Atividades 8

1. Arquivo N01.py: Para \(n=2,3,...,10,\) escolher aleatóriamente \(n+1\) pontos: \(\underline{p}_i,\) com abscissas distintas.
2. Arquivo N02.py:
   1. Calcular a matriz de Vandermonte das abscissas e seu determinante.
   2. Calcular o Polinômio Interpolador, \(P_n(x),\) usando Eliminação de Gauss.
   3. Calcular e verificar valores e residuos relativos do \(P_n(x)\) nas abscissas \(x_i.\)
3. Arquivo N03.py:
   1. Calcular as Funções de Lagrange, \(H_i(x),\) como Polinômios.
   2. Calcular o Polinômio Interpolador de Lagrange, \(P_n(x).\)
   3. Calcular e verificar valores e residuos relativos do \(P_n(x)\) nas abscissas \(x_i.\)

1. N01.py:

   Python Listing: N01.py. pss=[ #Add more... #Last list of 11 points [ [ 0.0, -8.0, ], [ 1.0, -5.0,], [ 2.0, 3.0,], [ 3.0, -8.0,], [ 4.0, 6.0,], [ 5.0, 2.0,], [ 6.0, 5.0,], [ 7.0, -4.0,], [ 8.0, 3.0,], [ 9.0, 1.0,], [10.0, -7.0], ], ] def Points_Split(ps): xs=[] ys=[] for i in range( len(ps) ): xs.append( ps[i][0] ) ys.append( ps[i][1] ) return xs,ys

2. N02.py:

   Python Listing: N02.py. from Polynomial import * from Interpolation import * from Lagrange import * from N01 import * for n in range( len(pss) ): xs,ys=Points_Split(pss[n]) print "---"*30,"\nList no",n,":",len(xs),"Points: Vandermonte\n","---"*30 #Question a V=Matrix_Vandermonte(xs) #Matrix_Print(V) #Question b P=Interpolation_Gauss(xs,ys) print "P Gauss=", Polynomia_Print(P) norm=Vector_Norm(ys) #Question c text=[ "i", "x", "\ty", "\tGauss", "\tResidual", ] print "\t".join(text) for i in range( len(xs) ): gauss=Polynomia_Calc(P,xs[i]) gauss_res = abs( (gauss -ys[i])/norm ) text=[ "%d" % i, "%.6f" % xs[i], "%.6f" % ys[i], "%.6f" % gauss, "%.6e" % gauss_res, ] print "\t".join(text)

   ------------------------------------------------------------------------------------------ List no 0 : 11 Points: Vandermonte ------------------------------------------------------------------------------------------ Gauss Forward, det 6.6586065841e+27 P Gauss= 0.000976x**10 -0.049177x**9 +1.064501x**8 -12.938269x**7 +96.825770x**6 -459.710156x**5 +1372.767028x**4 -2456.398826x**3 +2348.341726x**2 -886.903571x -8.000000 i x y Gauss Residual 0 0.000000 -8.000000 -8.000000 0.000000e+00 1 1.000000 -5.000000 -5.000000 5.618220e-11 2 2.000000 3.000000 3.000000 6.803620e-11 3 3.000000 -8.000000 -8.000000 7.269406e-11 4 4.000000 6.000000 6.000000 9.043581e-11 5 5.000000 2.000000 2.000000 1.261286e-10 6 6.000000 5.000000 5.000000 1.825464e-10 7 7.000000 -4.000000 -4.000000 2.138953e-10 8 8.000000 3.000000 3.000000 1.473769e-10 9 9.000000 1.000000 1.000000 1.186970e-10 10 10.000000 -7.000000 -7.000000 1.132541e-10 Output from: /usr/bin/python N02.py

3. N03.py:

   Python Listing: N03.py. from Polynomial import * from Interpolation import * from Lagrange import * from N01 import * for n in range( len(pss) ): print "---"*30,"\nList no",n,":",len(pss[n]),"Points: Lagrange\n","---"*30 xs,ys=Points_Split(pss[n]) #Question a for i in range( len(xs) ): Hi=Lagrange_H_P(i,xs) print "H_"+str(i)+"(x)=", Polynomia_Print(Hi) #Question b P=Lagrange_P(xs,ys) print "P Lagrange=", Polynomia_Print(P) #Question c text=[ "i", "x", "\ty", "\tLagrange Pol", "Residual", ] print "\t".join(text) norm=Vector_Norm(ys) for i in range( len(xs) ): lagrange=Polynomia_Calc(P,xs[i]) lagrange_res= abs( (lagrange-ys[i])/norm ) text=[ "%d" % i, "%.6f" % xs[i], "%.6f" % ys[i], "%.6f" % lagrange, "%.6e" % lagrange_res, ] print "\t".join(text)

   ------------------------------------------------------------------------------------------ List no 0 : 11 Points: Lagrange ------------------------------------------------------------------------------------------ H_0(x)= 0.000000x**10 -0.000015x**9 +0.000364x**8 -0.005002x**7 +0.043478x**6 -0.248582x**5 +0.941614x**4 -2.317433x**3 +3.514544x**2 -2.928968x +1.000000 H_1(x)= -0.000003x**10 +0.000149x**9 -0.003489x**8 +0.046528x**7 -0.388252x**6 +2.097569x**5 -7.318574x**4 +15.855754x**3 -19.289683x**2 +10.000000x H_2(x)= 0.000012x**10 -0.000657x**9 +0.015055x**8 -0.194965x**7 +1.566580x**6 -8.053038x**5 +26.266567x**4 -51.751339x**3 +54.651786x**2 -22.500000x H_3(x)= -0.000033x**10 +0.001720x**9 -0.038492x**8 +0.484722x**7 -3.763194x**6 +18.540278x**5 -57.372884x**4 +105.973280x**3 -103.825397x**2 +40.000000x H_4(x)= 0.000058x**10 -0.002951x**9 +0.064583x**8 -0.792014x**7 +5.962326x**6 -28.352951x**5 +84.327199x**4 -149.352083x**3 +140.645833x**2 -52.500000x H_5(x)= -0.000069x**10 +0.003472x**9 -0.074306x**8 +0.888889x**7 -6.512014x**6 +30.082639x**5 -86.873611x**4 +149.625000x**3 -137.540000x**2 +50.400000x H_6(x)= 0.000058x**10 -0.002836x**9 +0.059375x**8 -0.694097x**7 +4.965799x**6 -22.407465x**5 +63.294213x**4 -106.895602x**3 +96.680556x**2 -35.000000x H_7(x)= -0.000033x**10 +0.001587x**9 -0.032540x**8 +0.372421x**7 -2.610417x**6 +11.556944x**5 -32.095106x**4 +53.426190x**3 -47.761905x**2 +17.142857x H_8(x)= 0.000012x**10 -0.000583x**9 +0.011706x**8 -0.131424x**7 +0.905122x**6 -3.945226x**5 +10.810838x**4 -17.797768x**3 +15.772321x**2 -5.625000x H_9(x)= -0.000003x**10 +0.000127x**9 -0.002497x**8 +0.027546x**7 -0.186863x**6 +0.804051x**5 -2.179685x**4 +3.557165x**3 -3.130952x**2 +1.111111x H_10(x)= 0.000000x**10 -0.000012x**9 +0.000240x**8 -0.002604x**7 +0.017436x**6 -0.074219x**5 +0.199427x**4 -0.323165x**3 +0.282897x**2 -0.100000x P Lagrange= 0.000976x**10 -0.049177x**9 +1.064501x**8 -12.938269x**7 +96.825770x**6 -459.710156x**5 +1372.767028x**4 -2456.398826x**3 +2348.341726x**2 -886.903571x -8.000000 i x y Lagrange Pol Residual 0 0.000000 -8.000000 -8.000000 5.110892e-17 1 1.000000 -5.000000 -5.000000 1.303278e-14 2 2.000000 3.000000 3.000000 2.616266e-13 3 3.000000 -8.000000 -8.000000 1.020543e-12 4 4.000000 6.000000 6.000000 1.360673e-12 5 5.000000 2.000000 2.000000 3.663539e-12 6 6.000000 5.000000 5.000000 1.816048e-11 7 7.000000 -4.000000 -4.000000 3.584990e-11 8 8.000000 3.000000 3.000000 1.421434e-10 9 9.000000 1.000000 1.000000 2.776663e-10 10 10.000000 -7.000000 -7.000000 6.651324e-10 Output from: /usr/bin/python N03.py