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The algorithms described in this section do not require any derivative information to be supplied by the user. Any derivatives needed are approximated by finite differences. Note that if the finite-differencing step size chosen by these routines is inappropriate, an explicit user-supplied numerical derivative can always be used with the algorithms described in the previous section.
This is a version of the Hybrid algorithm which replaces calls to the
Jacobian function by its finite difference approximation. The finite
difference approximation is computed using gsl_multiroots_fdjac
with a relative step size of GSL_SQRT_DBL_EPSILON. Note that
this step size will not be suitable for all problems.
This is a finite difference version of the Hybrid algorithm without internal scaling.
The discrete Newton algorithm is the simplest method of solving a multidimensional system. It uses the Newton iteration
x -> x - J^{-1} f(x)
where the Jacobian matrix J is approximated by taking finite differences of the function f. The approximation scheme used by this implementation is,
J_{ij} = (f_i(x + \delta_j) - f_i(x)) / \delta_j
where \delta_j is a step of size \sqrt\epsilon |x_j| with \epsilon being the machine precision (\epsilon \approx 2.22 \times 10^-16). The order of convergence of Newton’s algorithm is quadratic, but the finite differences require n^2 function evaluations on each iteration. The algorithm may become unstable if the finite differences are not a good approximation to the true derivatives.
The Broyden algorithm is a version of the discrete Newton algorithm which attempts to avoids the expensive update of the Jacobian matrix on each iteration. The changes to the Jacobian are also approximated, using a rank-1 update,
J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df
where the vectors dx and df are the changes in x and f. On the first iteration the inverse Jacobian is estimated using finite differences, as in the discrete Newton algorithm.
This approximation gives a fast update but is unreliable if the changes are not small, and the estimate of the inverse Jacobian becomes worse as time passes. The algorithm has a tendency to become unstable unless it starts close to the root. The Jacobian is refreshed if this instability is detected (consult the source for details).
This algorithm is included only for demonstration purposes, and is not recommended for serious use.