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In computational mathematics, an iterative method attempts to solve a problem (for example an equation or system of equations) by finding successive approximations to the solution starting from an initial guess. This approach is in contrast to direct methods, which attempt to solve the problem in one-shot (like solving a linear system of equations Ax = b by finding the inverse of the matrix A). Iterative methods are useful for problems involving a large number of variables (sometimes of the order of millions), where direct methods would be prohibitively expensive and in some cases impossible even with the best available computing power. Wikipedia does not have an article with this exact name. ...
Guess values are more commonly called starting values or initial values. ...
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Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
It has been suggested that this article or section be merged with estimation. ...
In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
Newton's method One of the most familiar iterative methods is Newton's method. In numerical analysis, Newtons method (or the Newton–Raphson method or the Newton–Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...
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 x1 in the basin of attraction of x, and let xn+1 = f(xn) for n ≥ 1, and the sequence {xn}n ≥ 1 will converge to the solution x. In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...
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. In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ...
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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. In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...
The Jacobi method is an algorithm in linear algebra for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. ...
The Gauss-Seidel method is a technique used to solve a linear system of equations. ...
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). This article needs to be cleaned up to conform to a higher standard of quality. ...
In mathematics, an orthonormal basis of an inner product space V(i. ...
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive definite. ...
In mathematics, the generalized minimal residual method (usually abbreviated GMRES) is an iterative method for the numerical solution of a system of linear equations. ...
In mathematics, more specifically in numerical analysis, the biconjugate gradient method is an algorithm to solve systems of linear equations Unlike the conjugate gradient method, this algorithm does not require the matrix to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose . The algorithm Choose...
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. In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ...
Preconditioners The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods (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 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. (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...
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 equations, especially the elliptic type. Cornelius Lanczos, born Kornél Löwy (February 2, 1893–June 25, 1974), was a Hungarian mathematician and physicist. ...
In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...
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