@@ -1,2695 +1,2695 @@ -REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... - - -% Demonstration of the REDUCE SOLVE package. - -on fullroots; - - % To get complete solutions. - -% Simultaneous linear fractional equations. - -solve({(a*x+y)/(z-1)-3,y+b+z,x-y},{x,y,z}); - - - - 3*(b + 1) -{{x=--------------, - a + 4 - - - 3*(b + 1) - y=--------------, - a + 4 - - - a*b - b + 3 - z=----------------}} - a + 4 - - - -% Use of square-free factorization together with recursive use of -% quadratic and binomial solutions. - -solve((x**6-x**3-1)*(x**5-1)**2*x**2); - - -Unknown: x - - 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=1, - - x=0, - - 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 - - -multiplicities!*; - - -{2,2,2,2,2,2,1,1,1,1,1,1} - - - -% 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( - sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 -{x=-------------------------------------------------, - 2 - - - sqrt( - sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 - x=----------------------------------------------------, - 2 - - sqrt( - sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 - x=----------------------------------------------, - 2 - - - sqrt( - sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 - x=-------------------------------------------------, - 2 - - sqrt(sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 - x=----------------------------------------------, - 2 - - - sqrt(sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 - x=-------------------------------------------------, - 2 - - sqrt(sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 - x=-------------------------------------------, - 2 - - - sqrt(sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 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=sqrt( - i + 116) + 11, - - x= - sqrt( - i + 116) + 11, - - x=sqrt(i + 116) + 11, - - x= - sqrt(i + 116) + 11, - - x=4*sqrt(7) + 11, - - x= - 4*sqrt(7) + 11, - - x=2*sqrt(30) + 11, - - x= - 2*sqrt(30) + 11} - - - -% Recursive use of inverses, including multiple branches of rational -% fractional powers. - -solve(log(acos(asin(x**(2/3)-b)-1))+2,x); - - - 1 1 -{x=sqrt(sin(cos(----) + 1) + b)*(sin(cos(----) + 1) + b), - 2 2 - e e - - 1 1 - x= - sqrt(sin(cos(----) + 1) + b)*(sin(cos(----) + 1) + b)} - 2 2 - e e - - - -% Square-free factors that are unsolvable, being of fifth degree, -% transcendental, or without a defined inverse. - -operator f; - - - -solve((x-1)*(x+1)*(x-2)*(x+2)*(x-3)*(x*log(x)-1)*(f(x)-1),x); - - -{f(x) - 1=0, - - x=root_of(log(x_)*x_ - 1,x_,tag_2), - - x=3, - - x=2, - - x=1, - - x=-1, - - x=-2} - - -multiplicities!*; - - -{1,1,1,1,1,1,1} - - - -% Factors with more than one distinct top-level kernel, the first factor -% a cubic. (Cubic solution suppressed since it is too messy to be of -% much use). - -off fullroots; - - - -solve((x**(1/2)-(x-a)**(1/3))*(acos x-acos(2*x-b))* (2*log x - -log(x**2+x-c)-4),x); - - - 2 4 4 2 - e *(sqrt(4*c*e - 4*c + e ) - e ) -{x=-----------------------------------, - 4 - 2*(e - 1) - - 2 4 4 2 - - e *(sqrt(4*c*e - 4*c + e ) + e ) - x=--------------------------------------, - 4 - 2*(e - 1) - - 2 3 2 - x=root_of(a - 2*a*x_ - x_ + x_ ,x_,tag_7), - - x=b} - - -on fullroots; - - - -% Treatment of multiple-argument exponentials as polynomials. - -solve(a**(2*x)-3*a**x+2,x); - - - 2*arbint(3)*i*pi + log(2) -{x=---------------------------, - log(a) - - 2*arbint(2)*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 - - sqrt( - sqrt(5) - 3) -{x=----------------------, - sqrt(2) - - - sqrt( - sqrt(5) - 3) - x=-------------------------, - 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( - 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(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(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) - - i*(sqrt(5) - 1) - x=-----------------, - 2 - - i*( - sqrt(5) + 1) - x=--------------------} - 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); - - -{x=2*arbint(6)*pi + asin(a), - - x=2*arbint(6)*pi - asin(a) + pi, - - 2*arbint(5)*i*pi + log(b) - x=---------------------------, - log(2) - - 1/c 2*arbint(4)*pi 2*arbint(4)*pi - x=3 *(cos(----------------) + sin(----------------)*i)} - c c - - - -% Automatic restriction to principal branches. - -off allbranch; - - - -solve((sin x-a)*(2**x-b)*(x**c-3),x); - - -{x=asin(a), - - 1/c - x=3 , - - log(b) - x=--------} - log(2) - - - -% 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}); - - -{} - - - -% 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 2 3 -yy(1) := - xx(5)*a - xx(5) - xx(4) + xx(3)*a + xx(2)*a + xx(1) - a *b - - 2 2 2 2 - - 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 2 -yy(2) := b *( - xx(5)*b - xx(5)*b - xx(4)*b + xx(3) + xx(2)*b + xx(1)*b - a*b - - 3 2 - - 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 2 2 - + xx(1)*a*b - a*b *c - 3*a*b *c - 4*a*b*c - a*b - a*c - 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 + xx(2)*a - - 2 4 3 3 - - xx(2)*b + xx(1)*a + xx(1)*b - a - a*c - a - b - b *c - 3*b - - 2 2 - - 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 2 -yy(5) := - 3*xx(5) - xx(4) + 3*xx(3) + 2*xx(2) + xx(1) - 2*a - 3*b - 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}); - - - 2 2 - sqrt(a + b ) - a -{x= - 2*atan(-------------------), - b - - 2 2 - sqrt(a + b ) + a - x=2*atan(-------------------)} - b - - -solve ({a*sin(x+1) + b*cos(x+1)},{x}); - - - 2 2 - sqrt(a + b ) - a -{x= - 2*atan(-------------------) - 1, - b - - 2 2 - sqrt(a + b ) + a - x=2*atan(-------------------) - 1} - 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, - - 12 9 6 3 2 - r=sqrt(y - 4*y + 6*y - 4*y + y + 1)}, - - {y=arbcomplex(11), - - 6 3 - x=y - 2*y + 1, - - 12 9 6 3 2 - r= - sqrt(y - 4*y + 6*y - 4*y + y + 1)}} - - -solve ({e^x - e^(1/2 * x) - 7},{x}); - - - - sqrt(29) + 1 -{x=2*log(-----------------), - 2 - - sqrt(29) + 1 - x=2*log(--------------)} - 2 - - -% Generally not liftable. - - % variable inside and outside of sin. - - solve({sin x + x - 1/2},{x}); - - -{x=root_of(2*sin(x_) + 2*x_ - 1,x_,tag_12)} - - - % Variable inside and outside of exponential. - - solve({e^x - x**2},{x}); - - - - 1 -{x= - 2*lambert_w(------)} - 2 - - - % Variable inside trigonometrical functions with different forms. - - solve ({a*sin(x+1) + b*cos(x+2)},{x}); - - - 2 2 -{x=2*atan((cos(1)*a - sqrt(2*cos(2)*sin(1)*a*b - 2*cos(1)*sin(2)*a*b + a + b ) - - - sin(2)*b)/(cos(2)*b + sin(1)*a)), - - 2 2 - x=2*atan((cos(1)*a + sqrt(2*cos(2)*sin(1)*a*b - 2*cos(1)*sin(2)*a*b + a + b ) - - - sin(2)*b)/(cos(2)*b + sin(1)*a))} - - - % 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*b2*k - l ) - x3=-----------------, - 6*k*l - - 2 2 - 7*(45*b2*b4*k - 15*b4*l - 8*k*l ) - x4=------------------------------------, - 2 - 36*k*l - - 2 2 2 4 - 2205*b2*b4*b6*k - 108*b2*k*l - 735*b4*b6*l - 392*b6*k*l + 36*l - x5=--------------------------------------------------------------------, - 3 - 32*k*l - - 2 2 - x6=(11*(893025*b2*b4*b6*b9*k - 11520*b2*b4*k*l - 43740*b2*b9*k*l - - 2 4 2 4 - - 297675*b4*b6*b9*l + 3840*b4*l - 158760*b6*b9*k*l + 14580*b9*l - - 4 4 - + 2048*k*l ))/(11520*k*l ), - - 2 - x7=(47652707025*b11*b2*b4*b6*b9*k - 614718720*b11*b2*b4*k*l - - 2 2 - - 2334010140*b11*b2*b9*k*l - 15884235675*b11*b4*b6*b9*l - - 4 2 4 - + 204906240*b11*b4*l - 8471592360*b11*b6*b9*k*l + 778003380*b11*b9*l - - 4 - + 109283328*b11*k*l + 172398476250*b15*b2*b4*b6*b9*k - - 2 2 - - 2223936000*b15*b2*b4*k*l - 8444007000*b15*b2*b9*k*l - - 2 4 - - 57466158750*b15*b4*b6*b9*l + 741312000*b15*b4*l - - 2 4 4 - - 30648618000*b15*b6*b9*k*l + 2814669000*b15*b9*l + 395366400*b15*k*l - - 2 4 4 - - 172872000*b2*b4*b6*k*l + 8467200*b2*k*l + 57624000*b4*b6*l - - 4 6 3 - + 30732800*b6*k*l - 2822400*l )/(7729722000*b13*b15*k*l ), - - 2 - x8=(7*(972504225*b11*b2*b4*b6*b9*k - 12545280*b11*b2*b4*k*l - - 2 2 - - 47632860*b11*b2*b9*k*l - 324168075*b11*b4*b6*b9*l - - 4 2 4 - + 4181760*b11*b4*l - 172889640*b11*b6*b9*k*l + 15877620*b11*b9*l - - 4 2 4 - + 2230272*b11*k*l - 3528000*b2*b4*b6*k*l + 172800*b2*k*l - - 4 4 6 4 - + 1176000*b4*b6*l + 627200*b6*k*l - 57600*l ))/(24710400*b15*k*l )}} - - - -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}); - - - 8 8 7 3 7 2 7 7 6 4 -{{x1=(y*( - 8*x *y + 10*x + 9*x *y - 49*x *y + 85*x *y - 43*x + 23*x *y - - 6 3 6 2 6 6 5 5 5 4 - - 128*x *y + 266*x *y - 246*x *y + 77*x + 20*x *y - 145*x *y - - 5 3 5 2 5 5 4 6 4 5 - + 383*x *y - 512*x *y + 329*x *y - 75*x + 9*x *y - 84*x *y - - 4 4 4 3 4 2 4 4 3 7 - + 276*x *y - 469*x *y + 464*x *y - 233*x *y + 43*x + 3*x *y - - 3 6 3 5 3 4 3 3 3 2 3 - - 23*x *y + 97*x *y - 196*x *y + 245*x *y - 201*x *y + 87*x *y - - 3 2 8 2 7 2 6 2 5 2 4 - - 14*x - 2*x *y + 13*x *y - 25*x *y + 23*x *y - 10*x *y - - 2 3 2 2 2 2 9 8 7 - - 17*x *y + 31*x *y - 15*x *y + 2*x - 2*x*y + 10*x*y - 24*x*y - - 6 5 4 3 2 6 5 - + 41*x*y - 57*x*y + 53*x*y - 24*x*y + 2*x*y + x*y - 2*y + 7*y - - 4 3 2 10 10 9 2 9 9 - - 9*y + 5*y - y ))/(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x - - 8 3 8 2 8 8 7 4 7 3 7 2 - + x *y - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y - - 7 7 6 5 6 4 6 3 6 2 - + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y - - 6 6 5 6 5 5 5 4 5 3 - - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y - - 5 2 5 5 4 7 4 6 4 5 4 4 - - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y - - 4 3 4 2 4 4 3 8 3 7 - - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y - - 3 6 3 5 3 4 3 3 3 2 3 - + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y - - 2 9 2 8 2 7 2 6 2 5 2 4 - + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y - - 2 3 2 2 10 9 8 7 - + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y - - 6 5 4 3 9 8 7 6 - + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y - - 5 - + 2*y ), - - 10 10 9 2 9 9 8 3 8 2 8 - x2=(2*x *y - 2*x + 5*x *y - 12*x *y + 7*x - 8*x *y + 9*x *y + 2*x *y - - 8 7 4 7 3 7 2 7 7 6 5 - - x - 15*x *y + 65*x *y - 83*x *y + 52*x *y - 17*x + 5*x *y - - 6 4 6 3 6 2 6 6 5 6 5 5 - - 5*x *y - 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48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y - - 5 - + 2*y ), - - 9 9 8 2 8 8 7 3 7 2 7 - x4=(2*(2*x *y - 2*x + 4*x *y - 10*x *y + 6*x - 9*x *y + 21*x *y - 13*x *y - - 7 6 4 6 3 6 2 6 6 5 5 - + x - 18*x *y + 88*x *y - 130*x *y + 74*x *y - 14*x - 10*x *y - - 5 4 5 3 5 2 5 5 4 6 - + 74*x *y - 180*x *y + 191*x *y - 90*x *y + 15*x + 4*x *y - - 4 5 4 4 4 3 4 2 4 4 - - 18*x *y - 20*x *y + 105*x *y - 111*x *y + 47*x *y - 7*x - - 3 7 3 6 3 5 3 4 3 3 3 2 - + 16*x *y - 96*x *y + 188*x *y - 155*x *y + 44*x *y + 8*x *y - - 3 3 2 8 2 7 2 6 2 5 - - 6*x *y + x + 10*x *y - 62*x *y + 164*x *y - 219*x *y - - 2 4 2 3 2 2 2 9 8 7 - + 154*x *y - 56*x *y + 10*x *y - x *y + x*y - 13*x*y + 45*x*y - - 6 5 4 3 2 8 7 6 - - 72*x*y + 64*x*y - 35*x*y + 12*x*y - 2*x*y + 2*y - 7*y + 9*y - - 5 4 10 10 9 2 9 9 8 3 - - 5*y + y ))/(x*(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x + x *y - - 8 2 8 8 7 4 7 3 7 2 - - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y - - 7 7 6 5 6 4 6 3 6 2 - + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y - - 6 6 5 6 5 5 5 4 5 3 - - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y - - 5 2 5 5 4 7 4 6 4 5 - - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y - - 4 4 4 3 4 2 4 4 3 8 - + 90*x *y - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y - - 3 7 3 6 3 5 3 4 3 3 3 2 - - 118*x *y + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y - - 3 2 9 2 8 2 7 2 6 2 5 - + 2*x *y + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y - - 2 4 2 3 2 2 10 9 8 - - 86*x *y + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y - - 7 6 5 4 3 9 8 - - 121*x*y + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y - - 7 6 5 - + 15*y - 9*y + 2*y )), - - 10 10 9 2 9 9 8 3 8 2 8 - x5=(2*x *y - 2*x + 7*x *y - 16*x *y + 7*x - 3*x *y - 11*x *y + 21*x *y - - 8 7 4 7 3 7 2 7 7 6 5 - - x - 18*x *y + 60*x *y - 46*x *y + 23*x *y - 17*x - 4*x *y - - 6 4 6 3 6 2 6 6 5 6 5 5 - + 38*x *y - 70*x *y + 40*x *y - 36*x *y + 20*x + 14*x *y - 86*x *y - - 5 4 5 3 5 2 5 5 4 7 - + 164*x *y - 182*x *y + 114*x *y - 14*x *y - 8*x + 24*x *y - - 4 6 4 5 4 4 4 3 4 2 4 - - 167*x *y + 387*x *y - 455*x *y + 348*x *y - 164*x *y + 32*x *y - - 4 3 8 3 7 3 6 3 5 3 4 - + x + 21*x *y - 130*x *y + 339*x *y - 458*x *y + 370*x *y - - 3 3 3 2 3 2 9 2 8 2 7 - - 211*x *y + 81*x *y - 14*x *y + 5*x *y - 43*x *y + 140*x *y - - 2 6 2 5 2 4 2 3 2 2 2 - - 209*x *y + 165*x *y - 86*x *y + 42*x *y - 16*x *y + 2*x *y - - 9 8 7 6 5 4 3 2 - - 5*x*y + 20*x*y - 32*x*y + 16*x*y + 8*x*y - 9*x*y + x*y + x*y - - 9 8 7 6 5 4 3 10 10 - + 2*y - 6*y + 6*y + y - 6*y + 4*y - y )/(x*y*(2*x *y - 4*x - - 9 2 9 9 8 3 8 2 8 8 - + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y - 31*x - - 7 4 7 3 7 2 7 7 6 5 - - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x - 28*x *y - - 6 4 6 3 6 2 6 6 5 6 - + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x - 14*x *y - - 5 5 5 4 5 3 5 2 5 5 - + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y - 5*x - - 4 7 4 6 4 5 4 4 4 3 4 2 - + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y + 97*x *y - - 4 4 3 8 3 7 3 6 3 5 - - 24*x *y + x + 20*x *y - 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+ 11*x*y - x*y - 4*y + 10*y - 10*y + 5*y - y ))/(y*(2*x *y - - 10 9 2 9 9 8 3 8 2 8 - - 4*x + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y - - 8 7 4 7 3 7 2 7 7 - - 31*x - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x - - 6 5 6 4 6 3 6 2 6 6 - - 28*x *y + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x - - 5 6 5 5 5 4 5 3 5 2 5 - - 14*x *y + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y - - 5 4 7 4 6 4 5 4 4 4 3 - - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y - - 4 2 4 4 3 8 3 7 3 6 - + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y + 244*x *y - - 3 5 3 4 3 3 3 2 3 2 9 - - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y + 13*x *y - - 2 8 2 7 2 6 2 5 2 4 2 3 - - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y + 23*x *y - - 2 2 10 9 8 7 6 - - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y + 100*x*y - - 5 4 3 9 8 7 6 5 - - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y + 2*y )), - - 7 2 7 7 6 3 6 2 6 6 5 4 - x7=(6*(x *y - 2*x *y + x + x *y - 4*x *y + 5*x *y - 2*x - 6*x *y - - 5 3 5 2 5 5 4 5 4 4 - + 26*x *y - 38*x *y + 21*x *y - 3*x - 8*x *y + 49*x *y - - 4 3 4 2 4 4 3 6 3 5 3 4 - - 106*x *y + 101*x *y - 41*x *y + 5*x - x *y + 12*x *y - 42*x *y - - 3 3 3 2 3 3 2 7 2 6 2 5 - + 69*x *y - 52*x *y + 15*x *y - x + 4*x *y - 27*x *y + 59*x *y - - 2 4 2 3 2 2 2 8 7 6 - - 52*x *y + 14*x *y + 3*x *y - x *y + 3*x*y - 18*x*y + 39*x*y - - 5 4 3 2 7 6 5 4 3 - - 48*x*y + 34*x*y - 11*x*y + x*y + 2*y - 5*y + 6*y - 4*y + y ) - - 10 10 9 2 9 9 8 3 8 2 - )/(x*(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y - - 8 8 7 4 7 3 7 2 7 - + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y + 18*x *y - - 7 6 5 6 4 6 3 6 2 6 - + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y - 104*x *y - - 6 5 6 5 5 5 4 5 3 5 2 - + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y - 278*x *y - - 5 5 4 7 4 6 4 5 4 4 - + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y - - 4 3 4 2 4 4 3 8 3 7 - - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y - - 3 6 3 5 3 4 3 3 3 2 3 - + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y - - 2 9 2 8 2 7 2 6 2 5 - + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y - - 2 4 2 3 2 2 10 9 8 - - 86*x *y + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y - - 7 6 5 4 3 9 8 - - 121*x*y + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y - - 7 6 5 - + 15*y - 9*y + 2*y )), - - 9 8 2 8 8 7 3 7 2 7 - x8=(2*( - 2*x + x *y - 10*x *y + 13*x + 5*x *y - 24*x *y + 49*x *y - - 7 6 4 6 3 6 2 6 6 5 5 - - 30*x + 8*x *y - 41*x *y + 75*x *y - 78*x *y + 32*x + 7*x *y - - 5 4 5 3 5 2 5 5 4 6 4 5 - - 35*x *y + 61*x *y - 56*x *y + 41*x *y - 16*x - x *y + 9*x *y - - 4 4 4 3 4 2 4 4 3 7 3 6 - - 10*x *y + 15*x *y - 22*x *y + 6*x *y + 3*x - 10*x *y + 57*x *y - - 3 5 3 4 3 3 3 2 3 2 8 - - 107*x *y + 91*x *y - 55*x *y + 34*x *y - 10*x *y - 8*x *y - - 2 7 2 6 2 5 2 4 2 3 2 2 - + 46*x *y - 105*x *y + 116*x *y - 63*x *y + 23*x *y - 11*x *y - - 2 9 8 7 6 5 4 - + 2*x *y - 2*x*y + 16*x*y - 42*x*y + 54*x*y - 34*x*y + 6*x*y - - 3 2 8 7 6 5 4 3 10 - + x*y + x*y - 2*y + 6*y - 7*y + 3*y + y - y ))/(x*y*(2*x *y - - 10 9 2 9 9 8 3 8 2 8 - - 4*x + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y - - 8 7 4 7 3 7 2 7 7 - - 31*x - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x - - 6 5 6 4 6 3 6 2 6 6 - - 28*x *y + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x - - 5 6 5 5 5 4 5 3 5 2 5 - - 14*x *y + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y - - 5 4 7 4 6 4 5 4 4 4 3 - - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y - - 4 2 4 4 3 8 3 7 3 6 - + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y + 244*x *y - - 3 5 3 4 3 3 3 2 3 2 9 - - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y + 13*x *y - - 2 8 2 7 2 6 2 5 2 4 2 3 - - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y + 23*x *y - - 2 2 10 9 8 7 6 - - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y + 100*x*y - - 5 4 3 9 8 7 6 5 - - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y + 2*y )), - - 7 2 7 7 6 3 6 2 6 6 5 4 - x9=(6*( - 2*x *y + 2*x *y + 4*x - 4*x *y + 16*x *y - 6*x *y - 8*x + x *y - - 5 3 5 2 5 5 4 5 4 4 4 3 - + 18*x *y - 56*x *y + 26*x *y + 3*x + 4*x *y - 6*x *y - 40*x *y - - 4 2 4 4 3 6 3 5 3 4 3 3 - + 82*x *y - 38*x *y + 2*x - 6*x *y + 15*x *y - 9*x *y + 32*x *y - - 3 2 3 3 2 7 2 5 2 4 2 3 - - 46*x *y + 19*x *y - x + x *y - 5*x *y + 2*x *y - 7*x *y - - 2 2 2 8 7 6 5 4 - + 10*x *y - 3*x *y - 2*x*y + 9*x*y - 4*x*y - 16*x*y + 22*x*y - - 3 7 6 5 4 3 10 10 - - 9*x*y - 2*y + 2*y + 2*y - 4*y + 2*y ))/(y*(2*x *y - 4*x - - 9 2 9 9 8 3 8 2 8 8 - + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y - 31*x - - 7 4 7 3 7 2 7 7 6 5 - - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x - 28*x *y - - 6 4 6 3 6 2 6 6 5 6 - + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x - 14*x *y - - 5 5 5 4 5 3 5 2 5 5 - + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y - 5*x - - 4 7 4 6 4 5 4 4 4 3 4 2 - + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y + 97*x *y - - 4 4 3 8 3 7 3 6 3 5 - - 24*x *y + x + 20*x *y - 118*x *y + 244*x *y - 237*x *y - - 3 4 3 3 3 2 3 2 9 2 8 - + 117*x *y - 21*x *y - 7*x *y + 2*x *y + 13*x *y - 86*x *y - - 2 7 2 6 2 5 2 4 2 3 2 2 - + 228*x *y - 294*x *y + 204*x *y - 86*x *y + 23*x *y - 2*x *y - - 10 9 8 7 6 5 - + 4*x*y - 31*x*y + 84*x*y - 121*x*y + 100*x*y - 48*x*y - - 4 3 9 8 7 6 5 - + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*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,x1,x2,x3,x4,x5,x6,x7,x8,x9} - - 4352444991703786550093529782474564455970663240687 -{{x0=---------------------------------------------------, - 8420785423059099972039395927798127489505890997055 - - 459141297061698284317621371232198410031030658042 - x1=---------------------------------------------------, - 1684157084611819994407879185559625497901178199411 - - 1068462443128238131632235196977352568525519548284 - x2=---------------------------------------------------, - 1684157084611819994407879185559625497901178199411 - - 1645748379263608982132912334741766606871657041427 - x3=---------------------------------------------------, - 1684157084611819994407879185559625497901178199411 - - 25308331428404990886292916036626876985377936966579 - x4=----------------------------------------------------, - 42103927115295499860196979638990637447529454985275 - - 17958909252564152456194678743404876001526265937527 - x5=----------------------------------------------------, - 42103927115295499860196979638990637447529454985275 - - - 50670056205024448621117426699348037457452368820774 - x6=-------------------------------------------------------, - 42103927115295499860196979638990637447529454985275 - - - 11882862555847887107599498171234654114612212813799 - x7=-------------------------------------------------------, - 42103927115295499860196979638990637447529454985275 - - - 273286267131634194631661772113331181980867938658 - x8=-----------------------------------------------------, - 8420785423059099972039395927798127489505890997055 - - 46816360472823082478331070276129336252954604132203 - x9=----------------------------------------------------}} - 42103927115295499860196979638990637447529454985275 - - -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,x1,x2,x3,x4,x5,x6,x7,x8,x9} - - 4352444991703786550093529782474564455970663240687 -{{x0=---------------------------------------------------, - 8420785423059099972039395927798127489505890997055 - - 459141297061698284317621371232198410031030658042 - x1=---------------------------------------------------, - 1684157084611819994407879185559625497901178199411 - - 1068462443128238131632235196977352568525519548284 - x2=---------------------------------------------------, - 1684157084611819994407879185559625497901178199411 - - 1645748379263608982132912334741766606871657041427 - x3=---------------------------------------------------, - 1684157084611819994407879185559625497901178199411 - - 25308331428404990886292916036626876985377936966579 - x4=----------------------------------------------------, - 42103927115295499860196979638990637447529454985275 - - 17958909252564152456194678743404876001526265937527 - x5=----------------------------------------------------, - 42103927115295499860196979638990637447529454985275 - - - 50670056205024448621117426699348037457452368820774 - x6=-------------------------------------------------------, - 42103927115295499860196979638990637447529454985275 - - - 11882862555847887107599498171234654114612212813799 - x7=-------------------------------------------------------, - 42103927115295499860196979638990637447529454985275 - - - 273286267131634194631661772113331181980867938658 - x8=-----------------------------------------------------, - 8420785423059099972039395927798127489505890997055 - - 46816360472823082478331070276129336252954604132203 - x9=----------------------------------------------------}} - 42103927115295499860196979638990637447529454985275 - - - -% 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. - -% This one needs the Cramer code --- it takes forever otherwise. - -on cramer; - - - -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(14), - - k24=0, - - k25=0, - - arbcomplex(15)*a - k26=------------------, - c - - k27=0, - - k28=0, - - k29=0, - - k30=0, - - k31=arbcomplex(14), - - k32=0, - - k33=0, - - arbcomplex(15)*b - k34=------------------, - c - - k35=0, - - k36=0, - - k37=0, - - k38=0, - - k39=arbcomplex(14), - - k40=0, - - k41=0, - - k42=arbcomplex(15), - - k43=arbcomplex(16), - - k44=0, - - k45=0, - - k46=0, - - k47=0, - - k48=0, - - k49=0}} - - -off cramer; - - - -% 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: {a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5} - -{{b5=0, - - b4=0, - - a5=0, - - a4=0, - - b3=0, - - b1=arbcomplex(23), - - - 1 - a3=------, - b1 - - b2=0, - - a2=0, - - c5=0, - - c4=0, - - b0=arbcomplex(24), - - b0 + 2*b1 - c3=-----------, - b1 - - a0=arbcomplex(25), - - c1= - a0*b1 - b0 - 2*b1, - - c2=0, - - c0= - a0*b0 + b1}, - - {b5=0, - - b4=0, - - a5=0, - - a4=0, - - b3=-1, - - a3=0, - - b2=0, - - a2=0, - - c5=0, - - c4=0, - - a0=arbcomplex(17), - - c3=a0 + 2, - - b0=arbcomplex(18), - - b1=arbcomplex(19), - - c1= - a0*b1 - b0 - 2*b1, - - c2=0, - - c0= - a0*b0 + b1}, - - {b5=0, - - b4=0, - - a5=0, - - a4=0, - - b3=-1, - - b1=0, - - a3=0, - - b2=0, - - c5=0, - - c4=0, - - a0=arbcomplex(20), - - c3=a0 + 2, - - a2=arbcomplex(21), - - b0=arbcomplex(22), - - c1=a2 - b0, - - c2= - a2*b0, - - c0= - a0*b0}} - - - -% 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: {a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5} - -{} - - -% 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( - l1 + 2*l1*pz + l2 - pz ) -{{sp=------------------------------------, - l2 - - st=1, - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - sf=---------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - l1 - pz - cp=---------, - l2 - - 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - cf=---------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - s1=---------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) - c2=------------------------------------, - l2 - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - c1=------------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - 2 2 2 2 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) - py=--------------------------------------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - - l1 + pz - s2=------------, - l2 - - ct=0}, - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) - {sp=------------------------------------, - l2 - - st=1, - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - sf=---------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - l1 - pz - cp=---------, - l2 - - 2 2 2 2 - - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - cf=------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 2 - - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - s1=------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) - c2=------------------------------------, - l2 - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - c1=------------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - py - - 2 2 2 2 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) - =----------------------------------------------------------------------------- - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - , - - - l1 + pz - s2=------------, - l2 - - ct=0}, - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) - {sp=------------------------------------, - l2 - - st=-1, - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - sf=---------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - - l1 + pz - cp=------------, - l2 - - 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - cf=---------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 2 - - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - s1=------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz ) - c2=---------------------------------------, - l2 - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - c1=---------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - 2 2 2 2 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) - py=--------------------------------------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - - l1 + pz - s2=------------, - l2 - - ct=0}, - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) - {sp=------------------------------------, - l2 - - st=-1, - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - sf=---------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - - l1 + pz - cp=------------, - l2 - - 2 2 2 2 - - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - cf=------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - s1=---------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz ) - c2=---------------------------------------, - l2 - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - c1=---------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - py - - 2 2 2 2 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) - =----------------------------------------------------------------------------- - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - , - - - l1 + pz - s2=------------, - l2 - - ct=0}, - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz ) - {sp=---------------------------------------, - l2 - - st=1, - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - sf=------------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - l1 - pz - cp=---------, - l2 - - 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - cf=---------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - s1=---------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz ) - c2=---------------------------------------, - l2 - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - c1=---------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - py - - 2 2 2 2 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) - =----------------------------------------------------------------------------- - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - , - - - l1 + pz - s2=------------, - l2 - - ct=0}, - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz ) - {sp=---------------------------------------, - l2 - - st=1, - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - sf=------------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - l1 - pz - cp=---------, - l2 - - 2 2 2 2 - - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - cf=------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 2 - - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - s1=------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz ) - c2=---------------------------------------, - l2 - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - c1=---------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - 2 2 2 2 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) - py=--------------------------------------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - - l1 + pz - s2=------------, - l2 - - ct=0}, - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz ) - {sp=---------------------------------------, - l2 - - st=-1, - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - sf=------------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - - l1 + pz - cp=------------, - l2 - - 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - cf=---------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 2 - - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - s1=------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) - c2=------------------------------------, - l2 - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - c1=------------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - py - - 2 2 2 2 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) - =----------------------------------------------------------------------------- - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - , - - - l1 + pz - s2=------------, - l2 - - ct=0}, - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz ) - {sp=---------------------------------------, - l2 - - st=-1, - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - sf=------------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - - l1 + pz - cp=------------, - l2 - - 2 2 2 2 - - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - cf=------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) - s1=---------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) - c2=------------------------------------, - l2 - - 2 2 2 - - sqrt( - l1 + 2*l1*pz + l2 - pz )*px - c1=------------------------------------------, - 2 2 2 - l1 - 2*l1*pz - l2 + pz - - 2 2 2 2 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) - py=--------------------------------------------------------------------------, - 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + pz ) - - - l1 + pz - s2=------------, - l2 - - ct=0}} - - -% 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(26), - - c1*( - c1*beta*epsilon - c1*beta*theta + alpha*epsilon + alpha*theta) - c2=-----------------------------------------------------------------------, - c1*gamma*theta - epsilon*eta - - 2 - c1*( - c1 *beta*gamma + c1*alpha*gamma - c1*beta*eta + alpha*eta) - c3=-------------------------------------------------------------------}} - 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}); - - -{{y3=arbcomplex(28), - - 2 - y2=(99*y3 - 582*y3 - - 4 3 2 - + 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 - - 2 - )/(3*(9*y3 - 66*y3 + 25)), - - 2 - y1=(99*y3 - 582*y3 - - 4 3 2 - - 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 - - 2 - )/(3*(9*y3 - 66*y3 + 25))}, - - {y3=arbcomplex(27), - - 2 - y2=(99*y3 - 582*y3 - - 4 3 2 - - 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 - - 2 - )/(3*(9*y3 - 66*y3 + 25)), - - 2 - y1=(99*y3 - 582*y3 - - 4 3 2 - + 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 - - 2 - )/(3*(9*y3 - 66*y3 + 25))}, - - 11 11 11 - {y3=----,y2=----,y1=----}, - 3 3 3 - - - 5 - 5 - 5 - {y3=------,y2=------,y1=------}} - 3 3 3 - - -% Another nice nonlinear system. - -solve({y=x+t^2,x=y+u^2},{x,y,u,t}); - - -{{t=arbcomplex(31), - - u=t*i, - - y=arbcomplex(32), - - 2 - x=y - t }, - - {t=arbcomplex(29), - - u= - t*i, - - y=arbcomplex(30), - - 2 - x=y - t }} - - -% Example from Stan Kameny (relation between Gamma function values) -% containing surds in the coefficients. - -solve({x54=x14/4,x54*x34=sqrt pi/sqrt 2*x32,x32=x12/2, - x12=sqrt pi, x14*x34=pi*sqrt 2}); - - -Unknowns: {x12,x14,x32,x34,x54} - -{{x54=arbcomplex(33), - - sqrt(2)*pi - x34=------------, - 4*x54 - - x12=sqrt(pi), - - x14=4*arbcomplex(33), - - sqrt(pi) - x32=----------}} - 2 - - -% A system given by J. Hietarinta with complex coefficients. - -on complex; - - - -apu := {2*a - a6,2*b*c3 - 1,i - 2*x + 1,2*x**2 - 2*x + 1,n1 + 1}$ - - - -solve apu; - - -Unknowns: {a,a6,b,c3,n1,x} - -{{c3=arbcomplex(34), - - 1 - b=------, - 2*c3 - - a6 - a=----, - 2 - - n1=-1, - - 1 - x=-------}} - 1 - i - - -clear apu; - - - -off complex; - - - -% More examples that can now be solved. - -solve({e^(x+y)-1,x-y},{x,y}); - - -{{y=log(-1),x=log(-1)},{y=0,x=0}} - - -solve({e^(x+y)+sin x,x-y},{x,y}); - - - 2*y_ -{{y=root_of(e + sin(y_),y_,tag_14),x=y}} - % no algebraic solution exists. - -solve({e^(x+y)-1,x-y**2},{x,y}); - - - 2 2 -{{y=0,x=y },{y=-1,x=y }} - - -solve(e^(y^2) * e^y -1,y); - - -{y=0} - - -solve(e^(y^2 +y)-1,y); - - -{y=0} - - -solve(e^(y^2)-1,y); - - -{y=0} - - -solve(e^(y^2+1)-1,y); - - - 1 -{y=sqrt(log(---)), - e - - 1 - y= - sqrt(log(---))} - e - - -solve({e^(x+y+z)-1,x-y**2=1,x**2-z=2},{x,y,z}); - - - atanh(sqrt(5)) -{{y=2*cosh(----------------)*i, - 3 - - 4 2 - z=y + 2*y - 1, - - 2 - x=y + 1}, - - atanh(sqrt(5)) atanh(sqrt(5)) - {y= - cosh(----------------)*i + sqrt(3)*sinh(----------------), - 3 3 - - 4 2 - z=y + 2*y - 1, - - 2 - x=y + 1}, - - atanh(sqrt(5)) atanh(sqrt(5)) - {y= - (cosh(----------------)*i + sqrt(3)*sinh(----------------)), - 3 3 - - 4 2 - z=y + 2*y - 1, - - 2 - x=y + 1}, - - 4 2 2 - {y=0,z=y + 2*y - 1,x=y + 1}} - - -solve(e^(y^4+3y^2+y)-1,y); - - - 2/3 1/3 1/3 -{y=(sqrt( - 3*(sqrt(5) + 3) - 12*(sqrt(5) + 3) *2 + 2*sqrt( - - 2/3 2/3 1/3 1/3 1/6 - 9*(sqrt(5) + 3) *2 + (sqrt(5) + 3) *sqrt(15)*3 *3 - - 1/3 1/3 1/6 1/3 - + 3*(sqrt(5) + 3) *sqrt(3)*3 *3 + 12*(sqrt(5) + 3) - - 1/3 1/6 1/3 1/6 1/3 1/3 1/6 - + 2*6 *sqrt(15)*3 + 6*6 *sqrt(3)*3 + 6*2 )*3 *3 - - 2/3 1/3 1/3 - - 3*2 ) + (sqrt(5) + 3) *sqrt(3) - 2 *sqrt(3))/(2 - - 1/6 1/6 - *(sqrt(5) + 3) *2 *sqrt(3))} - - -% Transcendental equations proposed by Roger Germundsson -% - -eq1 := 2*asin(x) + asin(2*x) - PI/2; - - - 2*asin(2*x) + 4*asin(x) - pi -eq1 := ------------------------------ - 2 - -eq2 := 2*asin(x) - acos(3*x); - - -eq2 := - acos(3*x) + 2*asin(x) - -eq3 := acos(x) - atan(x); - - -eq3 := acos(x) - atan(x) - -eq4 := acos(2*x**2 - 4*x -x) - 2*asin(x); - - - 2 -eq4 := acos(2*x - 5*x) - 2*asin(x) - -eq5 := 2*atan(x) - atan( 2*x/(1-x**2) ); - - - 2*x -eq5 := atan(--------) + 2*atan(x) - 2 - x - 1 - - -sol1 := solve(eq1,x); - - - sqrt(3) - 1 -sol1 := {x=-------------} - 2 - -sol2 := solve(eq2,x); - - - sqrt(17) - 3 -sol2 := {x=--------------} - 4 - -sol3 := solve(eq3,x); - - - sqrt(sqrt(5) - 1) -sol3 := {x=-------------------} - sqrt(2) - -sol4 := solve(eq4,x); - - -sol4 := {} - -sol5 := solve(eq5,x); - - -sol5 := {x=arbcomplex(36)} - % This solution should be the open interval - % (-1,1). - -% Example 52 of M. Wester: the function has no real zero although -% REDUCE 3.5 and Maple tend to return 3/4. - -if solve(sqrt(x^2 +1) - x +2,x) neq {} then rederr "Illegal result"; - - - -% Using a root_of expression as an algebraic number. - -solve(x^5 - x - 1,x); - - - 5 -{x=root_of(x_ - x_ - 1,x_,tag_20)} - - -w:=rhs first ws; - - - 5 -w := root_of(x_ - x_ - 1,x_,tag_20) - - -w^5; - - - 5 -root_of(x_ - x_ - 1,x_,tag_20) + 1 - - -w^5-w; - - -1 - - -clear w; - - - -end; -(TIME: solve 66960 73789) +REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... + + +% Demonstration of the REDUCE SOLVE package. + +on fullroots; + + % To get complete solutions. + +% Simultaneous linear fractional equations. + +solve({(a*x+y)/(z-1)-3,y+b+z,x-y},{x,y,z}); + + + - 3*(b + 1) +{{x=--------------, + a + 4 + + - 3*(b + 1) + y=--------------, + a + 4 + + - a*b - b + 3 + z=----------------}} + a + 4 + + + +% Use of square-free factorization together with recursive use of +% quadratic and binomial solutions. + +solve((x**6-x**3-1)*(x**5-1)**2*x**2); + + +Unknown: x + + 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=1, + + x=0, + + 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 + + +multiplicities!*; + + +{2,2,2,2,2,2,1,1,1,1,1,1} + + + +% 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( - sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 +{x=-------------------------------------------------, + 2 + + - sqrt( - sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 + x=----------------------------------------------------, + 2 + + sqrt( - sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 + x=----------------------------------------------, + 2 + + - sqrt( - sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 + x=-------------------------------------------------, + 2 + + sqrt(sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 + x=----------------------------------------------, + 2 + + - sqrt(sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 + x=-------------------------------------------------, + 2 + + sqrt(sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 + x=-------------------------------------------, + 2 + + - sqrt(sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 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=sqrt( - i + 116) + 11, + + x= - sqrt( - i + 116) + 11, + + x=sqrt(i + 116) + 11, + + x= - sqrt(i + 116) + 11, + + x=4*sqrt(7) + 11, + + x= - 4*sqrt(7) + 11, + + x=2*sqrt(30) + 11, + + x= - 2*sqrt(30) + 11} + + + +% Recursive use of inverses, including multiple branches of rational +% fractional powers. + +solve(log(acos(asin(x**(2/3)-b)-1))+2,x); + + + 1 1 +{x=sqrt(sin(cos(----) + 1) + b)*(sin(cos(----) + 1) + b), + 2 2 + e e + + 1 1 + x= - sqrt(sin(cos(----) + 1) + b)*(sin(cos(----) + 1) + b)} + 2 2 + e e + + + +% Square-free factors that are unsolvable, being of fifth degree, +% transcendental, or without a defined inverse. + +operator f; + + + +solve((x-1)*(x+1)*(x-2)*(x+2)*(x-3)*(x*log(x)-1)*(f(x)-1),x); + + +{f(x) - 1=0, + + x=root_of(log(x_)*x_ - 1,x_,tag_2), + + x=3, + + x=2, + + x=1, + + x=-1, + + x=-2} + + +multiplicities!*; + + +{1,1,1,1,1,1,1} + + + +% Factors with more than one distinct top-level kernel, the first factor +% a cubic. (Cubic solution suppressed since it is too messy to be of +% much use). + +off fullroots; + + + +solve((x**(1/2)-(x-a)**(1/3))*(acos x-acos(2*x-b))* (2*log x + -log(x**2+x-c)-4),x); + + + 2 4 4 2 + e *(sqrt(4*c*e - 4*c + e ) - e ) +{x=-----------------------------------, + 4 + 2*(e - 1) + + 2 4 4 2 + - e *(sqrt(4*c*e - 4*c + e ) + e ) + x=--------------------------------------, + 4 + 2*(e - 1) + + 2 3 2 + x=root_of(a - 2*a*x_ - x_ + x_ ,x_,tag_7), + + x=b} + + +on fullroots; + + + +% Treatment of multiple-argument exponentials as polynomials. + +solve(a**(2*x)-3*a**x+2,x); + + + 2*arbint(3)*i*pi + log(2) +{x=---------------------------, + log(a) + + 2*arbint(2)*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 + + sqrt( - sqrt(5) - 3) +{x=----------------------, + sqrt(2) + + - sqrt( - sqrt(5) - 3) + x=-------------------------, + 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( - 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(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(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) + + i*(sqrt(5) - 1) + x=-----------------, + 2 + + i*( - sqrt(5) + 1) + x=--------------------} + 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); + + +{x=2*arbint(6)*pi + asin(a), + + x=2*arbint(6)*pi - asin(a) + pi, + + 2*arbint(5)*i*pi + log(b) + x=---------------------------, + log(2) + + 1/c 2*arbint(4)*pi 2*arbint(4)*pi + x=3 *(cos(----------------) + sin(----------------)*i)} + c c + + + +% Automatic restriction to principal branches. + +off allbranch; + + + +solve((sin x-a)*(2**x-b)*(x**c-3),x); + + +{x=asin(a), + + 1/c + x=3 , + + log(b) + x=--------} + log(2) + + + +% 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}); + + +{} + + + +% 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 2 3 +yy(1) := - xx(5)*a - xx(5) - xx(4) + xx(3)*a + xx(2)*a + xx(1) - a *b + + 2 2 2 2 + - 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 2 +yy(2) := b *( - xx(5)*b - xx(5)*b - xx(4)*b + xx(3) + xx(2)*b + xx(1)*b - a*b + + 3 2 + - 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 2 2 + + xx(1)*a*b - a*b *c - 3*a*b *c - 4*a*b*c - a*b - a*c - 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 + xx(2)*a + + 2 4 3 3 + - xx(2)*b + xx(1)*a + xx(1)*b - a - a*c - a - b - b *c - 3*b + + 2 2 + - 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 2 +yy(5) := - 3*xx(5) - xx(4) + 3*xx(3) + 2*xx(2) + xx(1) - 2*a - 3*b - 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}); + + + 2 2 + sqrt(a + b ) - a +{x= - 2*atan(-------------------), + b + + 2 2 + sqrt(a + b ) + a + x=2*atan(-------------------)} + b + + +solve ({a*sin(x+1) + b*cos(x+1)},{x}); + + + 2 2 + sqrt(a + b ) - a +{x= - 2*atan(-------------------) - 1, + b + + 2 2 + sqrt(a + b ) + a + x=2*atan(-------------------) - 1} + 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, + + 12 9 6 3 2 + r=sqrt(y - 4*y + 6*y - 4*y + y + 1)}, + + {y=arbcomplex(11), + + 6 3 + x=y - 2*y + 1, + + 12 9 6 3 2 + r= - sqrt(y - 4*y + 6*y - 4*y + y + 1)}} + + +solve ({e^x - e^(1/2 * x) - 7},{x}); + + + - sqrt(29) + 1 +{x=2*log(-----------------), + 2 + + sqrt(29) + 1 + x=2*log(--------------)} + 2 + + +% Generally not liftable. + + % variable inside and outside of sin. + + solve({sin x + x - 1/2},{x}); + + +{x=root_of(2*sin(x_) + 2*x_ - 1,x_,tag_12)} + + + % Variable inside and outside of exponential. + + solve({e^x - x**2},{x}); + + + - 1 +{x= - 2*lambert_w(------)} + 2 + + + % Variable inside trigonometrical functions with different forms. + + solve ({a*sin(x+1) + b*cos(x+2)},{x}); + + + 2 2 +{x=2*atan((cos(1)*a - sqrt(2*cos(2)*sin(1)*a*b - 2*cos(1)*sin(2)*a*b + a + b ) + + - sin(2)*b)/(cos(2)*b + sin(1)*a)), + + 2 2 + x=2*atan((cos(1)*a + sqrt(2*cos(2)*sin(1)*a*b - 2*cos(1)*sin(2)*a*b + a + b ) + + - sin(2)*b)/(cos(2)*b + sin(1)*a))} + + + % 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*b2*k - l ) + x3=-----------------, + 6*k*l + + 2 2 + 7*(45*b2*b4*k - 15*b4*l - 8*k*l ) + x4=------------------------------------, + 2 + 36*k*l + + 2 2 2 4 + 2205*b2*b4*b6*k - 108*b2*k*l - 735*b4*b6*l - 392*b6*k*l + 36*l + x5=--------------------------------------------------------------------, + 3 + 32*k*l + + 2 2 + x6=(11*(893025*b2*b4*b6*b9*k - 11520*b2*b4*k*l - 43740*b2*b9*k*l + + 2 4 2 4 + - 297675*b4*b6*b9*l + 3840*b4*l - 158760*b6*b9*k*l + 14580*b9*l + + 4 4 + + 2048*k*l ))/(11520*k*l ), + + 2 + x7=(47652707025*b11*b2*b4*b6*b9*k - 614718720*b11*b2*b4*k*l + + 2 2 + - 2334010140*b11*b2*b9*k*l - 15884235675*b11*b4*b6*b9*l + + 4 2 4 + + 204906240*b11*b4*l - 8471592360*b11*b6*b9*k*l + 778003380*b11*b9*l + + 4 + + 109283328*b11*k*l + 172398476250*b15*b2*b4*b6*b9*k + + 2 2 + - 2223936000*b15*b2*b4*k*l - 8444007000*b15*b2*b9*k*l + + 2 4 + - 57466158750*b15*b4*b6*b9*l + 741312000*b15*b4*l + + 2 4 4 + - 30648618000*b15*b6*b9*k*l + 2814669000*b15*b9*l + 395366400*b15*k*l + + 2 4 4 + - 172872000*b2*b4*b6*k*l + 8467200*b2*k*l + 57624000*b4*b6*l + + 4 6 3 + + 30732800*b6*k*l - 2822400*l )/(7729722000*b13*b15*k*l ), + + 2 + x8=(7*(972504225*b11*b2*b4*b6*b9*k - 12545280*b11*b2*b4*k*l + + 2 2 + - 47632860*b11*b2*b9*k*l - 324168075*b11*b4*b6*b9*l + + 4 2 4 + + 4181760*b11*b4*l - 172889640*b11*b6*b9*k*l + 15877620*b11*b9*l + + 4 2 4 + + 2230272*b11*k*l - 3528000*b2*b4*b6*k*l + 172800*b2*k*l + + 4 4 6 4 + + 1176000*b4*b6*l + 627200*b6*k*l - 57600*l ))/(24710400*b15*k*l )}} + + + +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}); + + + 8 8 7 3 7 2 7 7 6 4 +{{x1=(y*( - 8*x *y + 10*x + 9*x *y - 49*x *y + 85*x *y - 43*x + 23*x *y + + 6 3 6 2 6 6 5 5 5 4 + - 128*x *y + 266*x *y - 246*x *y + 77*x + 20*x *y - 145*x *y + + 5 3 5 2 5 5 4 6 4 5 + + 383*x *y - 512*x *y + 329*x *y - 75*x + 9*x *y - 84*x *y + + 4 4 4 3 4 2 4 4 3 7 + + 276*x *y - 469*x *y + 464*x *y - 233*x *y + 43*x + 3*x *y + + 3 6 3 5 3 4 3 3 3 2 3 + - 23*x *y + 97*x *y - 196*x *y + 245*x *y - 201*x *y + 87*x *y + + 3 2 8 2 7 2 6 2 5 2 4 + - 14*x - 2*x *y + 13*x *y - 25*x *y + 23*x *y - 10*x *y + + 2 3 2 2 2 2 9 8 7 + - 17*x *y + 31*x *y - 15*x *y + 2*x - 2*x*y + 10*x*y - 24*x*y + + 6 5 4 3 2 6 5 + + 41*x*y - 57*x*y + 53*x*y - 24*x*y + 2*x*y + x*y - 2*y + 7*y + + 4 3 2 10 10 9 2 9 9 + - 9*y + 5*y - y ))/(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x + + 8 3 8 2 8 8 7 4 7 3 7 2 + + x *y - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y + + 7 7 6 5 6 4 6 3 6 2 + + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y + + 6 6 5 6 5 5 5 4 5 3 + - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y + + 5 2 5 5 4 7 4 6 4 5 4 4 + - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y + + 4 3 4 2 4 4 3 8 3 7 + - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y + + 3 6 3 5 3 4 3 3 3 2 3 + + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y + + 2 9 2 8 2 7 2 6 2 5 2 4 + + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y + + 2 3 2 2 10 9 8 7 + + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y + + 6 5 4 3 9 8 7 6 + + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y + + 5 + + 2*y ), + + 10 10 9 2 9 9 8 3 8 2 8 + x2=(2*x *y - 2*x + 5*x *y - 12*x *y + 7*x - 8*x *y + 9*x *y + 2*x *y + + 8 7 4 7 3 7 2 7 7 6 5 + - x - 15*x *y + 65*x *y - 83*x *y + 52*x *y - 17*x + 5*x *y + + 6 4 6 3 6 2 6 6 5 6 5 5 + - 5*x *y - 20*x *y + 46*x *y - 54*x *y + 20*x + 23*x *y - 151*x *y + + 5 4 5 3 5 2 5 5 4 7 + + 321*x *y - 338*x *y + 166*x *y - 13*x *y - 8*x + 29*x *y + + 4 6 4 5 4 4 4 3 4 2 4 + - 207*x *y + 523*x *y - 676*x *y + 522*x *y - 222*x *y + 36*x *y + + 4 3 8 3 7 3 6 3 5 3 4 + + x + 16*x *y - 103*x *y + 300*x *y - 463*x *y + 433*x *y + + 3 3 3 2 3 2 9 2 7 2 6 2 5 + - 268*x *y + 98*x *y - 15*x *y - x *y + 22*x *y - 54*x *y + 60*x *y + + 2 4 2 3 2 2 2 10 9 8 + - 56*x *y + 44*x *y - 17*x *y + 2*x *y - 2*x*y + 10*x*y - 22*x*y + + 7 6 5 4 3 2 7 6 + + 34*x*y - 48*x*y + 48*x*y - 23*x*y + 2*x*y + x*y - 2*y + 7*y + + 5 4 3 10 10 9 2 9 9 + - 9*y + 5*y - y )/(x*(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x + + 8 3 8 2 8 8 7 4 7 3 + + x *y - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y + + 7 2 7 7 6 5 6 4 6 3 + - 105*x *y + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y + + 6 2 6 6 5 6 5 5 5 4 + + 308*x *y - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y + + 5 3 5 2 5 5 4 7 4 6 + + 401*x *y - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y + + 4 5 4 4 4 3 4 2 4 4 + + 14*x *y + 90*x *y - 149*x *y + 97*x *y - 24*x *y + x + + 3 8 3 7 3 6 3 5 3 4 + + 20*x *y - 118*x *y + 244*x *y - 237*x *y + 117*x *y + + 3 3 3 2 3 2 9 2 8 2 7 + - 21*x *y - 7*x *y + 2*x *y + 13*x *y - 86*x *y + 228*x *y + + 2 6 2 5 2 4 2 3 2 2 10 + - 294*x *y + 204*x *y - 86*x *y + 23*x *y - 2*x *y + 4*x*y + + 9 8 7 6 5 4 + - 31*x*y + 84*x*y - 121*x*y + 100*x*y - 48*x*y + 15*x*y + + 3 9 8 7 6 5 + - 3*x*y + 4*y - 12*y + 15*y - 9*y + 2*y )), + + 9 9 8 2 8 8 7 3 7 2 7 + x3=(2*x *y - 4*x + 8*x *y - 32*x *y + 26*x + 9*x *y - 70*x *y + 131*x *y + + 7 6 4 6 3 6 2 6 6 5 5 + - 66*x + 7*x *y - 73*x *y + 226*x *y - 253*x *y + 89*x + 11*x *y + + 5 4 5 3 5 2 5 5 4 6 + - 81*x *y + 244*x *y - 383*x *y + 280*x *y - 73*x + 13*x *y + + 4 5 4 4 4 3 4 2 4 4 + - 89*x *y + 235*x *y - 367*x *y + 360*x *y - 189*x *y + 39*x + + 3 7 3 6 3 5 3 4 3 3 3 2 + + 9*x *y - 59*x *y + 156*x *y - 227*x *y + 231*x *y - 171*x *y + + 3 3 2 8 2 7 2 6 2 5 2 4 + + 74*x *y - 13*x + 3*x *y - 21*x *y + 62*x *y - 78*x *y + 51*x *y + + 2 3 2 2 2 2 8 7 6 + - 35*x *y + 30*x *y - 14*x *y + 2*x - 5*x*y + 18*x*y - 22*x*y + + 5 4 3 2 8 7 6 5 4 + - x*y + 21*x*y - 13*x*y + x*y + x*y + 2*y - 6*y + 6*y + y - 6*y + + 3 2 10 10 9 2 9 9 8 3 + + 4*y - y )/(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x + x *y + + 8 2 8 8 7 4 7 3 7 2 + - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y + + 7 7 6 5 6 4 6 3 6 2 + + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y + + 6 6 5 6 5 5 5 4 5 3 + - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y + + 5 2 5 5 4 7 4 6 4 5 4 4 + - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y + + 4 3 4 2 4 4 3 8 3 7 + - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y + + 3 6 3 5 3 4 3 3 3 2 3 + + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y + + 2 9 2 8 2 7 2 6 2 5 2 4 + + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y + + 2 3 2 2 10 9 8 7 + + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y + + 6 5 4 3 9 8 7 6 + + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y + + 5 + + 2*y ), + + 9 9 8 2 8 8 7 3 7 2 7 + x4=(2*(2*x *y - 2*x + 4*x *y - 10*x *y + 6*x - 9*x *y + 21*x *y - 13*x *y + + 7 6 4 6 3 6 2 6 6 5 5 + + x - 18*x *y + 88*x *y - 130*x *y + 74*x *y - 14*x - 10*x *y + + 5 4 5 3 5 2 5 5 4 6 + + 74*x *y - 180*x *y + 191*x *y - 90*x *y + 15*x + 4*x *y + + 4 5 4 4 4 3 4 2 4 4 + - 18*x *y - 20*x *y + 105*x *y - 111*x *y + 47*x *y - 7*x + + 3 7 3 6 3 5 3 4 3 3 3 2 + + 16*x *y - 96*x *y + 188*x *y - 155*x *y + 44*x *y + 8*x *y + + 3 3 2 8 2 7 2 6 2 5 + - 6*x *y + x + 10*x *y - 62*x *y + 164*x *y - 219*x *y + + 2 4 2 3 2 2 2 9 8 7 + + 154*x *y - 56*x *y + 10*x *y - x *y + x*y - 13*x*y + 45*x*y + + 6 5 4 3 2 8 7 6 + - 72*x*y + 64*x*y - 35*x*y + 12*x*y - 2*x*y + 2*y - 7*y + 9*y + + 5 4 10 10 9 2 9 9 8 3 + - 5*y + y ))/(x*(2*x *y - 4*x + 8*x *y - 24*x *y + 20*x + x *y + + 8 2 8 8 7 4 7 3 7 2 + - 17*x *y + 47*x *y - 31*x - 24*x *y + 92*x *y - 105*x *y + + 7 7 6 5 6 4 6 3 6 2 + + 18*x *y + 15*x - 28*x *y + 172*x *y - 350*x *y + 308*x *y + + 6 6 5 6 5 5 5 4 5 3 + - 104*x *y + 4*x - 14*x *y + 103*x *y - 290*x *y + 401*x *y + + 5 2 5 5 4 7 4 6 4 5 + - 278*x *y + 83*x *y - 5*x + 6*x *y - 35*x *y + 14*x *y + + 4 4 4 3 4 2 4 4 3 8 + + 90*x *y - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y + + 3 7 3 6 3 5 3 4 3 3 3 2 + - 118*x *y + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y + + 3 2 9 2 8 2 7 2 6 2 5 + + 2*x *y + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y + + 2 4 2 3 2 2 10 9 8 + - 86*x *y + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y + + 7 6 5 4 3 9 8 + - 121*x*y + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + + 7 6 5 + + 15*y - 9*y + 2*y )), + + 10 10 9 2 9 9 8 3 8 2 8 + x5=(2*x *y - 2*x + 7*x *y - 16*x *y + 7*x - 3*x *y - 11*x *y + 21*x *y + + 8 7 4 7 3 7 2 7 7 6 5 + - x - 18*x *y + 60*x *y - 46*x *y + 23*x *y - 17*x - 4*x *y + + 6 4 6 3 6 2 6 6 5 6 5 5 + + 38*x *y - 70*x *y + 40*x *y - 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90*x *y + + 4 3 4 2 4 4 3 8 3 7 + - 149*x *y + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y + + 3 6 3 5 3 4 3 3 3 2 3 + + 244*x *y - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y + + 2 9 2 8 2 7 2 6 2 5 + + 13*x *y - 86*x *y + 228*x *y - 294*x *y + 204*x *y + + 2 4 2 3 2 2 10 9 8 + - 86*x *y + 23*x *y - 2*x *y + 4*x*y - 31*x*y + 84*x*y + + 7 6 5 4 3 9 8 + - 121*x*y + 100*x*y - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + + 7 6 5 + + 15*y - 9*y + 2*y )), + + 9 8 2 8 8 7 3 7 2 7 + x8=(2*( - 2*x + x *y - 10*x *y + 13*x + 5*x *y - 24*x *y + 49*x *y + + 7 6 4 6 3 6 2 6 6 5 5 + - 30*x + 8*x *y - 41*x *y + 75*x *y - 78*x *y + 32*x + 7*x *y + + 5 4 5 3 5 2 5 5 4 6 4 5 + - 35*x *y + 61*x *y - 56*x *y + 41*x *y - 16*x - x *y + 9*x *y + + 4 4 4 3 4 2 4 4 3 7 3 6 + - 10*x *y + 15*x *y - 22*x *y + 6*x *y + 3*x - 10*x *y + 57*x *y + + 3 5 3 4 3 3 3 2 3 2 8 + - 107*x *y + 91*x *y - 55*x *y + 34*x *y - 10*x *y - 8*x *y + + 2 7 2 6 2 5 2 4 2 3 2 2 + + 46*x *y - 105*x *y + 116*x *y - 63*x *y + 23*x *y - 11*x *y + + 2 9 8 7 6 5 4 + + 2*x *y - 2*x*y + 16*x*y - 42*x*y + 54*x*y - 34*x*y + 6*x*y + + 3 2 8 7 6 5 4 3 10 + + x*y + x*y - 2*y + 6*y - 7*y + 3*y + y - y ))/(x*y*(2*x *y + + 10 9 2 9 9 8 3 8 2 8 + - 4*x + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y + + 8 7 4 7 3 7 2 7 7 + - 31*x - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x + + 6 5 6 4 6 3 6 2 6 6 + - 28*x *y + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x + + 5 6 5 5 5 4 5 3 5 2 5 + - 14*x *y + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y + + 5 4 7 4 6 4 5 4 4 4 3 + - 5*x + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y + + 4 2 4 4 3 8 3 7 3 6 + + 97*x *y - 24*x *y + x + 20*x *y - 118*x *y + 244*x *y + + 3 5 3 4 3 3 3 2 3 2 9 + - 237*x *y + 117*x *y - 21*x *y - 7*x *y + 2*x *y + 13*x *y + + 2 8 2 7 2 6 2 5 2 4 2 3 + - 86*x *y + 228*x *y - 294*x *y + 204*x *y - 86*x *y + 23*x *y + + 2 2 10 9 8 7 6 + - 2*x *y + 4*x*y - 31*x*y + 84*x*y - 121*x*y + 100*x*y + + 5 4 3 9 8 7 6 5 + - 48*x*y + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*y + 2*y )), + + 7 2 7 7 6 3 6 2 6 6 5 4 + x9=(6*( - 2*x *y + 2*x *y + 4*x - 4*x *y + 16*x *y - 6*x *y - 8*x + x *y + + 5 3 5 2 5 5 4 5 4 4 4 3 + + 18*x *y - 56*x *y + 26*x *y + 3*x + 4*x *y - 6*x *y - 40*x *y + + 4 2 4 4 3 6 3 5 3 4 3 3 + + 82*x *y - 38*x *y + 2*x - 6*x *y + 15*x *y - 9*x *y + 32*x *y + + 3 2 3 3 2 7 2 5 2 4 2 3 + - 46*x *y + 19*x *y - x + x *y - 5*x *y + 2*x *y - 7*x *y + + 2 2 2 8 7 6 5 4 + + 10*x *y - 3*x *y - 2*x*y + 9*x*y - 4*x*y - 16*x*y + 22*x*y + + 3 7 6 5 4 3 10 10 + - 9*x*y - 2*y + 2*y + 2*y - 4*y + 2*y ))/(y*(2*x *y - 4*x + + 9 2 9 9 8 3 8 2 8 8 + + 8*x *y - 24*x *y + 20*x + x *y - 17*x *y + 47*x *y - 31*x + + 7 4 7 3 7 2 7 7 6 5 + - 24*x *y + 92*x *y - 105*x *y + 18*x *y + 15*x - 28*x *y + + 6 4 6 3 6 2 6 6 5 6 + + 172*x *y - 350*x *y + 308*x *y - 104*x *y + 4*x - 14*x *y + + 5 5 5 4 5 3 5 2 5 5 + + 103*x *y - 290*x *y + 401*x *y - 278*x *y + 83*x *y - 5*x + + 4 7 4 6 4 5 4 4 4 3 4 2 + + 6*x *y - 35*x *y + 14*x *y + 90*x *y - 149*x *y + 97*x *y + + 4 4 3 8 3 7 3 6 3 5 + - 24*x *y + x + 20*x *y - 118*x *y + 244*x *y - 237*x *y + + 3 4 3 3 3 2 3 2 9 2 8 + + 117*x *y - 21*x *y - 7*x *y + 2*x *y + 13*x *y - 86*x *y + + 2 7 2 6 2 5 2 4 2 3 2 2 + + 228*x *y - 294*x *y + 204*x *y - 86*x *y + 23*x *y - 2*x *y + + 10 9 8 7 6 5 + + 4*x*y - 31*x*y + 84*x*y - 121*x*y + 100*x*y - 48*x*y + + 4 3 9 8 7 6 5 + + 15*x*y - 3*x*y + 4*y - 12*y + 15*y - 9*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- + 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+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,x1,x2,x3,x4,x5,x6,x7,x8,x9} + + 4352444991703786550093529782474564455970663240687 +{{x0=---------------------------------------------------, + 8420785423059099972039395927798127489505890997055 + + 459141297061698284317621371232198410031030658042 + x1=---------------------------------------------------, + 1684157084611819994407879185559625497901178199411 + + 1068462443128238131632235196977352568525519548284 + x2=---------------------------------------------------, + 1684157084611819994407879185559625497901178199411 + + 1645748379263608982132912334741766606871657041427 + x3=---------------------------------------------------, + 1684157084611819994407879185559625497901178199411 + + 25308331428404990886292916036626876985377936966579 + x4=----------------------------------------------------, + 42103927115295499860196979638990637447529454985275 + + 17958909252564152456194678743404876001526265937527 + x5=----------------------------------------------------, + 42103927115295499860196979638990637447529454985275 + + - 50670056205024448621117426699348037457452368820774 + x6=-------------------------------------------------------, + 42103927115295499860196979638990637447529454985275 + + - 11882862555847887107599498171234654114612212813799 + x7=-------------------------------------------------------, + 42103927115295499860196979638990637447529454985275 + + - 273286267131634194631661772113331181980867938658 + x8=-----------------------------------------------------, + 8420785423059099972039395927798127489505890997055 + + 46816360472823082478331070276129336252954604132203 + x9=----------------------------------------------------}} + 42103927115295499860196979638990637447529454985275 + + + +% 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. + +% This one needs the Cramer code --- it takes forever otherwise. + +on cramer; + + + +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(14), + + k24=0, + + k25=0, + + arbcomplex(15)*a + k26=------------------, + c + + k27=0, + + k28=0, + + k29=0, + + k30=0, + + k31=arbcomplex(14), + + k32=0, + + k33=0, + + arbcomplex(15)*b + k34=------------------, + c + + k35=0, + + k36=0, + + k37=0, + + k38=0, + + k39=arbcomplex(14), + + k40=0, + + k41=0, + + k42=arbcomplex(15), + + k43=arbcomplex(16), + + k44=0, + + k45=0, + + k46=0, + + k47=0, + + k48=0, + + k49=0}} + + +off cramer; + + + +% 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: {a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5} + +{{b5=0, + + b4=0, + + a5=0, + + a4=0, + + b3=0, + + b1=arbcomplex(23), + + - 1 + a3=------, + b1 + + b2=0, + + a2=0, + + c5=0, + + c4=0, + + b0=arbcomplex(24), + + b0 + 2*b1 + c3=-----------, + b1 + + a0=arbcomplex(25), + + c1= - a0*b1 - b0 - 2*b1, + + c2=0, + + c0= - a0*b0 + b1}, + + {b5=0, + + b4=0, + + a5=0, + + a4=0, + + b3=-1, + + a3=0, + + b2=0, + + a2=0, + + c5=0, + + c4=0, + + a0=arbcomplex(17), + + c3=a0 + 2, + + b0=arbcomplex(18), + + b1=arbcomplex(19), + + c1= - a0*b1 - b0 - 2*b1, + + c2=0, + + c0= - a0*b0 + b1}, + + {b5=0, + + b4=0, + + a5=0, + + a4=0, + + b3=-1, + + b1=0, + + a3=0, + + b2=0, + + c5=0, + + c4=0, + + a0=arbcomplex(20), + + c3=a0 + 2, + + a2=arbcomplex(21), + + b0=arbcomplex(22), + + c1=a2 - b0, + + c2= - a2*b0, + + c0= - a0*b0}} + + + +% 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: {a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5} + +{} + + +% 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( - l1 + 2*l1*pz + l2 - pz ) +{{sp=------------------------------------, + l2 + + st=1, + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*px + sf=---------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + l1 - pz + cp=---------, + l2 + + 2 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + px + pz ) + cf=---------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + px + pz ) + s1=---------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz ) + c2=------------------------------------, + l2 + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*px + c1=------------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + 2 2 2 2 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) + py=--------------------------------------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + - l1 + pz + s2=------------, + l2 + + ct=0}, + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz ) + {sp=------------------------------------, + l2 + + st=1, + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*px + sf=---------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + l1 - pz + cp=---------, + l2 + + 2 2 2 2 + - sqrt(l1 - 2*l1*pz - l2 + px + pz ) + cf=------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 2 + - sqrt(l1 - 2*l1*pz - l2 + px + pz ) + s1=------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz ) + c2=------------------------------------, + l2 + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*px + c1=------------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + py + + 2 2 2 2 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) + =----------------------------------------------------------------------------- + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + , + + - l1 + pz + s2=------------, + l2 + + ct=0}, + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz ) + {sp=------------------------------------, + l2 + + st=-1, + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*px + sf=---------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + - l1 + pz + cp=------------, + l2 + + 2 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + px + pz ) + cf=---------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 2 + - sqrt(l1 - 2*l1*pz - l2 + px + pz ) + s1=------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz ) + c2=---------------------------------------, + l2 + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*px + c1=---------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + 2 2 2 2 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) + py=--------------------------------------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + - l1 + pz + s2=------------, + l2 + + ct=0}, + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz ) + {sp=------------------------------------, + l2 + + st=-1, + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*px + sf=---------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + - l1 + pz + cp=------------, + l2 + + 2 2 2 2 + - sqrt(l1 - 2*l1*pz - l2 + px + pz ) + cf=------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + px + pz ) + s1=---------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz ) + c2=---------------------------------------, + l2 + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*px + c1=---------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + py + + 2 2 2 2 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) + =----------------------------------------------------------------------------- + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + , + + - l1 + pz + s2=------------, + l2 + + ct=0}, + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz ) + {sp=---------------------------------------, + l2 + + st=1, + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*px + sf=------------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + l1 - pz + cp=---------, + l2 + + 2 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + px + pz ) + cf=---------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + px + pz ) + s1=---------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz ) + c2=---------------------------------------, + l2 + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*px + c1=---------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + py + + 2 2 2 2 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) + =----------------------------------------------------------------------------- + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + , + + - l1 + pz + s2=------------, + l2 + + ct=0}, + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz ) + {sp=---------------------------------------, + l2 + + st=1, + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*px + sf=------------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + l1 - pz + cp=---------, + l2 + + 2 2 2 2 + - sqrt(l1 - 2*l1*pz - l2 + px + pz ) + cf=------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 2 + - sqrt(l1 - 2*l1*pz - l2 + px + pz ) + s1=------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz ) + c2=---------------------------------------, + l2 + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*px + c1=---------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + 2 2 2 2 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) + py=--------------------------------------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + - l1 + pz + s2=------------, + l2 + + ct=0}, + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz ) + {sp=---------------------------------------, + l2 + + st=-1, + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*px + sf=------------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + - l1 + pz + cp=------------, + l2 + + 2 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + px + pz ) + cf=---------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 2 + - sqrt(l1 - 2*l1*pz - l2 + px + pz ) + s1=------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz ) + c2=------------------------------------, + l2 + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*px + c1=------------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + py + + 2 2 2 2 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) + =----------------------------------------------------------------------------- + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + , + + - l1 + pz + s2=------------, + l2 + + ct=0}, + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz ) + {sp=---------------------------------------, + l2 + + st=-1, + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*px + sf=------------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + - l1 + pz + cp=------------, + l2 + + 2 2 2 2 + - sqrt(l1 - 2*l1*pz - l2 + px + pz ) + cf=------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + px + pz ) + s1=---------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz ) + c2=------------------------------------, + l2 + + 2 2 2 + - sqrt( - l1 + 2*l1*pz + l2 - pz )*px + c1=------------------------------------------, + 2 2 2 + l1 - 2*l1*pz - l2 + pz + + 2 2 2 2 2 2 2 + sqrt( - l1 + 2*l1*pz + l2 - pz )*sqrt(l1 - 2*l1*pz - l2 + px + pz ) + py=--------------------------------------------------------------------------, + 2 2 2 + sqrt(l1 - 2*l1*pz - l2 + pz ) + + - l1 + pz + s2=------------, + l2 + + ct=0}} + + +% 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(26), + + c1*( - c1*beta*epsilon - c1*beta*theta + alpha*epsilon + alpha*theta) + c2=-----------------------------------------------------------------------, + c1*gamma*theta - epsilon*eta + + 2 + c1*( - c1 *beta*gamma + c1*alpha*gamma - c1*beta*eta + alpha*eta) + c3=-------------------------------------------------------------------}} + 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}); + + +{{y3=arbcomplex(28), + + 2 + y2=(99*y3 - 582*y3 + + 4 3 2 + + 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 + + 2 + )/(3*(9*y3 - 66*y3 + 25)), + + 2 + y1=(99*y3 - 582*y3 + + 4 3 2 + - 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 + + 2 + )/(3*(9*y3 - 66*y3 + 25))}, + + {y3=arbcomplex(27), + + 2 + y2=(99*y3 - 582*y3 + + 4 3 2 + - 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 + + 2 + )/(3*(9*y3 - 66*y3 + 25)), + + 2 + y1=(99*y3 - 582*y3 + + 4 3 2 + + 4*sqrt(243*y3 - 3348*y3 + 15282*y3 - 26100*y3 + 11875)*sqrt(2) - 125 + + 2 + )/(3*(9*y3 - 66*y3 + 25))}, + + 11 11 11 + {y3=----,y2=----,y1=----}, + 3 3 3 + + - 5 - 5 - 5 + {y3=------,y2=------,y1=------}} + 3 3 3 + + +% Another nice nonlinear system. + +solve({y=x+t^2,x=y+u^2},{x,y,u,t}); + + +{{t=arbcomplex(31), + + u=t*i, + + y=arbcomplex(32), + + 2 + x=y - t }, + + {t=arbcomplex(29), + + u= - t*i, + + y=arbcomplex(30), + + 2 + x=y - t }} + + +% Example from Stan Kameny (relation between Gamma function values) +% containing surds in the coefficients. + +solve({x54=x14/4,x54*x34=sqrt pi/sqrt 2*x32,x32=x12/2, + x12=sqrt pi, x14*x34=pi*sqrt 2}); + + +Unknowns: {x12,x14,x32,x34,x54} + +{{x54=arbcomplex(33), + + sqrt(2)*pi + x34=------------, + 4*x54 + + x12=sqrt(pi), + + x14=4*arbcomplex(33), + + sqrt(pi) + x32=----------}} + 2 + + +% A system given by J. Hietarinta with complex coefficients. + +on complex; + + + +apu := {2*a - a6,2*b*c3 - 1,i - 2*x + 1,2*x**2 - 2*x + 1,n1 + 1}$ + + + +solve apu; + + +Unknowns: {a,a6,b,c3,n1,x} + +{{c3=arbcomplex(34), + + 1 + b=------, + 2*c3 + + a6 + a=----, + 2 + + n1=-1, + + 1 + x=-------}} + 1 - i + + +clear apu; + + + +off complex; + + + +% More examples that can now be solved. + +solve({e^(x+y)-1,x-y},{x,y}); + + +{{y=log(-1),x=log(-1)},{y=0,x=0}} + + +solve({e^(x+y)+sin x,x-y},{x,y}); + + + 2*y_ +{{y=root_of(e + sin(y_),y_,tag_14),x=y}} + % no algebraic solution exists. + +solve({e^(x+y)-1,x-y**2},{x,y}); + + + 2 2 +{{y=0,x=y },{y=-1,x=y }} + + +solve(e^(y^2) * e^y -1,y); + + +{y=0} + + +solve(e^(y^2 +y)-1,y); + + +{y=0} + + +solve(e^(y^2)-1,y); + + +{y=0} + + +solve(e^(y^2+1)-1,y); + + + 1 +{y=sqrt(log(---)), + e + + 1 + y= - sqrt(log(---))} + e + + +solve({e^(x+y+z)-1,x-y**2=1,x**2-z=2},{x,y,z}); + + + atanh(sqrt(5)) +{{y=2*cosh(----------------)*i, + 3 + + 4 2 + z=y + 2*y - 1, + + 2 + x=y + 1}, + + atanh(sqrt(5)) atanh(sqrt(5)) + {y= - cosh(----------------)*i + sqrt(3)*sinh(----------------), + 3 3 + + 4 2 + z=y + 2*y - 1, + + 2 + x=y + 1}, + + atanh(sqrt(5)) atanh(sqrt(5)) + {y= - (cosh(----------------)*i + sqrt(3)*sinh(----------------)), + 3 3 + + 4 2 + z=y + 2*y - 1, + + 2 + x=y + 1}, + + 4 2 2 + {y=0,z=y + 2*y - 1,x=y + 1}} + + +solve(e^(y^4+3y^2+y)-1,y); + + + 2/3 1/3 1/3 +{y=(sqrt( - 3*(sqrt(5) + 3) - 12*(sqrt(5) + 3) *2 + 2*sqrt( + + 2/3 2/3 1/3 1/3 1/6 + 9*(sqrt(5) + 3) *2 + (sqrt(5) + 3) *sqrt(15)*3 *3 + + 1/3 1/3 1/6 1/3 + + 3*(sqrt(5) + 3) *sqrt(3)*3 *3 + 12*(sqrt(5) + 3) + + 1/3 1/6 1/3 1/6 1/3 1/3 1/6 + + 2*6 *sqrt(15)*3 + 6*6 *sqrt(3)*3 + 6*2 )*3 *3 + + 2/3 1/3 1/3 + - 3*2 ) + (sqrt(5) + 3) *sqrt(3) - 2 *sqrt(3))/(2 + + 1/6 1/6 + *(sqrt(5) + 3) *2 *sqrt(3))} + + +% Transcendental equations proposed by Roger Germundsson +% + +eq1 := 2*asin(x) + asin(2*x) - PI/2; + + + 2*asin(2*x) + 4*asin(x) - pi +eq1 := ------------------------------ + 2 + +eq2 := 2*asin(x) - acos(3*x); + + +eq2 := - acos(3*x) + 2*asin(x) + +eq3 := acos(x) - atan(x); + + +eq3 := acos(x) - atan(x) + +eq4 := acos(2*x**2 - 4*x -x) - 2*asin(x); + + + 2 +eq4 := acos(2*x - 5*x) - 2*asin(x) + +eq5 := 2*atan(x) - atan( 2*x/(1-x**2) ); + + + 2*x +eq5 := atan(--------) + 2*atan(x) + 2 + x - 1 + + +sol1 := solve(eq1,x); + + + sqrt(3) - 1 +sol1 := {x=-------------} + 2 + +sol2 := solve(eq2,x); + + + sqrt(17) - 3 +sol2 := {x=--------------} + 4 + +sol3 := solve(eq3,x); + + + sqrt(sqrt(5) - 1) +sol3 := {x=-------------------} + sqrt(2) + +sol4 := solve(eq4,x); + + +sol4 := {} + +sol5 := solve(eq5,x); + + +sol5 := {x=arbcomplex(36)} + % This solution should be the open interval + % (-1,1). + +% Example 52 of M. Wester: the function has no real zero although +% REDUCE 3.5 and Maple tend to return 3/4. + +if solve(sqrt(x^2 +1) - x +2,x) neq {} then rederr "Illegal result"; + + + +% Using a root_of expression as an algebraic number. + +solve(x^5 - x - 1,x); + + + 5 +{x=root_of(x_ - x_ - 1,x_,tag_20)} + + +w:=rhs first ws; + + + 5 +w := root_of(x_ - x_ - 1,x_,tag_20) + + +w^5; + + + 5 +root_of(x_ - x_ - 1,x_,tag_20) + 1 + + +w^5-w; + + +1 + + +clear w; + + + +end; +(TIME: solve 66960 73789)