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(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 timeconsuming 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. boundaryvalue problems (see ordinary differential equations).
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