The use of basic iterative methods for bounding a solution of a system of linear equations with an M-matrix and positive right-hand side

Kocurek, Martin

Displaying similar documents to “The use of basic iterative methods for bounding a solution of a system of linear equations with an M-matrix and positive right-hand side”

A new optimized iterative method for solving $M$ -matrix linear systems

Alireza Fakharzadeh Jahromi, Nafiseh Nasseri Shams (2022)

Applications of Mathematics

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In this paper, we present a new iterative method for solving a linear system, whose coefficient matrix is an $M$ -matrix. This method includes four parameters that are obtained by the accelerated overrelaxation (AOR) splitting and using the Taylor approximation. First, under some standard assumptions, we establish the convergence properties of the new method. Then, by minimizing the Frobenius norm of the iteration matrix, we find the optimal parameters. Meanwhile, numerical results on test...

On the optimality and sharpness of Laguerre's lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Yusaku Yamamoto (2017)

Applications of Mathematics

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Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix $A \in ℝ^{m \times m}$ play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on $Tr (A^{- 1})$ and $Tr (A^{- 2})$ have attracted attention recently, because they can be computed in $O (m)$ operations when $A$ is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that...

On the subspace projected approximate matrix method

Jan Brandts, Ricardo Reis da Silva (2015)

Applications of Mathematics

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We provide a comparative study of the Subspace Projected Approximate Matrix method, abbreviated SPAM, which is a fairly recent iterative method of computing a few eigenvalues of a Hermitian matrix $A$ . It falls in the category of inner-outer iteration methods and aims to reduce the costs of matrix-vector products with $A$ within its inner iteration. This is done by choosing an approximation $A_{0}$ of $A$ , and then, based on both $A$ and $A_{0}$ , to define a sequence ${(A_{k})}_{k = 0}^{n}$ of matrices that increasingly better...

Positive solution for a quasilinear equation with critical growth in $ℝ^{N}$

Lin Chen, Caisheng Chen, Zonghu Xiu (2016)

Annales Polonici Mathematici

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We study the existence of positive solutions of the quasilinear problem ⎧ $- Δ_{N} u + {V (x) | u |}^{N - 2} u = {f (u, | \nabla u |}^{N - 2} \nabla u)$ , $x \in ℝ^{N}$ , ⎨ ⎩ u(x) > 0, $x \in ℝ^{N}$ , where $Δ_{N} u = {d i v (| \nabla u |}^{N - 2} \nabla u)$ is the N-Laplacian operator, $V : ℝ^{N} \to ℝ$ is a continuous potential, $f : ℝ \times ℝ^{N} \to ℝ$ is a continuous function. The main result follows from an iterative method based on Mountain Pass techniques.

Ground states of supersymmetric matrix models

Gian Michele Graf (1998-1999)

Séminaire Équations aux dérivées partielles

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We consider supersymmetric matrix Hamiltonians. The existence of a zero-energy bound state, in particular for the $d = 9$ model, is of interest in M-theory. While we do not quite prove its existence, we show that the decay at infinity such a state would have is compatible with normalizability (and hence existence) in $d = 9$ . Moreover, it would be unique. Other values of $d$ , where the situation is somewhat different, shall also be addressed. The analysis is based on a Born-Oppenheimer approximation....

The new iteration methods for solving absolute value equations

Rashid Ali, Kejia Pan (2023)

Applications of Mathematics

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Many problems in operations research, management science, and engineering fields lead to the solution of absolute value equations. In this study, we propose two new iteration methods for solving absolute value equations $A x - | x | = b$ , where $A \in ℝ^{n \times n}$ is an $M$ -matrix or strictly diagonally dominant matrix, $b \in ℝ^{n}$ and $x \in ℝ^{n}$ is an unknown solution vector. Furthermore, we discuss the convergence of the proposed two methods under suitable assumptions. Numerical experiments are given to verify the feasibility, robustness...

An iterative construction for ordinary and very special hyperelliptic curves

Francis J. Sullivan (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si costruiscono famiglie di curve iperellittiche col $p$ —rango della varietà jacobiana uguale a zero. La costruzione sfrutta le proprietà elementari dell’operatore di Cartier e delle estensioni $p$ -cicliche dei corpi con la caratteristica $p$ maggiore di zero.

Investigating generalized quaternions with dual-generalized complex numbers

Nurten Gürses, Gülsüm Yeliz Şentürk, Salim Yüce (2023)

Mathematica Bohemica

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We aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values $α$ , $β$ and $𝔭$ . Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given.

A lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$ -matrix and its inverse

Wenlong Zeng, Jianzhou Liu (2022)

Czechoslovak Mathematical Journal

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We propose a lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$ -matrix and its inverse, in terms of an $S$ -type eigenvalues inclusion set and inequality scaling techniques. In addition, it is proved that the lower bound sequence converges. Several numerical experiments are given to demonstrate that the lower bound sequence is sharper than some existing ones in most cases.

G-matrices, $J$ -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_{1}$ and $D_{2}$ such that $A^{- T} = D_{1} A D_{2}$ , where $A^{- T}$ denotes the transpose of the inverse of $A$ . Denote by $J = diag (\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+ 1$ or $- 1$ . A nonsingular real matrix $Q$ is called $J$ -orthogonal if $Q^{T} J Q = J$ . Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

A tight bound of modified iterative hard thresholding algorithm for compressed sensing

Jinyao Ma, Haibin Zhang, Shanshan Yang, Jiaojiao Jiang (2023)

Applications of Mathematics

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We provide a theoretical study of the iterative hard thresholding with partially known support set (IHT-PKS) algorithm when used to solve the compressed sensing recovery problem. Recent work has shown that IHT-PKS performs better than the traditional IHT in reconstructing sparse or compressible signals. However, less work has been done on analyzing the performance guarantees of IHT-PKS. In this paper, we improve the current RIP-based bound of IHT-PKS algorithm from $δ_{3 s - 2 k} < \frac{1}{\sqrt{32}} \approx 0.1768$ to $δ_{3 s - 2 k} < \frac{\sqrt{5} - 1}{4} \approx 0.309$ , where $δ_{3 s - 2 k}$ is...

Nonlinear mappings preserving at least one eigenvalue

Constantin Costara, Dušan Repovš (2010)

Studia Mathematica

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We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either $F (x) = u x u^{- 1}$ or $F (x) = u x^{t} u^{- 1}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying...