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NEWTON'S METHOD One of the most familiar iterative methods is Newton's Method . ATTRACTIVE FIXED POINTS If an equation can be put into the form ''f''(''x'') = ''x'', and a solution x is an attractive Fixed Point of the function ''f'', then one may begin with a point ''x''1 in the basin of attraction of x, and let ''x''''n''+1 = ''f''(''x''''n'') for ''n'' ≥ 1, and the sequence {''x''''n''}''n'' ≥ 1 will converge to the solution x. LINEAR SYSTEMS In the case of a System Of Linear Equations , the two main classes of iterative methods are the stationary iterative methods, and the more general Krylov Subspace methods. Stationary iterative methods Stationary iterative methods solve a linear system with an Operator approximating the original one; and based on a measurement of the error (the residual), form a Correction Equation for which this process is repeated. While these methods are simple to derive, implement, and analyse, convergence is only guaranteed for a limited class of matrices. Examples of stationary iterative methods are the Jacobi Method and the Gauss-Seidel Method . Krylov subspace methods Krylov Subspace methods form an Orthogonal Basis of the sequence of successive matrix powers times the initial residual (the Krylov sequence). The approximations to the solution are then formed by minimizing the residual over the subspace formed. The prototypical method in this class is the Conjugate Gradient Method (CG). Other methods are the Generalized Minimal Residual Method (GMRES) and the Biconjugate Gradient Method (BiCG). Convergence Since these methods form a basis, it is evident that the method converges in ''N'' iterations, where ''N'' is the system size. However, in the presence of rounding errors this statement does not hold; moreover, in practice ''N'' can be very large, and the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, depending on a complicated function of the Spectrum of the operator. Preconditioners The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov Subspace Methods such as GMRES (alternatively, Preconditioned Krylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area. History Probably the first iterative method for solving a linear system appeared in a letter of Gauss to a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest. The theory of stationary iterative methods was solidly established with the work of D.M. Young starting in the 1950s. The Conjugate Gradient Method was also invented in the 1950s, with independent developments by Cornelius Lanczos , Magnus Hestenes and Eduard Stiefel , but its nature and applicability were misunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well for Partial Differential Equation s, especially the elliptic type. EXTERNAL LINKS
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