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pythonnumpylinear-algebraderivativetaylor-series

Why does my finite difference weights calculation for Taylor series give incorrect results?

I'm trying to calculate the weights for a finite-difference approximation of the first derivative f′(x)f'(x)f′(x) using the Taylor series expansion. I'm solving for weights a,b,c,d,ea, b, c, d, ea,b,c,d,e such that:

af(x+2Δx)+bf(x+Δx)+cf(x)+df(x−Δx)+ef(x−2Δx)a f(x+2\Delta x) + b f(x+\Delta x) + c f(x) + d f(x-\Delta x) + e f(x-2\Delta x)af(x+2Δx)+bf(x+Δx)+cf(x)+df(x−Δx)+ef(x−2Δx)

approximates f′(x)f'(x)f′(x). Here's what I did:

  1. I used the Taylor series expansion for f(x±kΔx)f(x \pm k\Delta x)f(x±kΔx), where k=1,2k = 1, 2k=1,2.

  2. I built a system of linear equations to enforce the following conditions:

    • Coefficients of f(x): a+b+c+d+e=0

    • Coefficients of f′(x): 2a+b−d−2e=1

    • Coefficients of f′′(x): 4a+b+d+4e=0

    • Coefficients of f(3)(x): 8a+b−d−8e=0

    • Coefficients of f(4)(x): 16a+b+d+16e=0

  3. I implemented the matrix equation A⋅z=bA \cdot z = bA⋅z=b in Python:

    import numpy as np

A = np.array([
    [1, 1, 1, 1, 1],      # Coefficients of f(x)
    [2, 1, 0, -1, -2],    # Coefficients of f'(x)
    [4, 1, 0, 1, 4],      # Coefficients of f''(x)
    [8, 1, 0, -1, 8],     # Coefficients of f'''(x)
    [16, 1, 0, 1, 16]     # Coefficients of f''''(x)
])

b = np.array([0, 1, 0, 0, 0])  # Targeting the first derivative

z = np.linalg.solve(A, b)
print(z)

  4. The Issue:

    The output I'm getting is:

    [0.25,0,-0,0,-0.25]

    However, the expected weights for the first derivative should be something like:

  5. [-1/12,2/3,0,-2/3,1/12]

  6. What I Tried:

    1. Double-checked the coefficients in matrix A to ensure they match the Taylor series expansion.

    2. Verified that the right-hand side vector b=[0,1,0,0,0] is correct for approximating f′(x).

    Despite this, the weights are incorrect. Am I missing something in the matrix setup or the Python implementation?

  7. Expected Behavior:

    I want the solution to match the theoretical weights for a central finite difference approximation of the first derivative f′(x) using five points.

Solution

  • You have a typo in the 4th line of the matrix A, the last element should be -8 instead of 8:

    A = np.array([
    [1, 1, 1, 1, 1],      # Coefficients of f(x)
    [2, 1, 0, -1, -2],    # Coefficients of f'(x)
    [4, 1, 0, 1, 4],      # Coefficients of f''(x)
    [8, 1, 0, -1, -8],     # Coefficients of f'''(x)
    [16, 1, 0, 1, 16]     # Coefficients of f''''(x)
])