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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 xn+1 = xnf(xn)/f ′(xn), n = 0,1,2,…

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

the iteration takes the mathematical form xn+1 = xnJ(xn)–1f(xn), n = 0,1,2,…

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

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).

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