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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