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r38/log/solve.rlg
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Tue Apr 15 00:32:24 2008 run on win32 % 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 x=root_of(( - a + 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}); 6 3 {{x=y - 2*y + 1, y=arbcomplex(12), 12 9 6 3 2 r=sqrt(y - 4*y + 6*y - 4*y + y + 1)}, 6 3 {x=y - 2*y + 1, y=arbcomplex(11), 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_13)} % 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 - 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 - 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 )), 9 9 8 2 8 8 7 3 7 2 x6=(2*(2*x *y - 4*x + 8*x *y - 24*x *y + 16*x - 2*x *y - 19*x *y 7 7 6 4 6 3 6 2 6 6 + 50*x *y - 23*x - 20*x *y + 71*x *y - 46*x *y - 15*x *y + 12*x 5 5 5 4 5 3 5 2 5 5 - 8*x *y + 82*x *y - 195*x *y + 155*x *y - 46*x *y + 2*x 4 6 4 5 4 4 4 3 4 2 4 + 8*x *y - 11*x *y - 81*x *y + 184*x *y - 142*x *y + 46*x *y 4 3 6 3 5 3 4 3 3 3 2 3 - 4*x - 21*x *y + 50*x *y + x *y - 60*x *y + 49*x *y - 14*x *y 3 2 8 2 7 2 6 2 5 2 4 2 3 + x + 6*x *y - 34*x *y + 82*x *y - 99*x *y + 54*x *y - 8*x *y 2 2 2 8 7 6 5 4 - 4*x *y + x *y - 6*x*y + 38*x*y - 79*x*y + 78*x*y - 41*x*y 3 2 7 6 5 4 3 10 + 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} {{a0=arbcomplex(25), a2=0, - 1 a3=------, b1 a4=0, a5=0, b0=arbcomplex(24), b1=arbcomplex(23), b2=0, b3=0, b4=0, b5=0, c0= - a0*b0 + b1, c1= - a0*b1 - b0 - 2*b1, c2=0, b0 + 2*b1 c3=-----------, b1 c4=0, c5=0}, {a0=arbcomplex(20), a2=arbcomplex(21), a3=0, a4=0, a5=0, b0=arbcomplex(22), b1=0, b2=0, b3=-1, b4=0, b5=0, c0= - a0*b0, c1=a2 - b0, c2= - a2*b0, c3=a0 + 2, c4=0, c5=0}, {a0=arbcomplex(17), a2=0, a3=0, a4=0, a5=0, b0=arbcomplex(18), b1=arbcomplex(19), b2=0, b3=-1, b4=0, b5=0, c0= - a0*b0 + b1, c1= - a0*b1 - b0 - 2*b1, c2=0, c3=a0 + 2, c4=0, c5=0}} % Problem 6. solve({2*a3*b3+a5*b3+a3*b5, a5*b3+2*a5*b5+a3*b5, a4*b4, a5*b3+a5*b5+a3*b5+a3*b3, b1, a3*b3, a2*b2, a5*b5, a5*b1+b5+a4*b3+a3*b4, a0*b2+b2+a4*b2+a2*b4+c2+a2*b0+a2*b1, b4+a4*b1, b3+a3*b1, a5*b3+a3*b5, a2*b1+b2, a4*b5+a5*b4, a2*b4+a4*b2, a0*b0+a0*b1+a0*b4+a3*b2+b0+b1+b4+a4*b0+a4*b1 +a2*b5+a4*b4+c1+c4+a5*b2+a2*b3+c0,-1+a3*b0+a0*b3+a0*b5+a5*b0 +b3+b5+a5*b4+a4*b3+a4*b5+a3*b4+a5*b1+a3*b1+c3+c5, a0*b5+a5*b0+a3*b4+2*a5*b4+a5*b1+b5+a4*b3+2*a4*b5+c5, a4*b0+2*a4*b4+a2*b5+b4+a4*b1+a5*b2+a0*b4+c4, c3+a0*b3+2*b3+b5+a4*b3+a3*b0+2*a3*b1+a5*b1+a3*b4, c1+a0*b1+2*b1+a4*b1+a2*b3+b0+a3*b2+b4}); Unknowns: {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 )*px {{c1=---------------------------------------, 2 2 2 l1 - 2*l1*pz - l2 + pz 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) c2=---------------------------------------, l2 2 2 2 2 sqrt(l1 - 2*l1*pz - l2 + px + pz ) s1=---------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) - l1 + pz s2=------------, l2 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 ) , 2 2 2 2 sqrt(l1 - 2*l1*pz - l2 + px + pz ) cf=---------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) ct=0, l1 - pz cp=---------, l2 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px sf=------------------------------------------, 2 2 2 l1 - 2*l1*pz - l2 + pz st=1, 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) sp=---------------------------------------}, 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 - sqrt( - l1 + 2*l1*pz + l2 - pz ) c2=---------------------------------------, l2 2 2 2 2 sqrt(l1 - 2*l1*pz - l2 + px + pz ) s1=---------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) - l1 + pz s2=------------, l2 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 ) , 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) cf=------------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) ct=0, - l1 + pz cp=------------, l2 2 2 2 sqrt( - l1 + 2*l1*pz + l2 - pz )*px sf=---------------------------------------, 2 2 2 l1 - 2*l1*pz - l2 + pz st=-1, 2 2 2 sqrt( - l1 + 2*l1*pz + l2 - pz ) sp=------------------------------------}, 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 - sqrt( - l1 + 2*l1*pz + l2 - pz ) c2=---------------------------------------, l2 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) s1=------------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) - l1 + pz s2=------------, l2 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 ) 2 2 2 2 sqrt(l1 - 2*l1*pz - l2 + px + pz ) cf=---------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) ct=0, - l1 + pz cp=------------, l2 2 2 2 sqrt( - l1 + 2*l1*pz + l2 - pz )*px sf=---------------------------------------, 2 2 2 l1 - 2*l1*pz - l2 + pz st=-1, 2 2 2 sqrt( - l1 + 2*l1*pz + l2 - pz ) sp=------------------------------------}, 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 - sqrt( - l1 + 2*l1*pz + l2 - pz ) c2=---------------------------------------, l2 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) s1=------------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) - l1 + pz s2=------------, l2 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 ) 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) cf=------------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) ct=0, l1 - pz cp=---------, l2 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px sf=------------------------------------------, 2 2 2 l1 - 2*l1*pz - l2 + pz st=1, 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) sp=---------------------------------------}, 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 sqrt( - l1 + 2*l1*pz + l2 - pz ) c2=------------------------------------, l2 2 2 2 2 sqrt(l1 - 2*l1*pz - l2 + px + pz ) s1=---------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) - l1 + pz s2=------------, l2 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 ) 2 2 2 2 sqrt(l1 - 2*l1*pz - l2 + px + pz ) cf=---------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) ct=0, l1 - pz cp=---------, l2 2 2 2 sqrt( - l1 + 2*l1*pz + l2 - pz )*px sf=---------------------------------------, 2 2 2 l1 - 2*l1*pz - l2 + pz st=1, 2 2 2 sqrt( - l1 + 2*l1*pz + l2 - pz ) sp=------------------------------------}, 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 sqrt( - l1 + 2*l1*pz + l2 - pz ) c2=------------------------------------, l2 2 2 2 2 sqrt(l1 - 2*l1*pz - l2 + px + pz ) s1=---------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) - l1 + pz s2=------------, l2 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 ) 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) cf=------------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) ct=0, - l1 + pz cp=------------, l2 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px sf=------------------------------------------, 2 2 2 l1 - 2*l1*pz - l2 + pz st=-1, 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) sp=---------------------------------------}, 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 sqrt( - l1 + 2*l1*pz + l2 - pz ) c2=------------------------------------, l2 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) s1=------------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) - l1 + pz s2=------------, l2 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 ) , 2 2 2 2 sqrt(l1 - 2*l1*pz - l2 + px + pz ) cf=---------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) ct=0, - l1 + pz cp=------------, l2 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz )*px sf=------------------------------------------, 2 2 2 l1 - 2*l1*pz - l2 + pz st=-1, 2 2 2 - sqrt( - l1 + 2*l1*pz + l2 - pz ) sp=---------------------------------------}, 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 sqrt( - l1 + 2*l1*pz + l2 - pz ) c2=------------------------------------, l2 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) s1=------------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) - l1 + pz s2=------------, l2 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 ) , 2 2 2 2 - sqrt(l1 - 2*l1*pz - l2 + px + pz ) cf=------------------------------------------, 2 2 2 sqrt(l1 - 2*l1*pz - l2 + pz ) ct=0, l1 - pz cp=---------, l2 2 2 2 sqrt( - l1 + 2*l1*pz + l2 - pz )*px sf=---------------------------------------, 2 2 2 l1 - 2*l1*pz - l2 + pz st=1, 2 2 2 sqrt( - l1 + 2*l1*pz + l2 - pz ) sp=------------------------------------}} l2 % 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}); 2 c1*( - c1 *beta*gamma + c1*alpha*gamma - c1*beta*eta + alpha*eta) {{c3=-------------------------------------------------------------------, c1*gamma*theta - epsilon*eta c1*( - c1*beta*epsilon - c1*beta*theta + alpha*epsilon + alpha*theta) c2=-----------------------------------------------------------------------, c1*gamma*theta - epsilon*eta c1=arbcomplex(26)}} solve( {( - 81*y1**2*y2**2 + 594*y1**2*y2 - 225*y1**2 + 594*y1*y2**2 - 3492* y1*y2 - 750*y1 - 225*y2**2 - 750*y2 + 14575)/81, ( - 81*y2**2*y3**2 + 594*y2**2*y3 - 225*y2**2 + 594*y2*y3**2 - 3492* y2*y3 - 750*y2 - 225*y3**2 - 750*y3 + 14575)/81, ( - 81*y1**2*y3**2 + 594*y1**2*y3 - 225*y1**2 + 594*y1*y3**2 - 3492* y1*y3 - 750*y1 - 225*y3**2 - 750*y3 + 14575)/81, (2*(81*y1**2*y2**2*y3 + 81*y1**2*y2*y3**2 - 594*y1**2*y2*y3 - 225*y1 **2*y2 - 225*y1**2*y3 + 1650*y1**2 + 81*y1*y2**2*y3**2 - 594*y1* y2**2*y3 - 225*y1*y2**2 - 594*y1*y2*y3**2 + 2592*y1*y2*y3 + 2550 *y1*y2 - 225*y1*y3**2 + 2550*y1*y3 - 3575*y1 - 225*y2**2*y3 + 1650*y2**2 - 225*y2*y3**2 + 2550*y2*y3 - 3575*y2 + 1650*y3**2 - 3575*y3 - 30250))/81}, {y1,y2,y3,y4}); 2 {{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)), 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)), y3=arbcomplex(27)}, 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)), 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)), y3=arbcomplex(28)}, 11 11 11 {y1=----,y2=----,y3=----}, 3 3 3 - 5 - 5 - 5 {y1=------,y2=------,y3=------}} 3 3 3 % Another nice nonlinear system. solve({y=x+t^2,x=y+u^2},{x,y,u,t}); 2 {{x=y - t , y=arbcomplex(32), u=t*i, t=arbcomplex(31)}, 2 {x=y - t , y=arbcomplex(30), u= - t*i, t=arbcomplex(29)}} % 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} {{x12=sqrt(pi), x14=4*arbcomplex(33), sqrt(pi) x32=----------, 2 sqrt(2)*pi x34=------------, 4*x54 x54=arbcomplex(33)}} % 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} a6 {{a=----, 2 1 b=------, 2*c3 c3=arbcomplex(34), 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}); {{x=log(-1),y=log(-1)},{x=0,y=0}} solve({e^(x+y)+sin x,x-y},{x,y}); 2*y_ {{x=y,y=root_of(e + sin(y_),y_,tag_15)}} % no algebraic solution exists. solve({e^(x+y)-1,x-y**2},{x,y}); 2 2 {{x=y ,y=0},{x=y ,y=-1}} 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); {y=i,y= - i} solve({e^(x+y+z)-1,x-y**2=1,x**2-z=2},{x,y,z}); 2 {{x=y + 1, 1 1 asinh(---) asinh(---) 2 2 y=sqrt(3)*cosh(------------)*i + sinh(------------), 3 3 4 2 z=y + 2*y - 1}, 2 {x=y + 1, 1 1 asinh(---) asinh(---) 2 2 y= - sqrt(3)*cosh(------------)*i + sinh(------------), 3 3 4 2 z=y + 2*y - 1}, 2 {x=y + 1, 1 asinh(---) 2 y= - 2*sinh(------------), 3 4 2 z=y + 2*y - 1}, 2 4 2 {x=y + 1,y=0,z=y + 2*y - 1}} solve(e^(y^4+3y^2+y)-1,y); 2/3 1/3 1/3 {y=(sqrt( - (sqrt(5) + 3) - 4*(sqrt(5) + 3) *2 + 2*sqrt( 2/3 2/3 1/3 3*(sqrt(5) + 3) *2 + (sqrt(5) + 3) *sqrt(5) 1/3 1/3 1/3 2/3 + 7*(sqrt(5) + 3) + 2*2 *sqrt(5) + 8*2 ) - 2 ) 1/3 1/3 1/6 1/6 + (sqrt(5) + 3) - 2 )/(2*(sqrt(5) + 3) *2 )} % Transcendental equations proposed by Roger Germundsson % <roger@isy.liu.se> 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_21)} w:=rhs first ws; 5 w := root_of(x_ - x_ - 1,x_,tag_21) w^5; 5 root_of(x_ - x_ - 1,x_,tag_21) + 1 w^5-w; 1 clear w; % The following examples come from Daniel Lichtblau of WRI and were % communicated by Laurent.Bernardin from ETH Zuerich. solve(x-Pi/2 = cos(x+Pi),x); {x=root_of(2*cos(x_) - pi + 2*x_,x_,tag_23)} solve(exp(x^2+x+2)-1,x); sqrt(7)*i - 1 {x=---------------} 2 solve(log(sqrt(1+z)/sqrt(z-1))=x,z); 2*x e + 1 {z=----------} 2*x e - 1 solve({exp(x+3*y-2)=7,3^(2*x-y+4)=2},{x,y}); x + 3*y 2 {{e - 7*e =0}, 2*x y {81*3 - 2*3 =0}} solve(a*3^(c*t)+b*3^((c+a)*t),t); - a log(------) b {t=-------------} log(3)*a solve(log(x+sqrt(x^2+a))=b,{x}); 2*b e - a {x=----------} b 2*e solve(z=log(w)/log(2)+w^2,w); 2 {w=root_of(log(w_) + log(2)*w_ - log(2)*z,w_,tag_26)} solve(w*2^(w^2)=5,w); 2 w_ {w=root_of(2 *w_ - 5,w_,tag_28)} solve(log(x/y)=1/y^2*(x+(1/x)),y); x 2 2 {y=root_of(log(----)*x*y_ - x - 1,y_,tag_30)} y_ solve(exp(z)=w*z^(-n),z); n z_ {z=root_of(z_ *e - w,z_,tag_32)} solve(-log(3)+log(2+y/3)/2-log(y/3)/2=(-I)/2*Pi,y); - 3 {y=------} 5 solve(-log(x)-log(y/x)/2+log(2+y/x)/2=(-3*I)/2*Pi,y); - 2*x {y=--------} 2 x + 1 solve((I+1)*log(x)+(3*I+3)*log(x+3)=7,x); i 3*i 4 i 3*i 3 i 3*i 2 {x=root_of(x_ *(x_ + 3) *x_ + 9*x_ *(x_ + 3) *x_ + 27*x_ *(x_ + 3) *x_ i 3*i 7 + 27*x_ *(x_ + 3) *x_ - e ,x_,tag_34)} solve(x+sqrt(x)=1,x); - sqrt(5) + 3 {x=----------------} 2 solve({cos(1/5+alpha+x)=5,cos(2/5+alpha-x)=6},{alpha,x}); 5*alpha - 5*x + 2 {{cos(-------------------) - 6=0}, 5 5*alpha + 5*x + 1 {cos(-------------------) - 5=0}} 5 end; Time for test: 3886 ms, plus GC time: 582 ms