Infinite powers of matrices and characteristic roots

Oldenburger, Rufus

doi:10.1215/s0012-7094-40-00627-5

Cited by 86 publications

(26 citation statements)

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“…The following result indicates an equivalent definition of convergence; see [3,Lemma 7.6.9], [14] [15], [17]. For other equivalent conditions, see, e.g., [16], [19], [20], [26].…”

Section: The Partial Ordermentioning

confidence: 99%

Comparison of Convergence of General Stationary Iterative Methods for Singular Matrices

Marek¹,

Szyld²

2002

SIAM J. Matrix Anal. & Appl.

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Abstract. New comparison theorems are presented comparing the asymptotic convergence factor of iterative methods for the solution of consistent (as well as inconsistent) singular systems of linear equations. The asymptotic convergence factor of the iteration matrix T is the quantity γ(T ) = max{|λ|, λ ∈ σ(T ), λ = 1}, where σ(T ) is the spectrum of T . In the new theorems, no restrictions are imposed on the projections associated with the two iteration matrices being compared. The splittings of the well-known example of Kaufman [SIAM J. Sci. Statist. Comput., 4 (1983), pp. 525-552] satisfy the hypotheses of the new theorems.

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“…The following result indicates an equivalent definition of convergence; see [3,Lemma 7.6.9], [14] [15], [17]. For other equivalent conditions, see, e.g., [16], [19], [20], [26].…”

Section: The Partial Ordermentioning

confidence: 99%

Comparison of Convergence of General Stationary Iterative Methods for Singular Matrices

Marek¹,

Szyld²

2002

SIAM J. Matrix Anal. & Appl.

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“…A matrix T 2 C n;n is convergent if and only if either qðT Þ < 1 or qðT Þ ¼ 1; 1 2 rðT Þ; cðT Þ < 1, and T has only elementary divisors corresponding to the eigenvalue 1, that is, NðI À T Þ \ RðI À T Þ ¼ f0g [4,39,57,61,63].…”

Section: Projected Iterative Algorithmsmentioning

confidence: 99%

Multicomponent transport algorithms for partially ionized mixtures

Giovangigli¹

2010

Journal of Computational Physics

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“…This paper is essentially inspired by a result which goes back to Hensel [7] and Oldenburger [14] (cf. also Meyer and Plemmons [10] and the monograph of Berman and Plemmons [2,Lemma 7.6.13] for any vector norm in ~.".…”

Section: P(t) Of T Satisfies P(t)>mentioning

confidence: 99%

On the solution of singular linear systems of algebraic equations by semiiterative methods

Eiermann

Marek²,

Niethammer

1988

Numer. Math.

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Summary. For a square matrix T~"'", where (l-T) is possibly singular, we investigate the solution of the linear fixed point problem x = Tx + e by applying semiiterative methods (SIM's) to the basic iteration Xo6~U", Xk 9 "=Txk-1 +c(k>l). Such problems arise if one splits the coefficient matrix of a linear system A x=b of algebraic equations according to A =M-N (M nonsingutar) which leads to x=M-~Nx+M-lb=:Tx+e. Even if x = Tx+e is consistent there are cases where the basic iteration fails to converge, namely if T possesses eigenvalues 2 + 1 with ]21 > 1, or if 2= 1 is an eigenvalue of T with nonlinear elementary divisors. In these cases -and also if x= Tx +e is incompatible -we derive necessary and sufficient conditions implying that a SIM tends to a vector ~ which can be described in terms of the Drazin inverse of (I -T). We further give conditions under which is a solution or a least squares solution of (I -T) x = c.

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Infinite powers of matrices and characteristic roots

Cited by 86 publications

References 0 publications

Comparison of Convergence of General Stationary Iterative Methods for Singular Matrices

Comparison of Convergence of General Stationary Iterative Methods for Singular Matrices

Multicomponent transport algorithms for partially ionized mixtures

On the solution of singular linear systems of algebraic equations by semiiterative methods

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