X(58) + X(59) + X(60) - 3

        651 N51 = -18 - X(61) * (X(61) + 1) + 2 * X(62)


        652 N52 = -25 + (X(61) - 1) ^ 2 + (X(62) - 6) ^ 2

        661 N53 = X(63) ^ 2 - 10 * X(63) + X(64) ^ 2 + 8

        663 N54 = X(63) * X(64) ^ 2 + X(63) - 10 * X(64) + 8


        671 X(67) = -10 + X(65) ^ 2 + 10 * X(66)

        673 N55 = -12 + 15 * X(65) + X(66) - 4 * X(67)
        675 N56 = 24 + X(66) - 25 * X(67)

        681 N57 = -7 + 10 * X(68) - 2 * X(69) ^ 2 + X(69) - 2 * X(70) ^ 2
        683 N58 = -12 + 8 * X(69) ^ 2 + 4 * X(70) ^ 2
        685 N59 = 8 + 8 * X(69) * X(70)



        691 N60 = -5 + 10 * X(71) - 2 * X(72) ^ 2 + X(72) - 2 * X(73) ^ 2
        693 N61 = -12 + 8 * X(72) ^ 2 + 4 * X(73) ^ 2
        695 N62 = 8 + 8 * X(72) * X(73)

        751 X(74) = -X(75) ^ 3 + 5 * X(75) ^ 2 + X(75) + 10
        755 N63 = -29 + X(74) + X(75) ^ 3 + X(75) ^ 2 - 14 * X(75)

        851 X(76) = -X(77) ^ 3 + 5 * X(77) ^ 2 + 2 * X(77) + 11


        855 N64 = -29 + X(76) + X(77) ^ 3 + X(77) ^ 2 - 14 * X(77)
        901 X(78) = -X(79) - X(80) - X(81)

        902 IF X(78) < .1 THEN GOTO 1670


        903 IF X(80) < .1 THEN GOTO 1670



        904 N66 = LOG(X(78)) + X(79) + LOG(X(80)) + X(81) + 2


        906 N67 = X(78) * X(79) * X(80) - X(81)

        909 N68 = X(78) - EXP(X(79)) + X(80) + EXP(X(81)) - 2

        921 X(84) = -2 + X(82) ^ 2 + X(83) ^ 2

        923 IF X(82) < .1 THEN GOTO 1670

        925 N69 = -1 + LOG(X(82)) - X(83) - X(84)




        927 N70 = X(82) + EXP(X(83)) + X(84) - 1



        1287 P = -ABS(N44) - ABS(N45) - ABS(N46) - ABS(N47) - ABS(N48) - ABS(N49) - ABS(N50) - ABS(N51) - ABS(N52) - ABS(N53) - ABS(N54) - ABS(N55) - ABS(N56) - ABS(N57) - ABS(N58) - ABS(N59) - ABS(N60) - ABS(N61) - ABS(N62) - ABS(N63) - ABS(N64) - ABS(N65) - ABS(N66) - ABS(N67) - ABS(N68) - ABS(N69) - ABS(N70) - ABS(N33) - ABS(N34) - ABS(N35) - ABS(N36) - ABS(N37) - ABS(N38) - ABS(N39) - ABS(N40) - ABS(N41) - ABS(N42) - ABS(N43)



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 44 TO 84



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

    1670 NEXT I
    1888 IF M < -2 THEN 1999



    1917 PRINT A(44), A(45), A(46), A(47), A(48), A(49), A(50), A(51), M, JJJJ


    1918 PRINT A(52), A(53), A(54), A(55), A(56), A(57), A(58), A(59), A(60), A(61), A(62), A(63), A(64), A(65), A(66), A(67), A(68), A(69), A(70), A(71), A(72), A(73), A(74), A(75), A(76), A(77), A(78), A(79), A(80), A(81), A(82), A(83), A(84)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ=25321 is shown below:

1      1      2      1      1
1      1      0      -2      -14300    
0      0      0      -1      0    
-1      6      1      -4      -2
10      1      1      1      1
1      1      1      -1      1
-1       1      15      -1      35
3        1      -1      1      -1
1      0      -1

1      1      2      1      1
1      1      0      -2      -556    
0      0      0      -1      0    
-1      6      1      -4      -2
10      1      1      1      1
1      1      1      -1      1
-1       1      15      -1      35
3        1      -1      1      -1
1      0      -1

1      1      2      1      1
1      1      0      -2      7806    
0      0      0      -1      0    
-1      6      1      -4      -2
10      1      1      1      1
1      1      1      -1      1
-1       1      15      -1      35
3        1      -1      1      -1
1      0      -1

1      1      1      1      1
1      1      1      0      25321    
0      0      0      0      -1    
1      6      1      -4      -2
10      1      1      1      1
1      1      1      -1      1
-1       1      15      -1      15
-1        1      -1      1      -1
1      0      -1

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

One can ignore the candidate solution at JJJJ=-14300 with A(76)=35 because this candidate solution does not satisfy the requirement -10<= Integers X(i)<= 20 for i = 44 through 84--see the problem statement above.  In contrast, the candidate solution at JJJJ=25321 satisfies all requirements.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [8], the wall-clock time through JJJJ=25321 was two hours.  

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

Simultaneously Solving in General Integers a Nonlinear System of 34 Simultaneous Equations in 34 General Integer Variables

Jsun Yui Wong

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

 X(74) = -X(75) ^ 3 + 5 * X(75) ^ 2 + X(75) + 10,
     
 -29 + X(74) + X(75) ^ 3 + X(75) ^ 2 - 14 * X(75) =0,

 X(76) =  -X(77) ^ 3 + 5 * X(77) ^ 2 + 2 * X(77) + 11,

  -29 + X(76) + X(77) ^ 3 + X(77) ^ 2 - 14 * X(77)=0,

 -5 + 10 * X(71) - 2 * X(72) ^ 2 + X(72) - 2 * X(73) ^ 2 = 0,

 -12 + 8 * X(72) ^ 2 + 4 * X(73) ^ 2 = 0,

  8 + 8 * X(72) * X(73) = 0,

  X(67) = -10 + X(65) ^ 2 + 10 * X(66),

  -12 + 15 * X(65) + X(66) - 4 * X(67) = 0,

   24 + X(66) - 25 * X(67) = 0,

   -7 + 10 * X(68) - 2 * X(69) ^ 2 + X(69) - 2 * X(70) ^ 2  = 0,

  -12 + 8 * X(69) ^ 2 + 4 * X(70) ^ 2 = 0,

  8 + 8 * X(69) * X(70) = 0,

 X(54) = -X(52) ^ 2,

 X(52) * COS(X(53)) - X(54) = 0,

 (EXP(X(52) + X(54))) * SIN(X(53) / 2) + X(54) = 0,

 X(55) ^ 5 + X(56) ^ 3 * X(57) ^ 4 + 1 = 0,

 X(55) * X(56) * X(57) = 0,

 X(57) ^ 4 - 1 = 0,

 X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51)  =  1,

 X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9,

  2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51)   =  9,
       
   X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51)   =  9,
 
   X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51)   =  9,

   X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51)   =  9,

   X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51)   =  9,

   X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51)   =  9,
   
   -18 - X(61) * (X(61) + 1) + 2 * X(62) =0,

   -25 + (X(61) - 1) ^ 2 + (X(62) - 6) ^ 2=0,

   X(63) ^ 2 - 10 * X(63) + X(64) ^ 2 + 8=0,

  X(63) * X(64) ^ 2 + X(63) - 10 * X(64) + 8=0,
   
 X(41) ^ 2 + X(42) - 37  = 0,                                        

 X(41) - X(42) ^ 2 - 5 = 0,

 X(41) + X(42) + X(43) - 3 = 0,

and

-10<= Integers X(i)<= 20 for i.= 44 through 77.

The first four equations are based on page 510 of  Burden and Faires [3].  The next nine equations are based on page 604 of Burden and Faires [2].  The next six equations come from Johnson and Riess [4]-- the first three from p. 194 and the other three from p. 198.  The next eight equations are based on pages 510 of Burden and Faires [3, p. 510].  The last seven equations come from p. 648 and p.649 of Burden, Faires, and Burden [1].  

One notes line 13, which is 13 A(KNEW) = -10 + FIX(RND * 30.98).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
    13 A(KNEW) = -10 + FIX(RND * 30.98)


    14 RANDOMIZE JJJJ

    16 M = -1D+317


    91 FOR KK = 44 TO 77


        94 A(KK) = A(KNEW)


    95 NEXT KK
    128 FOR I = 1 TO 20000


        129 FOR K = 44 TO 77


            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 44 + FIX(RND * 34)


            182 IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R
            187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
        191 NEXT IPP
        255 FOR J45 = 44 TO 77



            256 IF X(J45) < -10 THEN X(J45) = A(J45)



            257 IF X(J45) > 20 THEN X(J45) = A(J45)



        259 NEXT J45


        451 X(54) = -X(52) ^ 2


        453 N44 = X(52) * COS(X(53)) - X(54)
        455 N45 = (EXP(X(52) + X(54))) * SIN(X(53) / 2) + X(54)

        461 N46 = X(55) ^ 5 + X(56) ^ 3 * X(57) ^ 4 + 1



        463 N47 = X(55) * X(56) * X(57)



        465 N48 = X(57) ^ 4 - 1



        499 X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9



        501 N35 = 2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9

        504 N38 = X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9
        505 N39 = X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51) - 9
        506 N40 = X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51) - 9
        507 N41 = X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51) - 9
        508 N42 = X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51) - 9
        509 N43 = X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51) - 1


        601 X(58) = X(59) ^ 2 + 5



        602 N49 = X(58) ^ 2 + X(59) - 37


        604 N50 = X(58) + X(59) + X(60) - 3

        651 N51 = -18 - X(61) * (X(61) + 1) + 2 * X(62)


        652 N52 = -25 + (X(61) - 1) ^ 2 + (X(62) - 6) ^ 2

        661 N53 = X(63) ^ 2 - 10 * X(63) + X(64) ^ 2 + 8

        663 N54 = X(63) * X(64) ^ 2 + X(63) - 10 * X(64) + 8


        671 X(67) = -10 + X(65) ^ 2 + 10 * X(66)

        673 N55 = -12 + 15 * X(65) + X(66) - 4 * X(67)
        675 N56 = 24 + X(66) - 25 * X(67)

        681 N57 = -7 + 10 * X(68) - 2 * X(69) ^ 2 + X(69) - 2 * X(70) ^ 2
        683 N58 = -12 + 8 * X(69) ^ 2 + 4 * X(70) ^ 2
        685 N59 = 8 + 8 * X(69) * X(70)



        691 N60 = -5 + 10 * X(71) - 2 * X(72) ^ 2 + X(72) - 2 * X(73) ^ 2
        693 N61 = -12 + 8 * X(72) ^ 2 + 4 * X(73) ^ 2
        695 N62 = 8 + 8 * X(72) * X(73)

        751 X(74) = -X(75) ^ 3 + 5 * X(75) ^ 2 + X(75) + 10
        755 N63 = -29 + X(74) + X(75) ^ 3 + X(75) ^ 2 - 14 * X(75)

        851 X(76) = -X(77) ^ 3 + 5 * X(77) ^ 2 + 2 * X(77) + 11


        855 N64 = -29 + X(76) + X(77) ^ 3 + X(77) ^ 2 - 14 * X(77)



        1287 P = -ABS(N44) - ABS(N45) - ABS(N46) - ABS(N47) - ABS(N48) - ABS(N49) - ABS(N50) - ABS(N51) - ABS(N52) - ABS(N53) - ABS(N54) - ABS(N55) - ABS(N56) - ABS(N57) - ABS(N58) - ABS(N59) - ABS(N60) - ABS(N61) - ABS(N62) - ABS(N63) - ABS(N64) - ABS(N65) - ABS(N33) - ABS(N34) - ABS(N35) - ABS(N36) - ABS(N37) - ABS(N38) - ABS(N39) - ABS(N40) - ABS(N41) - ABS(N42) - ABS(N43)


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 44 TO 77


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

    1670 NEXT I
    1888 IF M < -.00001 THEN 1999


    1917 PRINT A(44), A(45), A(46), A(47), A(48), A(49), A(50), A(51), M, JJJJ


    1918 PRINT A(52), A(53), A(54), A(55), A(56), A(57), A(58), A(59), A(60), A(61), A(62), A(63), A(64), A(65), A(66), A(67), A(68), A(69), A(70), A(71), A(72), A(73), A(74), A(75), A(76), A(77)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [7]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-23926 is shown below:

1      1      1      1      1
1      1      1      0       -29730    
0      0      0      0      -1    
1      6      1      -4      -2
10      1      1      1      1
1      1      1      -1      1
-1       1      15      -1      35
3

1      1      1      1      1
1      1      1      0       -27552    
0      0      0      0      -1    
-1      6      1      -4      -2
10      1      1      1      1
1      1      1      -1      1
-1       1      15      -1      15
-1

1      1      1      1      1
1      1      1      0       -26175    
0      0      0      -1      0    
-1      6      1      -4      -2
10      1      1      1      1
1      1      1      -1      1
-1       1      15      -1      35
3

1      1      1      1      1
1      1      1      0       -24882    
0      0      0      -1      0    
1      6      1      -4      -2
10      1      1      1      1
1      1      1      -1      1
-1       1      15      -1      35
3

1      1      1      1      1
1      1      1      0       -23926    
0      0      0      0      -1    
-1      6      1      -4      -2
10      1      1      1      1
1      1      1      -1      1
-1       1      15      -1      15
-1

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

One can ignore the candidate solution at JJJJ=-29730 with A(76)=35 because this candidate solution does not satisfy the requirement -10<= Integers X(i)<= 20 for i.= 44 through 77--see the problem statement above.  In contrast, the candidate solution at JJJJ=-27552, for example, satisfies all requirements.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [7], the wall-clock time through JJJJ=-23926 was 22 minutes.  

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]  L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition.  Addison-Wesley Publishing Company, 1982
[5] 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.
[6] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[7] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

Simultaneously Solving in General Integers a Nonlinear System of 33 Simultaneous Equations in 51 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following problem:

 X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51)  =  1,


 X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9,


  2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51)   =  9,

       
   X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51)   =  9,

    X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51)   =  9,

     X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51)   =  9,

     X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51)   =  9,

     X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51)   =  9,
     
     
 X(41) ^ 2 + X(42) - 37  = 0,                                        


 X(41) - X(42) ^ 2 - 5 = 0,


 X(41) + X(42) + X(43) - 3 = 0,


 X(39) + X(40) ^ 3 - 5 * X(40) ^ 2 - X(40)   = 10,


 X(39) + X(40) ^ 3 + X(40) ^ 2 - 14 * X(40) = 29,

 - X(35) * (X(35) + 1) + 2 * X(36) = 18,

(X(35) - 1) ^ 2 + (X(36) - 6) ^ 2 = 25

 X(37) ^ 2 - 10 * X(37) + X(38) ^ 2 + 8 = 0,

 X(37) * X(38) ^ 2 + X(37) - 10 * X(38) + 8 = 0,

EXP(X(32) * X(33) * X(34)) = 1,

3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3 = 0,

3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3 = 6666.6666666,

(X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)  =  714,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
=  2902,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2  =  1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2  =  659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2  =  395,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1   =  348,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2   =  468,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2205.868,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2* X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2206.889,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2208.886,

-.5 * (X(1) * X(4) – X(2) * X(3) + X(3) * X(9) – X(5) * X(9) + X(5) * X(8) – X(6) * X(7)) = 0,

and

-5<= Integers X(i)<= 15 for i=1 through 51 except 0<= Integer X(13)<= 10.  

The first eight equations above are based on pages 510 of Burden and Faires [3, p. 510]. The next three equations come from page 649 of Burden, Faires, and Burden [2], and the next two equations are based on page 510 of Burden and Faires [3].  The next four equations come from Burden, Faires, and Burden [2, page 648]. Equation 18 through  equation 21 are based on page 100, page 118, 116, and page 111 of Hock and Schittkowski [8], respectively.  Equation 22 through equation 26 are based on page 147 of Asaadi [1].  Equation 27 is based on page 122 of Hock and Schittkowski [9]. Equation 30 through equation 32 are based on page 112 of the same book [9]. Equation 33, the last one, is based on page 117 of the same book [9].

One notes line 13, which is 13 A(KNEW) = -5 + FIX(RND * 20.98).


0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
    13 A(KNEW) = -5 + FIX(RND * 20.98)



    14 RANDOMIZE JJJJ

    16 M = -1D+317


    22 REM


    91 FOR KK = 1 TO 51



        94 A(KK) = A(KNEW)


    95 NEXT KK
    128 FOR I = 1 TO 16500000



        129 FOR K = 1 TO 51


            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 51)



            182 IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R
            187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
        191 NEXT IPP
        255 FOR J45 = 1 TO 51



            256 IF X(J45) < -5 THEN X(J45) = A(J45)



            257 IF X(J45) > 15 THEN X(J45) = A(J45)



        259 NEXT J45

        264 X(13) = 348 - (X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

        266 IF X(13) < 0 THEN 1670



        267 IF X(13) > 10 THEN 1670


        268 N11 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2205.868
        269 N12 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2206.889
        270 N13 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2208.886
        273 N14 = -.5 * (X(1) * X(4) - X(2) * X(3) + X(3) * X(9) - X(5) * X(9) + X(5) * X(8) - X(6) * X(7)) - 0

        275 N15 = X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

        276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2



        278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
        279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

        280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2


        281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2


        282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2



        283 N22 = -714 + (X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)

        284 N23 = -6666.6666666 + 3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3

        291 N24 = 0 + 3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3


        295 IF (X(32) * X(33) * X(34)) > 80 THEN GOTO 1670



        296 N25 = -1 + EXP(X(32) * X(33) * X(34))

        298 N26 = -18 - X(35) * (X(35) + 1) + 2 * X(36)


        299 N27 = -25 + (X(35) - 1) ^ 2 + (X(36) - 6) ^ 2

        301 N28 = X(37) ^ 2 - 10 * X(37) + X(38) ^ 2 + 8

        303 N29 = X(37) * X(38) ^ 2 + X(37) - 10 * X(38) + 8


        304 X(39) = 10 - X(40) ^ 3 + 5 * X(40) ^ 2 + X(40)



        305 N31 = -29 + X(39) + X(40) ^ 3 + X(40) ^ 2 - 14 * X(40)

        400 X(41) = X(42) ^ 2 + 5



        401 N32 = X(41) ^ 2 + X(42) - 37


        402 REM N33 = X(41) - X(42) ^ 2 - 5


        403 N34 = X(41) + X(42) + X(43) - 3



        499 X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9



        500 REM  X(44) = -2 * X(45) - X(46) - X(47) + 5


        501 N35 = 2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9


        503 REM   N37 =  X(44) +2* X(45) + X(46) + X(47)+x(  48   )+x(  49    )+x( 50    )+x( 51    )   - 9
        504 N38 = X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9
        505 N39 = X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51) - 9
        506 N40 = X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51) - 9
        507 N41 = X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51) - 9
        508 N42 = X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51) - 9
        509 N43 = X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51) - 1



        1287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21) - ABS(N22) - ABS(N23) - ABS(N24) - ABS(N25) - ABS(N26) - ABS(N27) - ABS(N28) - ABS(N29) - ABS(N30) - ABS(N31) - ABS(N32) - ABS(N33) - ABS(N34) - ABS(N35) - ABS(N36) - ABS(N37) - ABS(N38) - ABS(N39) - ABS(N40) - ABS(N41) - ABS(N42) - ABS(N43)



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 51


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

    1670 NEXT I
    1888 IF M < -1 THEN 1999


    1910 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10)
    1912 PRINT A(11), A(12), A(13), A(14), A(15), A(16), A(17), A(18), A(19), A(20)

    1914 PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), A(30)

    1915 PRINT A(31), A(32), A(33), A(34), A(35), A(36), A(37), A(38), A(39), A(40), A(41), A(42), A(43), A(44), A(45), A(46), A(47), A(48), A(49), A(50), A(51), M, JJJJ

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=-31512 is shown below:

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      -2      6      0      6
12      9       2      11         2  
3       2      1       1         0
0      0         4        9        -2
10       1      1       15        -1
6        1       -4       1       1
1       1       1        1         1
1       -9.020425E-04       -31704

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6       6      -2      6         6
8        15        2      14         0  
-1       5      1      1          0
0        13         6        0       -2
10       1       1       15         -1
6        1      -4      1         1
1       1       1       1         1
1       -9.020425E-04       -31648

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6       -2        6       0        6
9      4         -3        9        1  
5       -3        1      1         0
0      12        0       2        -2
10       1      1       15       -1
6        1       -4      1       1
1       1       1       1         1
1       -9.020425E-04       -31512

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=-31512 was eleven hours.

Acknowledgment

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

References
[1]  J. Asaadi, A Computational Comparison of Some Nonlinear Programs, Mathematical Programming, Vol. 4 (1973),  pp. 144-154.
[2] R. Burden, J. Faires, and A. Burden, Numerical Analysis, Tenth Edition.  Cengage Learning, 2016.
[3] R. Burden, and J. Faires, Numerical Analysis, Third Edition.  PWS Publishers, 1985.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] F. Glover, 1986. Future Paths for Integer ng and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[6] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[7] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[8] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. http://www.SciRP.org/journal/am.
[9] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[10] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
[11] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[12] 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.
[13] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[14] 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.
[15] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[16] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[17] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[18] 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.
[19] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/

Simultaneously Solving in General Integers a Nonlinear System of 29 Simultaneous Equations in 47 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following problem:

 2 * X(44) + X(45) + X(46) + X(47) - 5 = 0,

  X(44) + 2 * X(45) + X(46) + X(47) - 5 = 0,

  X(44) + X(45) + 2 * X(46) + X(47) - 5 = 0,

  X(44) * X(45) * X(46) * X(47) - 1 =0,
     
 X(41) ^ 2 + X(42) - 37  = 0,                                        


 X(41) - X(42) ^ 2 - 5 = 0,


 X(41) + X(42) + X(43) - 3 = 0,


 X(39) + X(40) ^ 3 - 5 * X(40) ^ 2 - X(40)   = 10,


 X(39) + X(40) ^ 3 + X(40) ^ 2 - 14 * X(40) = 29,

 - X(35) * (X(35) + 1) + 2 * X(36) = 18,

(X(35) - 1) ^ 2 + (X(36) - 6) ^ 2 = 25,

 X(37) ^ 2 - 10 * X(37) + X(38) ^ 2 + 8 = 0,

 X(37) * X(38) ^ 2 + X(37) - 10 * X(38) + 8 = 0,

EXP(X(32) * X(33) * X(34)) = 1,

3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3 = 0,

3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3 = 6666.6666666,

(X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)  =  714,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
=  2902,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2  =  1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2  =  659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2  =  395,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1   =  348,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2   =  468,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2205.868,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2* X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2206.889,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2208.886,
-.5 * (X(1) * X(4) – X(2) * X(3) + X(3) * X(9) – X(5) * X(9) + X(5) * X(8) – X(6) * X(7)) = 0, and

-5<= Integers X(i)<= 15 for i=1 through 47 except 0<= Integer X(13)<= 10.  

The first four equations above come from Burden and Faires [3, p. 510]. The next three equations come from page 649 of Burden, Faires, and Burden [2], and the next two equations are based on page 510 of Burden and Faires [3].   The next four equations come from Burden, Faires, and Burden [2, page 648]. Equation 14 through equation 17 are based on page 100, page 118, 116, and page 111 of Hock and Schittkowski [8], respectively.  Equation 18 through equation 22 are based on page 147 of Asaadi [1].  Equation 23 is based on page 122 of Hock and Schittkowski [9]. Equation 26 through equation 28 are based on page 112 of the same book [9]. Equation 29, the last one, is based on page 117 of the same book [9].

One notes line 13, which is 13 A(KNEW) = -5 + FIX(RND * 20.98).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
    13 A(KNEW) = -5 + FIX(RND * 20.98)


    14 RANDOMIZE JJJJ

    16 M = -1D+317


    22 REM


    91 FOR KK = 1 TO 47


        94 A(KK) = A(KNEW)


    95 NEXT KK
    128 FOR I = 1 TO 13000000


        129 FOR K = 1 TO 47


            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 47)



            182 IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R
            187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
        191 NEXT IPP
        255 FOR J45 = 1 TO 47


            256 IF X(J45) < -5 THEN X(J45) = A(J45)


            257 IF X(J45) > 15 THEN X(J45) = A(J45)


        259 NEXT J45

        264 X(13) = 348 - (X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

        266 IF X(13) < 0 THEN 1670


        267 IF X(13) > 10 THEN 1670


        268 N11 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2205.868
        269 N12 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2206.889
        270 N13 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2208.886
        273 N14 = -.5 * (X(1) * X(4) - X(2) * X(3) + X(3) * X(9) - X(5) * X(9) + X(5) * X(8) - X(6) * X(7)) - 0

        275 N15 = X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

        276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2


        278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
        279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

        280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2


        281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2


        282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2


        283 N22 = -714 + (X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)

        284 N23 = -6666.6666666 + 3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3

        291 N24 = 0 + 3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3


        295 IF (X(32) * X(33) * X(34)) > 80 THEN GOTO 1670


        296 N25 = -1 + EXP(X(32) * X(33) * X(34))

        298 N26 = -18 - X(35) * (X(35) + 1) + 2 * X(36)


        299 N27 = -25 + (X(35) - 1) ^ 2 + (X(36) - 6) ^ 2

        301 N28 = X(37) ^ 2 - 10 * X(37) + X(38) ^ 2 + 8

        303 N29 = X(37) * X(38) ^ 2 + X(37) - 10 * X(38) + 8


        304 N30 = -10 + X(39) + X(40) ^ 3 - 5 * X(40) ^ 2 - X(40)


        305 N31 = -29 + X(39) + X(40) ^ 3 + X(40) ^ 2 - 14 * X(40)

        400 X(41) = X(42) ^ 2 + 5


        401 N32 = X(41) ^ 2 + X(42) - 37


        402 REM N33 = X(41) - X(42) ^ 2 - 5


        403 N34 = X(41) + X(42) + X(43) - 3
        500 X(44) = -2 * X(45) - X(46) - X(47) + 5


        501 N35 = 2 * X(44) + X(45) + X(46) + X(47) - 5

        502 REM N36 = X(44) + 2 * X(45) + X(46) + X(47) - 5

        503 N37 = X(44) + X(45) + 2 * X(46) + X(47) - 5
        504 N38 = X(44) * X(45) * X(46) * X(47) - 1


        1287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21) - ABS(N22) - ABS(N23) - ABS(N24) - ABS(N25) - ABS(N26) - ABS(N27) - ABS(N28) - ABS(N29) - ABS(N30) - ABS(N31) - ABS(N32) - ABS(N33) - ABS(N34) - ABS(N35) - ABS(N36) - ABS(N37) - ABS(N38)


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 47


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

    1670 NEXT I
    1888 IF M < -30 THEN 1999


    1910 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10)
    1912 PRINT A(11), A(12), A(13), A(14), A(15), A(16), A(17), A(18), A(19), A(20)

    1914 PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), A(30)

    1915 PRINT A(31), A(32), A(33), A(34), A(35), A(36), A(37), A(38), A(39), A(40), A(41), A(42), A(43), A(44), A(45), A(46), A(47), M, JJJJ

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=-31862 is shown below:

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
5      12       5        9        1    
1      3       1      1       0
0      -2       0       3      -2
10       1      1       15      -1
6        1         -4      1      1
1       1       -9.020425E-04       -31997
   
6      6      6      6      6
6      6      6      6      6
6      6      6      3      5
6      -4      6     1        6  
6      11      5       8        0
2      3           1     1       0
0      0       1       -1       -2
10       1      1       15      -1
6        1       -4        1        1
1        1         -20.0009      -31979

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      6      6      0      6
13      9       0        10       0  
6       5       1          1       0
0       0       2       1      -2
10       1      1       15      -1
6        1         -4       1      1
1        1       -9.020425E-04       -31862

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=-31862 was two hours and fifteen minutes.

Acknowledgment

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

References
[1]  J. Asaadi, A Computational Comparison of Some Nonlinear Programs, Mathematical Programming, Vol. 4 (1973),  pp. 144-154.
[2] R. Burden, J. Faires, and A. Burden, Numerical Analysis, Tenth Edition.  Cengage Learning, 2016.
[3] R. Burden, and J. Faires, Numerical Analysis, Third Edition.  PWS Publishers, 1985.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] F. Glover, 1986. Future Paths for Integer ng and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[6] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[7] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[8] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. http://www.SciRP.org/journal/am.
[9] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[10] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
[11] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[12] 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.
[13] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[14] 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.
[15] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[16] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[17] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[18] 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.
[19] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/

Simultaneously Solving in General Integers a Nonlinear System of 18 Simultaneous Nonlinear Equations in 36 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following problem:

 - X(35) * (X(35) + 1) + 2 * X(36) = 18,

  (X(35) - 1) ^ 2 + (X(36) - 6) ^ 2 = 25

EXP(X(32) * X(33) * X(34)) = 1,

3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3 = 0,

3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3 = 6666.6666666,

(X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)  =  714,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
=  2902,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2  =  1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2  =  659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2  =  395,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1   =  348,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2   =  468,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2205.868,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2* X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2206.889,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2208.886,

-.5 * (X(1) * X(4) – X(2) * X(3) + X(3) * X(9) – X(5) * X(9) + X(5) * X(8) – X(6) * X(7)) = 0,

and

-5<= Integers X(i)<= 15 for i=1 through 36 except 0<= Integer X(13)<= 10.  

The first two equations come from Burden, Faires, and Burden [2, page 648].  Equations 3 through 6 are based on page 100, page 118, 116, and page 111 of Hock and Schittkowski [8], respectively. Equations 7 through 11 are based on page 147 of Asaadi [1].  Equation 12 is based on page 122 of Hock and Schittkowski [8], equations 15, and 16, and 17 are based on page 112 of the same book,  and equation 18 is based on page 117 of the same book, as well.

One notes line 13, which is 13 A(KNEW) = -5 + FIX(RND * 20.98).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
    13 A(KNEW) = -5 + FIX(RND * 20.98)


    14 RANDOMIZE JJJJ

    16 M = -1D+317


    22 REM


    91 FOR KK = 1 TO 36


        94 A(KK) = A(KNEW)


    95 NEXT KK
    128 FOR I = 1 TO 3000000



        129 FOR K = 1 TO 36



            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 36)



            182 IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R
            187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
        191 NEXT IPP
        255 FOR J45 = 1 TO 36



            256 IF X(J45) < -5 THEN X(J45) = A(J45)



            257 IF X(J45) > 15 THEN X(J45) = A(J45)



        259 NEXT J45

        264 X(13) = 348 - (X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

        266 IF X(13) < 0 THEN 1670


        267 IF X(13) > 10 THEN 1670



        268 N11 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2205.868
        269 N12 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2206.889
        270 N13 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2208.886
        273 N14 = -.5 * (X(1) * X(4) - X(2) * X(3) + X(3) * X(9) - X(5) * X(9) + X(5) * X(8) - X(6) * X(7)) - 0

        275 N15 = X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

        276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2




        278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
        279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

        280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2


        281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2


        282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2



        283 N22 = -714 + (X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)

        284 N23 = -6666.6666666 + 3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3

        291 N24 = 0 + 3 * X(30) + 1.E-6 * X(30) ^ 3 + 2 * X(31) + .522074E-6 * X(31) ^ 3


        295 IF (X(32) * X(33) * X(34)) > 80 THEN GOTO 1670



        296 N25 = -1 + EXP(X(32) * X(33) * X(34))

        298 N26 = -18 - X(35) * (X(35) + 1) + 2 * X(36)


        299 N27 = -25 + (X(35) - 1) ^ 2 + (X(36) - 6) ^ 2



        1287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21) - ABS(N22) - ABS(N23) - ABS(N24) - ABS(N25) - ABS(N26) - ABS(N27)



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 36



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

    1670 NEXT I
    1888 IF M < -100 THEN 1999



    1910 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10)
    1912 PRINT A(11), A(12), A(13), A(14), A(15), A(16), A(17), A(18), A(19), A(20)

    1914 PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), A(30)

    1915 PRINT A(31), A(32), A(33), A(34), A(35), A(36), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [15]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31920 is shown below:

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      6      6      0      6
10      10      0      14      1
6       5       1      1      0
0      -4       0       1       -2
10      -9.020425E-04      -31995

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      7      0     15      7
12     9      0      10       0
11      3    1      1       0
0      0     7      -2         -2
10     -22.0009      -31993

6      6      6      6      6
6      6      6      6      7
5      6      6      3      5
6      7      6      0      6
12      4      4      6       1
4       1      1      1       0
0       0     -1     -5         -2
10     -18.0009      -31988

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
11      11      2        8         2
2       -1       1      1      0
0      8      2      0         -2
2     -16.0009      -31982

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      6      6      7      -5
11      15      5        11     -1
3      0       1      1      0
0      0     6       7        -2
10     -11.0009      -31937

6      6      6      6      6
6      6      6      6      6
5      6      7      1        4
6      7      6      0      6
6      2      -1       4      -1
4      2       1      1      0
0      -4       3       0        -2
10     -17.0009      -31925

6      6      6      6      6
6      6      6      6      6
6      6      6      3        5
6      7      5        13        6
4      6      -4       4      -1
5      2      1      1      0
0      6       -4       0       -2
10     -18.0009      -31920

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 [15], the wall-clock time through JJJJ=-31920 was 20 minutes.

Acknowledgment

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

References
[1]  J. Asaadi, A Computational Comparison of Some Nonlinear Programs, Mathematical Programming, Vol. 4 (1973),  pp. 144-154.
[2] R. Burden, J. Faires, and A. Burden, Numerical Analysis, Tenth Edition.  Cengage Learning, 2016.
[3] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[4] F. Glover, 1986. Future Paths for Integer ng and Links to Artificial Intelligence. Computers and Operations Research, vol. 13, 5, 533-549
[5] F. Glover, 1989. Tabu Search, Part 1. ORSA Journal on Computing, vol. 1, number 3, 190-206.
[6] F. Glover, 1989. Tabu Search, Part 2. ORSA Journal on Computing, vol. 2, number 1, 4-32.
[7] Tianmin Han, Yuhuan Han, Solving large scale nonlinear equations by a new ODE numerical integration method, Applied Mathematics, 2010, 1, pp. 222-229. http://www.SciRP.org/journal/am.
[8] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1981.
[9] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
[10] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[11] 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.
[12] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[13] 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.
[14] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer, 1987.
[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[16] Wolfram Research, Inc., Diophantine Polynomial Systems. https:/reference.wolfram.com/language/tutorial/DiophantineReduce.html.
[17] 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.
[18] Jsun Yui Wong (2015, October 26). Testing the Domino Method of General Integer Nonlinear Programming with Brown’s Almost Linear System of Fifteen Equations, Second Edition. http://myblogsubstance.typepad.com/substance/2015/10/

Simultaneously Solving in General Integers a Nonlinear System of 14 Simultaneous Nonlinear Equations in 29 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following problem:

3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3 = 6666.6666666,

(X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)  =  714,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2 = 3155,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 = 3119,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2
=  2902,

 X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2  =  1590,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2  =  659,

X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2  =  395,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1 + X(13) ^ 1   =  348,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2   =  468,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2205.868,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2* X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2206.889,

10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) = 2208.886,

-.5 * (X(1) * X(4) – X(2) * X(3) + X(3) * X(9) – X(5) * X(9) + X(5) * X(8) – X(6) * X(7)) = 0,

and

0<= Integers X(i)<= 10 for i=1 through 29.  

Equation 1 and equation 2 above are based on page 116 and page 111 of Hock and Schittkowski [7], respectively. Equations 3, 4, 5, 6, and 7 are based on page 147 of Asaadi [1].  Equation 8 is based on page 122 of Hock and Schittkowski [7], equations 11, 12, and 13 are based on page 112 of the same book,  and equation 14 is based on page 117 of the same book, as well.

One notes line 13, which is 13 A(KNEW) = FIX(RND * 11).

0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
    13 A(KNEW) = FIX(RND * 11)


    14 RANDOMIZE JJJJ

    16 M = -1D+317


    22 REM


    91 FOR KK = 1 TO 29


        94 A(KK) = A(KNEW)


    95 NEXT KK
    128 FOR I = 1 TO 1000000


        129 FOR K = 1 TO 29




            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 29)




            182 IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R
            187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
        191 NEXT IPP
        255 FOR J45 = 1 TO 29




            256 IF X(J45) < 0 THEN X(J45) = A(J45)


            257 IF X(J45) > 10 THEN X(J45) = A(J45)




        259 NEXT J45

        264 X(13) = 348 - (X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 1 + X(11) ^ 1 + X(12) ^ 1)

        266 IF X(13) < 0 THEN 1670




        267 IF X(13) > 10 THEN 1670




        268 N11 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ (-.25) + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2205.868
        269 N12 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .125 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2206.889
        270 N13 = 10 * X(1) * X(2) ^ (-1) * X(4) ^ 2 * X(6) ^ (-3) * X(7) ^ .5 + 15 * X(1) ^ (-1) * X(2) ^ (-2) * X(3) * X(4) * X(5) ^ (-1) * X(7) ^ (-.5) + 20 * X(1) ^ (-2) * X(2) * X(4) ^ (-1) * X(5) ^ (-2) * X(6) + 25 * X(1) ^ 2 * X(2) ^ 2 * X(3) ^ (-1) * X(5) ^ .5 * X(6) ^ (-2) * X(7) - 2208.886
        273 N14 = -.5 * (X(1) * X(4) - X(2) * X(3) + X(3) * X(9) - X(5) * X(9) + X(5) * X(8) - X(6) * X(7)) - 0

        275 N15 = X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 + X(4) ^ 2 + X(5) ^ 2 + X(6) ^ 2 + X(7) ^ 2 + X(8) ^ 2 + X(9) ^ 2 + X(10) ^ 2 + X(11) ^ 2 + X(12) ^ 2 + X(13) ^ 2 - 468

        276 N16 = -395 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2




        278 N17 = -659 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2
        279 N18 = -1590 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2

        280 N19 = -2902 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2


        281 N20 = -3119 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2


        282 N21 = -3155 + X(1) ^ 2 + X(2) ^ 2 + X(1) * X(2) - 14 * X(1) - 16 * X(2) + (X(3) - 10) ^ 2 + 4 * (X(4) - 5) ^ 2 + (X(5) - 3) ^ 2 + 2 * (X(6) - 1) ^ 2 + 5 * X(7) ^ 2 + 7 * (X(8) - 11) ^ 2 + 2 * (X(9) - 10) ^ 2 + (X(10) - 7) ^ 2 + (X(11) - 9) ^ 2 + 10 * (X(12) - 1) ^ 2 + 5 * (X(13) - 7) ^ 2 + 4 * (X(14) - 14) ^ 2 + 27 * (X(15) - 1) ^ 2 + X(16) ^ 4 + (X(17) - 2) ^ 2 + 13 * (X(18) - 2) ^ 2 + (X(19) - 3) ^ 2 + X(20) ^ 2



        283 N22 = -714 + (X(21) - 10) ^ 2 + 5 * (X(22) - 12) ^ 2 + X(23) ^ 4 + 3 * (X(24) - 11) ^ 2 + 10 * X(25) ^ 6 + 7 * X(26) ^ 2 + X(27) ^ 4 - 4 * X(26) * X(27) - 10 * X(26) - 8 * X(27)

        284 N23 = -6666.6666666 + 3000 * X(28) + 1000 * X(28) ^ 3 + 2000 * X(29) + 666.6666666 * X(29) ^ 3



        287 P = -ABS(N11) - ABS(N12) - ABS(N13) - ABS(N14) - ABS(N15) - ABS(N16) - ABS(N17) - ABS(N18) - ABS(N19) - ABS(N20) - ABS(N21) - ABS(N22) - ABS(N23)



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 29



            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1666 REM   PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), a(28),a(29),M, JJJJ


    1670 NEXT I
    1888 IF M < -10 THEN 1999


    1910 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10)
    1912 PRINT A(11), A(12), A(13), A(14), A(15), A(16), A(17), A(18), A(19), A(20)

    1914 PRINT A(21), A(22), A(23), A(24), A(25), A(26), A(27), A(28), A(29), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [14]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31868 is shown below:

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
7      6      4      4      0
5      3      1      1      -9.020425E-04      
-31964

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
10     4     3    6    0
7      1      1      1      -9.020425E-04      
-31895

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
8      8      1    10    2
1      1      1      1      -9.020425E-04      
-31869

6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
6      6      6      6      6
5      6      4      7      0
1      4      1      1      -9.020425E-04      
-31868

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 [14], the wall-clock time through JJJJ=-31868 was 12 minutes.

Acknowledgment

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

References
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