# Newtons 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

*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*(

**) is the**

*x**n*×

*n*matrix whose

*i,j*th 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***), it is capable of generalization to other kinds of nonlinear problems, e.g. boundary-value problems (see ordinary differential equations).**

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**Newtons method**