Numerik_3

209 days ago by PatrickHammer

8. Die Funktion f (x) = 1/(1+x^2) werde im Intervall [−5, 5] durch ein Polynom p_n interpoliert. Für n = 20 und n = 30 bestimme man das jeweilige Interpolationspolynom p_n und stelle dieses graphisch dar. Dabei verwende man
a) gleichverteilte Stützstellen, siehe Aufgabe 5.
b) als Stützstellen die transformierten Nullstellen des Tschebyscheff-Polynoms T_n+1.
uniform_sampling(a,b,i,n)=a+(b-a)/n*i #cuts the interval into n equally sized parts. P=lambda s,f,a,b,n:sum([f(s(a,b,i,n))*prod([(x-s(a,b,j,n))/(s(a,b,i,n)-s(a,b,j,n)) for j in [0,..,n] if i!=j]) for i in [0,..,n]]) f(x)=1/(1+x^2); show(plot(f,(x,-5,5))); view(f) def doJob(n): L=P(uniform_sampling,f,-5,5,n) show(plot(L,(x,-5,5))); view(factor(L)) doJob(20); doJob(30) 
        

\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{x^{2} + 1}

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{128}{46917885625} \, x^{20} - \frac{12448}{46917885625} \, x^{18} + \frac{1744}{162345625} \, x^{16} - \frac{652468}{2759875625} \, x^{14} + \frac{1317033}{424596250} \, x^{12} - \frac{42652437}{1698385000} \, x^{10} + \frac{696884657}{5519751250} \, x^{8} - \frac{8646802423}{22079005000} \, x^{6} + \frac{565532401}{750686170} \, x^{4} - \frac{72496705}{75068617} \, x^{2} + 1

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{3486784401}{10692496448186020000000} \, x^{30} + \frac{483888190761}{10692496448186020000000} \, x^{28} - \frac{457543597509}{164499945356708000000} \, x^{26} + \frac{401295882069}{4012193789188000000} \, x^{24} - \frac{15039248859}{6419510062700800} \, x^{22} + \frac{1239494800644063}{32899989071341600000} \, x^{20} - \frac{914118872383961451}{2138499289637204000000} \, x^{18} + \frac{435982917499693443}{125794075861012000000} \, x^{16} - \frac{207177910906602303}{10281246584794250000} \, x^{14} + \frac{430804934717414949}{5140623292397125000} \, x^{12} - \frac{797252152648415563}{3212889557748203125} \, x^{10} + \frac{1681741109195371568}{3212889557748203125} \, x^{8} - \frac{1342404500478549689}{1670702570029065625} \, x^{6} + \frac{64072640971026561}{66828102801162625} \, x^{4} - \frac{15774463373569}{15817302438145} \, x^{2} + 1

\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{x^{2} + 1}

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{128}{46917885625} \, x^{20} - \frac{12448}{46917885625} \, x^{18} + \frac{1744}{162345625} \, x^{16} - \frac{652468}{2759875625} \, x^{14} + \frac{1317033}{424596250} \, x^{12} - \frac{42652437}{1698385000} \, x^{10} + \frac{696884657}{5519751250} \, x^{8} - \frac{8646802423}{22079005000} \, x^{6} + \frac{565532401}{750686170} \, x^{4} - \frac{72496705}{75068617} \, x^{2} + 1

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{3486784401}{10692496448186020000000} \, x^{30} + \frac{483888190761}{10692496448186020000000} \, x^{28} - \frac{457543597509}{164499945356708000000} \, x^{26} + \frac{401295882069}{4012193789188000000} \, x^{24} - \frac{15039248859}{6419510062700800} \, x^{22} + \frac{1239494800644063}{32899989071341600000} \, x^{20} - \frac{914118872383961451}{2138499289637204000000} \, x^{18} + \frac{435982917499693443}{125794075861012000000} \, x^{16} - \frac{207177910906602303}{10281246584794250000} \, x^{14} + \frac{430804934717414949}{5140623292397125000} \, x^{12} - \frac{797252152648415563}{3212889557748203125} \, x^{10} + \frac{1681741109195371568}{3212889557748203125} \, x^{8} - \frac{1342404500478549689}{1670702570029065625} \, x^{6} + \frac{64072640971026561}{66828102801162625} \, x^{4} - \frac{15774463373569}{15817302438145} \, x^{2} + 1
def doJob(n): chebyshev_zeros=lambda a,b,i,n: 5*cos((2*i+1)/(2*n+2)*pi) #5* because the formula delivers L=P(chebyshev_zeros,f,None,None,n) #zeros between -1 and 1 show(plot(L,(x,-5,5))); return L L1=doJob(20); L2=doJob(30); print "n=20:",L1,"n=30:",L2 
        
WARNING: Output truncated!  
full_output.txt





n=20: 1048576/22602081298828125*(x - 5*cos(41/42*pi))*(x -
5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x -
5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x -
5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x -
5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x -
5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x -
5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x + 5*sqrt(3))*x/((sqrt(3) -
2*cos(41/42*pi))*(sqrt(3) - 2*cos(13/14*pi))*(sqrt(3) -
2*cos(37/42*pi))*(sqrt(3) - 2*cos(11/14*pi))*(sqrt(3) -
2*cos(31/42*pi))*(sqrt(3) - 2*cos(29/42*pi))*(sqrt(3) -
2*cos(9/14*pi))*(sqrt(3) - 2*cos(25/42*pi))*(sqrt(3) -
2*cos(23/42*pi))*(sqrt(3) - 2*cos(19/42*pi))*(sqrt(3) -
2*cos(17/42*pi))*(sqrt(3) - 2*cos(5/14*pi))*(sqrt(3) -
2*cos(13/42*pi))*(sqrt(3) - 2*cos(11/42*pi))*(sqrt(3) -
2*cos(3/14*pi))*(sqrt(3) - 2*cos(5/42*pi))*(sqrt(3) -
2*cos(1/14*pi))*(sqrt(3) - 2*cos(1/42*pi))) +
1048576/22602081298828125*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x
- 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x -
5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x -
5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x -
5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x -
5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x -
5*cos(1/42*pi))*(2*x - 5*sqrt(3))*x/((sqrt(3) +
2*cos(41/42*pi))*(sqrt(3) + 2*cos(13/14*pi))*(sqrt(3) +
2*cos(37/42*pi))*(sqrt(3) + 2*cos(11/14*pi))*(sqrt(3) +
2*cos(31/42*pi))*(sqrt(3) + 2*cos(29/42*pi))*(sqrt(3) +
2*cos(9/14*pi))*(sqrt(3) + 2*cos(25/42*pi))*(sqrt(3) +
2*cos(23/42*pi))*(sqrt(3) + 2*cos(19/42*pi))*(sqrt(3) +
2*cos(17/42*pi))*(sqrt(3) + 2*cos(5/14*pi))*(sqrt(3) +
2*cos(13/42*pi))*(sqrt(3) + 2*cos(11/42*pi))*(sqrt(3) +
2*cos(3/14*pi))*(sqrt(3) + 2*cos(5/42*pi))*(sqrt(3) +
2*cos(1/14*pi))*(sqrt(3) + 2*cos(1/42*pi))) - 1/286102294921875*(x -
5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x -
5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x -
5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x -
5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x -
5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x -
5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x -
5*sqrt(3))*(2*x +
5*sqrt(3))/(cos(1/42*pi)*cos(1/14*pi)*cos(5/42*pi)*cos(3/14*pi)*cos(11/4\
2*pi)*cos(13/42*pi)*cos(5/14*pi)*cos(17/42*pi)*cos(19/42*pi)*cos(23/42*p\
i)*cos(25/42*pi)*cos(9/14*pi)*cos(29/42*pi)*cos(31/42*pi)*cos(11/14*pi)*\
cos(37/42*pi)*cos(13/14*pi)*cos(41/42*pi)) + 1/95367431640625*(x -
5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x -
5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x -
5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x -
5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x -
5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x -
5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x +
5*sqrt(3))*x/((25*cos(41/42*pi)^2 + 1)*(cos(13/14*pi) -
cos(41/42*pi))*(cos(37/42*pi) - cos(41/42*pi))*(cos(11/14*pi) -
cos(41/42*pi))*(cos(31/42*pi) - cos(41/42*pi))*(cos(29/42*pi) -
cos(41/42*pi))*(cos(9/14*pi) - cos(41/42*pi))*(cos(25/42*pi) -
cos(41/42*pi))*(cos(23/42*pi) - cos(41/42*pi))*(cos(19/42*pi) -
cos(41/42*pi))*(cos(17/42*pi) - cos(41/42*pi))*(cos(5/14*pi) -
cos(41/42*pi))*(cos(13/42*pi) - cos(41/42*pi))*(cos(11/42*pi) -
cos(41/42*pi))*(cos(3/14*pi) - cos(41/42*pi))*(cos(5/42*pi) -
cos(41/42*pi))*(cos(1/14*pi) - cos(41/42*pi))*(cos(1/42*pi) -
cos(41/42*pi))*(sqrt(3) - 2*cos(41/42*pi))*(sqrt(3) +
2*cos(41/42*pi))*cos(41/42*pi)) - 1/95367431640625*(x -
5*cos(41/42*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x -
5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x -
5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x -
5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x -
5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x -
5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x +
5*sqrt(3))*x/((25*cos(13/14*pi)^2 + 1)*(cos(13/14*pi) -
cos(41/42*pi))*(cos(37/42*pi) - cos(13/14*pi))*(cos(11/14*pi) -
cos(13/14*pi))*(cos(31/42*pi) - cos(13/14*pi))*(cos(29/42*pi) -
cos(13/14*pi))*(cos(9/14*pi) - cos(13/14*pi))*(cos(25/42*pi) -
cos(13/14*pi))*(cos(23/42*pi) - cos(13/14*pi))*(cos(19/42*pi) -
cos(13/14*pi))*(cos(17/42*pi) - cos(13/14*pi))*(cos(5/14*pi) -
cos(13/14*pi))*(cos(13/42*pi) - cos(13/14*pi))*(cos(11/42*pi) -
cos(13/14*pi))*(cos(3/14*pi) - cos(13/14*pi))*(cos(5/42*pi) -
cos(13/14*pi))*(cos(1/14*pi) - cos(13/14*pi))*(cos(1/42*pi) -
cos(13/14*pi))*(sqrt(3) - 2*cos(13/14*pi))*(sqrt(3) +
2*cos(13/14*pi))*cos(13/14*pi)) + 1/95367431640625*(x -
5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(11/14*pi))*(x -
5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x -
5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x -
5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x -
5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x -
5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x +
5*sqrt(3))*x/((25*cos(37/42*pi)^2 + 1)*(cos(37/42*pi) -
cos(41/42*pi))*(cos(37/42*pi) - cos(13/14*pi))*(cos(11/14*pi) -
cos(37/42*pi))*(cos(31/42*pi) - cos(37/42*pi))*(cos(29/42*pi) -
cos(37/42*pi))*(cos(9/14*pi) - cos(37/42*pi))*(cos(25/42*pi) -
cos(37/42*pi))*(cos(23/42*pi) - cos(37/42*pi))*(cos(19/42*pi) -
cos(37/42*pi))*(cos(17/42*pi) - cos(37/42*pi))*(cos(5/14*pi) -
cos(37/42*pi))*(cos(13/42*pi) - cos(37/42*pi))*(cos(11/42*pi) -
cos(37/42*pi))*(cos(3/14*pi) - cos(37/42*pi))*(cos(5/42*pi) -
cos(37/42*pi))*(cos(1/14*pi) - cos(37/42*pi))*(cos(1/42*pi) -
cos(37/42*pi))*(sqrt(3) - 2*cos(37/42*pi))*(sqrt(3) +
2*cos(37/42*pi))*cos(37/42*pi)) - 1/95367431640625*(x -
5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x -
5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x -
5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x -
5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x -
5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x -
5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x +
5*sqrt(3))*x/((25*cos(11/14*pi)^2 + 1)*(cos(11/14*pi) -
cos(41/42*pi))*(cos(11/14*pi) - cos(13/14*pi))*(cos(11/14*pi) -
cos(37/42*pi))*(cos(31/42*pi) - cos(11/14*pi))*(cos(29/42*pi) -
cos(11/14*pi))*(cos(9/14*pi) - cos(11/14*pi))*(cos(25/42*pi) -
cos(11/14*pi))*(cos(23/42*pi) - cos(11/14*pi))*(cos(19/42*pi) -
cos(11/14*pi))*(cos(17/42*pi) - cos(11/14*pi))*(cos(5/14*pi) -
cos(11/14*pi))*(cos(13/42*pi) - cos(11/14*pi))*(cos(11/42*pi) -
cos(11/14*pi))*(cos(3/14*pi) - cos(11/14*pi))*(cos(5/42*pi) -
cos(11/14*pi))*(cos(1/14*pi) - cos(11/14*pi))*(cos(1/42*pi) -
cos(11/14*pi))*(sqrt(3) - 2*cos(11/14*pi))*(sqrt(3) +
2*cos(11/14*pi))*cos(11/14*pi)) + 1/95367431640625*(x -
5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x -
5*cos(11/14*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x -
5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x -
5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x -
5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x -
5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x +
5*sqrt(3))*x/((25*cos(31/42*pi)^2 + 1)*(cos(31/42*pi) -
cos(41/42*pi))*(cos(31/42*pi) - cos(13/14*pi))*(cos(31/42*pi) -
cos(37/42*pi))*(cos(31/42*pi) - cos(11/14*pi))*(cos(29/42*pi) -
cos(31/42*pi))*(cos(9/14*pi) - cos(31/42*pi))*(cos(25/42*pi) -
cos(31/42*pi))*(cos(23/42*pi) - cos(31/42*pi))*(cos(19/42*pi) -
cos(31/42*pi))*(cos(17/42*pi) - cos(31/42*pi))*(cos(5/14*pi) -
cos(31/42*pi))*(cos(13/42*pi) - cos(31/42*pi))*(cos(11/42*pi) -
cos(31/42*pi))*(cos(3/14*pi) - cos(31/42*pi))*(cos(5/42*pi) -
cos(31/42*pi))*(cos(1/14*pi) - cos(31/42*pi))*(cos(1/42*pi) -
cos(31/42*pi))*(sqrt(3) - 2*cos(31/42*pi))*(sqrt(3) +
2*cos(31/42*pi))*cos(31/42*pi)) - 1/95367431640625*(x -
5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x -
5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(9/14*pi))*(x -
5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x -
5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x -
5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x -
5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x +
5*sqrt(3))*x/((25*cos(29/42*pi)^2 + 1)*(cos(29/42*pi) -
cos(41/42*pi))*(cos(29/42*pi) - cos(13/14*pi))*(cos(29/42*pi) -
cos(37/42*pi))*(cos(29/42*pi) - cos(11/14*pi))*(cos(29/42*pi) -
cos(31/42*pi))*(cos(9/14*pi) - cos(29/42*pi))*(cos(25/42*pi) -
cos(29/42*pi))*(cos(23/42*pi) - cos(29/42*pi))*(cos(19/42*pi) -
cos(29/42*pi))*(cos(17/42*pi) - cos(29/42*pi))*(cos(5/14*pi) -
cos(29/42*pi))*(cos(13/42*pi) - cos(29/42*pi))*(cos(11/42*pi) -
cos(29/42*pi))*(cos(3/14*pi) - cos(29/42*pi))*(cos(5/42*pi) -
cos(29/42*pi))*(cos(1/14*pi) - cos(29/42*pi))*(cos(1/42*pi) -
cos(29/42*pi))*(sqrt(3) - 2*cos(29/42*pi))*(sqrt(3) +
2*cos(29/42*pi))*cos(29/42*pi)) + 1/95367431640625*(x -
5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x -
5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x -
5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x -
5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x -
5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x -
5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x +
5*sqrt(3))*x/((25*cos(9/14*pi)^2 + 1)*(cos(9/14*pi) -
cos(41/42*pi))*(cos(9/14*pi) - cos(13/14*pi))*(cos(9/14*pi) -
cos(37/42*pi))*(cos(9/14*pi) - cos(11/14*pi))*(cos(9/14*pi) -
cos(31/42*pi))*(cos(9/14*pi) - cos(29/42*pi))*(cos(25/42*pi) -
cos(9/14*pi))*(cos(23/42*pi) - cos(9/14*pi))*(cos(19/42*pi) -
cos(9/14*pi))*(cos(17/42*pi) - cos(9/14*pi))*(cos(5/14*pi) -
cos(9/14*pi))*(cos(13/42*pi) - cos(9/14*pi))*(cos(11/42*pi) -
cos(9/14*pi))*(cos(3/14*pi) - cos(9/14*pi))*(cos(5/42*pi) -
cos(9/14*pi))*(cos(1/14*pi) - cos(9/14*pi))*(cos(1/42*pi) -
cos(9/14*pi))*(sqrt(3) - 2*cos(9/14*pi))*(sqrt(3) +
2*cos(9/14*pi))*cos(9/14*pi)) - 1/95367431640625*(x -
5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x -
5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x -
5*cos(9/14*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x -
5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x -
5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x -
5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x +
5*sqrt(3))*x/((25*cos(25/42*pi)^2 + 1)*(cos(25/42*pi) -
cos(41/42*pi))*(cos(25/42*pi) - cos(13/14*pi))*(cos(25/42*pi) -
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cos(31/42

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cos(11/62*pi))*(cos(1/62*pi) - cos(11/62*pi))*cos(11/62*pi)) +
1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x -
5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x -
5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x -
5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x -
5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x -
5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x -
5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x -
5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x -
5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(7/62*pi))*(x -
5*cos(5/62*pi))*(x - 5*cos(3/62*pi))*(x -
5*cos(1/62*pi))*x/((25*cos(9/62*pi)^2 + 1)*(cos(9/62*pi) -
cos(61/62*pi))*(cos(9/62*pi) - cos(59/62*pi))*(cos(9/62*pi) -
cos(57/62*pi))*(cos(9/62*pi) - cos(55/62*pi))*(cos(9/62*pi) -
cos(53/62*pi))*(cos(9/62*pi) - cos(51/62*pi))*(cos(9/62*pi) -
cos(49/62*pi))*(cos(9/62*pi) - cos(47/62*pi))*(cos(9/62*pi) -
cos(45/62*pi))*(cos(9/62*pi) - cos(43/62*pi))*(cos(9/62*pi) -
cos(41/62*pi))*(cos(9/62*pi) - cos(39/62*pi))*(cos(9/62*pi) -
cos(37/62*pi))*(cos(9/62*pi) - cos(35/62*pi))*(cos(9/62*pi) -
cos(33/62*pi))*(cos(9/62*pi) - cos(29/62*pi))*(cos(9/62*pi) -
cos(27/62*pi))*(cos(9/62*pi) - cos(25/62*pi))*(cos(9/62*pi) -
cos(23/62*pi))*(cos(9/62*pi) - cos(21/62*pi))*(cos(9/62*pi) -
cos(19/62*pi))*(cos(9/62*pi) - cos(17/62*pi))*(cos(9/62*pi) -
cos(15/62*pi))*(cos(9/62*pi) - cos(13/62*pi))*(cos(9/62*pi) -
cos(11/62*pi))*(cos(7/62*pi) - cos(9/62*pi))*(cos(5/62*pi) -
cos(9/62*pi))*(cos(3/62*pi) - cos(9/62*pi))*(cos(1/62*pi) -
cos(9/62*pi))*cos(9/62*pi)) - 1/931322574615478515625*(x -
5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x -
5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x -
5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x -
5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x -
5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x -
5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x -
5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x -
5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x -
5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(5/62*pi))*(x -
5*cos(3/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(7/62*pi)^2 +
1)*(cos(7/62*pi) - cos(61/62*pi))*(cos(7/62*pi) -
cos(59/62*pi))*(cos(7/62*pi) - cos(57/62*pi))*(cos(7/62*pi) -
cos(55/62*pi))*(cos(7/62*pi) - cos(53/62*pi))*(cos(7/62*pi) -
cos(51/62*pi))*(cos(7/62*pi) - cos(49/62*pi))*(cos(7/62*pi) -
cos(47/62*pi))*(cos(7/62*pi) - cos(45/62*pi))*(cos(7/62*pi) -
cos(43/62*pi))*(cos(7/62*pi) - cos(41/62*pi))*(cos(7/62*pi) -
cos(39/62*pi))*(cos(7/62*pi) - cos(37/62*pi))*(cos(7/62*pi) -
cos(35/62*pi))*(cos(7/62*pi) - cos(33/62*pi))*(cos(7/62*pi) -
cos(29/62*pi))*(cos(7/62*pi) - cos(27/62*pi))*(cos(7/62*pi) -
cos(25/62*pi))*(cos(7/62*pi) - cos(23/62*pi))*(cos(7/62*pi) -
cos(21/62*pi))*(cos(7/62*pi) - cos(19/62*pi))*(cos(7/62*pi) -
cos(17/62*pi))*(cos(7/62*pi) - cos(15/62*pi))*(cos(7/62*pi) -
cos(13/62*pi))*(cos(7/62*pi) - cos(11/62*pi))*(cos(7/62*pi) -
cos(9/62*pi))*(cos(5/62*pi) - cos(7/62*pi))*(cos(3/62*pi) -
cos(7/62*pi))*(cos(1/62*pi) - cos(7/62*pi))*cos(7/62*pi)) +
1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x -
5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x -
5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x -
5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x -
5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x -
5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x -
5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x -
5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x -
5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x -
5*cos(7/62*pi))*(x - 5*cos(3/62*pi))*(x -
5*cos(1/62*pi))*x/((25*cos(5/62*pi)^2 + 1)*(cos(5/62*pi) -
cos(61/62*pi))*(cos(5/62*pi) - cos(59/62*pi))*(cos(5/62*pi) -
cos(57/62*pi))*(cos(5/62*pi) - cos(55/62*pi))*(cos(5/62*pi) -
cos(53/62*pi))*(cos(5/62*pi) - cos(51/62*pi))*(cos(5/62*pi) -
cos(49/62*pi))*(cos(5/62*pi) - cos(47/62*pi))*(cos(5/62*pi) -
cos(45/62*pi))*(cos(5/62*pi) - cos(43/62*pi))*(cos(5/62*pi) -
cos(41/62*pi))*(cos(5/62*pi) - cos(39/62*pi))*(cos(5/62*pi) -
cos(37/62*pi))*(cos(5/62*pi) - cos(35/62*pi))*(cos(5/62*pi) -
cos(33/62*pi))*(cos(5/62*pi) - cos(29/62*pi))*(cos(5/62*pi) -
cos(27/62*pi))*(cos(5/62*pi) - cos(25/62*pi))*(cos(5/62*pi) -
cos(23/62*pi))*(cos(5/62*pi) - cos(21/62*pi))*(cos(5/62*pi) -
cos(19/62*pi))*(cos(5/62*pi) - cos(17/62*pi))*(cos(5/62*pi) -
cos(15/62*pi))*(cos(5/62*pi) - cos(13/62*pi))*(cos(5/62*pi) -
cos(11/62*pi))*(cos(5/62*pi) - cos(9/62*pi))*(cos(5/62*pi) -
cos(7/62*pi))*(cos(3/62*pi) - cos(5/62*pi))*(cos(1/62*pi) -
cos(5/62*pi))*cos(5/62*pi)) - 1/931322574615478515625*(x -
5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x -
5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x -
5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x -
5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x -
5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x -
5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x -
5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x -
5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x -
5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(7/62*pi))*(x -
5*cos(5/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(3/62*pi)^2 +
1)*(cos(3/62*pi) - cos(61/62*pi))*(cos(3/62*pi) -
cos(59/62*pi))*(cos(3/62*pi) - cos(57/62*pi))*(cos(3/62*pi) -
cos(55/62*pi))*(cos(3/62*pi) - cos(53/62*pi))*(cos(3/62*pi) -
cos(51/62*pi))*(cos(3/62*pi) - cos(49/62*pi))*(cos(3/62*pi) -
cos(47/62*pi))*(cos(3/62*pi) - cos(45/62*pi))*(cos(3/62*pi) -
cos(43/62*pi))*(cos(3/62*pi) - cos(41/62*pi))*(cos(3/62*pi) -
cos(39/62*pi))*(cos(3/62*pi) - cos(37/62*pi))*(cos(3/62*pi) -
cos(35/62*pi))*(cos(3/62*pi) - cos(33/62*pi))*(cos(3/62*pi) -
cos(29/62*pi))*(cos(3/62*pi) - cos(27/62*pi))*(cos(3/62*pi) -
cos(25/62*pi))*(cos(3/62*pi) - cos(23/62*pi))*(cos(3/62*pi) -
cos(21/62*pi))*(cos(3/62*pi) - cos(19/62*pi))*(cos(3/62*pi) -
cos(17/62*pi))*(cos(3/62*pi) - cos(15/62*pi))*(cos(3/62*pi) -
cos(13/62*pi))*(cos(3/62*pi) - cos(11/62*pi))*(cos(3/62*pi) -
cos(9/62*pi))*(cos(3/62*pi) - cos(7/62*pi))*(cos(3/62*pi) -
cos(5/62*pi))*(cos(1/62*pi) - cos(3/62*pi))*cos(3/62*pi)) +
1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x -
5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x -
5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x -
5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x -
5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x -
5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x -
5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x -
5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x -
5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x -
5*cos(7/62*pi))*(x - 5*cos(5/62*pi))*(x -
5*cos(3/62*pi))*x/((25*cos(1/62*pi)^2 + 1)*(cos(1/62*pi) -
cos(61/62*pi))*(cos(1/62*pi) - cos(59/62*pi))*(cos(1/62*pi) -
cos(57/62*pi))*(cos(1/62*pi) - cos(55/62*pi))*(cos(1/62*pi) -
cos(53/62*pi))*(cos(1/62*pi) - cos(51/62*pi))*(cos(1/62*pi) -
cos(49/62*pi))*(cos(1/62*pi) - cos(47/62*pi))*(cos(1/62*pi) -
cos(45/62*pi))*(cos(1/62*pi) - cos(43/62*pi))*(cos(1/62*pi) -
cos(41/62*pi))*(cos(1/62*pi) - cos(39/62*pi))*(cos(1/62*pi) -
cos(37/62*pi))*(cos(1/62*pi) - cos(35/62*pi))*(cos(1/62*pi) -
cos(33/62*pi))*(cos(1/62*pi) - cos(29/62*pi))*(cos(1/62*pi) -
cos(27/62*pi))*(cos(1/62*pi) - cos(25/62*pi))*(cos(1/62*pi) -
cos(23/62*pi))*(cos(1/62*pi) - cos(21/62*pi))*(cos(1/62*pi) -
cos(19/62*pi))*(cos(1/62*pi) - cos(17/62*pi))*(cos(1/62*pi) -
cos(15/62*pi))*(cos(1/62*pi) - cos(13/62*pi))*(cos(1/62*pi) -
cos(11/62*pi))*(cos(1/62*pi) - cos(9/62*pi))*(cos(1/62*pi) -
cos(7/62*pi))*(cos(1/62*pi) - cos(5/62*pi))*(cos(1/62*pi) -
cos(3/62*pi))*cos(1/62*pi))
WARNING: Output truncated!  
full_output.txt





n=20: 1048576/22602081298828125*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x + 5*sqrt(3))*x/((sqrt(3) - 2*cos(41/42*pi))*(sqrt(3) - 2*cos(13/14*pi))*(sqrt(3) - 2*cos(37/42*pi))*(sqrt(3) - 2*cos(11/14*pi))*(sqrt(3) - 2*cos(31/42*pi))*(sqrt(3) - 2*cos(29/42*pi))*(sqrt(3) - 2*cos(9/14*pi))*(sqrt(3) - 2*cos(25/42*pi))*(sqrt(3) - 2*cos(23/42*pi))*(sqrt(3) - 2*cos(19/42*pi))*(sqrt(3) - 2*cos(17/42*pi))*(sqrt(3) - 2*cos(5/14*pi))*(sqrt(3) - 2*cos(13/42*pi))*(sqrt(3) - 2*cos(11/42*pi))*(sqrt(3) - 2*cos(3/14*pi))*(sqrt(3) - 2*cos(5/42*pi))*(sqrt(3) - 2*cos(1/14*pi))*(sqrt(3) - 2*cos(1/42*pi))) + 1048576/22602081298828125*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*x/((sqrt(3) + 2*cos(41/42*pi))*(sqrt(3) + 2*cos(13/14*pi))*(sqrt(3) + 2*cos(37/42*pi))*(sqrt(3) + 2*cos(11/14*pi))*(sqrt(3) + 2*cos(31/42*pi))*(sqrt(3) + 2*cos(29/42*pi))*(sqrt(3) + 2*cos(9/14*pi))*(sqrt(3) + 2*cos(25/42*pi))*(sqrt(3) + 2*cos(23/42*pi))*(sqrt(3) + 2*cos(19/42*pi))*(sqrt(3) + 2*cos(17/42*pi))*(sqrt(3) + 2*cos(5/14*pi))*(sqrt(3) + 2*cos(13/42*pi))*(sqrt(3) + 2*cos(11/42*pi))*(sqrt(3) + 2*cos(3/14*pi))*(sqrt(3) + 2*cos(5/42*pi))*(sqrt(3) + 2*cos(1/14*pi))*(sqrt(3) + 2*cos(1/42*pi))) - 1/286102294921875*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))/(cos(1/42*pi)*cos(1/14*pi)*cos(5/42*pi)*cos(3/14*pi)*cos(11/42*pi)*cos(13/42*pi)*cos(5/14*pi)*cos(17/42*pi)*cos(19/42*pi)*cos(23/42*pi)*cos(25/42*pi)*cos(9/14*pi)*cos(29/42*pi)*cos(31/42*pi)*cos(11/14*pi)*cos(37/42*pi)*cos(13/14*pi)*cos(41/42*pi)) + 1/95367431640625*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(41/42*pi)^2 + 1)*(cos(13/14*pi) - cos(41/42*pi))*(cos(37/42*pi) - cos(41/42*pi))*(cos(11/14*pi) - cos(41/42*pi))*(cos(31/42*pi) - cos(41/42*pi))*(cos(29/42*pi) - cos(41/42*pi))*(cos(9/14*pi) - cos(41/42*pi))*(cos(25/42*pi) - cos(41/42*pi))*(cos(23/42*pi) - cos(41/42*pi))*(cos(19/42*pi) - cos(41/42*pi))*(cos(17/42*pi) - cos(41/42*pi))*(cos(5/14*pi) - cos(41/42*pi))*(cos(13/42*pi) - cos(41/42*pi))*(cos(11/42*pi) - cos(41/42*pi))*(cos(3/14*pi) - cos(41/42*pi))*(cos(5/42*pi) - cos(41/42*pi))*(cos(1/14*pi) - cos(41/42*pi))*(cos(1/42*pi) - cos(41/42*pi))*(sqrt(3) - 2*cos(41/42*pi))*(sqrt(3) + 2*cos(41/42*pi))*cos(41/42*pi)) - 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(13/14*pi)^2 + 1)*(cos(13/14*pi) - cos(41/42*pi))*(cos(37/42*pi) - cos(13/14*pi))*(cos(11/14*pi) - cos(13/14*pi))*(cos(31/42*pi) - cos(13/14*pi))*(cos(29/42*pi) - cos(13/14*pi))*(cos(9/14*pi) - cos(13/14*pi))*(cos(25/42*pi) - cos(13/14*pi))*(cos(23/42*pi) - cos(13/14*pi))*(cos(19/42*pi) - cos(13/14*pi))*(cos(17/42*pi) - cos(13/14*pi))*(cos(5/14*pi) - cos(13/14*pi))*(cos(13/42*pi) - cos(13/14*pi))*(cos(11/42*pi) - cos(13/14*pi))*(cos(3/14*pi) - cos(13/14*pi))*(cos(5/42*pi) - cos(13/14*pi))*(cos(1/14*pi) - cos(13/14*pi))*(cos(1/42*pi) - cos(13/14*pi))*(sqrt(3) - 2*cos(13/14*pi))*(sqrt(3) + 2*cos(13/14*pi))*cos(13/14*pi)) + 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(37/42*pi)^2 + 1)*(cos(37/42*pi) - cos(41/42*pi))*(cos(37/42*pi) - cos(13/14*pi))*(cos(11/14*pi) - cos(37/42*pi))*(cos(31/42*pi) - cos(37/42*pi))*(cos(29/42*pi) - cos(37/42*pi))*(cos(9/14*pi) - cos(37/42*pi))*(cos(25/42*pi) - cos(37/42*pi))*(cos(23/42*pi) - cos(37/42*pi))*(cos(19/42*pi) - cos(37/42*pi))*(cos(17/42*pi) - cos(37/42*pi))*(cos(5/14*pi) - cos(37/42*pi))*(cos(13/42*pi) - cos(37/42*pi))*(cos(11/42*pi) - cos(37/42*pi))*(cos(3/14*pi) - cos(37/42*pi))*(cos(5/42*pi) - cos(37/42*pi))*(cos(1/14*pi) - cos(37/42*pi))*(cos(1/42*pi) - cos(37/42*pi))*(sqrt(3) - 2*cos(37/42*pi))*(sqrt(3) + 2*cos(37/42*pi))*cos(37/42*pi)) - 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(11/14*pi)^2 + 1)*(cos(11/14*pi) - cos(41/42*pi))*(cos(11/14*pi) - cos(13/14*pi))*(cos(11/14*pi) - cos(37/42*pi))*(cos(31/42*pi) - cos(11/14*pi))*(cos(29/42*pi) - cos(11/14*pi))*(cos(9/14*pi) - cos(11/14*pi))*(cos(25/42*pi) - cos(11/14*pi))*(cos(23/42*pi) - cos(11/14*pi))*(cos(19/42*pi) - cos(11/14*pi))*(cos(17/42*pi) - cos(11/14*pi))*(cos(5/14*pi) - cos(11/14*pi))*(cos(13/42*pi) - cos(11/14*pi))*(cos(11/42*pi) - cos(11/14*pi))*(cos(3/14*pi) - cos(11/14*pi))*(cos(5/42*pi) - cos(11/14*pi))*(cos(1/14*pi) - cos(11/14*pi))*(cos(1/42*pi) - cos(11/14*pi))*(sqrt(3) - 2*cos(11/14*pi))*(sqrt(3) + 2*cos(11/14*pi))*cos(11/14*pi)) + 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(31/42*pi)^2 + 1)*(cos(31/42*pi) - cos(41/42*pi))*(cos(31/42*pi) - cos(13/14*pi))*(cos(31/42*pi) - cos(37/42*pi))*(cos(31/42*pi) - cos(11/14*pi))*(cos(29/42*pi) - cos(31/42*pi))*(cos(9/14*pi) - cos(31/42*pi))*(cos(25/42*pi) - cos(31/42*pi))*(cos(23/42*pi) - cos(31/42*pi))*(cos(19/42*pi) - cos(31/42*pi))*(cos(17/42*pi) - cos(31/42*pi))*(cos(5/14*pi) - cos(31/42*pi))*(cos(13/42*pi) - cos(31/42*pi))*(cos(11/42*pi) - cos(31/42*pi))*(cos(3/14*pi) - cos(31/42*pi))*(cos(5/42*pi) - cos(31/42*pi))*(cos(1/14*pi) - cos(31/42*pi))*(cos(1/42*pi) - cos(31/42*pi))*(sqrt(3) - 2*cos(31/42*pi))*(sqrt(3) + 2*cos(31/42*pi))*cos(31/42*pi)) - 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(29/42*pi)^2 + 1)*(cos(29/42*pi) - cos(41/42*pi))*(cos(29/42*pi) - cos(13/14*pi))*(cos(29/42*pi) - cos(37/42*pi))*(cos(29/42*pi) - cos(11/14*pi))*(cos(29/42*pi) - cos(31/42*pi))*(cos(9/14*pi) - cos(29/42*pi))*(cos(25/42*pi) - cos(29/42*pi))*(cos(23/42*pi) - cos(29/42*pi))*(cos(19/42*pi) - cos(29/42*pi))*(cos(17/42*pi) - cos(29/42*pi))*(cos(5/14*pi) - cos(29/42*pi))*(cos(13/42*pi) - cos(29/42*pi))*(cos(11/42*pi) - cos(29/42*pi))*(cos(3/14*pi) - cos(29/42*pi))*(cos(5/42*pi) - cos(29/42*pi))*(cos(1/14*pi) - cos(29/42*pi))*(cos(1/42*pi) - cos(29/42*pi))*(sqrt(3) - 2*cos(29/42*pi))*(sqrt(3) + 2*cos(29/42*pi))*cos(29/42*pi)) + 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(9/14*pi)^2 + 1)*(cos(9/14*pi) - cos(41/42*pi))*(cos(9/14*pi) - cos(13/14*pi))*(cos(9/14*pi) - cos(37/42*pi))*(cos(9/14*pi) - cos(11/14*pi))*(cos(9/14*pi) - cos(31/42*pi))*(cos(9/14*pi) - cos(29/42*pi))*(cos(25/42*pi) - cos(9/14*pi))*(cos(23/42*pi) - cos(9/14*pi))*(cos(19/42*pi) - cos(9/14*pi))*(cos(17/42*pi) - cos(9/14*pi))*(cos(5/14*pi) - cos(9/14*pi))*(cos(13/42*pi) - cos(9/14*pi))*(cos(11/42*pi) - cos(9/14*pi))*(cos(3/14*pi) - cos(9/14*pi))*(cos(5/42*pi) - cos(9/14*pi))*(cos(1/14*pi) - cos(9/14*pi))*(cos(1/42*pi) - cos(9/14*pi))*(sqrt(3) - 2*cos(9/14*pi))*(sqrt(3) + 2*cos(9/14*pi))*cos(9/14*pi)) - 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(25/42*pi)^2 + 1)*(cos(25/42*pi) - cos(41/42*pi))*(cos(25/42*pi) - cos(13/14*pi))*(cos(25/42*pi) - cos(37/42*pi))*(cos(25/42*pi) - cos(11/14*pi))*(cos(25/42*pi) - cos(31/42*pi))*(cos(25/42*pi) - cos(29/42*pi))*(cos(25/42*pi) - cos(9/14*pi))*(cos(23/42*pi) - cos(25/42*pi))*(cos(19/42*pi) - cos(25/42*pi))*(cos(17/42*pi) - cos(25/42*pi))*(cos(5/14*pi) - cos(25/42*pi))*(cos(13/42*pi) - cos(25/42*pi))*(cos(11/42*pi) - cos(25/42*pi))*(cos(3/14*pi) - cos(25/42*pi))*(cos(5/42*pi) - cos(25/42*pi))*(cos(1/14*pi) - cos(25/42*pi))*(cos(1/42*pi) - cos(25/42*pi))*(sqrt(3) - 2*cos(25/42*pi))*(sqrt(3) + 2*cos(25/42*pi))*cos(25/42*pi)) + 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(23/42*pi)^2 + 1)*(cos(23/42*pi) - cos(41/42*pi))*(cos(23/42*pi) - cos(13/14*pi))*(cos(23/42*pi) - cos(37/42*pi))*(cos(23/42*pi) - cos(11/14*pi))*(cos(23/42*pi) - cos(31/42*pi))*(cos(23/42*pi) - cos(29/42*pi))*(cos(23/42*pi) - cos(9/14*pi))*(cos(23/42*pi) - cos(25/42*pi))*(cos(19/42*pi) - cos(23/42*pi))*(cos(17/42*pi) - cos(23/42*pi))*(cos(5/14*pi) - cos(23/42*pi))*(cos(13/42*pi) - cos(23/42*pi))*(cos(11/42*pi) - cos(23/42*pi))*(cos(3/14*pi) - cos(23/42*pi))*(cos(5/42*pi) - cos(23/42*pi))*(cos(1/14*pi) - cos(23/42*pi))*(cos(1/42*pi) - cos(23/42*pi))*(sqrt(3) - 2*cos(23/42*pi))*(sqrt(3) + 2*cos(23/42*pi))*cos(23/42*pi)) - 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(19/42*pi)^2 + 1)*(cos(19/42*pi) - cos(41/42*pi))*(cos(19/42*pi) - cos(13/14*pi))*(cos(19/42*pi) - cos(37/42*pi))*(cos(19/42*pi) - cos(11/14*pi))*(cos(19/42*pi) - cos(31/42*pi))*(cos(19/42*pi) - cos(29/42*pi))*(cos(19/42*pi) - cos(9/14*pi))*(cos(19/42*pi) - cos(25/42*pi))*(cos(19/42*pi) - cos(23/42*pi))*(cos(17/42*pi) - cos(19/42*pi))*(cos(5/14*pi) - cos(19/42*pi))*(cos(13/42*pi) - cos(19/42*pi))*(cos(11/42*pi) - cos(19/42*pi))*(cos(3/14*pi) - cos(19/42*pi))*(cos(5/42*pi) - cos(19/42*pi))*(cos(1/14*pi) - cos(19/42*pi))*(cos(1/42*pi) - cos(19/42*pi))*(sqrt(3) - 2*cos(19/42*pi))*(sqrt(3) + 2*cos(19/42*pi))*cos(19/42*pi)) + 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(13/42*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(17/42*pi)^2 + 1)*(cos(17/42*pi) - 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cos(41/42*pi))*(cos(5/14*pi) - cos(13/14*pi))*(cos(5/14*pi) - cos(37/42*pi))*(cos(5/14*pi) - cos(11/14*pi))*(cos(5/14*pi) - cos(31/42*pi))*(cos(5/14*pi) - cos(29/42*pi))*(cos(5/14*pi) - cos(9/14*pi))*(cos(5/14*pi) - cos(25/42*pi))*(cos(5/14*pi) - cos(23/42*pi))*(cos(5/14*pi) - cos(19/42*pi))*(cos(5/14*pi) - cos(17/42*pi))*(cos(13/42*pi) - cos(5/14*pi))*(cos(11/42*pi) - cos(5/14*pi))*(cos(3/14*pi) - cos(5/14*pi))*(cos(5/42*pi) - cos(5/14*pi))*(cos(1/14*pi) - cos(5/14*pi))*(cos(1/42*pi) - cos(5/14*pi))*(sqrt(3) - 2*cos(5/14*pi))*(sqrt(3) + 2*cos(5/14*pi))*cos(5/14*pi)) + 1/95367431640625*(x - 5*cos(41/42*pi))*(x - 5*cos(13/14*pi))*(x - 5*cos(37/42*pi))*(x - 5*cos(11/14*pi))*(x - 5*cos(31/42*pi))*(x - 5*cos(29/42*pi))*(x - 5*cos(9/14*pi))*(x - 5*cos(25/42*pi))*(x - 5*cos(23/42*pi))*(x - 5*cos(19/42*pi))*(x - 5*cos(17/42*pi))*(x - 5*cos(5/14*pi))*(x - 5*cos(11/42*pi))*(x - 5*cos(3/14*pi))*(x - 5*cos(5/42*pi))*(x - 5*cos(1/14*pi))*(x - 5*cos(1/42*pi))*(2*x - 5*sqrt(3))*(2*x + 5*sqrt(3))*x/((25*cos(13/42*pi)^2 + 1)*(cos(13/42*pi) - cos(41/42*pi))*(cos(13/42*pi) - cos(13/14*pi))*(cos(13/42*pi) - cos(37/42*pi))*(cos(13/42*pi) - cos(11/14*pi))*(cos(13/42*pi) - cos(31/42

...

5/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(7/62*pi))*(x - 5*cos(5/62*pi))*(x - 5*cos(3/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(19/62*pi)^2 + 1)*(cos(19/62*pi) - cos(61/62*pi))*(cos(19/62*pi) - cos(59/62*pi))*(cos(19/62*pi) - cos(57/62*pi))*(cos(19/62*pi) - cos(55/62*pi))*(cos(19/62*pi) - cos(53/62*pi))*(cos(19/62*pi) - cos(51/62*pi))*(cos(19/62*pi) - cos(49/62*pi))*(cos(19/62*pi) - cos(47/62*pi))*(cos(19/62*pi) - cos(45/62*pi))*(cos(19/62*pi) - cos(43/62*pi))*(cos(19/62*pi) - cos(41/62*pi))*(cos(19/62*pi) - cos(39/62*pi))*(cos(19/62*pi) - cos(37/62*pi))*(cos(19/62*pi) - cos(35/62*pi))*(cos(19/62*pi) - cos(33/62*pi))*(cos(19/62*pi) - cos(29/62*pi))*(cos(19/62*pi) - cos(27/62*pi))*(cos(19/62*pi) - cos(25/62*pi))*(cos(19/62*pi) - cos(23/62*pi))*(cos(19/62*pi) - cos(21/62*pi))*(cos(17/62*pi) - cos(19/62*pi))*(cos(15/62*pi) - cos(19/62*pi))*(cos(13/62*pi) - cos(19/62*pi))*(cos(11/62*pi) - cos(19/62*pi))*(cos(9/62*pi) - cos(19/62*pi))*(cos(7/62*pi) - cos(19/62*pi))*(cos(5/62*pi) - cos(19/62*pi))*(cos(3/62*pi) - cos(19/62*pi))*(cos(1/62*pi) - cos(19/62*pi))*cos(19/62*pi)) + 1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(7/62*pi))*(x - 5*cos(5/62*pi))*(x - 5*cos(3/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(17/62*pi)^2 + 1)*(cos(17/62*pi) - cos(61/62*pi))*(cos(17/62*pi) - cos(59/62*pi))*(cos(17/62*pi) - cos(57/62*pi))*(cos(17/62*pi) - cos(55/62*pi))*(cos(17/62*pi) - cos(53/62*pi))*(cos(17/62*pi) - cos(51/62*pi))*(cos(17/62*pi) - cos(49/62*pi))*(cos(17/62*pi) - cos(47/62*pi))*(cos(17/62*pi) - cos(45/62*pi))*(cos(17/62*pi) - cos(43/62*pi))*(cos(17/62*pi) - cos(41/62*pi))*(cos(17/62*pi) - cos(39/62*pi))*(cos(17/62*pi) - cos(37/62*pi))*(cos(17/62*pi) - cos(35/62*pi))*(cos(17/62*pi) - cos(33/62*pi))*(cos(17/62*pi) - cos(29/62*pi))*(cos(17/62*pi) - cos(27/62*pi))*(cos(17/62*pi) - cos(25/62*pi))*(cos(17/62*pi) - cos(23/62*pi))*(cos(17/62*pi) - cos(21/62*pi))*(cos(17/62*pi) - cos(19/62*pi))*(cos(15/62*pi) - cos(17/62*pi))*(cos(13/62*pi) - cos(17/62*pi))*(cos(11/62*pi) - cos(17/62*pi))*(cos(9/62*pi) - cos(17/62*pi))*(cos(7/62*pi) - cos(17/62*pi))*(cos(5/62*pi) - cos(17/62*pi))*(cos(3/62*pi) - cos(17/62*pi))*(cos(1/62*pi) - cos(17/62*pi))*cos(17/62*pi)) - 1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(7/62*pi))*(x - 5*cos(5/62*pi))*(x - 5*cos(3/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(15/62*pi)^2 + 1)*(cos(15/62*pi) - cos(61/62*pi))*(cos(15/62*pi) - cos(59/62*pi))*(cos(15/62*pi) - cos(57/62*pi))*(cos(15/62*pi) - cos(55/62*pi))*(cos(15/62*pi) - cos(53/62*pi))*(cos(15/62*pi) - cos(51/62*pi))*(cos(15/62*pi) - cos(49/62*pi))*(cos(15/62*pi) - cos(47/62*pi))*(cos(15/62*pi) - cos(45/62*pi))*(cos(15/62*pi) - cos(43/62*pi))*(cos(15/62*pi) - cos(41/62*pi))*(cos(15/62*pi) - cos(39/62*pi))*(cos(15/62*pi) - cos(37/62*pi))*(cos(15/62*pi) - cos(35/62*pi))*(cos(15/62*pi) - cos(33/62*pi))*(cos(15/62*pi) - cos(29/62*pi))*(cos(15/62*pi) - cos(27/62*pi))*(cos(15/62*pi) - cos(25/62*pi))*(cos(15/62*pi) - cos(23/62*pi))*(cos(15/62*pi) - cos(21/62*pi))*(cos(15/62*pi) - cos(19/62*pi))*(cos(15/62*pi) - cos(17/62*pi))*(cos(13/62*pi) - cos(15/62*pi))*(cos(11/62*pi) - cos(15/62*pi))*(cos(9/62*pi) - cos(15/62*pi))*(cos(7/62*pi) - cos(15/62*pi))*(cos(5/62*pi) - cos(15/62*pi))*(cos(3/62*pi) - cos(15/62*pi))*(cos(1/62*pi) - cos(15/62*pi))*cos(15/62*pi)) + 1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(7/62*pi))*(x - 5*cos(5/62*pi))*(x - 5*cos(3/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(13/62*pi)^2 + 1)*(cos(13/62*pi) - cos(61/62*pi))*(cos(13/62*pi) - cos(59/62*pi))*(cos(13/62*pi) - cos(57/62*pi))*(cos(13/62*pi) - cos(55/62*pi))*(cos(13/62*pi) - cos(53/62*pi))*(cos(13/62*pi) - cos(51/62*pi))*(cos(13/62*pi) - cos(49/62*pi))*(cos(13/62*pi) - cos(47/62*pi))*(cos(13/62*pi) - cos(45/62*pi))*(cos(13/62*pi) - cos(43/62*pi))*(cos(13/62*pi) - cos(41/62*pi))*(cos(13/62*pi) - cos(39/62*pi))*(cos(13/62*pi) - cos(37/62*pi))*(cos(13/62*pi) - cos(35/62*pi))*(cos(13/62*pi) - cos(33/62*pi))*(cos(13/62*pi) - cos(29/62*pi))*(cos(13/62*pi) - cos(27/62*pi))*(cos(13/62*pi) - cos(25/62*pi))*(cos(13/62*pi) - cos(23/62*pi))*(cos(13/62*pi) - cos(21/62*pi))*(cos(13/62*pi) - cos(19/62*pi))*(cos(13/62*pi) - cos(17/62*pi))*(cos(13/62*pi) - cos(15/62*pi))*(cos(11/62*pi) - cos(13/62*pi))*(cos(9/62*pi) - cos(13/62*pi))*(cos(7/62*pi) - cos(13/62*pi))*(cos(5/62*pi) - cos(13/62*pi))*(cos(3/62*pi) - cos(13/62*pi))*(cos(1/62*pi) - cos(13/62*pi))*cos(13/62*pi)) - 1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(7/62*pi))*(x - 5*cos(5/62*pi))*(x - 5*cos(3/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(11/62*pi)^2 + 1)*(cos(11/62*pi) - cos(61/62*pi))*(cos(11/62*pi) - cos(59/62*pi))*(cos(11/62*pi) - cos(57/62*pi))*(cos(11/62*pi) - cos(55/62*pi))*(cos(11/62*pi) - cos(53/62*pi))*(cos(11/62*pi) - cos(51/62*pi))*(cos(11/62*pi) - cos(49/62*pi))*(cos(11/62*pi) - cos(47/62*pi))*(cos(11/62*pi) - cos(45/62*pi))*(cos(11/62*pi) - cos(43/62*pi))*(cos(11/62*pi) - cos(41/62*pi))*(cos(11/62*pi) - cos(39/62*pi))*(cos(11/62*pi) - cos(37/62*pi))*(cos(11/62*pi) - cos(35/62*pi))*(cos(11/62*pi) - cos(33/62*pi))*(cos(11/62*pi) - cos(29/62*pi))*(cos(11/62*pi) - cos(27/62*pi))*(cos(11/62*pi) - cos(25/62*pi))*(cos(11/62*pi) - cos(23/62*pi))*(cos(11/62*pi) - cos(21/62*pi))*(cos(11/62*pi) - cos(19/62*pi))*(cos(11/62*pi) - cos(17/62*pi))*(cos(11/62*pi) - cos(15/62*pi))*(cos(11/62*pi) - cos(13/62*pi))*(cos(9/62*pi) - cos(11/62*pi))*(cos(7/62*pi) - cos(11/62*pi))*(cos(5/62*pi) - cos(11/62*pi))*(cos(3/62*pi) - cos(11/62*pi))*(cos(1/62*pi) - cos(11/62*pi))*cos(11/62*pi)) + 1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(7/62*pi))*(x - 5*cos(5/62*pi))*(x - 5*cos(3/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(9/62*pi)^2 + 1)*(cos(9/62*pi) - cos(61/62*pi))*(cos(9/62*pi) - cos(59/62*pi))*(cos(9/62*pi) - cos(57/62*pi))*(cos(9/62*pi) - cos(55/62*pi))*(cos(9/62*pi) - cos(53/62*pi))*(cos(9/62*pi) - cos(51/62*pi))*(cos(9/62*pi) - cos(49/62*pi))*(cos(9/62*pi) - cos(47/62*pi))*(cos(9/62*pi) - cos(45/62*pi))*(cos(9/62*pi) - cos(43/62*pi))*(cos(9/62*pi) - cos(41/62*pi))*(cos(9/62*pi) - cos(39/62*pi))*(cos(9/62*pi) - cos(37/62*pi))*(cos(9/62*pi) - cos(35/62*pi))*(cos(9/62*pi) - cos(33/62*pi))*(cos(9/62*pi) - cos(29/62*pi))*(cos(9/62*pi) - cos(27/62*pi))*(cos(9/62*pi) - cos(25/62*pi))*(cos(9/62*pi) - cos(23/62*pi))*(cos(9/62*pi) - cos(21/62*pi))*(cos(9/62*pi) - cos(19/62*pi))*(cos(9/62*pi) - cos(17/62*pi))*(cos(9/62*pi) - cos(15/62*pi))*(cos(9/62*pi) - cos(13/62*pi))*(cos(9/62*pi) - cos(11/62*pi))*(cos(7/62*pi) - cos(9/62*pi))*(cos(5/62*pi) - cos(9/62*pi))*(cos(3/62*pi) - cos(9/62*pi))*(cos(1/62*pi) - cos(9/62*pi))*cos(9/62*pi)) - 1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(5/62*pi))*(x - 5*cos(3/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(7/62*pi)^2 + 1)*(cos(7/62*pi) - cos(61/62*pi))*(cos(7/62*pi) - cos(59/62*pi))*(cos(7/62*pi) - cos(57/62*pi))*(cos(7/62*pi) - cos(55/62*pi))*(cos(7/62*pi) - cos(53/62*pi))*(cos(7/62*pi) - cos(51/62*pi))*(cos(7/62*pi) - cos(49/62*pi))*(cos(7/62*pi) - cos(47/62*pi))*(cos(7/62*pi) - cos(45/62*pi))*(cos(7/62*pi) - cos(43/62*pi))*(cos(7/62*pi) - cos(41/62*pi))*(cos(7/62*pi) - cos(39/62*pi))*(cos(7/62*pi) - cos(37/62*pi))*(cos(7/62*pi) - cos(35/62*pi))*(cos(7/62*pi) - cos(33/62*pi))*(cos(7/62*pi) - cos(29/62*pi))*(cos(7/62*pi) - cos(27/62*pi))*(cos(7/62*pi) - cos(25/62*pi))*(cos(7/62*pi) - cos(23/62*pi))*(cos(7/62*pi) - cos(21/62*pi))*(cos(7/62*pi) - cos(19/62*pi))*(cos(7/62*pi) - cos(17/62*pi))*(cos(7/62*pi) - cos(15/62*pi))*(cos(7/62*pi) - cos(13/62*pi))*(cos(7/62*pi) - cos(11/62*pi))*(cos(7/62*pi) - cos(9/62*pi))*(cos(5/62*pi) - cos(7/62*pi))*(cos(3/62*pi) - cos(7/62*pi))*(cos(1/62*pi) - cos(7/62*pi))*cos(7/62*pi)) + 1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(7/62*pi))*(x - 5*cos(3/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(5/62*pi)^2 + 1)*(cos(5/62*pi) - cos(61/62*pi))*(cos(5/62*pi) - cos(59/62*pi))*(cos(5/62*pi) - cos(57/62*pi))*(cos(5/62*pi) - cos(55/62*pi))*(cos(5/62*pi) - cos(53/62*pi))*(cos(5/62*pi) - cos(51/62*pi))*(cos(5/62*pi) - cos(49/62*pi))*(cos(5/62*pi) - cos(47/62*pi))*(cos(5/62*pi) - cos(45/62*pi))*(cos(5/62*pi) - cos(43/62*pi))*(cos(5/62*pi) - cos(41/62*pi))*(cos(5/62*pi) - cos(39/62*pi))*(cos(5/62*pi) - cos(37/62*pi))*(cos(5/62*pi) - cos(35/62*pi))*(cos(5/62*pi) - cos(33/62*pi))*(cos(5/62*pi) - cos(29/62*pi))*(cos(5/62*pi) - cos(27/62*pi))*(cos(5/62*pi) - cos(25/62*pi))*(cos(5/62*pi) - cos(23/62*pi))*(cos(5/62*pi) - cos(21/62*pi))*(cos(5/62*pi) - cos(19/62*pi))*(cos(5/62*pi) - cos(17/62*pi))*(cos(5/62*pi) - cos(15/62*pi))*(cos(5/62*pi) - cos(13/62*pi))*(cos(5/62*pi) - cos(11/62*pi))*(cos(5/62*pi) - cos(9/62*pi))*(cos(5/62*pi) - cos(7/62*pi))*(cos(3/62*pi) - cos(5/62*pi))*(cos(1/62*pi) - cos(5/62*pi))*cos(5/62*pi)) - 1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(7/62*pi))*(x - 5*cos(5/62*pi))*(x - 5*cos(1/62*pi))*x/((25*cos(3/62*pi)^2 + 1)*(cos(3/62*pi) - cos(61/62*pi))*(cos(3/62*pi) - cos(59/62*pi))*(cos(3/62*pi) - cos(57/62*pi))*(cos(3/62*pi) - cos(55/62*pi))*(cos(3/62*pi) - cos(53/62*pi))*(cos(3/62*pi) - cos(51/62*pi))*(cos(3/62*pi) - cos(49/62*pi))*(cos(3/62*pi) - cos(47/62*pi))*(cos(3/62*pi) - cos(45/62*pi))*(cos(3/62*pi) - cos(43/62*pi))*(cos(3/62*pi) - cos(41/62*pi))*(cos(3/62*pi) - cos(39/62*pi))*(cos(3/62*pi) - cos(37/62*pi))*(cos(3/62*pi) - cos(35/62*pi))*(cos(3/62*pi) - cos(33/62*pi))*(cos(3/62*pi) - cos(29/62*pi))*(cos(3/62*pi) - cos(27/62*pi))*(cos(3/62*pi) - cos(25/62*pi))*(cos(3/62*pi) - cos(23/62*pi))*(cos(3/62*pi) - cos(21/62*pi))*(cos(3/62*pi) - cos(19/62*pi))*(cos(3/62*pi) - cos(17/62*pi))*(cos(3/62*pi) - cos(15/62*pi))*(cos(3/62*pi) - cos(13/62*pi))*(cos(3/62*pi) - cos(11/62*pi))*(cos(3/62*pi) - cos(9/62*pi))*(cos(3/62*pi) - cos(7/62*pi))*(cos(3/62*pi) - cos(5/62*pi))*(cos(1/62*pi) - cos(3/62*pi))*cos(3/62*pi)) + 1/931322574615478515625*(x - 5*cos(61/62*pi))*(x - 5*cos(59/62*pi))*(x - 5*cos(57/62*pi))*(x - 5*cos(55/62*pi))*(x - 5*cos(53/62*pi))*(x - 5*cos(51/62*pi))*(x - 5*cos(49/62*pi))*(x - 5*cos(47/62*pi))*(x - 5*cos(45/62*pi))*(x - 5*cos(43/62*pi))*(x - 5*cos(41/62*pi))*(x - 5*cos(39/62*pi))*(x - 5*cos(37/62*pi))*(x - 5*cos(35/62*pi))*(x - 5*cos(33/62*pi))*(x - 5*cos(29/62*pi))*(x - 5*cos(27/62*pi))*(x - 5*cos(25/62*pi))*(x - 5*cos(23/62*pi))*(x - 5*cos(21/62*pi))*(x - 5*cos(19/62*pi))*(x - 5*cos(17/62*pi))*(x - 5*cos(15/62*pi))*(x - 5*cos(13/62*pi))*(x - 5*cos(11/62*pi))*(x - 5*cos(9/62*pi))*(x - 5*cos(7/62*pi))*(x - 5*cos(5/62*pi))*(x - 5*cos(3/62*pi))*x/((25*cos(1/62*pi)^2 + 1)*(cos(1/62*pi) - cos(61/62*pi))*(cos(1/62*pi) - cos(59/62*pi))*(cos(1/62*pi) - cos(57/62*pi))*(cos(1/62*pi) - cos(55/62*pi))*(cos(1/62*pi) - cos(53/62*pi))*(cos(1/62*pi) - cos(51/62*pi))*(cos(1/62*pi) - cos(49/62*pi))*(cos(1/62*pi) - cos(47/62*pi))*(cos(1/62*pi) - cos(45/62*pi))*(cos(1/62*pi) - cos(43/62*pi))*(cos(1/62*pi) - cos(41/62*pi))*(cos(1/62*pi) - cos(39/62*pi))*(cos(1/62*pi) - cos(37/62*pi))*(cos(1/62*pi) - cos(35/62*pi))*(cos(1/62*pi) - cos(33/62*pi))*(cos(1/62*pi) - cos(29/62*pi))*(cos(1/62*pi) - cos(27/62*pi))*(cos(1/62*pi) - cos(25/62*pi))*(cos(1/62*pi) - cos(23/62*pi))*(cos(1/62*pi) - cos(21/62*pi))*(cos(1/62*pi) - cos(19/62*pi))*(cos(1/62*pi) - cos(17/62*pi))*(cos(1/62*pi) - cos(15/62*pi))*(cos(1/62*pi) - cos(13/62*pi))*(cos(1/62*pi) - cos(11/62*pi))*(cos(1/62*pi) - cos(9/62*pi))*(cos(1/62*pi) - cos(7/62*pi))*(cos(1/62*pi) - cos(5/62*pi))*(cos(1/62*pi) - cos(3/62*pi))*cos(1/62*pi))
9. Die Funktion f(x) = cos(x) werde im Intervall [0, π] in paarweise verschiedenen Stützstellen durch das Interpolationspolynom p_n interpoliert. x in [0,π] a) Papier is the king. b) Es seien die Stützstellen gleichmäßig verteilt. Man bestimme die Anzahl der Stützstellen, sodass der maximale Interpolationsfehler auf dem Intervall [0, π] kleiner als 10−6 ist. c) Als Stützstellen werden die Nullstellen des Tschebyscheff-Polynoms T_n+1 verwendet. Man bestimme die Anzahl der Stützstellen, sodass der maximale Interpolationsfehler auf dem Intervall [0,pi] kleiner als 10^-6 ist.
var('x') def gooody(n): g(x)=abs(1/factorial(n+1)*1*prod([x-uniform_sampling(0,pi,i,n) for i in [0,..,n]])) return max([g(a.rhs()) for a in solve(g.derivative()==0,x)]) N(gooody(9)) 
        
3.17524473663985e-7
3.17524473663985e-7
N(gooody(8)) 
        
3.01681748009828e-6
3.01681748009828e-6