The problem of multidimensional root finding requires the simultaneous solution of @math{n} equations, @math{f_i}, in @math{n} variables, @math{x_i},
In general there are no bracketing methods available for @math{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,
where @math{x}, @math{f} are vector quantities and @math{J} is the Jacobian matrix @c{$J_{ij} = \partial f_i / \partial x_j$} @math{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 @math{|f|} on each step proposed by Newton's method, or taking steepest-descent steps in the direction of the negative gradient of @math{|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,
The evaluation of the Jacobian matrix can be problematic, either because programming the derivatives is intractable or because computation of the @math{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 @math{J} or @c{$J^{-1}$}
@math{J^{-1}} by approximate methods.