Function File: Df = jacobs (x, f)
Function File: Df = jacobs (x, f, hook)

Calculate the jacobian of a function using the complex step method.

Let f be a user-supplied function. Given a point x at which we seek for the Jacobian, the function jacobs returns the Jacobian matrix d(f(1), …, df(end))/d(x(1), …, x(n)). The function uses the complex step method and thus can be applied to real analytic functions.

The optional argument hook is a structure with additional options. hook can have the following fields:

  • h - can be used to define the magnitude of the complex step and defaults to 1e-20; steps larger than 1e-3 are not allowed.
  • fixed - is a logical vector internally usable by some optimization functions; it indicates for which elements of x no gradient should be computed, but zero should be returned.

For example: 

f = @(x) [x(1)^2 + x(2); x(2)*exp(x(1))];
Df = jacobs ([1, 2], f)

Demonstration 1

The following code

 # Relative error against several h-values
 k = 3:20; h = 10 .^ (-k); x = 0.3*pi;
 err = zeros (1, numel (k));
 for count = 1 : numel (k)
   err(count) = abs (jacobs (x, @sin,	struct ("h", h(count))) - cos (x)) / abs (cos (x)) + eps;
 endfor
 loglog (h, err); grid minor;
 xlabel ("h"); ylabel ("|Df(x) - cos(x)| / |cos(x)|")
 title ("f(x)=sin(x), f'(x)=cos(x) at x = 0.3pi")

Produces the following figure

Figure 1

Package: optim