Convergence Theory of the CR Method for Linear Singular Systems

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  • 特異な係数行列をもつ連立一次方程式に対するCR法の収束性
  • トクイ ナ ケイスウ ギョウレツ オ モツ レンリツ 1ジ ホウテイシキ ニ タイスル CRホウ ノ シュウソクセイ

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The convergence rate of the residual vector of the conjugate residual (CR) method is well known for a linear system Ax=b, where A is nonsingular. In this paper, we consider the convergence theory of the CR method for a linear system, where the coefficient matrix is singular. First, when we give a certain condition, we show that the algorithm of the CR method can be decomposed into components in the range space of A, which we denote by R(A), and the orthogonal complement space of R(A). Secondly, we present a bound of the residual norms of the CR method in R(A). These two results imply that we can derive an estimate of the error bound for a linear singular system. Moreover, we show necessary and sufficient conditions for the convergence of the CR method starting with an arbitrary right-hand side vector. As a byproduct, the residual norm of the CR method for symmetric positive semi-definite coefficient matrices is also analyzed.

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