Sat May 30 16:12:05 PDT 1992
REDUCE 3.4.1, 15-Jul-92 ...

1: 1: 
2: 2: 
3: 3: 
Time: 17 ms

4: 4: % Demonstration of the REDUCE SOLVE package.

on fullroots;

   % To get complete solutions.

% Simultaneous linear fractional equations.

solve({(a*x+y)/(z-1)-3,y+b+z,x-y},{x,y,z});


      - 3*(B + 1)
{{X=--------------,
        A + 4

      - 3*(B + 1)
  Y=--------------,
        A + 4

      - A*B - B + 3
  Z=----------------}}
         A + 4



% Use of square-free factorization together with recursive use of
% quadratic and binomial solutions.

solve((x**6-x**3-1)*(x**5-1)**2*x**2);


Unknown: X

                       1/3
     - ( - SQRT(5) + 1)   *(SQRT(3)*I + 1)
{X=----------------------------------------,
                       1/3
                    2*2

                    1/3
    ( - SQRT(5) + 1)   *(SQRT(3)*I - 1)
 X=-------------------------------------,
                     1/3
                  2*2

                    1/3
    ( - SQRT(5) + 1)
 X=---------------------,
            1/3
           2

                    1/3
     - (SQRT(5) + 1)   *(SQRT(3)*I + 1)
 X=-------------------------------------,
                     1/3
                  2*2

                 1/3
    (SQRT(5) + 1)   *(SQRT(3)*I - 1)
 X=----------------------------------,
                    1/3
                 2*2

                 1/3
    (SQRT(5) + 1)
 X=------------------,
           1/3
          2

 X=1,

     - 2*SQRT(SQRT(5) - 5) - SQRT(10) - SQRT(2)
 X=---------------------------------------------,
                     4*SQRT(2)

    2*SQRT(SQRT(5) - 5) - SQRT(10) - SQRT(2)
 X=------------------------------------------,
                   4*SQRT(2)

    2*SQRT( - SQRT(5) - 5) + SQRT(10) - SQRT(2)
 X=---------------------------------------------,
                     4*SQRT(2)

     - 2*SQRT( - SQRT(5) - 5) + SQRT(10) - SQRT(2)
 X=------------------------------------------------,
                      4*SQRT(2)

 X=0}


multiplicities!*;


{1,1,1,1,1,1,2,2,2,2,2,2}



% A singular equation without and with a consistent inhomogeneous term.

solve(a,x);


{}


solve(0,x);


{X=ARBCOMPLEX(1)}


off solvesingular;



solve(0,x);


{}



% Use of DECOMPOSE to solve high degree polynomials.

solve(x**8-8*x**7+34*x**6-92*x**5+175*x**4-236*x**3+226*x**2-140*x+46);


Unknown: X

     - SQRT( - 2*SQRT( - 4*SQRT(3) - 3) - 6) + 2
{X=----------------------------------------------,
                         2

    SQRT( - 2*SQRT( - 4*SQRT(3) - 3) - 6) + 2
 X=-------------------------------------------,
                        2

    SQRT(2*SQRT( - 4*SQRT(3) - 3) - 6) + 2
 X=----------------------------------------,
                      2

     - SQRT(2*SQRT( - 4*SQRT(3) - 3) - 6) + 2
 X=-------------------------------------------,
                        2

    SQRT(2*SQRT(4*SQRT(3) - 3) - 6) + 2
 X=-------------------------------------,
                     2

     - SQRT(2*SQRT(4*SQRT(3) - 3) - 6) + 2
 X=----------------------------------------,
                      2

     - SQRT( - 2*SQRT(4*SQRT(3) - 3) - 6) + 2
 X=-------------------------------------------,
                        2

    SQRT( - 2*SQRT(4*SQRT(3) - 3) - 6) + 2
 X=----------------------------------------}
                      2


solve(x**8-88*x**7+2924*x**6-43912*x**5+263431*x**4-218900*x**3+
           65690*x**2-7700*x+234,x);


{X= - 4*SQRT(7) + 11,

 X=4*SQRT(7) + 11,

 X= - 2*SQRT(30) + 11,

 X=2*SQRT(30) + 11,

 X= - SQRT( - I + 116) + 11,

 X=SQRT( - I + 116) + 11,

 X=SQRT(I + 116) + 11,

 X= - SQRT(I + 116) + 11}



% Recursive use of inverses, including multiple branches of rational
% fractional powers.

solve(log(acos(asin(x**(2/3)-b)-1))+2,x);


             1             3/2
{X=(SIN(COS(----) + 1) + B)   ,
              2
             E

                1             3/2
 X= - (SIN(COS(----) + 1) + B)   }
                 2
                E



% Square-free factors that are unsolvable, being of fifth degree,
% transcendental, or without a defined inverse.

operator f;



solve((x-1)*(x+1)*(x-2)*(x+2)*(x-3)*(x*log(x)-1)*(f(x)-1),x);


{F(X) - 1=0,

 X=ROOT_OF(LOG(X_)*X_ - 1,X_),

 X=-1,

 X=-2,

 X=3,

 X=2,

 X=1}


multiplicities!*;


{1,1,1,1,1,1,1}



% Factors with more than one distinct top-level kernel, the first factor
% requiring the cubic formula. (SOLVE also uses the quartic formula, but
% the output is usually unavoidably too messy to be of much use).

solve((x**(1/2)-(x-a)**(1/3))*(acos x-acos(2*x-b))* (2*log x
          -log(x**2+x-c)-4),x);


                      1/3
{X=ROOT_OF(( - A + X_)    - SQRT(X_),X_),

     2            4          4     2
    E *(SQRT(4*C*E  - 4*C + E ) - E )
 X=-----------------------------------,
                   4
               2*(E  - 1)

        2            4          4     2
     - E *(SQRT(4*C*E  - 4*C + E ) + E )
 X=--------------------------------------,
                     4
                 2*(E  - 1)

 X=B}



% Treatment of multiple-argument exponentials as polynomials.

solve(a**(2*x)-3*a**x+2,x);


    2*ARBINT(2)*I*PI
{X=------------------,
         LOG(A)

    LOG(2) + 2*ARBINT(3)*I*PI
 X=---------------------------}
             LOG(A)



% A 12th degree reciprocal polynomial that is irreductible over the
% integers, having a reduced polynomial that is also reciprocal.
% (Reciprocal polynomials are those that have symmetric or antisymmetric
% coefficient patterns.) We also demonstrate suppression of automatic
% integer root extraction.

solve(x**12-4*x**11+12*x**10-28*x**9+45*x**8-68*x**7+69*x**6-68*x**5+
45*x**4-28*x**3+12*x**2-4*x+1);


Unknown: X

     - 2*SQRT( - SQRT(3)*I - 9) - SQRT(6)*I + SQRT(2)
{X=---------------------------------------------------,
                        4*SQRT(2)

    2*SQRT( - SQRT(3)*I - 9) - SQRT(6)*I + SQRT(2)
 X=------------------------------------------------,
                      4*SQRT(2)

    2*SQRT(SQRT(3)*I - 9) + SQRT(6)*I + SQRT(2)
 X=---------------------------------------------,
                     4*SQRT(2)

     - 2*SQRT(SQRT(3)*I - 9) + SQRT(6)*I + SQRT(2)
 X=------------------------------------------------,
                      4*SQRT(2)

     - SQRT( - SQRT(5) - 3)
 X=-------------------------,
            SQRT(2)

    SQRT( - SQRT(5) - 3)
 X=----------------------,
          SQRT(2)

     - SQRT(SQRT(5) - 3)
 X=----------------------,
          SQRT(2)

    SQRT(SQRT(5) - 3)
 X=-------------------,
         SQRT(2)

     - 2*SQRT( - 3*SQRT(5) - 1) - SQRT(10) + 3*SQRT(2)
 X=----------------------------------------------------,
                        4*SQRT(2)

    2*SQRT( - 3*SQRT(5) - 1) - SQRT(10) + 3*SQRT(2)
 X=-------------------------------------------------,
                       4*SQRT(2)

    2*SQRT(3*SQRT(5) - 1) + SQRT(10) + 3*SQRT(2)
 X=----------------------------------------------,
                     4*SQRT(2)

     - 2*SQRT(3*SQRT(5) - 1) + SQRT(10) + 3*SQRT(2)
 X=-------------------------------------------------}
                       4*SQRT(2)



% The treatment of factors with non-unique inverses by introducing
% unique new real or integer indeterminant kernels.

solve((sin x-a)*(2**x-b)*(x**c-3),x);


    1/C       2*ARBINT(4)*PI            2*ARBINT(4)*PI
{X=3   *(SIN(----------------)*I + COS(----------------)),
                    C                         C

    LOG(B) + 2*ARBINT(5)*I*PI
 X=---------------------------,
             LOG(2)

 X=ASIN(A) + 2*ARBINT(6)*PI,

 X= - ASIN(A) + 2*ARBINT(6)*PI + PI}



% Automatic restriction to principal branches.

off allbranch;



solve((sin x-a)*(2**x-b)*(x**c-3),x);


    1/C
{X=3   ,

    LOG(B)
 X=--------,
    LOG(2)

 X=ASIN(A)}



% Regular system of linear equations.

solve({2*x1+x2+3*x3-9,x1-2*x2+x3+2,3*x1+2*x2+2*x3-7}, {x1,x2,x3});


{{X1=-1,X2=2,X3=3}}



% Underdetermined system of linear equations.

on solvesingular;



solve({x1-4*x2+2*x3+1,2*x1-3*x2-x3-5*x4+7,3*x1-7*x2+x3-5*x4+8},
      {x1,x2,x3,x4});


{{X1=4*ARBCOMPLEX(8) + 2*ARBCOMPLEX(7) - 5,

  X2=ARBCOMPLEX(8) + ARBCOMPLEX(7) - 1,

  X3=ARBCOMPLEX(7),

  X4=ARBCOMPLEX(8)}}



% Inconsistent system of linear equations.

solve({2*x1+3*x2-x3-2,7*x1+4*x2+2*x3-8,3*x1-2*x2+4*x3-5},
      {x1,x2,x3});


***** SOLVE given inconsistent equations 



% Overdetermined system of linear equations.

solve({x1-x2+x3-12,2*x1+3*x2-x3-13,3*x2+4*x3-5,-3*x1+x2+4*x3+20},
      {x1,x2,x3});


{{X1=9,X2=-1,X3=2}}



% Degenerate system of linear equations.

operator xx,yy;



yy(1) := -a**2*b**3-3*a**2*b**2-3*a**2*b+a**2*(xx(3)-2)-a*b-a*c+a*(xx(2)
         -xx(5))-xx(4)-xx(5)+xx(1)-1;


                                             2
YY(1) :=  - XX(5)*A - XX(5) - XX(4) + XX(3)*A  + XX(2)*A + XX(1)

             2  3      2  2      2        2
          - A *B  - 3*A *B  - 3*A *B - 2*A  - A*B - A*C - 1


yy(2) := -a*b**3-b**5+b**4*(-xx(4)-xx(5)+xx(1)-5)-b**3*c+b**3*(xx(2)
         -xx(5)-3)+b**2*(xx(3)-1);


          2            2                    2
YY(2) := B *( - XX(5)*B  - XX(5)*B - XX(4)*B  + XX(3) + XX(2)*B

                       2          3      2
              + XX(1)*B  - A*B - B  - 5*B  - B*C - 3*B - 1)


yy(3) := -a*b**3*c-3*a*b**2*c-4*a*b*c+a*b*(-xx(4)-xx(5)+xx(1)-1)
         +a*c*(xx(3)-1)-b**2*c-b*c**2+b*c*(xx(2)-xx(5));


YY(3) :=  - XX(5)*A*B - XX(5)*B*C - XX(4)*A*B + XX(3)*A*C + XX(2)*B*C

                           3          2
          + XX(1)*A*B - A*B *C - 3*A*B *C - 4*A*B*C - A*B - A*C

             2        2
          - B *C - B*C


yy(4) := -a**2-a*c+a*(xx(2)-xx(4)-2*xx(5)+xx(1)-1)-b**4-b**3*c-3*b**3
         -3*b**2*c-2*b**2-2*b*c+b*(xx(3)-xx(2)-xx(4)+xx(1)-2)
         +c*(xx(3)-1);


YY(4) :=  - 2*XX(5)*A - XX(4)*A - XX(4)*B + XX(3)*B + XX(3)*C

                                                     2              4
          + XX(2)*A - XX(2)*B + XX(1)*A + XX(1)*B - A  - A*C - A - B

             3        3      2        2
          - B *C - 3*B  - 3*B *C - 2*B  - 2*B*C - 2*B - C


yy(5) := -2*a-3*b**3-9*b**2-11*b-2*c+3*xx(3)+2*xx(2)-xx(4)-3*xx(5)+xx(1)
         -4;


                                                                   3
YY(5) :=  - 3*XX(5) - XX(4) + 3*XX(3) + 2*XX(2) + XX(1) - 2*A - 3*B

               2
          - 9*B  - 11*B - 2*C - 4


soln  :=  solve({yy(1),yy(2),yy(3),yy(4),yy(5)},
                {xx(1),xx(2),xx(3),xx(4),xx(5)});


SOLN := {{XX(1)=ARBCOMPLEX(10) + ARBCOMPLEX(9) + 1,

          XX(2)=ARBCOMPLEX(10) + A + B + C,

                 3      2
          XX(3)=B  + 3*B  + 3*B + 1,

          XX(4)=ARBCOMPLEX(9),

          XX(5)=ARBCOMPLEX(10)}}


for i  :=  1:5 do xx(i) := part(soln,1,i,2);



for i  :=  1:5 do write yy(i);


0

0

0

0

0



%  Single equations liftable to polynomial systems.

solve ({a*sin x + b*cos x},{x});


{X=ROOT_OF(SIN(X_)*A + COS(X_)*B,X_)}


solve ({a*sin(x+1) + b*cos(x+1)},{x});


{X=ROOT_OF(SIN(X_ + 1)*A + COS(X_ + 1)*B,X_)}

 
% Intersection of 2 curves: system with a free parameter.

solve ({sqrt(x^2 + y^2)=r,0=sqrt(x)+ y**3-1},{x,y,r});


{{Y=ARBCOMPLEX(12),

     6      3
  X=Y  - 2*Y  + 1,

          12      9      6      3    2
  R=SQRT(Y   - 4*Y  + 6*Y  - 4*Y  + Y  + 1)},

 {Y=ARBCOMPLEX(11),

     6      3
  X=Y  - 2*Y  + 1,

             12      9      6      3    2
  R= - SQRT(Y   - 4*Y  + 6*Y  - 4*Y  + Y  + 1)}}


%  Not yet soluble.

solve ({e^x - e^(1/2 * x) - 7},{x});


            X_/2    X_
{X=ROOT_OF(E     - E   + 7,X_)}


% Generally not liftable.
  
   % variable inside and outside of sin.

   solve({sin x + x - 1/2},{x});


{X=ROOT_OF(2*SIN(X_) + 2*X_ - 1,X_)}

 
   % Variable inside and outside of exponential.

   solve({e^x - x**2},{x});


            X_     2
{X=ROOT_OF(E   - X_ ,X_)}


   % Variable inside trigonometrical functions with different forms.

   solve ({a*sin(x+1) + b*cos(x+2)},{x});


{X=ROOT_OF(SIN(X_ + 1)*A + COS(X_ + 2)*B,X_)}

 
   % Undetermined exponents.

   solve({x^a - 2},{x});


    1/A
{X=2   }

 

% Example taken from M.L. Griss, ACM Trans. Math. Softw. 2 (1976) 1.

e1 := x1 - l/(3*k)$



e2 := x2 - 1$



e3 := x3 - 35*b6/(6*l)*x4 + 33*b11/(2*l)*x6 - 715*b15/(14*l)*x8$



e4 := 14*k/(3*l)*x1 - 7*b4/(2*l)*x3 + x4$



e5 := x5 - 891*b11/(40*l)*x6 +3861*b15/(56*l)*x8$



e6 := -88*k/(15*l)*x1 + 22*b4/(5*l)*x3 - 99*b9/(8*l)*x5 +x6$



e7 := -768*k/(5005*b13)*x1 + 576*b4/(5005*b13)*x3 -
      324*b9/(1001*b13)*x5 + x7 - 16*l/(715*b13)*x8$



e8 := 7*l/(143*b15)*x1 + 49*b6/(429*b15)*x4 - 21*b11/(65*b15)*x6 +
      x8 - 7*b2/(143*b15)$



solve({e1,e2,e3,e4,e5,e6,e7,e8},{x1,x2,x3,x4,x5,x6,x7,x8});


       L
{{X1=-----,
      3*K

  X2=1,

                   2
      5*(3*K*B2 - L )
  X3=-----------------,
           6*K*L

                 2                    2
      7*( - 8*K*L  + 45*K*B4*B2 - 15*L *B4)
  X4=---------------------------------------,
                           2
                     36*K*L

                2             2                            4
  X5=( - 392*K*L *B6 - 108*K*L *B2 + 2205*K*B6*B4*B2 + 36*L

              2               3
       - 735*L *B6*B4)/(32*K*L ),

                  4             2                  2
  X6=(11*(2048*K*L  - 158760*K*L *B6*B9 - 11520*K*L *B4*B2

                      2                                      4
           - 43740*K*L *B9*B2 + 893025*K*B6*B4*B9*B2 + 3840*L *B4

                    4              2                      4
           + 14580*L *B9 - 297675*L *B6*B4*B9))/(11520*K*L ),

                  4                   4                    4
  X7=(30732800*K*L *B6 + 109283328*K*L *B11 + 395366400*K*L *B15

                    4                    2
       + 8467200*K*L *B2 - 8471592360*K*L *B6*B11*B9

                        2                          2
       - 30648618000*K*L *B6*B15*B9 - 172872000*K*L *B6*B4*B2

                      2                           2
       - 614718720*K*L *B11*B4*B2 - 2334010140*K*L *B11*B9*B2

                       2                           2
       - 2223936000*K*L *B15*B4*B2 - 8444007000*K*L *B15*B9*B2

       + 47652707025*K*B6*B11*B4*B9*B2

                                                   6
       + 172398476250*K*B6*B15*B4*B9*B2 - 2822400*L

                   4                    4
       + 57624000*L *B6*B4 + 204906240*L *B11*B4

                    4                     4
       + 778003380*L *B11*B9 + 741312000*L *B15*B4

                     4                       2
       + 2814669000*L *B15*B9 - 15884235675*L *B6*B11*B4*B9

                      2                              3
       - 57466158750*L *B6*B15*B4*B9)/(7729722000*K*L *B15*B13),

                   4                 4                 4
  X8=(7*(627200*K*L *B6 + 2230272*K*L *B11 + 172800*K*L *B2

                         2                        2
          - 172889640*K*L *B6*B11*B9 - 3528000*K*L *B6*B4*B2

                        2                         2
          - 12545280*K*L *B11*B4*B2 - 47632860*K*L *B11*B9*B2

                                                 6            4
          + 972504225*K*B6*B11*B4*B9*B2 - 57600*L  + 1176000*L *B6*B4

                     4                    4
          + 4181760*L *B11*B4 + 15877620*L *B11*B9

                       2                             4
          - 324168075*L *B6*B11*B4*B9))/(24710400*K*L *B15)}}



f1 := x1 - x*x2 - y*x3 + 1/2*x**2*x4 + x*y*x5 + 1/2*y**2*x6 +
      1/6*x**3*x7 + 1/2*x*y*(x - y)*x8 - 1/6*y**3*x9$



f2 := x1 - y*x3 + 1/2*y**2*x6 - 1/6*y**3*x9$



f3 := x1 + y*x2 - y*x3 + 1/2*y**2*x4 - y**2*x5 + 1/2*y**2*x6 +
      1/6*y**3*x7 + 1/2*y**3*x8 - 1/6*y**3*x9$



f4 := x1 + (1 - x)*x2 - x*x3 + 1/2*(1 - x)**2*x4 - y*(1 - x)*x5 +
      1/2*y**2*x6 + 1/6*(1 - x)**3*x7 + 1/2*y*(1 - x - y)*(1 - x)*x8
      - 1/6*y**3*x9$



f5 := x1 + (1 - x - y)*x2 + 1/2*(1 - x - y)**2*x4 +
      1/6*(1 - x - y)**3*x7$



f6 := x1 + (1 - x - y)*x3 + 1/2*(1 - x - y)*x6 +
      1/6*(1 - x - y)**3*x9$



f7 := x1 - x*x2 + (1 - y)*x3 + 1/2*x*x4 - x*(1 - y)*x5 +
      1/2*(1 - y)**2*x6 - 1/6*x**3*x7 + 1/2*x*(1 - y)*(1 - y + x)*x8
      + 1/6*(1-y)**3*x9$



f8 := x1 - x*x2 + x*x3 + 1/2*x**2*x4 - x**2*x5 + 1/2*x**2*x6 +
      1/6*x**3*x7 - 1/2*x**3*x8 + 1/6*x**3*x9$



f9 := x1 - x*x2 + 1/2*x**2*x4 + 1/6*x**3*x7$



solve({f1,f2,f3,f4,f5,f6,f7,f8,f9},{x1,x2,x3,x4,x5,x6,x7,x8,x9});


{{X1=0,X2=0,X3=0,X4=0,X5=0,X6=0,X7=0,X8=0,X9=0}}


solve({f1 - 1,f2,f3,f4,f5,f6,f7,f8,f9},{x1,x2,x3,x4,x5,x6,x7,x8,x9});


               9         9      8  3       8  2        8         8
{{X1=(Y*( - 8*X *Y + 10*X  + 9*X *Y  - 57*X *Y  + 103*X *Y - 53*X

                7  4        7  3        7  2        7          7
          + 32*X *Y  - 186*X *Y  + 400*X *Y  - 374*X *Y + 120*X

                6  5        6  4        6  3         6  2        6
          + 43*X *Y  - 296*X *Y  + 777*X *Y  - 1024*X *Y  + 652*X *Y

                 6       5  6        5  5        5  4         5  3
          - 152*X  + 29*X *Y  - 249*X *Y  + 804*X *Y  - 1364*X *Y

                  5  2        5          5       4  7        4  6
          + 1305*X *Y  - 637*X *Y + 118*X  + 12*X *Y  - 116*X *Y

                 4  5        4  4         4  3        4  2        4
          + 457*X *Y  - 941*X *Y  + 1178*X *Y  - 898*X *Y  + 363*X *Y

                4    3  8       3  7       3  6        3  5
          - 57*X  + X *Y  - 13*X *Y  + 95*X *Y  - 270*X *Y

                 3  4        3  3        3  2        3         3
          + 431*X *Y  - 463*X *Y  + 319*X *Y  - 116*X *Y + 16*X

               2  9       2  8       2  7       2  6       2  5
          - 4*X *Y  + 25*X *Y  - 62*X *Y  + 89*X *Y  - 90*X *Y

                2  4       2  3       2  2       2        2        10
          + 46*X *Y  + 24*X *Y  - 44*X *Y  + 18*X *Y - 2*X  - 2*X*Y

                  9         8         7          6          5
          + 12*X*Y  - 34*X*Y  + 65*X*Y  - 100*X*Y  + 117*X*Y

                  4         3        2            7      6       5
          - 86*X*Y  + 31*X*Y  - 2*X*Y  - X*Y - 2*Y  + 9*Y  - 16*Y

                4      3    2       11        11       10  2
          + 14*Y  - 6*Y  + Y ))/(2*X  *Y - 4*X   + 10*X  *Y

               10         10      9  3       9  2       9         9
         - 30*X  *Y + 24*X   + 9*X *Y  - 49*X *Y  + 91*X *Y - 51*X

               8  4       8  3       8  2       8         8
         - 23*X *Y  + 74*X *Y  - 41*X *Y  - 60*X *Y + 46*X

               7  5        7  4        7  3        7  2        7
         - 52*X *Y  + 288*X *Y  - 547*X *Y  + 431*X *Y  - 107*X *Y

               7       6  6        6  5        6  4         6  3
         - 11*X  - 42*X *Y  + 303*X *Y  - 812*X *Y  + 1059*X *Y

                6  2        6        6      5  7       5  6
         - 690*X *Y  + 191*X *Y - 9*X  - 8*X *Y  + 82*X *Y

                5  5        5  4        5  3        5  2        5
         - 379*X *Y  + 781*X *Y  - 828*X *Y  + 458*X *Y  - 112*X *Y

              5       4  8        4  7        4  6        4  5
         + 6*X  + 26*X *Y  - 159*X *Y  + 293*X *Y  - 161*X *Y

                4  4        4  3        4  2       4      4
         - 122*X *Y  + 225*X *Y  - 128*X *Y  + 27*X *Y - X

               3  9        3  8        3  7        3  6        3  5
         + 33*X *Y  - 224*X *Y  + 590*X *Y  - 775*X *Y  + 558*X *Y

                3  4       3  3      3  2      3         2  10
         - 224*X *Y  + 37*X *Y  + 7*X *Y  - 2*X *Y + 17*X *Y

                2  9        2  8        2  7        2  6        2  5
         - 130*X *Y  + 398*X *Y  - 643*X *Y  + 598*X *Y  - 338*X *Y

                2  4       2  3      2  2        11         10
         + 124*X *Y  - 28*X *Y  + 2*X *Y  + 4*X*Y   - 35*X*Y

                  9          8          7          6         5
         + 119*X*Y  - 217*X*Y  + 236*X*Y  - 157*X*Y  + 65*X*Y

                 4        3      10       9       8       7       6
         - 18*X*Y  + 3*X*Y  + 4*Y   - 16*Y  + 27*Y  - 24*Y  + 11*Y

              5
         - 2*Y ),

         11        11      10  2       10        10      9  3
  X2=(2*X  *Y - 2*X   + 7*X  *Y  - 16*X  *Y + 9*X   - 3*X *Y

            9  2       9        9       8  4       8  3       8  2
       - 8*X *Y  + 21*X *Y - 8*X  - 23*X *Y  + 82*X *Y  - 90*X *Y

             8         8       7  5       7  4        7  3
       + 49*X *Y - 16*X  - 10*X *Y  + 75*X *Y  - 168*X *Y

              7  2        7         7       6  6        6  5
       + 181*X *Y  - 123*X *Y + 37*X  + 28*X *Y  - 161*X *Y

              6  4        6  3       6  2       6         6
       + 306*X *Y  - 272*X *Y  + 66*X *Y  + 61*X *Y - 28*X

             5  7        5  6        5  5         5  4         5  3
       + 52*X *Y  - 381*X *Y  + 995*X *Y  - 1335*X *Y  + 1026*X *Y

              5  2       5        5       4  8        4  7
       - 401*X *Y  + 41*X *Y + 9*X  + 45*X *Y  - 339*X *Y

               4  6         4  5         4  4         4  3
       + 1030*X *Y  - 1662*X *Y  + 1631*X *Y  - 1012*X *Y

              4  2       4      4       3  9        3  8        3  7
       + 356*X *Y  - 50*X *Y - X  + 15*X *Y  - 119*X *Y  + 425*X *Y

              3  6        3  5        3  4        3  3        3  2
       - 817*X *Y  + 956*X *Y  - 757*X *Y  + 410*X *Y  - 130*X *Y

             3        2  10       2  9       2  7       2  6
       + 17*X *Y - 3*X *Y   + 11*X *Y  - 42*X *Y  + 66*X *Y

             2  5       2  4       2  3       2  2      2          11
       - 68*X *Y  + 77*X *Y  - 59*X *Y  + 20*X *Y  - 2*X *Y - 2*X*Y

               10         9         8         7          6         5
       + 12*X*Y   - 32*X*Y  + 56*X*Y  - 84*X*Y  + 103*X*Y  - 80*X*Y

               4        3      2      8      7       6       5      4
       + 30*X*Y  - 2*X*Y  - X*Y  - 2*Y  + 9*Y  - 16*Y  + 14*Y  - 6*Y

          3         11        11       10  2       10         10
       + Y )/(X*(2*X  *Y - 4*X   + 10*X  *Y  - 30*X  *Y + 24*X

                       9  3       9  2       9         9       8  4
                  + 9*X *Y  - 49*X *Y  + 91*X *Y - 51*X  - 23*X *Y

                        8  3       8  2       8         8       7  5
                  + 74*X *Y  - 41*X *Y  - 60*X *Y + 46*X  - 52*X *Y

                         7  4        7  3        7  2        7
                  + 288*X *Y  - 547*X *Y  + 431*X *Y  - 107*X *Y

                        7       6  6        6  5        6  4
                  - 11*X  - 42*X *Y  + 303*X *Y  - 812*X *Y

                          6  3        6  2        6        6
                  + 1059*X *Y  - 690*X *Y  + 191*X *Y - 9*X

                       5  7       5  6        5  5        5  4
                  - 8*X *Y  + 82*X *Y  - 379*X *Y  + 781*X *Y

                         5  3        5  2        5        5
                  - 828*X *Y  + 458*X *Y  - 112*X *Y + 6*X

                        4  8        4  7        4  6        4  5
                  + 26*X *Y  - 159*X *Y  + 293*X *Y  - 161*X *Y

                         4  4        4  3        4  2       4      4
                  - 122*X *Y  + 225*X *Y  - 128*X *Y  + 27*X *Y - X

                        3  9        3  8        3  7        3  6
                  + 33*X *Y  - 224*X *Y  + 590*X *Y  - 775*X *Y

                         3  5        3  4       3  3      3  2
                  + 558*X *Y  - 224*X *Y  + 37*X *Y  + 7*X *Y

                       3         2  10        2  9        2  8
                  - 2*X *Y + 17*X *Y   - 130*X *Y  + 398*X *Y

                         2  7        2  6        2  5        2  4
                  - 643*X *Y  + 598*X *Y  - 338*X *Y  + 124*X *Y

                        2  3      2  2        11         10
                  - 28*X *Y  + 2*X *Y  + 4*X*Y   - 35*X*Y

                           9          8          7          6
                  + 119*X*Y  - 217*X*Y  + 236*X*Y  - 157*X*Y

                          5         4        3      10       9
                  + 65*X*Y  - 18*X*Y  + 3*X*Y  + 4*Y   - 16*Y

                        8       7       6      5
                  + 27*Y  - 24*Y  + 11*Y  - 2*Y )),

         10        10       9  2       9         9       8  3
  X3=(2*X  *Y - 4*X   + 10*X *Y  - 38*X *Y + 30*X  + 17*X *Y

              8  2        8         8       7  4        7  3
       - 110*X *Y  + 189*X *Y - 92*X  + 16*X *Y  - 152*X *Y

              7  2        7          7       6  5        6  4
       + 427*X *Y  - 450*X *Y + 155*X  + 18*X *Y  - 161*X *Y

              6  3        6  2        6          6       5  6
       + 543*X *Y  - 862*X *Y  + 622*X *Y - 162*X  + 24*X *Y

              5  5        5  4        5  3         5  2        5
       - 181*X *Y  + 560*X *Y  - 994*X *Y  + 1023*X *Y  - 542*X *Y

              5       4  7        4  6        4  5        4  4
       + 112*X  + 22*X *Y  - 161*X *Y  + 480*X *Y  - 829*X *Y

              4  3        4  2        4         4       3  8
       + 958*X *Y  - 720*X *Y  + 302*X *Y - 52*X  + 12*X *Y

             3  7        3  6        3  5        3  4        3  3
       - 89*X *Y  + 277*X *Y  - 461*X *Y  + 509*X *Y  - 437*X *Y

              3  2        3         3      2  9       2  8
       + 275*X *Y  - 101*X *Y + 15*X  + 3*X *Y  - 29*X *Y

              2  7        2  6        2  5       2  4       2  3
       + 101*X *Y  - 162*X *Y  + 128*X *Y  - 65*X *Y  + 52*X *Y

             2  2       2        2        9         8         7
       - 43*X *Y  + 17*X *Y - 2*X  - 5*X*Y  + 25*X*Y  - 46*X*Y

               6         5         4         3      2            9
       + 27*X*Y  + 23*X*Y  - 40*X*Y  + 18*X*Y  - X*Y  - X*Y + 2*Y

            8       7      6      5       4      3    2      11
       - 8*Y  + 12*Y  - 5*Y  - 7*Y  + 10*Y  - 5*Y  + Y )/(2*X  *Y

              11       10  2       10         10      9  3       9  2
         - 4*X   + 10*X  *Y  - 30*X  *Y + 24*X   + 9*X *Y  - 49*X *Y

               9         9       8  4       8  3       8  2       8
         + 91*X *Y - 51*X  - 23*X *Y  + 74*X *Y  - 41*X *Y  - 60*X *Y

               8       7  5        7  4        7  3        7  2
         + 46*X  - 52*X *Y  + 288*X *Y  - 547*X *Y  + 431*X *Y

                7         7       6  6        6  5        6  4
         - 107*X *Y - 11*X  - 42*X *Y  + 303*X *Y  - 812*X *Y

                 6  3        6  2        6        6      5  7
         + 1059*X *Y  - 690*X *Y  + 191*X *Y - 9*X  - 8*X *Y

               5  6        5  5        5  4        5  3        5  2
         + 82*X *Y  - 379*X *Y  + 781*X *Y  - 828*X *Y  + 458*X *Y

                5        5       4  8        4  7        4  6
         - 112*X *Y + 6*X  + 26*X *Y  - 159*X *Y  + 293*X *Y

                4  5        4  4        4  3        4  2       4
         - 161*X *Y  - 122*X *Y  + 225*X *Y  - 128*X *Y  + 27*X *Y

            4       3  9        3  8        3  7        3  6
         - X  + 33*X *Y  - 224*X *Y  + 590*X *Y  - 775*X *Y

                3  5        3  4       3  3      3  2      3
         + 558*X *Y  - 224*X *Y  + 37*X *Y  + 7*X *Y  - 2*X *Y

               2  10        2  9        2  8        2  7        2  6
         + 17*X *Y   - 130*X *Y  + 398*X *Y  - 643*X *Y  + 598*X *Y

                2  5        2  4       2  3      2  2        11
         - 338*X *Y  + 124*X *Y  - 28*X *Y  + 2*X *Y  + 4*X*Y

                 10          9          8          7          6
         - 35*X*Y   + 119*X*Y  - 217*X*Y  + 236*X*Y  - 157*X*Y

                 5         4        3      10       9       8       7
         + 65*X*Y  - 18*X*Y  + 3*X*Y  + 4*Y   - 16*Y  + 27*Y  - 24*Y

               6      5
         + 11*Y  - 2*Y ),

            10        10      9  2       9        9      8  3
  X4=(2*(2*X  *Y - 2*X   + 6*X *Y  - 14*X *Y + 8*X  - 5*X *Y

               8  2      8        8       7  4        7  3
          + 7*X *Y  + 3*X *Y - 5*X  - 27*X *Y  + 118*X *Y

                 7  2       7         7       6  5        6  4
          - 164*X *Y  + 88*X *Y - 15*X  - 28*X *Y  + 180*X *Y

                 6  3        6  2        6         6      5  6
          - 398*X *Y  + 395*X *Y  - 178*X *Y + 29*X  - 6*X *Y

                5  5        5  4        5  3        5  2        5
          + 66*X *Y  - 274*X *Y  + 476*X *Y  - 392*X *Y  + 152*X *Y

                5       4  7        4  6        4  5       4  4
          - 22*X  + 20*X *Y  - 118*X *Y  + 186*X *Y  - 30*X *Y

                 4  3        4  2       4        4       3  8
          - 172*X *Y  + 166*X *Y  - 60*X *Y + 8*X  + 26*X *Y

                 3  7        3  6        3  5        3  4       3  3
          - 174*X *Y  + 448*X *Y  - 562*X *Y  + 353*X *Y  - 92*X *Y

               3  2      3      3       2  9       2  8        2  7
          - 4*X *Y  + 6*X *Y - X  + 11*X *Y  - 85*X *Y  + 271*X *Y

                 2  6        2  5        2  4       2  3       2  2
          - 455*X *Y  + 437*X *Y  - 245*X *Y  + 78*X *Y  - 13*X *Y

             2        10         9         8          7          6
          + X *Y + X*Y   - 14*X*Y  + 60*X*Y  - 124*X*Y  + 145*X*Y

                   5         4         3        2      9      8
          - 104*X*Y  + 48*X*Y  - 14*X*Y  + 2*X*Y  + 2*Y  - 9*Y

                7       6      5    4          11        11
          + 16*Y  - 14*Y  + 6*Y  - Y ))/(X*(2*X  *Y - 4*X

                  10  2       10         10      9  3       9  2
            + 10*X  *Y  - 30*X  *Y + 24*X   + 9*X *Y  - 49*X *Y

                  9         9       8  4       8  3       8  2
            + 91*X *Y - 51*X  - 23*X *Y  + 74*X *Y  - 41*X *Y

                  8         8       7  5        7  4        7  3
            - 60*X *Y + 46*X  - 52*X *Y  + 288*X *Y  - 547*X *Y

                   7  2        7         7       6  6        6  5
            + 431*X *Y  - 107*X *Y - 11*X  - 42*X *Y  + 303*X *Y

                   6  4         6  3        6  2        6        6
            - 812*X *Y  + 1059*X *Y  - 690*X *Y  + 191*X *Y - 9*X

                 5  7       5  6        5  5        5  4        5  3
            - 8*X *Y  + 82*X *Y  - 379*X *Y  + 781*X *Y  - 828*X *Y

                   5  2        5        5       4  8        4  7
            + 458*X *Y  - 112*X *Y + 6*X  + 26*X *Y  - 159*X *Y

                   4  6        4  5        4  4        4  3
            + 293*X *Y  - 161*X *Y  - 122*X *Y  + 225*X *Y

                   4  2       4      4       3  9        3  8
            - 128*X *Y  + 27*X *Y - X  + 33*X *Y  - 224*X *Y

                   3  7        3  6        3  5        3  4
            + 590*X *Y  - 775*X *Y  + 558*X *Y  - 224*X *Y

                  3  3      3  2      3         2  10        2  9
            + 37*X *Y  + 7*X *Y  - 2*X *Y + 17*X *Y   - 130*X *Y

                   2  8        2  7        2  6        2  5
            + 398*X *Y  - 643*X *Y  + 598*X *Y  - 338*X *Y

                   2  4       2  3      2  2        11         10
            + 124*X *Y  - 28*X *Y  + 2*X *Y  + 4*X*Y   - 35*X*Y

                     9          8          7          6         5
            + 119*X*Y  - 217*X*Y  + 236*X*Y  - 157*X*Y  + 65*X*Y

                    4        3      10       9       8       7
            - 18*X*Y  + 3*X*Y  + 4*Y   - 16*Y  + 27*Y  - 24*Y

                  6      5
            + 11*Y  - 2*Y )),

         11        11      10  2       10        10      9  3
  X5=(2*X  *Y - 2*X   + 9*X  *Y  - 20*X  *Y + 9*X   + 4*X *Y

             9  2       9        9       8  4       8  3       8  2
       - 34*X *Y  + 44*X *Y - 8*X  - 21*X *Y  + 52*X *Y  - 14*X *Y

          8         8       7  5        7  4        7  3        7  2
       + X *Y - 16*X  - 22*X *Y  + 116*X *Y  - 176*X *Y  + 109*X *Y

             7         7       6  6       6  5       6  4       6  3
       - 76*X *Y + 37*X  + 10*X *Y  - 44*X *Y  + 56*X *Y  - 72*X *Y

             6  2       6         6       5  7        5  6
       + 38*X *Y  + 42*X *Y - 28*X  + 38*X *Y  - 267*X *Y

              5  5        5  4        5  3        5  2       5
       + 637*X *Y  - 801*X *Y  + 644*X *Y  - 292*X *Y  + 38*X *Y

            5       4  8        4  7        4  6         4  5
       + 9*X  + 45*X *Y  - 321*X *Y  + 893*X *Y  - 1300*X *Y

               4  4        4  3        4  2       4      4       3  9
       + 1173*X *Y  - 723*X *Y  + 277*X *Y  - 45*X *Y - X  + 26*X *Y

              3  8        3  7         3  6        3  5        3  4
       - 194*X *Y  + 609*X *Y  - 1006*X *Y  + 993*X *Y  - 667*X *Y

              3  3        3  2       3        2  10       2  9
       + 334*X *Y  - 111*X *Y  + 16*X *Y + 5*X *Y   - 53*X *Y

              2  8        2  7        2  6        2  5        2  4
       + 203*X *Y  - 381*X *Y  + 390*X *Y  - 243*X *Y  + 119*X *Y

             2  3       2  2      2          10         9         8
       - 57*X *Y  + 19*X *Y  - 2*X *Y - 5*X*Y   + 27*X*Y  - 58*X*Y

               7        6         5         4      3      2      10
       + 54*X*Y  - 7*X*Y  - 23*X*Y  + 14*X*Y  - X*Y  - X*Y  + 2*Y

            9       8      7      6       5      4    3
       - 8*Y  + 12*Y  - 5*Y  - 7*Y  + 10*Y  - 5*Y  + Y )/(X*Y*(

              11        11       10  2       10         10      9  3
           2*X  *Y - 4*X   + 10*X  *Y  - 30*X  *Y + 24*X   + 9*X *Y

                  9  2       9         9       8  4       8  3
            - 49*X *Y  + 91*X *Y - 51*X  - 23*X *Y  + 74*X *Y

                  8  2       8         8       7  5        7  4
            - 41*X *Y  - 60*X *Y + 46*X  - 52*X *Y  + 288*X *Y

                   7  3        7  2        7         7       6  6
            - 547*X *Y  + 431*X *Y  - 107*X *Y - 11*X  - 42*X *Y

                   6  5        6  4         6  3        6  2
            + 303*X *Y  - 812*X *Y  + 1059*X *Y  - 690*X *Y

                   6        6      5  7       5  6        5  5
            + 191*X *Y - 9*X  - 8*X *Y  + 82*X *Y  - 379*X *Y

                   5  4        5  3        5  2        5        5
            + 781*X *Y  - 828*X *Y  + 458*X *Y  - 112*X *Y + 6*X

                  4  8        4  7        4  6        4  5
            + 26*X *Y  - 159*X *Y  + 293*X *Y  - 161*X *Y

                   4  4        4  3        4  2       4      4
            - 122*X *Y  + 225*X *Y  - 128*X *Y  + 27*X *Y - X

                  3  9        3  8        3  7        3  6
            + 33*X *Y  - 224*X *Y  + 590*X *Y  - 775*X *Y

                   3  5        3  4       3  3      3  2      3
            + 558*X *Y  - 224*X *Y  + 37*X *Y  + 7*X *Y  - 2*X *Y

                  2  10        2  9        2  8        2  7
            + 17*X *Y   - 130*X *Y  + 398*X *Y  - 643*X *Y

                   2  6        2  5        2  4       2  3      2  2
            + 598*X *Y  - 338*X *Y  + 124*X *Y  - 28*X *Y  + 2*X *Y

                   11         10          9          8          7
            + 4*X*Y   - 35*X*Y   + 119*X*Y  - 217*X*Y  + 236*X*Y

                     6         5         4        3      10       9
            - 157*X*Y  + 65*X*Y  - 18*X*Y  + 3*X*Y  + 4*Y   - 16*Y

                  8       7       6      5
            + 27*Y  - 24*Y  + 11*Y  - 2*Y )),

            10        10       9  2       9         9      8  3
  X6=(2*(2*X  *Y - 4*X   + 10*X *Y  - 30*X *Y + 20*X  + 6*X *Y

                8  2       8         8       7  4       7  3
          - 51*X *Y  + 90*X *Y - 39*X  - 22*X *Y  + 54*X *Y

                7  2       7         7       6  5        6  4
          + 23*X *Y  - 88*X *Y + 35*X  - 28*X *Y  + 173*X *Y

                 6  3        6  2       6         6       5  5
          - 312*X *Y  + 186*X *Y  - 19*X *Y - 10*X  + 79*X *Y

                 5  4        5  3        5  2       5        5
          - 358*X *Y  + 534*X *Y  - 343*X *Y  + 94*X *Y - 6*X

               4  7       4  6       4  5        4  4        4  3
          + 8*X *Y  - 40*X *Y  - 20*X *Y  + 266*X *Y  - 386*X *Y

                 4  2       4        4      3  8       3  7
          + 237*X *Y  - 64*X *Y + 5*X  + 6*X *Y  - 55*X *Y

                 3  6        3  5      3  4        3  3       3  2
          + 153*X *Y  - 148*X *Y  - 7*X *Y  + 101*X *Y  - 67*X *Y

                3      3      2  9       2  8        2  7        2  6
          + 16*X *Y - X  + 6*X *Y  - 46*X *Y  + 154*X *Y  - 260*X *Y

                 2  5        2  4       2  3      2  2    2
          + 231*X *Y  - 103*X *Y  + 15*X *Y  + 4*X *Y  - X *Y

                 9         8          7          6          5
          - 6*X*Y  + 44*X*Y  - 121*X*Y  + 167*X*Y  - 129*X*Y

                  4         3      2      8       7       6       5
          + 57*X*Y  - 13*X*Y  + X*Y  - 4*Y  + 14*Y  - 20*Y  + 15*Y

               4    3          11        11       10  2       10
          - 6*Y  + Y ))/(Y*(2*X  *Y - 4*X   + 10*X  *Y  - 30*X  *Y

                  10      9  3       9  2       9         9
            + 24*X   + 9*X *Y  - 49*X *Y  + 91*X *Y - 51*X

                  8  4       8  3       8  2       8         8
            - 23*X *Y  + 74*X *Y  - 41*X *Y  - 60*X *Y + 46*X

                  7  5        7  4        7  3        7  2        7
            - 52*X *Y  + 288*X *Y  - 547*X *Y  + 431*X *Y  - 107*X *Y

                  7       6  6        6  5        6  4         6  3
            - 11*X  - 42*X *Y  + 303*X *Y  - 812*X *Y  + 1059*X *Y

                   6  2        6        6      5  7       5  6
            - 690*X *Y  + 191*X *Y - 9*X  - 8*X *Y  + 82*X *Y

                   5  5        5  4        5  3        5  2
            - 379*X *Y  + 781*X *Y  - 828*X *Y  + 458*X *Y

                   5        5       4  8        4  7        4  6
            - 112*X *Y + 6*X  + 26*X *Y  - 159*X *Y  + 293*X *Y

                   4  5        4  4        4  3        4  2       4
            - 161*X *Y  - 122*X *Y  + 225*X *Y  - 128*X *Y  + 27*X *Y

               4       3  9        3  8        3  7        3  6
            - X  + 33*X *Y  - 224*X *Y  + 590*X *Y  - 775*X *Y

                   3  5        3  4       3  3      3  2      3
            + 558*X *Y  - 224*X *Y  + 37*X *Y  + 7*X *Y  - 2*X *Y

                  2  10        2  9        2  8        2  7
            + 17*X *Y   - 130*X *Y  + 398*X *Y  - 643*X *Y

                   2  6        2  5        2  4       2  3      2  2
            + 598*X *Y  - 338*X *Y  + 124*X *Y  - 28*X *Y  + 2*X *Y

                   11         10          9          8          7
            + 4*X*Y   - 35*X*Y   + 119*X*Y  - 217*X*Y  + 236*X*Y

                     6         5         4        3      10       9
            - 157*X*Y  + 65*X*Y  - 18*X*Y  + 3*X*Y  + 4*Y   - 16*Y

                  8       7       6      5
            + 27*Y  - 24*Y  + 11*Y  - 2*Y )),

          8  2      8      8      7  3      7  2      7        7
  X7=(6*(X *Y  - 2*X *Y + X  + 2*X *Y  - 7*X *Y  + 8*X *Y - 3*X

               6  4       6  3       6  2       6      6       5  5
          - 5*X *Y  + 21*X *Y  - 29*X *Y  + 14*X *Y - X  - 14*X *Y

                5  4        5  3        5  2       5        5
          + 81*X *Y  - 170*X *Y  + 160*X *Y  - 65*X *Y + 8*X

               4  6       4  5        4  4        4  3        4  2
          - 9*X *Y  + 69*X *Y  - 197*X *Y  + 276*X *Y  - 194*X *Y

                4        4      3  7       3  6      3  5       3  4
          + 61*X *Y - 6*X  + 3*X *Y  - 14*X *Y  + 5*X *Y  + 59*X *Y

                 3  3       3  2       3      3      2  8       2  7
          - 107*X *Y  + 70*X *Y  - 17*X *Y + X  + 7*X *Y  - 49*X *Y

                 2  6        2  5        2  4       2  3      2  2
          + 125*X *Y  - 159*X *Y  + 100*X *Y  - 22*X *Y  - 3*X *Y

             2          9         8         7         6         5
          + X *Y + 3*X*Y  - 21*X*Y  + 59*X*Y  - 92*X*Y  + 88*X*Y

                  4         3      2      8      7       6       5
          - 49*X*Y  + 13*X*Y  - X*Y  + 2*Y  - 7*Y  + 11*Y  - 10*Y

               4    3          11        11       10  2       10
          + 5*Y  - Y ))/(X*(2*X  *Y - 4*X   + 10*X  *Y  - 30*X  *Y

                  10      9  3       9  2       9         9
            + 24*X   + 9*X *Y  - 49*X *Y  + 91*X *Y - 51*X

                  8  4       8  3       8  2       8         8
            - 23*X *Y  + 74*X *Y  - 41*X *Y  - 60*X *Y + 46*X

                  7  5        7  4        7  3        7  2        7
            - 52*X *Y  + 288*X *Y  - 547*X *Y  + 431*X *Y  - 107*X *Y

                  7       6  6        6  5        6  4         6  3
            - 11*X  - 42*X *Y  + 303*X *Y  - 812*X *Y  + 1059*X *Y

                   6  2        6        6      5  7       5  6
            - 690*X *Y  + 191*X *Y - 9*X  - 8*X *Y  + 82*X *Y

                   5  5        5  4        5  3        5  2
            - 379*X *Y  + 781*X *Y  - 828*X *Y  + 458*X *Y

                   5        5       4  8        4  7        4  6
            - 112*X *Y + 6*X  + 26*X *Y  - 159*X *Y  + 293*X *Y

                   4  5        4  4        4  3        4  2       4
            - 161*X *Y  - 122*X *Y  + 225*X *Y  - 128*X *Y  + 27*X *Y

               4       3  9        3  8        3  7        3  6
            - X  + 33*X *Y  - 224*X *Y  + 590*X *Y  - 775*X *Y

                   3  5        3  4       3  3      3  2      3
            + 558*X *Y  - 224*X *Y  + 37*X *Y  + 7*X *Y  - 2*X *Y

                  2  10        2  9        2  8        2  7
            + 17*X *Y   - 130*X *Y  + 398*X *Y  - 643*X *Y

                   2  6        2  5        2  4       2  3      2  2
            + 598*X *Y  - 338*X *Y  + 124*X *Y  - 28*X *Y  + 2*X *Y

                   11         10          9          8          7
            + 4*X*Y   - 35*X*Y   + 119*X*Y  - 217*X*Y  + 236*X*Y

                     6         5         4        3      10       9
            - 157*X*Y  + 65*X*Y  - 18*X*Y  + 3*X*Y  + 4*Y   - 16*Y

                  8       7       6      5
            + 27*Y  - 24*Y  + 11*Y  - 2*Y )),

               10    9  2       9         9      8  3       8  2
  X8=(2*( - 2*X   + X *Y  - 12*X *Y + 15*X  + 6*X *Y  - 35*X *Y

                8         8       7  4       7  3        7  2
          + 72*X *Y - 43*X  + 13*X *Y  - 70*X *Y  + 148*X *Y

                 7         7       6  5       6  4        6  3
          - 157*X *Y + 62*X  + 15*X *Y  - 84*X *Y  + 177*X *Y

                 6  2        6         6      5  6       5  5
          - 209*X *Y  + 151*X *Y - 48*X  + 6*X *Y  - 33*X *Y

                5  4        5  3       5  2       5         5
          + 86*X *Y  - 102*X *Y  + 75*X *Y  - 51*X *Y + 19*X

                4  7       4  6        4  5        4  4       4  3
          - 11*X *Y  + 67*X *Y  - 126*X *Y  + 116*X *Y  - 92*X *Y

                4  2       4        4       3  8        3  7
          + 62*X *Y  - 13*X *Y - 3*X  - 18*X *Y  + 113*X *Y

                 3  6        3  5        3  4        3  3       3  2
          - 269*X *Y  + 314*X *Y  - 209*X *Y  + 112*X *Y  - 55*X *Y

                3         2  9       2  8        2  7        2  6
          + 12*X *Y - 10*X *Y  + 70*X *Y  - 193*X *Y  + 275*X *Y

                 2  5       2  4       2  3       2  2      2
          - 213*X *Y  + 92*X *Y  - 33*X *Y  + 14*X *Y  - 2*X *Y

                 10         9         8          7         6
          - 2*X*Y   + 18*X*Y  - 60*X*Y  + 102*X*Y  - 95*X*Y

                  5        4      3      2      9      8       7
          + 43*X*Y  - 4*X*Y  - X*Y  - X*Y  - 2*Y  + 8*Y  - 13*Y

                6      5      4    3            11        11
          + 10*Y  - 2*Y  - 2*Y  + Y ))/(X*Y*(2*X  *Y - 4*X

                  10  2       10         10      9  3       9  2
            + 10*X  *Y  - 30*X  *Y + 24*X   + 9*X *Y  - 49*X *Y

                  9         9       8  4       8  3       8  2
            + 91*X *Y - 51*X  - 23*X *Y  + 74*X *Y  - 41*X *Y

                  8         8       7  5        7  4        7  3
            - 60*X *Y + 46*X  - 52*X *Y  + 288*X *Y  - 547*X *Y

                   7  2        7         7       6  6        6  5
            + 431*X *Y  - 107*X *Y - 11*X  - 42*X *Y  + 303*X *Y

                   6  4         6  3        6  2        6        6
            - 812*X *Y  + 1059*X *Y  - 690*X *Y  + 191*X *Y - 9*X

                 5  7       5  6        5  5        5  4        5  3
            - 8*X *Y  + 82*X *Y  - 379*X *Y  + 781*X *Y  - 828*X *Y

                   5  2        5        5       4  8        4  7
            + 458*X *Y  - 112*X *Y + 6*X  + 26*X *Y  - 159*X *Y

                   4  6        4  5        4  4        4  3
            + 293*X *Y  - 161*X *Y  - 122*X *Y  + 225*X *Y

                   4  2       4      4       3  9        3  8
            - 128*X *Y  + 27*X *Y - X  + 33*X *Y  - 224*X *Y

                   3  7        3  6        3  5        3  4
            + 590*X *Y  - 775*X *Y  + 558*X *Y  - 224*X *Y

                  3  3      3  2      3         2  10        2  9
            + 37*X *Y  + 7*X *Y  - 2*X *Y + 17*X *Y   - 130*X *Y

                   2  8        2  7        2  6        2  5
            + 398*X *Y  - 643*X *Y  + 598*X *Y  - 338*X *Y

                   2  4       2  3      2  2        11         10
            + 124*X *Y  - 28*X *Y  + 2*X *Y  + 4*X*Y   - 35*X*Y

                     9          8          7          6         5
            + 119*X*Y  - 217*X*Y  + 236*X*Y  - 157*X*Y  + 65*X*Y

                    4        3      10       9       8       7
            - 18*X*Y  + 3*X*Y  + 4*Y   - 16*Y  + 27*Y  - 24*Y

                  6      5
            + 11*Y  - 2*Y )),

               8  2      8        8      7  3       7  2      7
  X9=(6*( - 2*X *Y  + 2*X *Y + 4*X  - 6*X *Y  + 20*X *Y  - 4*X *Y

                7      6  4       6  3       6  2       6         6
          - 12*X  - 3*X *Y  + 38*X *Y  - 78*X *Y  + 24*X *Y + 11*X

               5  5       5  4        5  3        5  2       5      5
          + 5*X *Y  + 11*X *Y  - 114*X *Y  + 164*X *Y  - 61*X *Y - X

               4  6      4  5       4  4        4  3        4  2
          - 2*X *Y  + 5*X *Y  - 43*X *Y  + 154*X *Y  - 166*X *Y

                4        4      3  7       3  6       3  5       3  4
          + 59*X *Y - 3*X  - 5*X *Y  + 21*X *Y  - 29*X *Y  + 43*X *Y

                3  3       3  2       3      3    2  8      2  7
          - 85*X *Y  + 75*X *Y  - 23*X *Y + X  - X *Y  + 8*X *Y

               2  6      2  5       2  4      2  3       2  2
          - 9*X *Y  - 9*X *Y  + 13*X *Y  + 8*X *Y  - 13*X *Y

               2          9         8         7         6         5
          + 3*X *Y - 2*X*Y  + 11*X*Y  - 15*X*Y  - 10*X*Y  + 40*X*Y

                  4         3      8      7      5      4      3
          - 35*X*Y  + 11*X*Y  - 2*Y  + 4*Y  - 6*Y  + 6*Y  - 2*Y ))/(Y

             11        11       10  2       10         10      9  3
        *(2*X  *Y - 4*X   + 10*X  *Y  - 30*X  *Y + 24*X   + 9*X *Y

                 9  2       9         9       8  4       8  3
           - 49*X *Y  + 91*X *Y - 51*X  - 23*X *Y  + 74*X *Y

                 8  2       8         8       7  5        7  4
           - 41*X *Y  - 60*X *Y + 46*X  - 52*X *Y  + 288*X *Y

                  7  3        7  2        7         7       6  6
           - 547*X *Y  + 431*X *Y  - 107*X *Y - 11*X  - 42*X *Y

                  6  5        6  4         6  3        6  2
           + 303*X *Y  - 812*X *Y  + 1059*X *Y  - 690*X *Y

                  6        6      5  7       5  6        5  5
           + 191*X *Y - 9*X  - 8*X *Y  + 82*X *Y  - 379*X *Y

                  5  4        5  3        5  2        5        5
           + 781*X *Y  - 828*X *Y  + 458*X *Y  - 112*X *Y + 6*X

                 4  8        4  7        4  6        4  5        4  4
           + 26*X *Y  - 159*X *Y  + 293*X *Y  - 161*X *Y  - 122*X *Y

                  4  3        4  2       4      4       3  9
           + 225*X *Y  - 128*X *Y  + 27*X *Y - X  + 33*X *Y

                  3  8        3  7        3  6        3  5
           - 224*X *Y  + 590*X *Y  - 775*X *Y  + 558*X *Y

                  3  4       3  3      3  2      3         2  10
           - 224*X *Y  + 37*X *Y  + 7*X *Y  - 2*X *Y + 17*X *Y

                  2  9        2  8        2  7        2  6
           - 130*X *Y  + 398*X *Y  - 643*X *Y  + 598*X *Y

                  2  5        2  4       2  3      2  2        11
           - 338*X *Y  + 124*X *Y  - 28*X *Y  + 2*X *Y  + 4*X*Y

                   10          9          8          7          6
           - 35*X*Y   + 119*X*Y  - 217*X*Y  + 236*X*Y  - 157*X*Y

                   5         4        3      10       9       8
           + 65*X*Y  - 18*X*Y  + 3*X*Y  + 4*Y   - 16*Y  + 27*Y

                 7       6      5
           - 24*Y  + 11*Y  - 2*Y ))}}



% The following examples were discussed in Char, B.W., Fee, G.J.,
% Geddes, K.O., Gonnet, G.H., Monagan, M.B., Watt, S.M., "On the
% Design and Performance of the Maple System", Proc. 1984 Macsyma
% Users' Conference, G.E., Schenectady, NY, 1984, 199-219.

% Problem 1.

solve({ -22319*x0+25032*x1-83247*x2+67973*x3+54189*x4
       -67793*x5+81135*x6+22293*x7+27327*x8+96599*x9-15144,
       79815*x0+37299*x1-28495*x2-52463*x3+25708*x4 -55333*x5-
       2742*x6+83127*x7-29417*x8-43202*x9+93314, -29065*x0-77803*x1-
       49717*x2-64748*x3-68324*x4 -50162*x5-64222*x6-
       4716*x7+30737*x8+22971*x9+90348, 62470*x0+59658*x1-
       46120*x2+58376*x3-28208*x4 -74506*x5+28491*x6+21099*x7+29149*x8-
       20387*x9+36254, -98233*x0-26263*x1-63227*x2+34307*x3+92294*x4
       +10148*x5+3192*x6+24044*x7-83764*x8-1121*x9+13871,
       -20427*x0+62666*x1+27330*x2-78670*x3+9036*x4 +56024*x5-4525*x6-
       50589*x7-62127*x8-32846*x9+38466,
       -85609*x0+5424*x1+86992*x2+59651*x3-60859*x4 -55984*x5-
       6061*x6+44417*x7+92421*x8+6701*x9-9459,
       -68255*x0+19652*x1+92650*x2-93032*x3-30191*x4 -31075*x5-
       89060*x6+12150*x7-78089*x8-12462*x9+1027, 55526*x0-
       91202*x1+91329*x2-25919*x3-98215*x4 +30554*x5+913*x6-
       35751*x7+17948*x8-58850*x9+66583, 40612*x0+84364*x1-
       83317*x2+10658*x3+37213*x4 +50489*x5+72040*x6-
       21227*x7+60772*x8+95114*x9-68533});


Unknowns: {X9,X7,X5,X8,X6,X4,X3,X2,X1,X0}

      46816360472823082478331070276129336252954604132203
{{X9=----------------------------------------------------,
      42103927115295499860196979638990637447529454985275

       - 11882862555847887107599498171234654114612212813799
  X7=-------------------------------------------------------,
       42103927115295499860196979638990637447529454985275

      17958909252564152456194678743404876001526265937527
  X5=----------------------------------------------------,
      42103927115295499860196979638990637447529454985275

       - 273286267131634194631661772113331181980867938658
  X8=-----------------------------------------------------,
       8420785423059099972039395927798127489505890997055

       - 50670056205024448621117426699348037457452368820774
  X6=-------------------------------------------------------,
       42103927115295499860196979638990637447529454985275

      25308331428404990886292916036626876985377936966579
  X4=----------------------------------------------------,
      42103927115295499860196979638990637447529454985275

      1645748379263608982132912334741766606871657041427
  X3=---------------------------------------------------,
      1684157084611819994407879185559625497901178199411

      1068462443128238131632235196977352568525519548284
  X2=---------------------------------------------------,
      1684157084611819994407879185559625497901178199411

      459141297061698284317621371232198410031030658042
  X1=---------------------------------------------------,
      1684157084611819994407879185559625497901178199411

      4352444991703786550093529782474564455970663240687
  X0=---------------------------------------------------}}
      8420785423059099972039395927798127489505890997055


solve({ -22319*x0+25032*x1-83247*x2+67973*x3+54189*x4
        -67793*x5+81135*x6+22293*x7+27327*x8+96599*x9-15144,
        79815*x0+37299*x1-28495*x2-52463*x3+25708*x4 -55333*x5-
        2742*x6+83127*x7-29417*x8-43202*x9+93314, -29065*x0-77803*x1-
        49717*x2-64748*x3-68324*x4 -50162*x5-64222*x6-
        4716*x7+30737*x8+22971*x9+90348, 62470*x0+59658*x1-
        46120*x2+58376*x3-28208*x4-74506*x5+28491*x6+21099*x7+29149*x8-
        20387*x9+36254,-98233*x0-26263*x1-63227*x2+34307*x3+92294*x4
        +10148*x5+3192*x6+24044*x7-83764*x8-1121*x9+13871,
        -20427*x0+62666*x1+27330*x2-78670*x3+9036*x4 +56024*x5-4525*x6-
        50589*x7-62127*x8-32846*x9+38466,
        -85609*x0+5424*x1+86992*x2+59651*x3-60859*x4 -55984*x5-
        6061*x6+44417*x7+92421*x8+6701*x9-9459,
        -68255*x0+19652*x1+92650*x2-93032*x3-30191*x4 -31075*x5-
        89060*x6+12150*x7-78089*x8-12462*x9+1027, 55526*x0-
        91202*x1+91329*x2-25919*x3-98215*x4 +30554*x5+913*x6-
        35751*x7+17948*x8-58850*x9+66583, 40612*x0+84364*x1-
        83317*x2+10658*x3+37213*x4 +50489*x5+72040*x6-
        21227*x7+60772*x8+95114*x9-68533});


Unknowns: {X9,X7,X5,X8,X6,X4,X3,X2,X1,X0}

      46816360472823082478331070276129336252954604132203
{{X9=----------------------------------------------------,
      42103927115295499860196979638990637447529454985275

       - 11882862555847887107599498171234654114612212813799
  X7=-------------------------------------------------------,
       42103927115295499860196979638990637447529454985275

      17958909252564152456194678743404876001526265937527
  X5=----------------------------------------------------,
      42103927115295499860196979638990637447529454985275

       - 273286267131634194631661772113331181980867938658
  X8=-----------------------------------------------------,
       8420785423059099972039395927798127489505890997055

       - 50670056205024448621117426699348037457452368820774
  X6=-------------------------------------------------------,
       42103927115295499860196979638990637447529454985275

      25308331428404990886292916036626876985377936966579
  X4=----------------------------------------------------,
      42103927115295499860196979638990637447529454985275

      1645748379263608982132912334741766606871657041427
  X3=---------------------------------------------------,
      1684157084611819994407879185559625497901178199411

      1068462443128238131632235196977352568525519548284
  X2=---------------------------------------------------,
      1684157084611819994407879185559625497901178199411

      459141297061698284317621371232198410031030658042
  X1=---------------------------------------------------,
      1684157084611819994407879185559625497901178199411

      4352444991703786550093529782474564455970663240687
  X0=---------------------------------------------------}}
      8420785423059099972039395927798127489505890997055



% The next two problems give the current routines some trouble and
% have therefore been commented out.

% Problem 2.

comment
solve({ 81*x30-96*x21-45, -36*x4+59*x29+26,
       -59*x26+5*x3-33, -81*x19-92*x23-21*x17-9, -46*x29-
       13*x22+22*x24+83, 47*x4-47*x14-15*x26-40, 83*x30+70*x17+56*x10-
       31, 10*x27-90*x9+52*x21+52, -33*x20-97*x26+20*x6-76,
       97*x16+41*x8-13*x12+66, 16*x16-52*x10-73*x28+49, -28*x1-53*x24-
       x27-67, -22*x26-29*x24+73*x10+8, 88*x18+61*x19-98*x9-55, 99*x28-
       91*x26+26*x21-95, -6*x18+25*x7-77*x2+99, 28*x13-50*x17-52*x14-64,
       -50*x20+26*x11+93*x2+77, -70*x8+74*x19-94*x26+86, -18*x18-2*x16-
       79*x23+91, 36*x26-13*x11-53*x25-5, 10*x7+57*x16-85*x10-14,
       -3*x27+44*x4+52*x22-1, 21*x11+20*x25-30*x4-83, 70*x2-97*x19-
       41*x26-50, -51*x8+95*x12-85*x26+45, 83*x30+41*x12+50*x2+53,
       -4*x26+69*x8-58*x5-95, 59*x27-78*x30-66*x23+16, -10*x20-36*x11-
       60*x1-59});



% Problem 3.
comment
solve({ 115*x40+566*x41-378*x42+11401086415/6899901,
       560*x0-45*x1-506*x2-11143386403/8309444, -621*x1-
       328*x2+384*x3+1041841/64675, -856*x2+54*x3+869*x4-41430291/24700,
       596*x3-608*x4-560*x5-10773384/11075,
       -61*x4+444*x5+924*x6+4185100079/11278780, 67*x5-95*x6-
       682*x7+903866812/6618863, 196*x6+926*x7-930*x8-
       2051864151/2031976, -302*x7-311*x8-890*x9-14210414139/27719792,
       121*x8-781*x9-125*x10-4747129093/39901584, 10*x9+555*x10-
       912*x11+32476047/3471829, -151*x38+732*x39-
       397*x40+327281689/173242, 913*x10-259*x11-982*x12-
       18080663/5014020, 305*x11+9*x12-357*x13+1500752933/1780680,
       179*x12-588*x13+665*x14+8128189/51832, 406*x13+843*x14-
       833*x15+201925713/97774, 107*x14+372*x15+505*x16-
       5161192791/3486415, 720*x15-212*x16+607*x17-31529295571/7197760,
       951*x16-685*x17+148*x18+1034546543/711104, -654*x17-
       899*x18+543*x19+1942961717/1646560,
       -448*x18+673*x19+702*x20+856422818/1286375, 396*x19-
       196*x20+218*x21-4386267866/21303625, -233*x20-796*x21-373*x22-
       85246365829/57545250, 921*x21-368*x22+730*x23-
       93446707622/51330363, -424*x22+378*x23+727*x24-
       6673617931/3477462, -633*x23+565*x24-208*x25+8607636805/4092942,
       971*x24+170*x25-865*x26-25224505/18354, 937*x25+333*x26-463*x27-
       339307103/1025430, 494*x26-8*x27-50*x28+57395804/34695,
       530*x27+631*x28-193*x29-8424597157/680022,
       -435*x28+252*x29+916*x30+196828511/19593, 327*x29+403*x30-
       845*x31+8458823325/5927971, 246*x30+881*x31-
       394*x32+13624765321/156546826, 946*x31+169*x32-43*x33-
       53594199271/126093183, -146*x32+503*x33-
       363*x34+66802797635/15234909, -132*x33-
       686*x34+376*x35+8167530636/902635, -38*x34-188*x35-
       583*x36+1814153743/1124240, 389*x35+562*x36-688*x37-
       12251043951/5513560, -769*x37-474*x38-89*x39-2725415872/1235019,
       -625*x36-122*x37+468*x38+7725682775/4506736,
       839*x39+936*x40+703*x41+1912091857/1000749,
       -314*x41+102*x42+790*x43+7290073150/8132873, -905*x42-
       454*x43+524*x44-10110944527/4538233, 379*x43+518*x44-328*x45-
       2071620692/519645, 284*x44-979*x45+690*x46-915987532/16665,
       198*x45-650*x46-763*x47+548801657/11220, 974*x46+12*x47+410*x48-
       3831097561/51051, -498*x47-135*x48-230*x49-18920705/9282,
       665*x48+156*x49+34*x0-27714736/156585, -519*x49-366*x0-730*x1-
       2958446681/798985});



% Problem 4.

solve({ -b*k8/a+c*k8/a, -b*k11/a+c*k11/a,
       -b*k10/a+c*k10/a+k2,
        -k3-b*k9/a+c*k9/a, -b*k14/a+c*k14/a, -b*k15/a+c*k15/a,
        -b*k18/a+c*k18/a-k2, -b*k17/a+c*k17/a, -b*k16/a+c*k16/a+k4,
        -b*k13/a+c*k13/a-b*k21/a+c*k21/a+b*k5/a-c*k5/a,
        b*k44/a-c*k44/a, -b*k45/a+c*k45/a, -b*k20/a+c*k20/a,
        -b*k44/a+c*k44/a, b*k46/a-c*k46/a,
        b**2*k47/a**2-2*b*c*k47/a**2+c**2*k47/a**2,
        k3, -k4, -b*k12/a+c*k12/a-a*k6/b+c*k6/b,
        -b*k19/a+c*k19/a+a*k7/c-b*k7/c, b*k45/a-c*k45/a,
        -b*k46/a+c*k46/a, -k48+c*k48/a+c*k48/b-c**2*k48/(a*b),
        -k49+b*k49/a+b*k49/c-b**2*k49/(a*c), a*k1/b-c*k1/b,
        a*k4/b-c*k4/b, a*k3/b-c*k3/b+k9, -k10+a*k2/b-c*k2/b,
        a*k7/b-c*k7/b, -k9, k11, b*k12/a-c*k12/a+a*k6/b-c*k6/b,
        a*k15/b-c*k15/b, k10+a*k18/b-c*k18/b,
        -k11+a*k17/b-c*k17/b, a*k16/b-c*k16/b,
        -a*k13/b+c*k13/b+a*k21/b-c*k21/b+a*k5/b-c*k5/b,
        -a*k44/b+c*k44/b, a*k45/b-c*k45/b,
        a*k14/c-b*k14/c+a*k20/b-c*k20/b, a*k44/b-c*k44/b,
        -a*k46/b+c*k46/b, -k47+c*k47/a+c*k47/b-c**2*k47/(a*b),
        a*k19/b-c*k19/b, -a*k45/b+c*k45/b, a*k46/b-c*k46/b,
        a**2*k48/b**2-2*a*c*k48/b**2+c**2*k48/b**2,
        -k49+a*k49/b+a*k49/c-a**2*k49/(b*c), k16, -k17,
        -a*k1/c+b*k1/c, -k16-a*k4/c+b*k4/c, -a*k3/c+b*k3/c,
        k18-a*k2/c+b*k2/c, b*k19/a-c*k19/a-a*k7/c+b*k7/c,
        -a*k6/c+b*k6/c, -a*k8/c+b*k8/c, -a*k11/c+b*k11/c+k17,
        -a*k10/c+b*k10/c-k18, -a*k9/c+b*k9/c,
        -a*k14/c+b*k14/c-a*k20/b+c*k20/b,
        -a*k13/c+b*k13/c+a*k21/c-b*k21/c-a*k5/c+b*k5/c,
        a*k44/c-b*k44/c, -a*k45/c+b*k45/c, -a*k44/c+b*k44/c,
        a*k46/c-b*k46/c, -k47+b*k47/a+b*k47/c-b**2*k47/(a*c),
        -a*k12/c+b*k12/c, a*k45/c-b*k45/c, -a*k46/c+b*k46/c,
        -k48+a*k48/b+a*k48/c-a**2*k48/(b*c),
        a**2*k49/c**2-2*a*b*k49/c**2+b**2*k49/c**2, k8, k11, -k15,
        k10-k18, -k17, k9, -k16, -k29, k14-k32, -k21+k23-k31,
        -k24-k30, -k35, k44, -k45, k36, k13-k23+k39, -k20+k38,
        k25+k37, b*k26/a-c*k26/a-k34+k42, -2*k44, k45, k46,
        b*k47/a-c*k47/a, k41, k44, -k46, -b*k47/a+c*k47/a,
        k12+k24, -k19-k25, -a*k27/b+c*k27/b-k33, k45, -k46,
        -a*k48/b+c*k48/b, a*k28/c-b*k28/c+k40, -k45, k46,
        a*k48/b-c*k48/b, a*k49/c-b*k49/c, -a*k49/c+b*k49/c,
        -k1, -k4, -k3, k15, k18-k2, k17, k16, k22, k25-k7,
        k24+k30, k21+k23-k31, k28, -k44, k45, -k30-k6, k20+k32,
        k27+b*k33/a-c*k33/a, k44, -k46, -b*k47/a+c*k47/a, -k36,
        k31-k39-k5, -k32-k38, k19-k37, k26-a*k34/b+c*k34/b-k42,
        k44, -2*k45, k46, a*k48/b-c*k48/b, a*k35/c-b*k35/c-k41,
        -k44, k46, b*k47/a-c*k47/a, -a*k49/c+b*k49/c, -k40, k45,
        -k46, -a*k48/b+c*k48/b, a*k49/c-b*k49/c, k1, k4, k3, -k8,
        -k11, -k10+k2, -k9, k37+k7, -k14-k38, -k22, -k25-k37, -k24+k6,
        -k13-k23+k39, -k28+b*k40/a-c*k40/a, k44, -k45, -k27, -k44,
        k46, b*k47/a-c*k47/a, k29, k32+k38, k31-k39+k5, -k12+k30,
        k35-a*k41/b+c*k41/b, -k44, k45, -k26+k34+a*k42/c-b*k42/c,
        k44, k45, -2*k46, -b*k47/a+c*k47/a, -a*k48/b+c*k48/b,
        a*k49/c-b*k49/c, k33, -k45, k46, a*k48/b-c*k48/b,
        -a*k49/c+b*k49/c },
       {k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14,
        k15, k16, k17, k18, k19, k20, k21, k22, k23, k24, k25, k26,
        k27, k28, k29, k30, k31, k32, k33, k34, k35, k36, k37, k38,
        k39, k40, k41, k42, k43, k44, k45, k46, k47, k48, k49});


{{K1=0,

  K2=0,

  K3=0,

  K4=0,

  K5=0,

  K6=0,

  K7=0,

  K8=0,

  K9=0,

  K10=0,

  K11=0,

  K12=0,

  K13=0,

  K14=0,

  K15=0,

  K16=0,

  K17=0,

  K18=0,

  K19=0,

  K20=0,

  K21=0,

  K22=0,

  K23=ARBCOMPLEX(13),

  K24=0,

  K25=0,

       ARBCOMPLEX(14)*A
  K26=------------------,
              C

  K27=0,

  K28=0,

  K29=0,

  K30=0,

  K31=ARBCOMPLEX(13),

  K32=0,

  K33=0,

       ARBCOMPLEX(14)*B
  K34=------------------,
              C

  K35=0,

  K36=0,

  K37=0,

  K38=0,

  K39=ARBCOMPLEX(13),

  K40=0,

  K41=0,

  K42=ARBCOMPLEX(14),

  K43=ARBCOMPLEX(15),

  K44=0,

  K45=0,

  K46=0,

  K47=0,

  K48=0,

  K49=0}}



% Problem 5.

solve ({2*a3*b3+a5*b3+a3*b5, a5*b3+2*a5*b5+a3*b5,
        a5*b5, a2*b2, a4*b4, a5*b1+b5+a4*b3+a3*b4,
        a5*b3+a5*b5+a3*b5+a3*b3, a0*b2+b2+a4*b2+a2*b4+c2+a2*b0+a2*b1,
        a0*b0+a0*b1+a0*b4+a3*b2+b0+b1+b4+a4*b0+a4*b1+a2*b5+a4*b4+c1+c4
        +a5*b2+a2*b3+c0,
        -1+a3*b0+a0*b3+a0*b5+a5*b0+b3+b5+a5*b4+a4*b3+a4*b5+a3*b4+a5*b1
        +a3*b1+c3+c5,
        b4+a4*b1, a5*b3+a3*b5, a2*b1+b2, a4*b5+a5*b4, a2*b4+a4*b2,
        a0*b5+a5*b0+a3*b4+2*a5*b4+a5*b1+b5+a4*b3+2*a4*b5+c5,
        a4*b0+2*a4*b4+a2*b5+b4+a4*b1+a5*b2+a0*b4+c4,
        c3+a0*b3+2*b3+b5+a4*b3+a3*b0+2*a3*b1+a5*b1+a3*b4,
        c1+a0*b1+2*b1+a4*b1+a2*b3+b0+a3*b2+b4});


Unknowns: {C2,C0,C5,C4,C3,B5,A5,A3,A2,A0,C1,B0,B1,A4,B3,B2,B4}

{{B4=0,

  A4=0,

  A5=0,

  B5=0,

  B3=0,

  B1=ARBCOMPLEX(22),

       - 1
  A3=------,
       B1

  B2=0,

  A2=0,

  A0=ARBCOMPLEX(23),

  B0=ARBCOMPLEX(24),

  C1= - A0*B1 - B0 - 2*B1,

      B0 + 2*B1
  C3=-----------,
         B1

  C4=0,

  C5=0,

  C0= - A0*B0 + B1,

  C2=0},

 {B4=0,

  A4=0,

  A5=0,

  B5=0,

  B3=-1,

  B1=0,

  A3=0,

  B2=0,

  A2=ARBCOMPLEX(19),

  B0=ARBCOMPLEX(20),

  C1=A2 - B0,

  A0=ARBCOMPLEX(21),

  C3=A0 + 2,

  C4=0,

  C5=0,

  C0= - A0*B0,

  C2= - A2*B0},

 {B4=0,

  A4=0,

  A5=0,

  B5=0,

  B3=-1,

  A3=0,

  B2=0,

  A2=0,

  A0=ARBCOMPLEX(16),

  B0=ARBCOMPLEX(17),

  B1=ARBCOMPLEX(18),

  C1= - A0*B1 - B0 - 2*B1,

  C3=A0 + 2,

  C4=0,

  C5=0,

  C0= - A0*B0 + B1,

  C2=0}}



% Problem 6.

solve({2*a3*b3+a5*b3+a3*b5, a5*b3+2*a5*b5+a3*b5,
       a4*b4, a5*b3+a5*b5+a3*b5+a3*b3, b1, a3*b3, a2*b2, a5*b5,
       a5*b1+b5+a4*b3+a3*b4, a0*b2+b2+a4*b2+a2*b4+c2+a2*b0+a2*b1,
       b4+a4*b1, b3+a3*b1, a5*b3+a3*b5, a2*b1+b2, a4*b5+a5*b4,
       a2*b4+a4*b2, a0*b0+a0*b1+a0*b4+a3*b2+b0+b1+b4+a4*b0+a4*b1
       +a2*b5+a4*b4+c1+c4+a5*b2+a2*b3+c0,-1+a3*b0+a0*b3+a0*b5+a5*b0
       +b3+b5+a5*b4+a4*b3+a4*b5+a3*b4+a5*b1+a3*b1+c3+c5,
       a0*b5+a5*b0+a3*b4+2*a5*b4+a5*b1+b5+a4*b3+2*a4*b5+c5,
       a4*b0+2*a4*b4+a2*b5+b4+a4*b1+a5*b2+a0*b4+c4,
       c3+a0*b3+2*b3+b5+a4*b3+a3*b0+2*a3*b1+a5*b1+a3*b4,
       c1+a0*b1+2*b1+a4*b1+a2*b3+b0+a3*b2+b4});


Unknowns: {C2,C0,C5,C4,C3,B5,A5,A3,A2,A0,C1,B0,B1,A4,B3,B2,B4}

{}


% Example cited by Bruno Buchberger
%        in R.Janssen: Trends in Computer Algebra,
%     Springer, 1987
% Geometry of a simple robot,
%   l1,l2   length of arms
%   ci,si   cos and sin of rotation angles


solve( { c1*c2 -cf*ct*cp + sf*sp,
         s1*c2 - sf*ct*cp - cf*sp,
         s2 + st*cp,
         -c1*s2 - cf*ct*sp + sf*cp,
         -s1*s2 + sf*ct*sp - cf*cp,
         c2 - st*sp,
         s1 - cf*st,
         -c1 - sf*st,
         ct,
         l2*c1*c2 - px,
         l2*s1*c2 - py,
         l2*s2 + l1 - pz,
         c1**2 + s1**2 -1,
         c2**2 + s2**2 -1,
         cf**2 + sf**2 -1,
         ct**2 + st**2 -1,
         cp**2 + sp**2 -1},
      {c1,c2,s1,s2,py,cf,ct,cp,sf,st,sp});


             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )
{{SP=---------------------------------,
                    L2

  ST=1,

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  SF=---------------------------------------,
              2     2               2
            L2  - L1  + 2*L1*PZ - PZ

      L1 - PZ
  CP=---------,
        L2

  CT=0,

             2     2     2               2
      SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  CF=---------------------------------------,
                2     2               2
         SQRT(L2  - L1  + 2*L1*PZ - PZ )

       - L1 + PZ
  S2=------------,
          L2

             2     2     2               2
      SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  S1=---------------------------------------,
                2     2               2
         SQRT(L2  - L1  + 2*L1*PZ - PZ )

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )
  C2=---------------------------------,
                    L2

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  C1=------------------------------------,
            2     2               2
          L2  - L1  + 2*L1*PZ - PZ

            2     2     2               2
  PY=SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )},

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )
 {SP=---------------------------------,
                    L2

  ST=1,

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  SF=---------------------------------------,
              2     2               2
            L2  - L1  + 2*L1*PZ - PZ

      L1 - PZ
  CP=---------,
        L2

  CT=0,

                2     2     2               2
       - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  CF=------------------------------------------,
                 2     2               2
          SQRT(L2  - L1  + 2*L1*PZ - PZ )

       - L1 + PZ
  S2=------------,
          L2

                2     2     2               2
       - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  S1=------------------------------------------,
                 2     2               2
          SQRT(L2  - L1  + 2*L1*PZ - PZ )

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )
  C2=---------------------------------,
                    L2

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  C1=------------------------------------,
            2     2               2
          L2  - L1  + 2*L1*PZ - PZ

               2     2     2               2
  PY= - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )},

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )
 {SP=------------------------------------,
                      L2

  ST=1,

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  SF=------------------------------------,
            2     2               2
          L2  - L1  + 2*L1*PZ - PZ

      L1 - PZ
  CP=---------,
        L2

  CT=0,

             2     2     2               2
      SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  CF=---------------------------------------,
                2     2               2
         SQRT(L2  - L1  + 2*L1*PZ - PZ )

       - L1 + PZ
  S2=------------,
          L2

             2     2     2               2
      SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  S1=---------------------------------------,
                2     2               2
         SQRT(L2  - L1  + 2*L1*PZ - PZ )

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )
  C2=------------------------------------,
                      L2

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  C1=---------------------------------------,
              2     2               2
            L2  - L1  + 2*L1*PZ - PZ

               2     2     2               2
  PY= - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )},

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )
 {SP=------------------------------------,
                      L2

  ST=1,

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  SF=------------------------------------,
            2     2               2
          L2  - L1  + 2*L1*PZ - PZ

      L1 - PZ
  CP=---------,
        L2

  CT=0,

                2     2     2               2
       - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  CF=------------------------------------------,
                 2     2               2
          SQRT(L2  - L1  + 2*L1*PZ - PZ )

       - L1 + PZ
  S2=------------,
          L2

                2     2     2               2
       - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  S1=------------------------------------------,
                 2     2               2
          SQRT(L2  - L1  + 2*L1*PZ - PZ )

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )
  C2=------------------------------------,
                      L2

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  C1=---------------------------------------,
              2     2               2
            L2  - L1  + 2*L1*PZ - PZ

            2     2     2               2
  PY=SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )},

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )
 {SP=---------------------------------,
                    L2

  ST=-1,

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  SF=---------------------------------------,
              2     2               2
            L2  - L1  + 2*L1*PZ - PZ

       - L1 + PZ
  CP=------------,
          L2

  CT=0,

             2     2     2               2
      SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  CF=---------------------------------------,
                2     2               2
         SQRT(L2  - L1  + 2*L1*PZ - PZ )

       - L1 + PZ
  S2=------------,
          L2

                2     2     2               2
       - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  S1=------------------------------------------,
                 2     2               2
          SQRT(L2  - L1  + 2*L1*PZ - PZ )

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )
  C2=------------------------------------,
                      L2

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  C1=---------------------------------------,
              2     2               2
            L2  - L1  + 2*L1*PZ - PZ

            2     2     2               2
  PY=SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )},

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )
 {SP=---------------------------------,
                    L2

  ST=-1,

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  SF=---------------------------------------,
              2     2               2
            L2  - L1  + 2*L1*PZ - PZ

       - L1 + PZ
  CP=------------,
          L2

  CT=0,

                2     2     2               2
       - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  CF=------------------------------------------,
                 2     2               2
          SQRT(L2  - L1  + 2*L1*PZ - PZ )

       - L1 + PZ
  S2=------------,
          L2

             2     2     2               2
      SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  S1=---------------------------------------,
                2     2               2
         SQRT(L2  - L1  + 2*L1*PZ - PZ )

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )
  C2=------------------------------------,
                      L2

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  C1=---------------------------------------,
              2     2               2
            L2  - L1  + 2*L1*PZ - PZ

               2     2     2               2
  PY= - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )},

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )
 {SP=------------------------------------,
                      L2

  ST=-1,

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  SF=------------------------------------,
            2     2               2
          L2  - L1  + 2*L1*PZ - PZ

       - L1 + PZ
  CP=------------,
          L2

  CT=0,

             2     2     2               2
      SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  CF=---------------------------------------,
                2     2               2
         SQRT(L2  - L1  + 2*L1*PZ - PZ )

       - L1 + PZ
  S2=------------,
          L2

                2     2     2               2
       - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  S1=------------------------------------------,
                 2     2               2
          SQRT(L2  - L1  + 2*L1*PZ - PZ )

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )
  C2=---------------------------------,
                    L2

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  C1=------------------------------------,
            2     2               2
          L2  - L1  + 2*L1*PZ - PZ

               2     2     2               2
  PY= - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )},

                2     2               2
       - SQRT(L2  - L1  + 2*L1*PZ - PZ )
 {SP=------------------------------------,
                      L2

  ST=-1,

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  SF=------------------------------------,
            2     2               2
          L2  - L1  + 2*L1*PZ - PZ

       - L1 + PZ
  CP=------------,
          L2

  CT=0,

                2     2     2               2
       - SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  CF=------------------------------------------,
                 2     2               2
          SQRT(L2  - L1  + 2*L1*PZ - PZ )

       - L1 + PZ
  S2=------------,
          L2

             2     2     2               2
      SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )
  S1=---------------------------------------,
                2     2               2
         SQRT(L2  - L1  + 2*L1*PZ - PZ )

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )
  C2=---------------------------------,
                    L2

             2     2               2
      SQRT(L2  - L1  + 2*L1*PZ - PZ )*PX
  C1=------------------------------------,
            2     2               2
          L2  - L1  + 2*L1*PZ - PZ

            2     2     2               2
  PY=SQRT(L2  - PX  - L1  + 2*L1*PZ - PZ )}}


% Steady state computation of a prototypical chemical
% reaction network (the "Edelstein" network)
 
solve(
 { alpha * c1 - beta * c1**2 - gamma*c1*c2 + epsilon*c3,
   -gamma*c1*c2 + (epsilon+theta)*c3 -eta *c2,
   gamma*c1*c2 + eta*c2 - (epsilon+theta) * c3},
  {c3,c2,c1});


{{C1=ARBCOMPLEX(25),

  C2=(C1*( - C1*BETA*EPSILON - C1*BETA*THETA + ALPHA*EPSILON

           + ALPHA*THETA))/(C1*GAMMA*THETA - EPSILON*ETA),

  C3=(C1

             2
      *( - C1 *BETA*GAMMA - C1*BETA*ETA + C1*ALPHA*GAMMA + ALPHA*ETA)

      )/(C1*GAMMA*THETA - EPSILON*ETA)}}


solve(
{( - 81*y1**2*y2**2 + 594*y1**2*y2 - 225*y1**2 + 594*y1*y2**2 - 3492*
y1*y2 - 750*y1 - 225*y2**2 - 750*y2 + 14575)/81,
( - 81*y2**2*y3**2 + 594*y2**2*y3 - 225*y2**2 + 594*y2*y3**2 - 3492*
y2*y3 - 750*y2 - 225*y3**2 - 750*y3 + 14575)/81,
( - 81*y1**2*y3**2 + 594*y1**2*y3 - 225*y1**2 + 594*y1*y3**2 - 3492*
y1*y3 - 750*y1 - 225*y3**2 - 750*y3 + 14575)/81,
(2*(81*y1**2*y2**2*y3 + 81*y1**2*y2*y3**2 - 594*y1**2*y2*y3 - 225*y1
**2*y2 - 225*y1**2*y3 + 1650*y1**2 + 81*y1*y2**2*y3**2 - 594*y1*
y2**2*y3 - 225*y1*y2**2 - 594*y1*y2*y3**2 + 2592*y1*y2*y3 + 2550
*y1*y2 - 225*y1*y3**2 + 2550*y1*y3 - 3575*y1 - 225*y2**2*y3 +
1650*y2**2 - 225*y2*y3**2 + 2550*y2*y3 - 3575*y2 + 1650*y3**2 -
3575*y3 - 30250))/81}, {y1,y2,y3,y4});


       2   2         2            2            2
{{81*Y2 *Y3  - 594*Y2 *Y3 + 225*Y2  - 594*Y2*Y3  + 3492*Y2*Y3

                    2
   + 750*Y2 + 225*Y3  + 750*Y3 - 14575=0,

          2                               2
  27*Y1*Y3  - 198*Y1*Y3 + 75*Y1 + 27*Y2*Y3  - 198*Y2*Y3 + 75*Y2

           2
   - 198*Y3  + 1164*Y3 + 250=0},

 {Y3=ARBCOMPLEX(27),

           2
  Y2=(99*Y3  - 582*Y3

                      4          3           2
       + 4*SQRT(486*Y3  - 6696*Y3  + 30564*Y3  - 52200*Y3 + 23750)

                      2
       - 125)/(3*(9*Y3  - 66*Y3 + 25)),

           2
  Y1=(99*Y3  - 582*Y3

                      4          3           2
       - 4*SQRT(486*Y3  - 6696*Y3  + 30564*Y3  - 52200*Y3 + 23750)

                      2
       - 125)/(3*(9*Y3  - 66*Y3 + 25))},

 {Y3=ARBCOMPLEX(26),

           2
  Y2=(99*Y3  - 582*Y3

                      4          3           2
       - 4*SQRT(486*Y3  - 6696*Y3  + 30564*Y3  - 52200*Y3 + 23750)

                      2
       - 125)/(3*(9*Y3  - 66*Y3 + 25)),

           2
  Y1=(99*Y3  - 582*Y3

                      4          3           2
       + 4*SQRT(486*Y3  - 6696*Y3  + 30564*Y3  - 52200*Y3 + 23750)

                      2
       - 125)/(3*(9*Y3  - 66*Y3 + 25))},

       - 5       - 5       - 5
 {Y3=------,Y2=------,Y1=------},
       3         3         3

      11      11      11
 {Y3=----,Y2=----,Y1=----}}
      3       3       3


% Another nice nonlinear system.

solve({y=x+t^2,x=y+u^2},{x,y,u,t});


{{T=ARBCOMPLEX(30),

  U=T*I,

  Y=ARBCOMPLEX(31),

         2
  X=Y - T },

 {T=ARBCOMPLEX(28),

  U= - T*I,

  Y=ARBCOMPLEX(29),

         2
  X=Y - T }}


end;

5: 5: 
Time: 46155 ms  plus GC time: 4641 ms
6: 6: 
Quitting
Sat May 30 16:12:59 PDT 1992
