MediumCalculusca-234

Hessian Trace by Finite Differences

Problem

Estimate the trace of the Hessian of f(x) = sum(coeffs[i] * x_i^3) with central second differences.

Each diagonal entry is approximated by (f(x + h e_i) - 2 f(x) + f(x - h e_i)) / h^2, and the trace is their sum. Given the coefficients and the point x, return the estimate using h = 1e-3; the formula is exact for this cubic function.

Hint

The trace of the Hessian is the Laplacian, and the analytic value here is 6 * sum(coeffs[i] * x_i).

Test cases

case 1

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

expected: 18

case 2

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

expected: -6

case 3

in: [[1],[0]]

expected: 0

case 4

in: [[1,2,3],[1,1,1]]

expected: 36

Starter codePython

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

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