test

notebook

Published

January 1, 2020

Install packages for gene46100

# %pip install seaborn matplotlib 
# %pip install scikit-learn plotnine tqdm pandas
# %pip install numpy ## gets installed with scikit-learn
# %pip install tqdm
# %pip install torch
# %pip install torchvision torchmetrics

import numpy as np
import matplotlib.pyplot as plt

def periodic_u(x_values):
    """
    Calculates the periodic function u(x) = x, with period L=1.
    This is equivalent to u(x) = x - floor(x) for an array of x values.
    Args:
        x_values (np.array): Input x values.
    Returns:
        np.array: Values of the periodic function.
    """
    return x_values - np.floor(x_values)

def fourier_series_approx(x_values, n_terms_approx):
    """
    Calculates the Fourier series approximation of u(x).
    The formula derived is: u_approx(x) = 1/2 - (1/pi) * sum_{n=1}^{N} (sin(2*pi*n*x) / n)
    Args:
        x_values (np.array): Input x values.
        n_terms_approx (int): Number of terms (N) to include in the summation.
    Returns:
        np.array: Values of the Fourier series approximation.
    """
    # Initialize an array of zeros with the same shape as x_values to store the sum
    series_sum_val = np.zeros_like(x_values, dtype=float)

    # Sum the series term by term
    for n_val in range(1, n_terms_approx + 1):
        series_sum_val += (np.sin(2 * np.pi * n_val * x_values)) / n_val

    # Apply the full formula
    approximation = 0.5 - (1 / np.pi) * series_sum_val
    return approximation

# --- Parameters for plotting ---
N_TERMS_IN_SERIES = 100  # Number of terms in the Fourier series approximation
X_PLOT_MIN = 0           # Minimum x value for plotting
X_PLOT_MAX = 3           # Maximum x value for plotting (to show 3 periods)
NUM_POINTS_PLOT = 1000   # Number of points for generating smooth plot lines

# --- Generate x values for plotting ---
# Using linspace to create an array of evenly spaced x values
x_plot_values = np.linspace(X_PLOT_MIN, X_PLOT_MAX, NUM_POINTS_PLOT)

# --- Calculate y values for the original periodic function ---
y_original_periodic = periodic_u(x_plot_values)

# --- Calculate y values for the Fourier series approximation ---
y_fourier_approximation = fourier_series_approx(x_plot_values, N_TERMS_IN_SERIES)

# --- Plotting the results ---
plt.figure(figsize=(14, 8)) # Create a figure with a specified size for better readability

# Plot the original periodic function
plt.plot(x_plot_values, y_original_periodic, label='Original periodic function $u(x) = x - \\lfloor x \\rfloor$', color='dodgerblue', linewidth=2.5)

# Plot the Fourier series approximation
plt.plot(x_plot_values, y_fourier_approximation, label=f'Fourier Series Approx. ({N_TERMS_IN_SERIES} terms)', color='red', linestyle='--', linewidth=2)

# --- Adding plot enhancements ---
plt.title(f'Original Function vs. Fourier Series Approximation\n($u(x)=x$, period $L=1$, {N_TERMS_IN_SERIES} terms)', fontsize=18)
plt.xlabel('$x$', fontsize=16)
plt.ylabel('$u(x)$', fontsize=16)
plt.legend(fontsize=12, loc='upper right')
plt.grid(True, linestyle=':', alpha=0.6) # Add a grid for easier value reading
plt.ylim(-0.25, 1.25) # Set y-axis limits to focus on the function's range
plt.axhline(0, color='black', linewidth=0.75) # Add a horizontal line at y=0
plt.axvline(0, color='black', linewidth=0.75) # Add a vertical line at x=0

# Add text annotations for clarity if needed, e.g., pointing out Gibbs phenomenon
# plt.annotate('Gibbs Phenomenon', xy=(1, 0.9), xytext=(1.2, 0.7),
#              arrowprops=dict(facecolor='black', shrink=0.05), fontsize=10)

# Display the plot
plt.show()

--- title: test date: 2020-01-01 eval: true freeze: true categories: - notebook jupyter: kernelspec: name: "conda-env-gene46100-py" language: "python" display_name: "gene46100" --- Install packages for gene46100 ```{python} # %pip install seaborn matplotlib # %pip install scikit-learn plotnine tqdm pandas # %pip install numpy ## gets installed with scikit-learn # %pip install tqdm # %pip install torch # %pip install torchvision torchmetrics ``` ```{python} import numpy as np import matplotlib.pyplot as plt def periodic_u(x_values): """ Calculates the periodic function u(x) = x, with period L=1. This is equivalent to u(x) = x - floor(x) for an array of x values. Args: x_values (np.array): Input x values. Returns: np.array: Values of the periodic function. """ return x_values - np.floor(x_values) def fourier_series_approx(x_values, n_terms_approx): """ Calculates the Fourier series approximation of u(x). The formula derived is: u_approx(x) = 1/2 - (1/pi) * sum_{n=1}^{N} (sin(2*pi*n*x) / n) Args: x_values (np.array): Input x values. n_terms_approx (int): Number of terms (N) to include in the summation. Returns: np.array: Values of the Fourier series approximation. """ # Initialize an array of zeros with the same shape as x_values to store the sum series_sum_val = np.zeros_like(x_values, dtype=float) # Sum the series term by term for n_val in range(1, n_terms_approx + 1): series_sum_val += (np.sin(2 * np.pi * n_val * x_values)) / n_val # Apply the full formula approximation = 0.5 - (1 / np.pi) * series_sum_val return approximation # --- Parameters for plotting --- N_TERMS_IN_SERIES = 100 # Number of terms in the Fourier series approximation X_PLOT_MIN = 0 # Minimum x value for plotting X_PLOT_MAX = 3 # Maximum x value for plotting (to show 3 periods) NUM_POINTS_PLOT = 1000 # Number of points for generating smooth plot lines # --- Generate x values for plotting --- # Using linspace to create an array of evenly spaced x values x_plot_values = np.linspace(X_PLOT_MIN, X_PLOT_MAX, NUM_POINTS_PLOT) # --- Calculate y values for the original periodic function --- y_original_periodic = periodic_u(x_plot_values) # --- Calculate y values for the Fourier series approximation --- y_fourier_approximation = fourier_series_approx(x_plot_values, N_TERMS_IN_SERIES) # --- Plotting the results --- plt.figure(figsize=(14, 8)) # Create a figure with a specified size for better readability # Plot the original periodic function plt.plot(x_plot_values, y_original_periodic, label='Original periodic function $u(x) = x - \\lfloor x \\rfloor$', color='dodgerblue', linewidth=2.5) # Plot the Fourier series approximation plt.plot(x_plot_values, y_fourier_approximation, label=f'Fourier Series Approx. ({N_TERMS_IN_SERIES} terms)', color='red', linestyle='--', linewidth=2) # --- Adding plot enhancements --- plt.title(f'Original Function vs. Fourier Series Approximation\n($u(x)=x$, period $L=1$, {N_TERMS_IN_SERIES} terms)', fontsize=18) plt.xlabel('$x$', fontsize=16) plt.ylabel('$u(x)$', fontsize=16) plt.legend(fontsize=12, loc='upper right') plt.grid(True, linestyle=':', alpha=0.6) # Add a grid for easier value reading plt.ylim(-0.25, 1.25) # Set y-axis limits to focus on the function's range plt.axhline(0, color='black', linewidth=0.75) # Add a horizontal line at y=0 plt.axvline(0, color='black', linewidth=0.75) # Add a vertical line at x=0 # Add text annotations for clarity if needed, e.g., pointing out Gibbs phenomenon # plt.annotate('Gibbs Phenomenon', xy=(1, 0.9), xytext=(1.2, 0.7), # arrowprops=dict(facecolor='black', shrink=0.05), fontsize=10) # Display the plot plt.show() ``` ```{python} ```