low-effort.py

import numpy as np
from scipy.sparse import diags
from scipy.sparse.linalg import spsolve

def finite_difference_fit(x, y, order=2):
    """
    Implement an arbitrary order one-dimensional finite difference scheme
    to fit a given set of data points.
    
    Parameters:
    x (array): x-coordinates of data points
    y (array): y-coordinates of data points
    order (int): Order of the finite difference scheme (default: 2)
    
    Returns:
    array: Fitted y values
    """
    n = len(x)
    
    # Calculate step size (assuming uniform grid)
    h = x[1] - x[0]
    
    # Create finite difference coefficients
    coeffs = np.zeros(order + 1)
    for i in range(order + 1):
        coeffs[i] = (-1)**(i) * np.math.comb(order, i)
    
    # Create sparse matrix for finite difference operator
    diagonals = [coeffs[i] * np.ones(n - i) for i in range(order + 1)]
    offsets = list(range(order + 1))
    A = diags(diagonals, offsets, shape=(n, n))
    
    # Apply boundary conditions (zero derivative at boundaries)
    A[0, :order+1] = 0
    A[0, 0] = 1
    A[-1, -order-1:] = 0
    A[-1, -1] = 1
    
    # Solve the system
    b = y.copy()
    b[0] = y[0]
    b[-1] = y[-1]
    
    y_fit = spsolve(A, b)
    
    return y_fit

# Example usage
if __name__ == "__main__":
    # Generate some sample data
    x = np.linspace(0, 10, 100)
    y = np.sin(x) + 0.1 * np.random.randn(len(x))
    
    # Fit using 2nd order finite difference
    y_fit = finite_difference_fit(x, y, order=2)
    
    # Plot results
    import matplotlib.pyplot as plt
    plt.figure(figsize=(10, 6))
    plt.plot(x, y, 'o', label='Data')
    plt.plot(x, y_fit, label='Fitted')
    plt.legend()
    plt.xlabel('x')
    plt.ylabel('y')
    plt.title('1D Finite Difference Fit')
    plt.show()