## Quick Reference

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

*x*

_{0}is an approximation to the solution. For the system

*f*(

*x*) = 0,

*f*= (

*f*

_{1},

*f*

_{2},…,

*f*

*)*

_{n}^{T},

*x*= (

*x*

_{1},

*x*

_{2},…,

*x*

*)*

_{n}^{T}, the iteration takes the mathematical form

*x*

*=*

_{n+1}*x*

*−*

_{n}*J*(

*x*

*)*

_{n}^{−1}

*f*(

*x*

*),*

_{n}*n*= 0,1,2,… where

*J*(

*x*) is the

*n*′

*n*matrix whose

*i*,

*j*th element is ∂

*f*

*(*

_{i}*x*)/∂

*x*

*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}*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).

*f*(*x*) = 0

*x** _{n+1}* =

*x*

*−*

_{n}*f*(

*x*

*)/*

_{n}*f*′(

*x*

*),*

_{n}*n* = 0,1,2,…

*f*(*x*) = 0,

*f* = (*f*_{1},*f*_{2},…,*f** _{n}*)

^{T},

*x* = (*x*_{1},*x*_{2},…,*x** _{n}*)

^{T},

*x** _{n+1}* =

*x*

*−*

_{n}*J*(

*x*

*)*

_{n}^{−1}

*f*(

*x*

*),*

_{n}*n* = 0,1,2,…

∂*f** _{i}*(

*x*)/∂

*x*

_{j}
*Subjects:*
Computing.

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