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- W2022123449 abstract "The concept of a regular splitting of a real nonsingular matrix, introduced by R. Varga, is used in iterative methods for solving linear systems. The purpose of this paper is to extend this concept to rectangular linear systems [ ( * )qquad Ax = b, ] in two directions. Firstly, this is accomplished by replacing $A^{ - 1} $ by $A^dag $, the Moore–Penrose inverse of A, and, secondly, by considering matrices that leave a cone invariant.Let $A in R^{m times n} $. The splitting $A = M - N$ is called proper if $R(A) = R(M)$ and $N(A) = N(M)$. Such is the case when A and M are nonsingular. Consider the iteration [ ( * * )qquad x^{i + 1} = M^ dag Nx^i + M^ dag b. ] It is shown that for a proper splitting, $rho (M^ dag N)$ (the spectral radius of $M^ dag N$) is less than l, if and only if the iteration (**) converges to $A^ dag b$, the best least squares approximate solution of the system (*). This approach has the advantage of avoiding the normal system $A^T Ax = A^T b$ in solving (*). Necessary an..." @default.
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- W2022123449 date "1974-03-01" @default.
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- W2022123449 title "Cones and Iterative Methods for Best Least Squares Solutions of Linear Systems" @default.
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- W2022123449 doi "https://doi.org/10.1137/0711015" @default.
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