Animations

- link to interactive notebooks session.

# https://www.geeksforgeeks.org/using-matplotlib-for-animations/

import numpy as np
from IPython.display import HTML
from matplotlib import pyplot as plt
from matplotlib.animation import FuncAnimation

import numerical_function_spaces.orlicz_spaces as osm

Define Orlicz function

def Orlicz_function(u):
    Phi = np.zeros(len(u))
    for i in range(len(u)):
        n = -1
        while True:
            if u[i] > n and u[i] <= n + 1:  # below two conjugated functions?
                Phi[i] = (n + 1) * u[i] - (n + 1) * n / 2
                # Phi[i] = n * u[i] - (n) * (n + 1) / 2
                break
            n = n + 1
    return Phi

Define sequence

\[\begin{equation*} x_n(t)=\chi_{\left[0,\frac{n}{100}\right)}(t) \end{equation*}\]

for \(n \in \left\{1, 2, \dots,200\right\}.\)

def x_n(n):
    x = np.array([[1], [n / 100]])
    return x

n = range(1,201)  # used in FuncAnimation as frames 

kappa() function

# execution took around 1 minute
fig = plt.figure()
ax = plt.axes(xlim=(0, 15),
              ylim=(0, 5))

p_norm = 1

k_min = 0.1
k_max = 15
dk = 0.01

k = np.arange(k_min, k_max, dk)
kappa = np.zeros(len(k))

line, = ax.plot([], [], linewidth=2, label=f'$\\kappa_{{p, x_n}}(k)$')
line1, = ax.plot([], [], marker="o", color="crimson", linewidth=5, ms=5, label="$||x_n||$")


def animate(n):
    x = x_n(n)
    for i in range(len(k)):
        kappa[i] = osm.kappa(Orlicz_function, x=x, k=k[i], p_norm=p_norm)
    line.set_data(k, kappa)
    k_s = osm.p_Amemiya_norm_with_stars(Orlicz_function, x=x, p_norm=p_norm,
                                        k_min=k_min, k_max=k_max, dk=dk)
    line1.set_data([k_s[1], k_s[2]], [k_s[0], k_s[0]])
    ax.legend()
    ax.set_title(f"$p={p_norm}, n={n}$")
    return line,


anim = FuncAnimation(fig, animate,
                     frames=n, interval=200, blit=True)
# anim.save('im.gif', writer="pillow")
# anim.save(filename="html_example.html", writer="html")
plt.close()
HTML(anim.to_jshtml())

Once Loop Reflect

alpha() function

# execution took around 2 minutes
fig = plt.figure()
ax = plt.axes(xlim=(0, 15),
              ylim=(-2, 15))

p_norm = 1

k_min = 0.1
k_max = 15
dk = 0.01

u_max = 20  # for p_plus and Psi and alpha function
du = 0.01  # for p_plus and Psi and alpha function

p_plus = osm.right_side_derivative(Orlicz_function, u_max=u_max, du=du)
Psi = osm.conjugate_function(Orlicz_function, u_max=u_max, du=du)

line, = ax.plot([], [], linewidth=2, label=f'$\\alpha_{{p, x_n}}(k)$')
line1, = ax.plot([], [], marker="o", color="crimson", linewidth=5, ms=5,
                 label="$K_{{\\alpha, p}}\left(x_n\\right)$")

ax.axhline(y=0, linestyle="--", linewidth=1.3, label="0", color="green")


def animate(n):
    x = x_n(n)
    k, alpha = osm.array_for_alpha(Orlicz_function, u_max=u_max, du=du, p_norm=p_norm, x=x,
                                   p_plus=p_plus,
                                   Psi=Psi,
                                   k_min=k_min, k_max=k_max, dk=dk)
    line.set_data(k, alpha)
    k_s = osm.p_Amemiya_norm_with_stars(Orlicz_function, x=x, p_norm=p_norm)
    line1.set_data([k_s[1], k_s[2]], [0, 0])
    ax.legend()
    ax.set_title(f"$p={p_norm}, n={n}$")
    return line,


anim = FuncAnimation(fig, animate,
                     frames=n, interval=200, blit=True)
# anim.save('im.gif', writer="pillow")
# anim.save(filename="html_example.html", writer="html")
plt.close()
HTML(anim.to_jshtml())

Once Loop Reflect

tau() function

For \(p\_norm=1\) functions \(tau()\) and \(alpha()\) are almost the same (shifted up and down by 1).

# execution took around 2 minutes
fig = plt.figure()
ax = plt.axes(xlim=(0, 15),
              ylim=(-2, 15))

p_norm = 1

k_min = 0.1
k_max = 15
dk = 0.01

u_max = 20  # for p_plus and Psi and alpha function
du = 0.01  # for p_plus and Psi and alpha function

p_plus = osm.right_side_derivative(Orlicz_function, u_max=u_max, du=du)
Psi = osm.conjugate_function(Orlicz_function, u_max=u_max, du=du)

line, = ax.plot([], [], linewidth=2, label=f'$\\tau_{{p, x_n}}(k)$')
line1, = ax.plot([], [], marker="o", color="crimson", linewidth=5, ms=5,
                 label="$K_{{\\tau, p}}\left(x_n\\right)$")

ax.axhline(y=1, linestyle="--", linewidth=1.3, label="1", color="green")


def animate(n):
    x = x_n(n)
    k, tau = osm.array_for_tau(Orlicz_function, u_max=u_max, du=du, p_norm=p_norm, x=x,
                               p_plus=p_plus,
                               Psi=Psi,
                               k_min=k_min, k_max=k_max, dk=dk)
    line.set_data(k, tau)
    k_s = osm.p_Amemiya_norm_with_stars(Orlicz_function, x=x, p_norm=p_norm)
    line1.set_data([k_s[1], k_s[2]], [1, 1])
    ax.legend()
    ax.set_title(f"$p={p_norm}, n={n}$")
    return line,


anim = FuncAnimation(fig, animate,
                     frames=n, interval=200, blit=True)
# anim.save('im.gif', writer="pillow")
# anim.save(filename="html_example.html", writer="html")
plt.close()
HTML(anim.to_jshtml())

Once Loop Reflect

We can animate relative to a parameter p_norm

def Orlicz_function(u):
    Phi = np.zeros(len(u))
    for i in range(len(u)):
        n = -1
        while True:
            if u[i] > n and u[i] <= n + 1:  # below two conjugated functions?
                Phi[i] = (n + 1) * u[i] - (n + 1) * n / 2
                # Phi[i] = n * u[i] - (n) * (n + 1) / 2
                break
            n = n + 1
    return Phi


x = np.array([[1], [1]])  # \chi_{[0,1]}

# execution took around 30 seconds
fig = plt.figure()
ax = plt.axes(xlim=(0, 7),
              ylim=(0.5, 4))

p_norm = list(np.arange(1, 2, 0.05))
p_norm.extend(list(np.arange(2, 5, 0.2)))
p_norm.extend(list(np.arange(5, 10, 0.5)))
p_norm.extend(list(np.arange(10, 50, 1)))
# p_norm.extend(list(np.arange(50,200,5)))

p_norm.append(np.inf)

k_min = 0.1
k_max = 15
dk = 0.01

k = np.arange(k_min, k_max, dk)
kappa = np.zeros(len(k))

line, = ax.plot([], [], linewidth=2, label=f'$\\kappa_{{p, x_n}}(k)$')
line1, = ax.plot([], [], marker="o", color="crimson", linewidth=5, ms=5, label="$||x_n||$")


def animate(n):
    for i in range(len(k)):
        kappa[i] = osm.kappa(Orlicz_function, x=x, k=k[i], p_norm=n)
    line.set_data(k, kappa)
    k_s = osm.p_Amemiya_norm_with_stars(Orlicz_function, x=x, p_norm=n,
                                        k_min=k_min, k_max=k_max, dk=dk)
    line1.set_data([k_s[1], k_s[2]], [k_s[0], k_s[0]])
    ax.legend()
    ax.set_title(f"$p={round(n, 2)}$")
    return line,


anim = FuncAnimation(fig, animate,
                     frames=p_norm, interval=200, blit=True)
# anim.save('im.gif', writer="pillow")
# anim.save(filename="html_example.html", writer="html")
plt.close()
HTML(anim.to_jshtml())

Once Loop Reflect

# execution took around 1 minute
fig = plt.figure()
ax = plt.axes(xlim=(0, 5),
              ylim=(-2, 30))

k_min = 0.1
k_max = 15
dk = 0.01

u_max = 20
du = 0.01

p_plus = osm.right_side_derivative(Orlicz_function, u_max=u_max, du=du)
Psi = osm.conjugate_function(Orlicz_function, u_max=u_max, du=du)

p_norm = list(np.arange(1, 2, 0.05))
p_norm.extend(list(np.arange(2, 5, 0.2)))
p_norm.extend(list(np.arange(5, 10, 0.5)))
p_norm.extend(list(np.arange(10, 50, 1)))
# p_norm.extend(list(np.arange(50,200,5)))

p_norm.append(np.inf)

line, = ax.plot([], [], linewidth=2, label=f'$\\alpha_{{p, x_n}}(k)$')
line1, = ax.plot([], [], marker="o", color="crimson", linewidth=5, ms=5,
                 label="$K_{{\\alpha, p}}\left(x_n\\right)$")

ax.axhline(y=0, linestyle="--", linewidth=1.3, label="0", color="green")


def animate(n):
    k, alpha = osm.array_for_alpha(Orlicz_function, u_max=u_max, du=du, p_norm=n, x=x,
                                   p_plus=p_plus,
                                   Psi=Psi,
                                   k_min=k_min, k_max=k_max, dk=dk)
    line.set_data(k, alpha)
    k_s = osm.p_Amemiya_norm_with_stars(Orlicz_function, x=x, p_norm=n)
    line1.set_data([k_s[1], k_s[2]], [0, 0])
    ax.legend()
    ax.set_title(f"$p={round(n, 2)}$")
    return line,


anim = FuncAnimation(fig, animate,
                     frames=p_norm, interval=200, blit=True)
# anim.save('im.gif', writer="pillow")
# anim.save(filename="html_example.html", writer="html")
plt.close()
HTML(anim.to_jshtml())

Once Loop Reflect

# execution took around 1 minute
fig = plt.figure()
ax = plt.axes(xlim=(0, 5),
              ylim=(-1, 15))

k_min = 0.1
k_max = 15
dk = 0.01

u_max = 20
du = 0.01

p_plus = osm.right_side_derivative(Orlicz_function, u_max=u_max, du=du)
Psi = osm.conjugate_function(Orlicz_function, u_max=u_max, du=du)

p_norm = list(np.arange(1, 2, 0.05))
p_norm.extend(list(np.arange(2, 5, 0.2)))
p_norm.extend(list(np.arange(5, 10, 0.5)))
p_norm.extend(list(np.arange(10, 50, 1)))
# p_norm.extend(list(np.arange(50,200,5)))

p_norm.append(np.inf)

line, = ax.plot([], [], linewidth=2, label=f'$\\tau_{{p, x_n}}(k)$')
line1, = ax.plot([], [], marker="o", color="crimson", linewidth=5, ms=5,
                 label="$K_{{\\alpha, p}}\left(x_n\\right)$")

ax.axhline(y=1, linestyle="--", linewidth=1.3, label="1", color="green")


def animate(n):
    k, tau = osm.array_for_tau(Orlicz_function, u_max=u_max, du=du, p_norm=n, x=x,
                               p_plus=p_plus,
                               Psi=Psi,
                               k_min=k_min, k_max=k_max, dk=dk)
    line.set_data(k, tau)
    k_s = osm.p_Amemiya_norm_with_stars(Orlicz_function, x=x, p_norm=n)
    line1.set_data([k_s[1], k_s[2]], [1, 1])
    ax.legend()
    ax.set_title(f"$p={round(n, 2)}$")
    return line,


anim = FuncAnimation(fig, animate,
                     frames=p_norm, interval=200, blit=True)
# anim.save('im.gif', writer="pillow")
# anim.save(filename="html_example.html", writer="html")
plt.close()
HTML(anim.to_jshtml())

Once Loop Reflect