Hessian-Vector Product Finite Difference

Problem

Approximate a Hessian-vector product by central finite differences for f(x1, x2) = x1^2 + 3*x1*x2 + 2*x2^2.

The analytic gradient is grad(x) = [2*x1 + 3*x2, 3*x1 + 4*x2]. Return (grad(x + h*v) - grad(x - h*v)) / (2*h) as a two-element list, where v is the direction vector and h the difference step.

Hint

Perturb x along v in both directions and difference the gradients.

Test cases

case 1

in: [[1,2],[3,-1],0.0001]

expected: [2.9999999999930083,4.999999999988347]

case 2

in: [[0,0],[1,0],0.0001]

expected: [2,3]

case 3

in: [[-2,0.5],[0,1],0.00001]

expected: [2.9999999999752442,4.000000000004]

case 4

in: [[1,1],[1,1],0.001]

expected: [4.999999999999449,6.999999999999673]

Starter codePython

def hvp_finite_diff(x, v, h):
    # Your code here
    pass

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