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
|
import numpy as np
import pandas as pd
import scipy.special
import seaborn as sns
import seaborn_image as isns
import matplotlib.pyplot as plt
import toolz
# Basis functions for 2D Fourier
@np.vectorize
@toolz.curry
def b(m, n, mu, nu):
return np.exp(2j * np.pi * (m * mu + n * nu))
N = 100
H = 4
basis = []
for m in range(H):
for n in range(H + 1):
mu = np.linspace(0, 1, N)
nu = np.linspace(0, 1, N)
muv, nuv = np.meshgrid(mu, nu)
bs = np.real(b(m, n, muv, nuv))
basis.append(bs)
g = isns.ImageGrid(basis, col_wrap=(H + 1), cbar=False)
g.fig.savefig("flat-basis-functions.pdf")
plt.tight_layout()
plt.show()
# Legendre Polynomials
N = 100
sns.set_theme("talk", "darkgrid", font_scale=.75)
fig, axes = plt.subplots(3, 4, figsize=(12, 8))
x = np.linspace(-1, 1, N)
for n in range(np.prod(axes.shape)):
p = scipy.special.lpmv(0, n, x)
# axes.ravel()[n].plot(x, p)
sns.lineplot(ax=axes.ravel()[n], x=x, y=p)
plt.tight_layout()
fig.savefig("legendre-polynomials.pdf")
plt.show()
# Associated Legendre Polynomials
N = 100
sns.set_theme("talk", "darkgrid", font_scale=.75)
fig, axes = plt.subplots(2, 3, figsize=(12, 8))
x = np.linspace(-1, 1, N)
for m in range(3 * 2):
for n in range(5):
p = scipy.special.lpmv(m, m + n, x)
sns.lineplot(ax=axes.ravel()[m], x=x, y=p)
# sns.lineplot(x=x, y=p)
plt.tight_layout()
fig.savefig("associated-legendre-polynomials.pdf")
plt.show()
|
