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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Kaczmarz method ： ウィキペディア英語版 Kaczmarz method The Kaczmarz method or Kaczmarz's algorithm is an iterative algorithm for solving linear equation systems $A\; x\; =\; b$. It was first discovered by the Polish mathematician Stefan Kaczmarz, and was rediscovered in the field of image reconstruction from projections by Richard Gordon, Robert Bender, and Gabor Herman in 1970, where it is called the Algebraic Reconstruction Technique (ART). ART includes the positivity constraint, making it nonlinear. The Kaczmarz method is applicable to any linear system of equations, but its computational advantage relative to other methods depends on the system being sparse. It has been demonstrated to be superior, in some biomedical imaging applications, to other methods such as the filtered backprojection method. It has many applications ranging from computed tomography (CT) to signal processing. It can be obtained also by applying to the hyperplanes, described by the linear system, the method of successive projections onto convex sets (POCS). ==Algorithm 1: Kaczmarz algorithm== Let $A\; x\; =\; b$ be a linear system, let $m$ the number of rows of A, $a\_$ be the $i$th row of complex-valued matrix $A$, and let $x^$ be arbitrary complex-valued initial approximation to the solution of $Ax=b$. For $k=0,1,...$ compute: :$$ x^ = x^ + \frac, x^ \rangle} \overline denotes complex conjugation of $a\_i$. If the linear system is consistent, $x^k$ converges to the minimum-norm solution, provided that the iterations start with the zero vector. A more general algorithm can be defined using a relaxation parameter $\backslash lambda^k$ : $$ x^ = x^ + \lambda^k \frac, x^ \rangle} \overline{a_{i}} There are versions of the method that converge to a regularized weighted least squares solution when applied to a system of inconsistent equations and, at least as far as initial behavior is concerned, at a lesser cost than other iterative methods, such as the conjugate gradient method.〔See and references therein.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kaczmarz method」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.