Cálculo Numérico, Semestre 2017.2, Prova I

from math import *

from Vector import *
from Matrix import *
from Gauss import *

#Question a
vs=[
    [-2.0,1.0,1.0,1.0,],
    [1.0,-2.0,1.0,1.0,],
    [1.0,1.0,-2.0,1.0,],
    [1.0,1.0,1.0,-2.0,],
]

V=[
    [-2.0,1.0,1.0,1.0,],
    [1.0,-2.0,1.0,1.0,],
    [1.0,1.0,-2.0,1.0,],
    [1.0,1.0,1.0,-2.0,],
]

#Question d
I=Matrix_Identity(4)

VV=Matrix_Copy(V)
V_inv=Inverse(VV)

print "V^{-1}=",
Matrix_Print(V_inv)

VxV=Matrices_Mult(V,V_inv)
R=Matrices_Sub(I,VxV)

print "R=",
Matrix_Print(R)

print "residual=",Matrix_Norm(R)/Matrix_Norm(V)

#Question c
for i in range( len(vs) ):
    VV=Matrix_Copy(V)
    xi=list(vs[i])
    
    Gauss(VV,xi)
    
    print "x_%d=" % i,
    Vector_Print(xi)
    
    vv=Matrix_Mult_Vector(V,xi)
    vv=Vectors_Sub(vv,vs[i])
    
    print "vv=",
    Vector_Print(vv)
    

    print "residual=%.6e" % (  Vector_Norm(vv)/Vector_Norm(vs[i])  )

Gauss Forward, det -27.0
Matrix_Gauss_Test, Residual: 8.326673e-17
V^{-1}= [
    [-0.000000,0.333333,0.333333,0.333333]
    [0.333333,0.000000,0.333333,0.333333]
    [0.333333,0.333333,0.000000,0.333333]
    [0.333333,0.333333,0.333333,0.000000]
]
R= [
    [0.000000,0.000000,0.000000,0.000000]
    [0.000000,0.000000,0.000000,0.000000]
    [0.000000,0.000000,0.000000,0.000000]
    [0.000000,0.000000,0.000000,0.000000]
]
residual= 3.14718645319e-17
Gauss Forward, det -27.0
x_0= [1.000000,0.000000,0.000000,0.000000]
vv= [0.000000,0.000000,0.000000,0.000000]
residual=0.000000e+00
Gauss Forward, det -27.0
x_1= [0.000000,1.000000,0.000000,0.000000]
vv= [0.000000,0.000000,0.000000,0.000000]
residual=0.000000e+00
Gauss Forward, det -27.0
x_2= [0.000000,0.000000,1.000000,0.000000]
vv= [0.000000,0.000000,0.000000,0.000000]
residual=0.000000e+00
Gauss Forward, det -27.0
x_3= [0.000000,0.000000,0.000000,1.000000]
vv= [0.000000,0.000000,0.000000,0.000000]
residual=0.000000e+00

from math import *

from Polynomial import *

#Question a

Ps=[
    [-1.0, 1.0],
    [ 1.0,-1.0, 1.0],
    [-1.0, 1.0,-1.0, 1.0],
    [ 1.0,-1.0, 1.0,-1.0, 1.0],
    [-1.0, 1.0,-1.0, 1.0,-1.0,1.0],
]

xs=[
    -5.0,-4.0,-3.0,-2.0,-1.0,
    0.0,
    1.0,2.0,3.0,4.0,5.0,
]

def Question_AA():
    text=["i"]
    spec=["l"]
    for j in range( len(xs) ):
        text.append( "%.6f" %xs[j] )
        spec.append("l")

    print "\\begin{tabular}{"+"".join(spec)+"}"

    print " & ".join(text)+"\\\\"+"\\hline"

    for i in range( len(Ps) ):
        Pi=Ps[i]
        text=["%d" %i]
    
        for j in range( len(xs) ):
            text.append( "%.6f" % Polynomia_Calc(Pi,xs[j]) )
        print " & ".join(text)+"\\\\"

    print "\\end{tabular}"
    
def Question_A():
    text=["i"]
    for j in range( len(xs) ):
        text.append( "%.6f" %xs[j] )

    print "\t".join(text)

    for i in range( len(Ps) ):
        Pi=Ps[i]
        text=["%d" %i]
    
        for j in range( len(xs) ):
            text.append( "%.6f" % Polynomia_Calc(Pi,xs[j]) )
        print "\t".join(text)

def Question_B():
    for i in range( len(Ps) ):
        Pi=Ps[i]
        for j in range( len(Ps) ):
            Pj=Ps[j]

            Qij=Polynomias_Mult(Pi,Pj)

            print "Q_%d_%d=" % (i,j),
            Polynomia_Print(Qij)

def Polynomia_Derive(P):
    Q=[]
    for i in range(  1,len(P)   ):
        Q.append(i*P[i])

    return Q

def Question_C():
    for i in range( len(Ps) ):
        Pi=Ps[i]
        dPi=Polynomia_Derive(Pi)
        Polynomia_Print(dPi)

        

Question_A()
Question_B()
Question_C()

i    -5.000000    -4.000000    -3.000000    -2.000000    -1.000000    0.000000    1.000000    2.000000    3.000000    4.000000    5.000000
0    -6.000000    -5.000000    -4.000000    -3.000000    -2.000000    -1.000000    0.000000    1.000000    2.000000    3.000000    4.000000
1    31.000000    21.000000    13.000000    7.000000    3.000000    1.000000    1.000000    3.000000    7.000000    13.000000    21.000000
2    -156.000000    -85.000000    -40.000000    -15.000000    -4.000000    -1.000000    0.000000    5.000000    20.000000    51.000000    104.000000
3    781.000000    341.000000    121.000000    31.000000    5.000000    1.000000    1.000000    11.000000    61.000000    205.000000    521.000000
4    -3906.000000    -1365.000000    -364.000000    -63.000000    -6.000000    -1.000000    0.000000    21.000000    182.000000    819.000000    2604.000000
Q_0_0= 1.000000x**2 -2.000000x +1.000000
Q_0_1= 1.000000x**3 -2.000000x**2 +2.000000x -1.000000
Q_0_2= 1.000000x**4 -2.000000x**3 +2.000000x**2 -2.000000x +1.000000
Q_0_3= 1.000000x**5 -2.000000x**4 +2.000000x**3 -2.000000x**2 +2.000000x -1.000000
Q_0_4= 1.000000x**6 -2.000000x**5 +2.000000x**4 -2.000000x**3 +2.000000x**2 -2.000000x +1.000000
Q_1_0= 1.000000x**3 -2.000000x**2 +2.000000x -1.000000
Q_1_1= 1.000000x**4 -2.000000x**3 +3.000000x**2 -2.000000x +1.000000
Q_1_2= 1.000000x**5 -2.000000x**4 +3.000000x**3 -3.000000x**2 +2.000000x -1.000000
Q_1_3= 1.000000x**6 -2.000000x**5 +3.000000x**4 -3.000000x**3 +3.000000x**2 -2.000000x +1.000000
Q_1_4= 1.000000x**7 -2.000000x**6 +3.000000x**5 -3.000000x**4 +3.000000x**3 -3.000000x**2 +2.000000x -1.000000
Q_2_0= 1.000000x**4 -2.000000x**3 +2.000000x**2 -2.000000x +1.000000
Q_2_1= 1.000000x**5 -2.000000x**4 +3.000000x**3 -3.000000x**2 +2.000000x -1.000000
Q_2_2= 1.000000x**6 -2.000000x**5 +3.000000x**4 -4.000000x**3 +3.000000x**2 -2.000000x +1.000000
Q_2_3= 1.000000x**7 -2.000000x**6 +3.000000x**5 -4.000000x**4 +4.000000x**3 -3.000000x**2 +2.000000x -1.000000
Q_2_4= 1.000000x**8 -2.000000x**7 +3.000000x**6 -4.000000x**5 +4.000000x**4 -4.000000x**3 +3.000000x**2 -2.000000x +1.000000
Q_3_0= 1.000000x**5 -2.000000x**4 +2.000000x**3 -2.000000x**2 +2.000000x -1.000000
Q_3_1= 1.000000x**6 -2.000000x**5 +3.000000x**4 -3.000000x**3 +3.000000x**2 -2.000000x +1.000000
Q_3_2= 1.000000x**7 -2.000000x**6 +3.000000x**5 -4.000000x**4 +4.000000x**3 -3.000000x**2 +2.000000x -1.000000
Q_3_3= 1.000000x**8 -2.000000x**7 +3.000000x**6 -4.000000x**5 +5.000000x**4 -4.000000x**3 +3.000000x**2 -2.000000x +1.000000
Q_3_4= 1.000000x**9 -2.000000x**8 +3.000000x**7 -4.000000x**6 +5.000000x**5 -5.000000x**4 +4.000000x**3 -3.000000x**2 +2.000000x -1.000000
Q_4_0= 1.000000x**6 -2.000000x**5 +2.000000x**4 -2.000000x**3 +2.000000x**2 -2.000000x +1.000000
Q_4_1= 1.000000x**7 -2.000000x**6 +3.000000x**5 -3.000000x**4 +3.000000x**3 -3.000000x**2 +2.000000x -1.000000
Q_4_2= 1.000000x**8 -2.000000x**7 +3.000000x**6 -4.000000x**5 +4.000000x**4 -4.000000x**3 +3.000000x**2 -2.000000x +1.000000
Q_4_3= 1.000000x**9 -2.000000x**8 +3.000000x**7 -4.000000x**6 +5.000000x**5 -5.000000x**4 +4.000000x**3 -3.000000x**2 +2.000000x -1.000000
Q_4_4= 1.000000x**10 -2.000000x**9 +3.000000x**8 -4.000000x**7 +5.000000x**6 -6.000000x**5 +5.000000x**4 -4.000000x**3 +3.000000x**2 -2.000000x +1.000000
1.000000
2.000000x -1.000000
3.000000x**2 -2.000000x +1.000000
4.000000x**3 -3.000000x**2 +2.000000x -1.000000
5.000000x**4 -4.000000x**3 +3.000000x**2 -2.000000x +1.000000

from math import *

from Lagrange import *

xs=[-3.0,-2.0,-1.0,0.0,1.0,2.0,3.0]
ys=[5.0,1.0,-4.0,-1.0,3.0,2.0,1.0]


#Question a
print "Question a"

Hs=[]
for i in range( len(xs) ):
    Hi=Lagrange_H_P(i,xs)

    print "H_"+str(i)+"=",
    Polynomia_Print(Hi)
    
    Hs.append(Hi)

#Question b

print "Question b"

text=["x" ]
for j in range( len(xs) ):
    text.append( "%.6f" % xs[j] )
print "\t".join(text)

for i in range( len(Hs) ):
    Hi=Hs[i]
    
    text=["H_"+str(i) ]
    for j in range( len(xs) ):
        value=Polynomia_Calc(Hi,xs[j])
        text.append( "%.6f" % value )

    print "\t".join(text)

#Question c
print "Question c"
   
P=Lagrange_P(xs,ys)

text=["x" ]
for j in range( len(xs) ):
    text.append( "%.6f" % xs[j] )
print "\t".join(text)

text=["y" ]
for j in range( len(xs) ):
    value=Polynomia_Calc(P,xs[j])
    text.append( "%.6f" % value )
print "\t".join(text)

Question a
H_0= 0.001389x**6 -0.004167x**5 -0.006944x**4 +0.020833x**3 +0.005556x**2 -0.016667x
H_1= -0.008333x**6 +0.016667x**5 +0.083333x**4 -0.166667x**3 -0.075000x**2 +0.150000x
H_2= 0.020833x**6 -0.020833x**5 -0.270833x**4 +0.270833x**3 +0.750000x**2 -0.750000x
H_3= -0.027778x**6 +0.388889x**4 -1.361111x**2 +1.000000
H_4= 0.020833x**6 +0.020833x**5 -0.270833x**4 -0.270833x**3 +0.750000x**2 +0.750000x
H_5= -0.008333x**6 -0.016667x**5 +0.083333x**4 +0.166667x**3 -0.075000x**2 -0.150000x
H_6= 0.001389x**6 +0.004167x**5 -0.006944x**4 -0.020833x**3 +0.005556x**2 +0.016667x
Question b
x    -3.000000    -2.000000    -1.000000    0.000000    1.000000    2.000000    3.000000
H_0    1.000000    -0.000000    -0.000000    0.000000    0.000000    0.000000    0.000000
H_1    -0.000000    1.000000    -0.000000    0.000000    0.000000    0.000000    -0.000000
H_2    0.000000    0.000000    1.000000    0.000000    0.000000    0.000000    0.000000
H_3    -0.000000    0.000000    0.000000    1.000000    0.000000    0.000000    -0.000000
H_4    0.000000    0.000000    0.000000    0.000000    1.000000    0.000000    0.000000
H_5    -0.000000    -0.000000    -0.000000    -0.000000    -0.000000    1.000000    -0.000000
H_6    0.000000    0.000000    0.000000    0.000000    -0.000000    -0.000000    1.000000
Question c
x    -3.000000    -2.000000    -1.000000    0.000000    1.000000    2.000000    3.000000
y    5.000000    1.000000    -4.000000    -1.000000    3.000000    2.000000    1.000000

from math import *

from Integration import *


#Question a

def f_0(x):
    return e**(-x)

def f_1(x):
    return x*e**(-x)

def f_2(x):
    return x*x*e**(-x)

def f_3(x):
    return x*x*x*e**(-x)

def f_4(x):
    return x*x*x*x*e**(-x)

#Question b
def F_0(x):
    return -e**(-x)

def F_1(x):
    return -(x+1.0)*e**(-x)

def F_2(x):
    return -(x**2+2.0*x+2.0)*e**(-x)

def F_3(x):
    return -(x**3+3.0*x**2+6.0*x+6.0)*e**(-x)

def F_4(x):
    return -(x**4+4.0*x**3+12.0*x**2+24.0*x+24.0)*e**(-x)

fs=[f_0,f_1,f_2,f_3,f_4]
Fs=[F_0,F_1,F_2,F_3,F_4]


a=0.0
b=1.0
ni=10
for n in range( len(fs) ):
    fn=fs[n]
    Fn=Fs[n]

    print "f_"+str(n)+":"
    text=[
        "ni",
        "Ana",
        "Trapeziod",
        "r",
        "Simpson",
        "r",
        ]

    print "\t".join(text)

    analytical=Fn(b)-Fn(a)

    for ni in range(1,11):
        trapeziod=Trapezoids_R(fn,a,b,10*ni)
        simpson=Simpsons_R(fn,a,b,5*ni)
    
        text=[
            "%d" % (10*ni),
            "%.6f" % analytical,
            "%.6f" % trapeziod,
            "%.6e" % abs( (trapeziod-analytical)/analytical ),
            "%.6f" % simpson,
            "%.6e" % abs( (simpson-analytical)/analytical ),
        ]

        print "\t".join(text)

f_0:
ni    Ana    Trapeziod    r    Simpson    r
10    0.632121    0.632647    8.331945e-04    0.632121    5.548949e-07
20    0.632121    0.632252    2.083247e-04    0.632121    3.471189e-08
30    0.632121    0.632179    9.259088e-05    0.632121    6.857803e-09
40    0.632121    0.632153    5.208279e-05    0.632121    2.169977e-09
50    0.632121    0.632142    3.333311e-05    0.632121    8.888463e-10
60    0.632121    0.632135    2.314804e-05    0.632121    4.286551e-10
70    0.632121    0.632131    1.700674e-05    0.632121    2.313796e-10
80    0.632121    0.632129    1.302080e-05    0.632121    1.356313e-10
90    0.632121    0.632127    1.028804e-05    0.632121    8.467472e-11
100    0.632121    0.632126    8.333319e-06    0.632121    5.555448e-11
f_1:
ni    Ana    Trapeziod    r    Simpson    r
10    0.264241    0.263408    3.152496e-03    0.264240    4.751650e-06
20    0.264241    0.264033    7.883470e-04    0.264241    2.973914e-07
30    0.264241    0.264149    3.503948e-04    0.264241    5.875911e-08
40    0.264241    0.264189    1.971007e-04    0.264241    1.859343e-08
50    0.264241    0.264208    1.261455e-04    0.264241    7.616186e-09
60    0.264241    0.264218    8.760145e-05    0.264241    3.673012e-09
70    0.264241    0.264224    6.436043e-05    0.264241    1.982627e-09
80    0.264241    0.264228    4.927604e-05    0.264241    1.162190e-09
90    0.264241    0.264231    3.893421e-05    0.264241    7.255538e-10
100    0.264241    0.264233    3.153673e-05    0.264241    4.760378e-10
f_2:
ni    Ana    Trapeziod    r    Simpson    r
10    0.160603    0.160909    1.903980e-03    0.160606    1.941701e-05
20    0.160603    0.160679    4.769074e-04    0.160603    1.216635e-06
30    0.160603    0.160637    2.120340e-04    0.160603    2.404355e-07
40    0.160603    0.160622    1.192839e-04    0.160603    7.608777e-08
50    0.160603    0.160615    7.634610e-05    0.160603    3.116792e-08
60    0.160603    0.160611    5.301978e-05    0.160603    1.503144e-08
70    0.160603    0.160609    3.895404e-05    0.160603    8.113801e-09
80    0.160603    0.160608    2.982455e-05    0.160603    4.756238e-09
90    0.160603    0.160607    2.356527e-05    0.160603    2.969328e-09
100    0.160603    0.160606    1.908799e-05    0.160603    1.948193e-09
f_3:
ni    Ana    Trapeziod    r    Simpson    r
10    0.113929    0.114543    5.390801e-03    0.113925    3.611625e-05
20    0.113929    0.114082    1.345996e-03    0.113929    2.272130e-06
30    0.113929    0.113997    5.980801e-04    0.113929    4.493611e-07
40    0.113929    0.113967    3.363924e-04    0.113929    1.422411e-07
50    0.113929    0.113953    2.152829e-04    0.113929    5.827341e-08
60    0.113929    0.113946    1.494989e-04    0.113929    2.810553e-08
70    0.113929    0.113941    1.098346e-04    0.113929    1.517164e-08
80    0.113929    0.113939    8.409143e-05    0.113929    8.893708e-09
90    0.113929    0.113937    6.644224e-05    0.113929    5.552457e-09
100    0.113929    0.113935    5.381800e-05    0.113929    3.643040e-09
f_4:
ni    Ana    Trapeziod    r    Simpson    r
10    0.087836    0.088756    1.047122e-02    0.087836    3.277019e-06
20    0.087836    0.088066    2.617686e-03    0.087836    1.603420e-07
30    0.087836    0.087939    1.163407e-03    0.087836    3.003663e-08
40    0.087836    0.087894    6.544145e-04    0.087836    9.322396e-09
50    0.087836    0.087873    4.188247e-04    0.087836    3.784044e-09
60    0.087836    0.087862    2.908503e-04    0.087836    1.815847e-09
70    0.087836    0.087855    2.136859e-04    0.087836    9.772070e-10
80    0.087836    0.087851    1.636032e-04    0.087836    5.716976e-10
90    0.087836    0.087848    1.292667e-04    0.087836    3.564255e-10
100    0.087836    0.087846    1.047060e-04    0.087836    2.336200e-10