, f, epsilon=None, args=(), kwargs={}, return_grad=False)[source]

Calculate Hessian with finite difference derivative approximation


x : array_like

value at which function derivative is evaluated

f : function

function of one array f(x, *args, **kwargs)

epsilon : float or array-like, optional

Stepsize used, if None, then stepsize is automatically chosen according to EPS**(1/3)*x.

args : tuple

Arguments for function f.

kwargs : dict

Keyword arguments for function f.

return_grad : bool

Whether or not to also return the gradient


hess : ndarray

array of partial second derivatives, Hessian

grad : nparray

Gradient if return_grad == True


Equation (8) in Ridout. Computes the Hessian as:

1/(2*d_j*d_k) * ((f(x + d[j]*e[j] + d[k]*e[k]) - f(x + d[j]*e[j])) -
           (f(x + d[k]*e[k]) - f(x)) +
           (f(x - d[j]*e[j] - d[k]*e[k]) - f(x + d[j]*e[j])) -
           (f(x - d[k]*e[k]) - f(x)))

where e[j] is a vector with element j == 1 and the rest are zero and d[i] is epsilon[i].


Ridout, M.S. (2009) Statistical applications of the complex-step method
of numerical differentiation. The American Statistician, 63, 66-74