The problem of multidimensional root finding requires the simultaneous
solution of *n* equations, *f_i*, in *n* variables,
*x_i*,

f_i (x_1, ..., x_n) = 0 for i = 1 ... n.

In general there are no bracketing methods available for *n*
dimensional systems, and no way of knowing whether any solutions
exist. All algorithms proceed from an initial guess using a variant of
the Newton iteration,

x -> x' = x - J^{-1} f(x)

where *x*, *f* are vector quantities and *J* is the
Jacobian matrix *J_{ij} = d f_i / d x_j*.
Additional strategies can be used to enlarge the region of
convergence. These include requiring a decrease in the norm *|f|* on
each step proposed by Newton’s method, or taking steepest-descent steps in
the direction of the negative gradient of *|f|*.

Several root-finding algorithms are available within a single framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

- initialize solver state,
`s`, for algorithm`T` - update
`s`using the iteration`T` - test
`s`for convergence, and repeat iteration if necessary

The evaluation of the Jacobian matrix can be problematic, either because
programming the derivatives is intractable or because computation of the
*n^2* terms of the matrix becomes too expensive. For these reasons
the algorithms provided by the library are divided into two classes according
to whether the derivatives are available or not.

The state for solvers with an analytic Jacobian matrix is held in a
`gsl_multiroot_fdfsolver`

struct. The updating procedure requires
both the function and its derivatives to be supplied by the user.

The state for solvers which do not use an analytic Jacobian matrix is
held in a `gsl_multiroot_fsolver`

struct. The updating procedure
uses only function evaluations (not derivatives). The algorithms
estimate the matrix *J* or *J^{-1}* by approximate methods.