test_roots = -> run_test [ "roots(x)", "0", "roots(2^x-y,y)", "2^x", "roots(x^2)", "0", "roots(x^3)", "0", "roots(2 x)", "0", "roots(2 x^2)", "0", "roots(i*x^2-13*i*x+36*i)", "[4,9]", "roots(2 x^3)", "0", "roots(6+11*x+6*x^2+x^3)", "[-3,-2,-1]", "roots(a*x^2+b*x+c)", #"[-b/(2*a)-(-4*a*c+b^2)^(1/2)/(2*a),-b/(2*a)+(-4*a*c+b^2)^(1/2)/(2*a)]", "[-1/2*(b^2/(a^2)-4*c/a)^(1/2)-b/(2*a),1/2*(b^2/(a^2)-4*c/a)^(1/2)-b/(2*a)]", "roots(3+7*x+5*x^2+x^3)", "[-3,-1]", "roots(x^3+x^2+x+1)", "[-1,-i,i]", "roots(x^2==1)", "[-1,1]", "roots(3 x + 12 == 24)", "4", "y=roots(x^2+b*x+c/k)[1]", "", "y^2+b*y+c/k", "0", "y=roots(x^2+b*x+c/k)[2]", "", "y^2+b*y+c/k", "0", "y=roots(a*x^2+b*x+c/4)[1]", "", "a*y^2+b*y+c/4", "0", "y=roots(a*x^2+b*x+c/4)[2]", "", "a*y^2+b*y+c/4", "0", "y=quote(y)", "", # -------------------------------------------- # examples covering https://github.com/davidedc/Algebrite/issues/130 "roots(-2*z+x^2+y^2,z)", "1/2*x^2+1/2*y^2", "roots(-2*z+x^2+y^2,z)", "1/2*x^2+1/2*y^2", "roots(x+y+z)", "-y-z", "roots(x+y+z,z)", "-x-y", "roots(x+y+a,z)", "Stop: roots: 1st argument is not a polynomial in the variable z", "clearall", "", # -------------------------------------------- # some more tests with 3rd degree polynomials # including use of cubic formula. # Only the ones marked "DOES use cubic formula" # actually do so, all other examples manage to # be reduced via factoring. # -------------------------------------------- "roots(x^3 + x^2 + x + 1)", "[-1,-i,i]", "roots(2*x^3 + 3*x^2 - 11*x - 6)", "[-3,-1/2,2]", "roots(x^3 - 7*x^2 + 4*x + 12)", "[-1,2,6]", "roots(x^3 + 1)", "[-1,1/2-1/2*i*3^(1/2),1/2+1/2*i*3^(1/2)]", # also: "[-1,-(-1)^(2/3),(-1)^(1/3))", "roots(x^3 - 1)", "[1,-1/2-1/2*i*3^(1/2),-1/2+1/2*i*3^(1/2)]", # also: "[1,-(-1)^(1/3),(-1)^(2/3))", "clearall", "", # DOES use cubic formula "thePoly = x^3 + d", "", "roots(thePoly)", # also OK: # "[-d^(1/3),1/2*d^(1/3)*(1-i*3^(1/2)),1/2*d^(1/3)*(1+i*3^(1/2)))", # or "[-(-1)^(2/3)*d^(1/3),-d^(1/3),(-1)^(1/3)*d^(1/3))", "[1/2*d^(1/3)-1/2*i*3^(1/2)*d^(1/3),1/2*d^(1/3)+1/2*i*3^(1/2)*d^(1/3),-d^(1/3)]", "and((simplify(subst(last[1],x,thePoly)) == 0),(simplify(subst(last[2],x,thePoly)) == 0),(simplify(subst(last[3],x,thePoly)) == 0))", "1", # DOES use cubic formula # the actual format of this solution might change, the important thing # is that the next few tests work, where we plug in the # symbolic solutions in the polynomial again and we check that we # get the zeroes. "clearall", "", "thePoly = a*x^3 + b*x^2 + c*x + d", "", "roots(thePoly)", "[-1/3*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)-b^2/(3*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3))-b/(3*a)+c/(a*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)),(-1/3*a*b*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)-1/2*a*c+1/6*b^2-1/2*i*3^(1/2)*a*c-1/6*i*3^(1/2)*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(2/3)+1/6*i*3^(1/2)*b^2+1/6*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(2/3))/(a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)),(-1/3*a*b*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)-1/2*a*c+1/6*b^2+1/2*i*3^(1/2)*a*c+1/6*i*3^(1/2)*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(2/3)-1/6*i*3^(1/2)*b^2+1/6*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(2/3))/(a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3))]", "and((simplify(subst(last[1],x,thePoly)) == 0),(simplify(subst(last[2],x,thePoly)) == 0),(simplify(subst(last[3],x,thePoly)) == 0))", "1", "roots(x^3 + 6x - 20)", "[2,-1-3*i,-1+3*i]", "roots(x^3 - 6x - 40)", "[4,-2-i*2^(1/2)*3^(1/2),-2+i*2^(1/2)*3^(1/2)]", "roots(x^3 + x^2 - 5x - 5)", "[-1,-5^(1/2),5^(1/2)]", "roots(x^3 - 8x + 3)", "[-3,3/2-1/2*5^(1/2),3/2+1/2*5^(1/2)]", "roots(x^3 - 8x - 3)", "[3,-3/2-1/2*5^(1/2),-3/2+1/2*5^(1/2)]", "roots(x^3 - 18x + 35)", "[-5,5/2-1/2*i*3^(1/2),5/2+1/2*i*3^(1/2)]", "clearall", "", # DOES use cubic formula "thePoly = x^3 - 3x + 1", "", "roots(thePoly)", "[-(-1)^(1/9)+(-1)^(8/9),1/2*cos(1/9*pi)-1/2*cos(8/9*pi)+1/2*i*sin(1/9*pi)-1/2*i*sin(8/9*pi)-3^(1/2)*cos(11/18*pi),1/2*cos(1/9*pi)-1/2*cos(8/9*pi)+1/2*i*sin(1/9*pi)-1/2*i*sin(8/9*pi)+3^(1/2)*cos(11/18*pi)]", # also: "[(3+1/3*(27/2+27/2*i*3^(1/2))^(2/3)-3*i*3^(1/2)+1/3*i*3^(1/2)*(27/2+27/2*i*3^(1/2))^(2/3))/(2*(27/2+27/2*i*3^(1/2))^(1/3)),(3+1/3*(27/2+27/2*i*3^(1/2))^(2/3)+3*i*3^(1/2)-1/3*i*3^(1/2)*(27/2+27/2*i*3^(1/2))^(2/3))/(2*(27/2+27/2*i*3^(1/2))^(1/3)),(-3-1/3*(27/2+27/2*i*3^(1/2))^(2/3))/((27/2+27/2*i*3^(1/2))^(1/3)))", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))", "1", "clearall", "", # DOES use cubic formula "thePoly = x^3 - 3x - 1", "", "roots(thePoly)", "[-(-1)^(2/9)+(-1)^(7/9),1/2*cos(2/9*pi)-1/2*cos(7/9*pi)+1/2*i*sin(2/9*pi)-1/2*i*sin(7/9*pi)-3^(1/2)*cos(13/18*pi),1/2*cos(2/9*pi)-1/2*cos(7/9*pi)+1/2*i*sin(2/9*pi)-1/2*i*sin(7/9*pi)+3^(1/2)*cos(13/18*pi)]", # also: "[(3+1/3*(-27/2+27/2*i*3^(1/2))^(2/3)-3*i*3^(1/2)+1/3*i*3^(1/2)*(-27/2+27/2*i*3^(1/2))^(2/3))/(2*(-27/2+27/2*i*3^(1/2))^(1/3)),(3+1/3*(-27/2+27/2*i*3^(1/2))^(2/3)+3*i*3^(1/2)-1/3*i*3^(1/2)*(-27/2+27/2*i*3^(1/2))^(2/3))/(2*(-27/2+27/2*i*3^(1/2))^(1/3)),(-3-1/3*(-27/2+27/2*i*3^(1/2))^(2/3))/((-27/2+27/2*i*3^(1/2))^(1/3)))", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))", "1", "roots(x^3 - 15x - 4)", "[4,-2-3^(1/2),-2+3^(1/2)]", "roots(2*x^3 - 4x^2 - 22*x + 24)", "[-3,1,4]", "clearall", "", # DOES use cubic formula "thePoly = 3*x^3 - 10*x^2 - 14*x + 27", "", "roots(thePoly)", "[1/3*(10/3-226/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))-(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3)),1/3*(10/3+113/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))+1/2*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3)-113*i*3^(1/2)/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))+1/2*i*3^(1/2)*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3)),1/3*(10/3+113/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))+1/2*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3)+113*i*3^(1/2)/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))-1/2*i*3^(1/2)*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(10^(-13))),(abs(float(subst(float(last[2]),x,thePoly))) < float(10^(-13))), (abs(float(subst(float(last[3]),x,thePoly))) < float(10^(-13))))", "1", "clearall", "", # DOES use cubic formula "thePoly = 1*x^3 + 6*x^2 - 12*x + 8", "", "roots(thePoly)", "[-2+2^(1/3)+2^(2/3)-i*2^(1/3)*3^(1/2)+i*2^(2/3)*3^(1/2),-2+2^(1/3)+2^(2/3)+i*2^(1/3)*3^(1/2)-i*2^(2/3)*3^(1/2),-2*(1+2^(1/3)+2^(2/3))]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-14))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-14))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-14))))", "1", "roots(1*x^3 + 6*x^2 + 12*x + 8)", "-2", "clearall", "", # DOES use cubic formula "thePoly = 1*x^3 + 0*x^2 + 12*x - 10", "", "roots(thePoly)", "[(-6+1/6*(-135+27*89^(1/2))^(2/3)-6*i*3^(1/2)-1/6*i*3^(1/2)*(-135+27*89^(1/2))^(2/3))/((-135+27*89^(1/2))^(1/3)),(-6+1/6*(-135+27*89^(1/2))^(2/3)+6*i*3^(1/2)+1/6*i*3^(1/2)*(-135+27*89^(1/2))^(2/3))/((-135+27*89^(1/2))^(1/3)),(12-1/3*(-135+27*89^(1/2))^(2/3))/((-135+27*89^(1/2))^(1/3))]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-14))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-14))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-14))))", "1", "roots(1*x^3 + 0*x^2 - 18*x + 35)", "[-5,5/2-1/2*i*3^(1/2),5/2+1/2*i*3^(1/2)]", "clearall", "", # DOES use cubic formula "thePoly = 1*x^3 + 0*x^2 - 3*x - 6", "", "roots(thePoly)", "[(3+1/3*(-81+54*2^(1/2))^(2/3)-3*i*3^(1/2)+1/3*i*3^(1/2)*(-81+54*2^(1/2))^(2/3))/(2*(-81+54*2^(1/2))^(1/3)),(3+1/3*(-81+54*2^(1/2))^(2/3)+3*i*3^(1/2)-1/3*i*3^(1/2)*(-81+54*2^(1/2))^(2/3))/(2*(-81+54*2^(1/2))^(1/3)),(-3-1/3*(-81+54*2^(1/2))^(2/3))/((-81+54*2^(1/2))^(1/3))]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-14))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-14))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-14))))", "1", "roots(2*x^3 - 30*x^2 + 162*x - 350)", "[7,4-3*i,4+3*i]", "roots(1*x^3 - 4*x^2 - 6*x + 5)", "[5,-1/2-1/2*5^(1/2),-1/2+1/2*5^(1/2)]", "clearall", "", # DOES use cubic formula "thePoly = 3*x^3 + 6*x^2 + 4*x + 9", "", "roots(thePoly)", "[1/3*(-2-73^(1/3)),1/3*(-2+1/2*73^(1/3)-1/2*i*3^(1/2)*73^(1/3)),1/3*(-2+1/2*73^(1/3)+1/2*i*3^(1/2)*73^(1/3))]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-14))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-14))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-14))))", "1", "clearall", "", # DOES use cubic formula "thePoly = 3*x^3 + 21*x^2 + 2*x + 3", "", "roots(thePoly)", "[1/3*(-7-47/((671/2+1/2*34949^(1/2))^(1/3))-(671/2+1/2*34949^(1/2))^(1/3)),1/3*(-7+47/(2*(671/2+1/2*34949^(1/2))^(1/3))+1/2*(671/2+1/2*34949^(1/2))^(1/3)-47*i*3^(1/2)/(2*(671/2+1/2*34949^(1/2))^(1/3))+1/2*i*3^(1/2)*(671/2+1/2*34949^(1/2))^(1/3)),1/3*(-7+47/(2*(671/2+1/2*34949^(1/2))^(1/3))+1/2*(671/2+1/2*34949^(1/2))^(1/3)+47*i*3^(1/2)/(2*(671/2+1/2*34949^(1/2))^(1/3))-1/2*i*3^(1/2)*(671/2+1/2*34949^(1/2))^(1/3))]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(10^(-12))))", "1", "clearall", "", # DOES use cubic formula "thePoly = 3*x^3 - 6*x^2 + 4*x - 5", "", "roots(thePoly)", # also these ones could be sort of OK: # "[2/3-1/3*(-1)^(1/3)*37^(1/3),2/3+1/6*(-1)^(1/3)*37^(1/3)-(-1)^(5/6)*37^(1/3)/(2*3^(1/2)),2/3+1/6*(-1)^(1/3)*37^(1/3)+(-1)^(5/6)*37^(1/3)/(2*3^(1/2)))", # "[2/3-1/3*(-1)^(1/3)*37^(1/3),2/3-1/6*37^(1/3)+i*37^(1/3)/(2*3^(1/2)),2/3+1/3*37^(1/3))", # "[1/3*(2-(-1)^(1/3)*37^(1/3)),1/3*(2-1/2*37^(1/3)+1/2*i*3^(1/2)*37^(1/3)),1/3*(2+37^(1/3)))", "[2/3-1/3*(-1)^(1/3)*37^(1/3),1/3*(2-1/2*37^(1/3)+1/2*i*3^(1/2)*37^(1/3)),1/3*(2+37^(1/3))]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-14))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-14))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-14))))", "1", "roots(8*x^3 - 36*x^2 + 54*x - 27)", "3/2", "roots(3*x^3 - 5*x^2 - 1*x - 2)", "[2,-1/6-1/6*i*11^(1/2),-1/6+1/6*i*11^(1/2)]", "clearall", "", # DOES use cubic formula "thePoly = 3*x^3 - 6*x^2 + 4*x - i", "", "roots(thePoly)", "[1/3*(2-(8-9*i)^(1/3)),1/3*(2+1/2*(8-9*i)^(1/3)-1/2*i*3^(1/2)*(8-9*i)^(1/3)),1/3*(2+1/2*(8-9*i)^(1/3)+1/2*i*3^(1/2)*(8-9*i)^(1/3))]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))", "1", "clearall", "", # DOES use cubic formula "thePoly = x^3+i", "", "roots(thePoly)", # also these could be OK: # "[1/2*(-1)^(1/6)-1/2*(-1)^(2/3)*3^(1/2),-(-1)^(1/6),1/2*(-1)^(1/6)*(1+i*3^(1/2)))", # "[-1/2*i+1/2*3^(1/2),-(-1)^(1/6),i)", # "[-(-1)^(1/6),-(-1)^(5/6),i)", "[-1/2*i-1/2*3^(1/2),-1/2*i+1/2*3^(1/2),i]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))", "1", "clearall", "", # DOES use cubic formula "thePoly = x^3-i", "", "roots(thePoly)", # "[-i,1/2*(i-3^(1/2)),1/2*(i+3^(1/2)))", # "[-i,(-1)^(1/6),(-1)^(5/6))", "[-3/4*i-1/2*(-1)^(5/6)-1/4*3^(1/2),3/4*i-1/2*(-1)^(5/6)+1/4*3^(1/2),(-1)^(5/6)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))", "1", # some quartics "clearall", "", "thePoly = x^4 + 1", "", "theRoots = roots(thePoly)", "", "theRoots", # "[-(-1)^(1/4),-(-1)^(3/4),(-1)^(1/4),(-1)^(3/4))", "[-1/2*2^(1/2)-1/2*i*2^(1/2),-1/2*2^(1/2)+1/2*i*2^(1/2),1/2*2^(1/2)-1/2*i*2^(1/2),1/2*2^(1/2)+1/2*i*2^(1/2)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-15))))", "1", "clearall", "", "thePoly = 4*x^4 - 1*x^3 + 4*x^2 + 3*x + 5", "", "theRoots = roots(thePoly)", "", "theRoots", "[1/16-1/2*(-125/96-447/(256*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2))-265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)-265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)+1/2*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2),1/16-1/2*(-125/96+447/(256*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2))-265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)-265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)-1/2*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2),1/16+1/2*(-125/96-447/(256*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2))-265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)-265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)+1/2*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2),1/16+1/2*(-125/96+447/(256*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2))-265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)-265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)-1/2*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = x^5 + 1", "", "theRoots = roots(thePoly)", "", "theRoots", # "[-1,-(-1)^(2/5),-(-1)^(4/5),(-1)^(1/5),(-1)^(3/5))", "[-1,cos(1/5*pi)+i*sin(1/5*pi),cos(3/5*pi)+i*sin(3/5*pi),-cos(2/5*pi)-i*sin(2/5*pi),-cos(4/5*pi)-i*sin(4/5*pi)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[5]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = a*x^5 + k", "", "theRoots = roots(thePoly)", "", "theRoots[1] = simplify(theRoots[1])", "", "theRoots[1]", "-(-1)^(2/5)*((k/a)^(2/5))^(1/2)", "theRoots[2] = simplify(theRoots[2])", "", "theRoots[2]", "-(-1)^(4/5)*((k/a)^(2/5))^(1/2)", "theRoots[3] = circexp(theRoots[3])", "", "theRoots[3]", "exp(1/5*i*pi)*(k/a)^(1/5)", "theRoots[4] = circexp(theRoots[4])", "", "theRoots[4]", "exp(3/5*i*pi)*(k/a)^(1/5)", "theRoots[5] = simplify(theRoots[5])", "", "theRoots[5]", "-(k/a)^(1/5)", # unfortunately the comparison here doesn't work, # due to rounding errors float() produces expressions that still hve # a and k, albeit with really small # coefficients, and hence the "<" comparison finds variables with # undefined values and # fails. #"and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[5]),x,thePoly))) < float(2*10^(-12))))", #"1", "clearall", "", "thePoly = x^3 - 7*x^2 + 41*x - 87", "", "theRoots = roots(thePoly)", "", "theRoots", "[3,2-5*i,2+5*i]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = 398683376+1720835*x+2320*x^2+x^3", "", "theRoots = roots(thePoly)", "", # root 1 ~= -961.79, root 2 ~= -895.12, root 3 ~= -463.09 # all three of them have a very small "error" imaginary part ~= 10^-13 "theRoots", "[-2320/3-219895/(3*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))-1/3*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3),-2320/3+219895/(6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))+1/6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3)-219895*i*3^(1/2)/(6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))+1/6*i*3^(1/2)*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3),-2320/3+219895/(6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))+1/6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3)+219895*i*3^(1/2)/(6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))-1/6*i*3^(1/2)*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3)]", # the error here is particularly high because of the big coefficients "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-7))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-7))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-7))))", "1", "clearall", "", "thePoly = x^4 - 1*x^3 + 4*x^2 + 3*x + 5", "", "theRoots = roots(thePoly)", "", "theRoots", "[-1/2-1/2*i*3^(1/2),-1/2+1/2*i*3^(1/2),1-2*i,1+2*i]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = x^4 - 2*x^3 - 7*x^2 + 8*x + 12", "", "theRoots = roots(thePoly)", "", "theRoots", "[-2,-1,2,3]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = x^4+8*x^2+3", "", "theRoots = roots(thePoly)", "", "theRoots", "[-(-4-13^(1/2))^(1/2),-(-4+13^(1/2))^(1/2),(-4-13^(1/2))^(1/2),(-4+13^(1/2))^(1/2)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = -1*x^3-1*x^2+10*x - 8", "", "theRoots = roots(thePoly)", "", "theRoots", "[-4,1,2]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = -3-9*x+3*x^2+x^3", "", "theRoots = roots(thePoly)", "", # these solutions are slightly verbose but they are essentially good, # take into account that ((108+108*i*3^(1/2))^(1/3)) is # really just a compact version (in terms of number of nodes) # for 3*2^(2/3)*(1+i*sqrt(3))^(1/3) # (just take out 108 from the radical, and 108 is 2x2x3x3x3) # so essentially these are written a little redundantly but # they actually are in pretty good form. "theRoots", # "[-1-12/((108+108*i*3^(1/2))^(1/3))-1/3*(108+108*i*3^(1/2))^(1/3),-1+6/((108+108*i*3^(1/2))^(1/3))+1/6*(108+108*i*3^(1/2))^(1/3)-6*i*3^(1/2)/((108+108*i*3^(1/2))^(1/3))+1/6*i*3^(1/2)*(108+108*i*3^(1/2))^(1/3),-1+6/((108+108*i*3^(1/2))^(1/3))+1/6*(108+108*i*3^(1/2))^(1/3)+6*i*3^(1/2)/((108+108*i*3^(1/2))^(1/3))-1/6*i*3^(1/2)*(108+108*i*3^(1/2))^(1/3))", "[-1+cos(1/9*pi)-cos(8/9*pi)+i*sin(1/9*pi)-i*sin(8/9*pi)-2*3^(1/2)*cos(11/18*pi),-1+cos(1/9*pi)-cos(8/9*pi)+i*sin(1/9*pi)-i*sin(8/9*pi)+2*3^(1/2)*cos(11/18*pi),-1-2*cos(1/9*pi)+2*cos(8/9*pi)-2*i*sin(1/9*pi)+2*i*sin(8/9*pi)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = x^4 + 8*x^3 + 12*x^2 + (2*30^(1/2) -16)*x + 4*30^(1/2)-28", "", "theRoots = roots(thePoly)", "", "theRoots", "[-2-1/2*(18-4*5^(1/2))^(1/2)+3^(1/2)/(2^(1/2)),-2-1/2*(18+4*5^(1/2))^(1/2)-1/2*2^(1/2)*3^(1/2),-2+1/2*(18-4*5^(1/2))^(1/2)+3^(1/2)/(2^(1/2)),-2+1/2*(18+4*5^(1/2))^(1/2)-1/2*2^(1/2)*3^(1/2)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = x^3 + x - 2", "", "theRoots = roots(thePoly)", "", "theRoots", "[1,-1/2-1/2*i*7^(1/2),-1/2+1/2*i*7^(1/2)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = x^3 + x^2 - 7", "", "theRoots = roots(thePoly)", "", # note how we can't use "last" here because the assignment returns nothing "and(abs(float(subst(theRoots[1],x,thePoly))) < float(2*10^(-12)),abs(float(subst(theRoots[2],x,thePoly))) < float(2*10^(-12)),abs(float(subst(theRoots[3],x,thePoly))) < float(2*10^(-12)))", "1", # some quartics "clearall", "", "thePoly = x^4 + 8*x^2 + 3", "", "theRoots = roots(thePoly)", "", "theRoots", "[-(-4-13^(1/2))^(1/2),-(-4+13^(1/2))^(1/2),(-4-13^(1/2))^(1/2),(-4+13^(1/2))^(1/2)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(8*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(8*10^(-15))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-15))))", "1", "clearall", "", "thePoly = x^4 - 10*x^3 + 21*x^2 + 40*x - 100", "", "theRoots = roots(thePoly)", "", "theRoots", "[-2,2,5]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = 2*x^4 - 8*x^3 + 2*x^2 + 24*x - 14", "", "theRoots = roots(thePoly)", "", "clearall", "", "thePoly = x^4 - 4*x^3 + x^2 + 12*x - 7", "", "theRoots = roots(thePoly)", "", "theRoots", "[-1/2-1/2*5^(1/2),-1/2+1/2*5^(1/2),5/2-1/2*i*3^(1/2),5/2+1/2*i*3^(1/2)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = 2*x^4 - 8*x^3 + 2*x^2 + 24*x - 14", "", "theRoots = roots(thePoly)", "", "theRoots", "[-1/2-1/2*5^(1/2),-1/2+1/2*5^(1/2),5/2-1/2*i*3^(1/2),5/2+1/2*i*3^(1/2)]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", "1", "clearall", "", "thePoly = x^4 - 9*x^3 + 22*x^2 + 28*x - 120", "", "theRoots = roots(thePoly)", "", "theRoots", "[-2,3,4-2*i,4+2*i]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", "1", #"thePoly = 4* x^4 - 9*x^3 + 22*x^2 + 28*x - 120", #"", # # these are really ugly - sympy or wolfram alpha don't give clean symbolic solutions either #"theRoots = roots(thePoly)", #"", # #"and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", #"1", # #"thePoly = -20*x^4 + 5*x^3 + 17*x^2 - 29*x + 87", #"", # # these are really ugly - sympy or wolfram alpha don't give clean symbolic solutions either #"theRoots = roots(thePoly)", #"", # #"and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))", #"1", # "clearall", "", "thePoly = x^4 + 2*x^3 - 41*x^2 - 42*x + 360", "", "theRoots = roots(thePoly)", "", "theRoots", "[-6,-4,3,5]", "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-15))))", "1", # clean up "thePoly = quote(thePoly)", "", "theRoots = quote(theRoots)", "", ]