An Introduction to Applied Matrix Analysis

Jin, Xiao Qing; Vong, Seakweng

doi:10.1142/9932

Cited by 7 publications

(4 citation statements)

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“…is proposed where c( Ã) is the Chan's circulant approximation [5,13,19] to the symmetric positive definite Toeplitz matrix Ã, and s( B) is the Strang's circulant approximation [3,4,13] to the Toeplitz matrix…”

Section: Mixed-type Circulant Preconditionermentioning

confidence: 99%

“…Besides the properties of AB, an effective preconditioner is proposed for solving linear systems with coefficient matrix AB. By approximating Ã and B by using Chan's circulant matrix [5,13,19] and Strang's circulant matrix [3,4,13] respectively, a mixedtype circulant preconditioner is obtained. Since circulant matrices can be diagonalized by Fourier matrix so that the proposed preconditioner can be inverted efficiently via fast Fourier transform (FFT) [8].…”

Section: Introductionmentioning

confidence: 99%

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A Mixed-Type Circulant Preconditioner for a Nonlocal Elastic Model

Lei¹

2023

EAJAM

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A linear system from finite difference discretization of a generalized nonlocal elastic model was studied, where the model is composed of a Riesz potential operator with a fractional differential operator. Some properties of the coefficient matrix are proven theoretically and it is found that the linear system is very ill-conditioned when the parameter in the long-range hydrodynamic interactions is close to zero. Therefore, the usual Krylov subspace method with the Strang-Strang circulant preconditioner loses the power of preconditioning so that the iterative method converges slowly. Here the problem is fixed by utilizing a mixed-type circulant preconditioner which is obtained by both Strang's and Chan's circulant approximations. The invertibility of the preconditioner and a small-norm-low-rank decomposition of the difference matrix of the coefficient matrix and the preconditioner are shown theoretically under certain conditions. Numerical examples are given to illustrate the efficiency of the proposed fast solver.

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Section: Mixed-type Circulant Preconditionermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Mixed-Type Circulant Preconditioner for a Nonlocal Elastic Model

Lei¹

2023

EAJAM

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“…The closed‐loop system (5) can be rewritten in the following Kronecker product 42,43 form:

\{\begin{array}{l} \overset{x^{*}}{true} false(t false) = prefix- x^{*} false(t false) + z false(t false) \\ {\overset{z}{true}}_{i} false(t false) = false(boldU prefix- boldG prefix- boldW prefix- boldW Λ false(t false) false) z false(t false) prefix- boldH x^{*} false(t false) + boldC \overset{f}{true} false(x^{*} false(t false) false) + boldD \overset{f}{true} false(x^{*} false(t prefix- τ false(t false) false) false) \\ + boldF \int_{t prefix- τ}^{t} W false(t prefix- s false) \overset{f}{true} false(x^{*} false(s false) false) d s . \end{array}

We also need the following lemmas:…”

Section: Preliminariesmentioning

confidence: 99%

Exponential synchronization of coupled inertial neural networks with mixed delays via weighted integral inequalities

Vong

Shi

Yao

2020

Intl J Robust & Nonlinear

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We study the exponential synchronization of coupled inertial neural networks with both discrete-time delay and distributed delay by quantized pinning controllers. Novel integral inequalities, which generalize the Jensen-based inequality, are developed by choosing appropriate weight functions in our recent work. An exponential synchronization criterion is established by applying these inequalities to analyzing a Lyapunov-Krasovskii functional which takes mixed delays into account. Numerical simulations show that the criterion can reduce conservativeness when designing parameters of the controller.

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“…Then for P, F ∈ B m× n , (P • F) i j = 1, if p i j = 1 and f i j = 1 0, otherwise Definition 1.4. (Frobenius inner product [5]) Consider the vector space R n× n over the field R. Then the Frobenius inner product , F : R n× n × R n× n → R is defined by,…”

Section: Introductionmentioning

confidence: 99%

A note on Frobenius inner product and the $m$-distance matrices of a tree

Asharaf¹,

Bindhu²

2020

Malaya J. Mat

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The m-distance matrix D m of a simple connected undirected graph has an important role in computing the distance matrix D of the graph from the powers of the adjacency matrix using Hadamard product. This paper shows that for an undirected tree T with diameter d, {D 0 .D 1 ,. .. , D d } is an orthogonal basis for the space spanned by the binary equivalent matrices of the first d + 1 powers of the adjacency matrix of T and it gives an invertible conversion matrix for finding the m-distance matrix of T using Frobenius inner product on matrices.

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An Introduction to Applied Matrix Analysis

Cited by 7 publications

References 0 publications

A Mixed-Type Circulant Preconditioner for a Nonlocal Elastic Model

A Mixed-Type Circulant Preconditioner for a Nonlocal Elastic Model

Exponential synchronization of coupled inertial neural networks with mixed delays via weighted integral inequalities

A note on Frobenius inner product and the $m$-distance matrices of a tree

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