Tuesday, December 13, 2016

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.

No comments:

Post a Comment