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Newton's method

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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 nn 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).

f(x) = 0

xn+1 = xnf(xn)/f′(xn),

n = 0,1,2,…

f(x) = 0,

f = (f1,f2,…,fn)T,

x = (x1,x2,…,xn)T,

xn+1 = xnJ(xn)−1f(xn),

n = 0,1,2,…

fi(x)/∂xj

Subjects: Computing.

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