HardCalculusca-164

Hessian in 1D Finite Difference

Problem

The five-point central stencil for the second derivative is

f''(x) ≈ (-f(x+2h) + 16*f(x+h) - 30*f(x) + 16*f(x-h) - f(x-2h)) / (12*h^2)

which is exact for polynomials up to degree 5. Apply it to f(x) = sum(coeffs[i] * x^i) at x with h = 1e-3.

Hint

The weights -1, 16, -30, 16, -1 over 12 h^2 give fourth-order accuracy.

case 1

in: [[0,0,1],1]

expected: 2

case 2

in: [[0,0,0,0,1],1]

expected: 12

case 3

in: [[0,0,0,1],2]

expected: 12

case 4

in: [[5],3]

expected: 0

Starter codePython

def hessian_1d_5point(coeffs, x, h=1e-3):
    # Your code here
    pass

4 test cases · run your code against them in the editor