Newton's method

Newton's method An iterative technique for solving one or more nonlinear equations. For the single equation f(x) = 0

the iteration is x_n+1 = x_n – f(x_n)/f ′(x_n), n = 0,1,2,…

where x₀ is an approximation to the solution. For the system f(x) = 0, f = (f₁,f₂,…,f_n)^T, x = (x₁,x₂,…,x_n)^T,

the iteration takes the mathematical form x_n+1 = x_n – J(x_n)^–1f(x_n), n = 0,1,2,…

where J(x) is the n×n matrix whose i,jth element is f_i(x)/x_j

In practice each iteration is carried out by solving a system of linear equations. Subject to appropriate conditions the iteration converges quadratically (ultimately an approximate squaring of the error occurs). The disadvantage of the method is that a constant recalculation of J may be too time-consuming and so the method is most often used in a modified form, e.g. with approximate derivatives. Since Newton's method is derived by a linearization of f(x), it is capable of generalization to other kinds of nonlinear problems, e.g. boundary-value problems (see ordinary differential equations).