Sat Jun 29 13:45:44 PDT 1991
REDUCE 3.4, 15-Jul-91 ...
1: 1:
2: 2:
3: 3: % Demonstration of the REDUCE SOLVE package.
% 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,
LOG(X)*X - 1=0,
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
{( - A + X) - SQRT(X)=0,
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});
- A
{{X=SOL({COS(X)=---------------},{X}),
2 2
SQRT(A + B )
B
X=SOL({SIN(X)=---------------},{X})},
2 2
SQRT(A + B )
A
{X=SOL({COS(X)=---------------},{X}),
2 2
SQRT(A + B )
- B
X=SOL({SIN(X)=---------------},{X})}}
2 2
SQRT(A + B )
solve ({a*sin(x+1) + b*cos(x+1)},{x});
2 2
- SQRT(A + B ) - A
{{X=SOL({COS(X + 1)=----------------------},{X}),
2 2
SQRT(A + B )
2 2
- SQRT(A + B ) + B
X=SOL({SIN(X + 1)=----------------------},{X})},
2 2
SQRT(A + B )
2 2
- SQRT(A + B ) + A
{X=SOL({COS(X + 1)=----------------------},{X}),
2 2
SQRT(A + B )
2 2
- SQRT(A + B ) - B
X=SOL({SIN(X + 1)=----------------------},{X})}}
2 2
SQRT(A + B )
% 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,
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,
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
{ - E + E - 7=0}
% Generally not liftable.
% variable inside and outside of sin.
solve({sin x + x - 1/2},{x});
- 2*SIN(X) + 1
{X=-----------------}
2
% Variable inside and outside of exponential.
solve({e^x - x**2},{x});
X 2
{E - X =0}
% Variable inside trigonometrical functions with different forms.
solve ({a*sin(x+1) + b*cos(x+2)},{x});
{SIN(X + 1)*A + COS(X + 2)*B=0}
% 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: {X0,X9,X7,X5,X8,X6,X4,X3,X2,X1}
4352444991703786550093529782474564455970663240687
{{X0=---------------------------------------------------,
8420785423059099972039395927798127489505890997055
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
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: {X0,X9,X7,X5,X8,X6,X4,X3,X2,X1}
4352444991703786550093529782474564455970663240687
{{X0=---------------------------------------------------,
8420785423059099972039395927798127489505890997055
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
% 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=-1,
B1=0,
A3=0,
B2=0,
A2=ARBCOMPLEX(22),
B0=ARBCOMPLEX(23),
C1=A2 - B0,
A0=ARBCOMPLEX(24),
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(19),
B0=ARBCOMPLEX(20),
B1=ARBCOMPLEX(21),
C1= - A0*B1 - B0 - 2*B1,
C3=A0 + 2,
C4=0,
C5=0,
C0= - A0*B0 + B1,
C2=0},
{B4=0,
A4=0,
A5=0,
B5=0,
B3=0,
B1=ARBCOMPLEX(16),
- 1
A3=------,
B1
B2=0,
A2=0,
A0=ARBCOMPLEX(17),
B0=ARBCOMPLEX(18),
C1= - A0*B1 - B0 - 2*B1,
B0 + 2*B1
C3=-----------,
B1
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 2
{2*(81*Y1 *Y2 *Y3 + 81*Y1 *Y2*Y3 - 594*Y1 *Y2*Y3 - 225*Y1 *Y2
2 2 2 2 2
- 225*Y1 *Y3 + 1650*Y1 + 81*Y1*Y2 *Y3 - 594*Y1*Y2 *Y3
2 2
- 225*Y1*Y2 - 594*Y1*Y2*Y3 + 2592*Y1*Y2*Y3 + 2550*Y1*Y2
2 2 2
- 225*Y1*Y3 + 2550*Y1*Y3 - 3575*Y1 - 225*Y2 *Y3 + 1650*Y2
2 2
- 225*Y2*Y3 + 2550*Y2*Y3 - 3575*Y2 + 1650*Y3 - 3575*Y3 - 30250
)=0,
2 2 2 2 2
- 81*Y1 *Y3 + 594*Y1 *Y3 - 225*Y1 + 594*Y1*Y3 - 3492*Y1*Y3
2
- 750*Y1 - 225*Y3 - 750*Y3 + 14575=0,
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 2 2 2
- 81*Y1 *Y2 + 594*Y1 *Y2 - 225*Y1 + 594*Y1*Y2 - 3492*Y1*Y2
2
- 750*Y1 - 225*Y2 - 750*Y2 + 14575=0}
end;
4: 4:
Quitting
Sat Jun 29 13:47:50 PDT 1991