numeric_methods/task2.py at master · olehpidhi/numeric_methods · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
import numpy as npy
import matplotlib.pyplot as plt
import numpy.linalg as linalg


def f(x):
    return x * x * npy.cos(x + 1)
N = 20
a = 0
b = 2.0
h = (b - a) / N
finite_differences = []
data = []
factorials = []
xValues = []

data.append(f(a))
xValues.append(a)
factorials.append(1)
for i in xrange(1, N + 1):
    data.append(f(a + i * h))
    xValues.append(a + i * h)
    factorials.append(factorials[i-1] * i)

finite_differences.append(data)

for i in xrange(1, len(data)):
    finite_differences.append([])
    for j in xrange(0, len(finite_differences[i - 1]) - 1):
        finite_differences[i].append(finite_differences[i-1][j+1] - finite_differences[i-1][j])


def forward_newton(x):

    accumulator = f(a)
    t = (x - a) / h
    numerator = 1
    for i in xrange(1, N + 1):
        numerator *= t
        accumulator += finite_differences[i][0] * numerator / factorials[i]
        t -= 1
    return accumulator

newton = []
newtonX = []

h1 = (b - a) / 30
for i in xrange(0, 31):
    newtonX.append(a + i * h1)
    newton.append(forward_newton(a + i * h1))

n = N // 2


def forward_gauss(x):
    t = (x - (b-a) / 2.) / h
    accumulator = finite_differences[0][n]
    numeratorOdd = t
    numeratorEven = t * (t-1)
    i = 1
    while i <= n:
        accumulator += (finite_differences[2 * i - 1][n - i + 1] * numeratorOdd) / factorials[i * 2 - 1]
        accumulator += (finite_differences[2 * i][n - i] * numeratorEven) / factorials[i * 2]
        numeratorEven *= (t + i - 1) * (t - i + 1)
        numeratorOdd *= (t + i - 1) * (t - i)
        i += 1
    return accumulator

gauss = []
gaussX = []

for i in xrange(0, 21):
    gaussX.append(a + i * h)
    gauss.append(forward_gauss(a + i * h))


plt.plot(xValues, data)
plt.plot(newtonX, newton)
plt.plot(gaussX, gauss)


# TASK 3. Least squares method
def phi(x,i):
    return x**i;


def scalar_phi(i, j):
    result = 0
    for k in range(N + 1):
        result += phi(xValues[k], i) * phi(xValues[k], j)
    return result


def scalar(i):
    result = 0
    for k in range(N + 1):
        result += f(xValues[k]) * phi(xValues[k], i)
    return result

matrix = []
for i in range(N + 1):
    matrix.append([])
    for j in range(N + 1):
        matrix[i].append(scalar_phi(i, j))
b = []
for i in range(N + 1):
    b.append(scalar(i))

solution = linalg.solve(matrix, b)


def F(x):
    result = 0
    for i in xrange(N + 1):
        result += solution[i] * phi(x,i)
    return result

FValues = []

for i in xrange(N+1):
    FValues.append(F(xValues[i]))


def Norm(func):
    result = 0
    for i in range(N):
        result += (func(xValues[i + 1]) ** 2 + func(xValues[i]) ** 2) * (xValues[i + 1] - xValues[i]) / 2.
    return npy.sqrt(result)

print(abs(Norm(f) - Norm(F)))
plt.plot(xValues, FValues)
plt.show()