taylor_series_hessian.jl

using TaylorSeries, StaticArrays, ForwardDiff


f( (x, y) ) = x^2 + 5x*y + 2y^2

H = zeros(2, 2)

t = Taylor1(2)



a, b, = 1, 2

H_target = ForwardDiff.hessian(f, SA[a, b])

xx = a + t
yy = b + 0t

H[1, 1] = 2 * f( (xx, yy) )[2]

H[2, 2] = 2 * f( (a + 0t, b + t))[2]




v = SA[1, 1]
xx = SA[a, b] .+ t.*v

f(xx)


f(xx)[2] - 0.5*(H[1, 1] + H[2, 2])

## H[1, 1] + 2 H[1, 2] + H[2, 2]

# 2nd order term is 0.5*∑ v_i H_{ij} v_j
# If v_i == v_j == 1 then this is ∑ H_{ij} / 2

# So H[1, 2] = 2nd_order_term - H[1, 1] - H[1, 2]