Atividades 4

1. Implemente as defs das funções no item anterior. Implemente também suas coeficientes de Taylor, \(a_n.\) Salvar em Arquivo N01.py.

   Python Listing: N01.py. from math import * def Factorial(n): if (n==0): return 1 elif (n==1): return 1 else: return n*Factorial(n-1) def f1(x): return e**x def Text1(): return 'e**x' def a1(n): return 1.0/(1.0*Factorial(n)) def f2(x): return cos(x) def Text2(): return 'cos(x)' def a2(n): if ( (n%2)==1): return 0.0 p=n/2 return (-1.0)**p/(1.0*Factorial(n))

   Output from: /usr/bin/python N01.py

2. Para \(x=0.5\) e as funções no item anterior, calcule os valores do polinômio de Taylor de ordem \(0,1,...,10.\) Salvar em Arquivo N02.py.

   Python Listing: N02.py. from math import * from Polynomial import * from Residual import * from N01 import * def Calc_Taylor_Pol(a,n): P=[] for i in range(n+1): P.append( a(i) ) return P fs=[f1,f2] ass=[a1,a2] texts=[Text1,Text2] x=0.5 for i in range( len(fs) ): f=fs[i] a=ass[i] text=texts[i] print text() print "x\t\tn\tAnalytical\tApprox\t\tAbsolute\tRelative" for n in range(10): P=Calc_Taylor_Pol(a,n) #Polynomia_Print(P) f_ana=f(x) f_tay=Polynomia_Calc(P,x) print "%.6f\t%d\t%.6f\t%.6f\t%.6e\t%.6e" % (x,n,f_ana,f_tay,R_Abs(f_ana,f_tay),R_Rel(f_ana,f_tay))

   e**x x n Analytical Approx Absolute Relative 0.500000 0 1.648721 1.000000 6.487213e-01 3.934693e-01 0.500000 1 1.648721 1.500000 1.487213e-01 9.020401e-02 0.500000 2 1.648721 1.625000 2.372127e-02 1.438768e-02 0.500000 3 1.648721 1.645833 2.887937e-03 1.751623e-03 0.500000 4 1.648721 1.648438 2.837707e-04 1.721156e-04 0.500000 5 1.648721 1.648698 2.335403e-05 1.416494e-05 0.500000 6 1.648721 1.648720 1.652645e-06 1.002380e-06 0.500000 7 1.648721 1.648721 1.025454e-07 6.219691e-08 0.500000 8 1.648721 1.648721 5.664166e-09 3.435490e-09 0.500000 9 1.648721 1.648721 2.818770e-10 1.709670e-10 cos(x) x n Analytical Approx Absolute Relative 0.500000 0 0.877583 1.000000 1.224174e-01 1.394939e-01 0.500000 1 0.877583 1.000000 1.224174e-01 1.394939e-01 0.500000 2 0.877583 0.875000 2.582562e-03 2.942814e-03 0.500000 3 0.877583 0.875000 2.582562e-03 2.942814e-03 0.500000 4 0.877583 0.877604 2.160478e-05 2.461851e-05 0.500000 5 0.877583 0.877604 2.160478e-05 2.461851e-05 0.500000 6 0.877583 0.877582 9.661260e-08 1.100895e-07 0.500000 7 0.877583 0.877582 9.661260e-08 1.100895e-07 0.500000 8 0.877583 0.877583 2.686054e-10 3.060742e-10 0.500000 9 0.877583 0.877583 2.686054e-10 3.060742e-10 Output from: /usr/bin/python N02.py