Some applications of partial orderings to iterative methods for rectangular linear systems
TLDR
In this paper, the notions of K -semipositivity, K -monotonicity and K -positive subinverses, introduced by Vandergraft for square matrices, were extended to rectangular matrices.About:
This article is published in Linear Algebra and its Applications.The article was published on 1978-01-01 and is currently open access. It has received 7 citations till now. The article focuses on the topics: Matrix (mathematics) & Square matrix.read more
Citations
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Convergent nonnegative matrices and iterative methods for consistent linear systems
TL;DR: This transformation is parameter-dependent, and in the case where all the eigenvalues of the iteration matrix are real, it is shown how to choose this parameter so that the asymptotic convergence rate of the new schemes is optimal.
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Computing generalized inverses of matrices by iterative methods based on splittings of matrices
Yong-Lin Chen,Xue-Yuan Tan +1 more
TL;DR: This paper proves the result that if A has the splitting A=M-N and if the generalized inverses A^(^2^)"T","S and M-N exist, then the iterationX"j"+"1=M^( ^2^)",T","SNX" j+M+S+T=T,S=S if and only if X"j+M^ (^2 ^)"T),"Sconverges to A^
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Cone reachability for linear differential systems
Michael Neumann,Ronald J. Stern +1 more
TL;DR: In this article, the authors consider the problem of characterizing the closure of the set XA(K) of all initial points x(0) ∈ Rn which give rise to solutions x(t) that enter (and remain in) K.
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Semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices
Yong-Lin Chen,Xue-Yuan Tan +1 more
TL;DR: In this paper, iterations and extrapolated iterations for the consistent rectangular or sigular linear systems are simultaneously studied and the efficient and convenient construction methods of the semiconvergent iteration matrices G and G"@w are presented.
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3-part splittings for singular and rectangular linear systems
TL;DR: In this article, a 3-part splitting of A into M − Q 1 − Q 2 is proposed to obtain a solution to the linear system Ax = b, A ϵ C m × n, or for obtaining an approximate solution to (∗) if the system is not solvable.
References
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Positive operators and an inertia theorem
TL;DR: In this article, a generalization of Perron-Frobenius is used to prove an inertia theorem, which contains Lyapunov's theorem and Stein's theorem as special cases.
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Spectral properties of matrices which have invariant cones
TL;DR: Matrix theorem giving necessary and sufficient conditions for leaving cone invariant, considering finite dimensional spaces as mentioned in this paper, considering finite-dimensional spaces, is used to define the conditions for cone invariance.
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On recurring theorems on diagonal dominance
TL;DR: In this paper, the authors give a number of equivalent conditions which reproduce, and in some cases strengthen, many consequences of recent generalizations of the property of diagonally dominant n x n complex matrices.
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Cones and Iterative Methods for Best Least Squares Solutions of Linear Systems
Abraham Berman,Plemmons Robert J +1 more
TL;DR: The concept of a regular splitting of a real nonsingular matrix, introduced by R. Varga, is used in iterative methods for solving linear systems as discussed by the authors, and it is shown that for a proper splitting, the spectral radius is less than l, if and only if the iteration converges to the best least squares approximate solution of the system.