Sunday, December 4, 2016

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.
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