The new iteration methods for solving absolute value equations

Rashid Ali; Kejia Pan

Displaying similar documents to “The new iteration methods for solving absolute value equations”

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...

Computing the greatest $𝐗$ -eigenvector of a matrix in max-min algebra

Ján Plavka (2016)

Kybernetika

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A vector $x$ is said to be an eigenvector of a square max-min matrix $A$ if $A \otimes x = x$ . An eigenvector $x$ of $A$ is called the greatest $𝐗$ -eigenvector of $A$ if $x \in 𝐗 = {x; \underset{̲}{x} \leq x \leq \bar{x}}$ and $y \leq x$ for each eigenvector $y \in 𝐗$ . A max-min matrix $A$ is called strongly $𝐗$ -robust if the orbit $x, A \otimes x, A^{2} \otimes x, \dots$ reaches the greatest $𝐗$ -eigenvector with any starting vector of $𝐗$ . We suggest an $O (n^{3})$ algorithm for computing the greatest $𝐗$ -eigenvector of $A$ and study the strong $𝐗$ -robustness. The necessary and sufficient conditions for strong $𝐗$ -robustness are introduced...

Distance matrices perturbed by Laplacians

Balaji Ramamurthy, Ravindra Bhalchandra Bapat, Shivani Goel (2020)

Applications of Mathematics

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Let $T$ be a tree with $n$ vertices. To each edge of $T$ we assign a weight which is a positive definite matrix of some fixed order, say, $s$ . Let $D_{i j}$ denote the sum of all the weights lying in the path connecting the vertices $i$ and $j$ of $T$ . We now say that $D_{i j}$ is the distance between $i$ and $j$ . Define $D : = [D_{i j}]$ , where $D_{i i}$ is the $s \times s$ null matrix and for $i \neq j$ , $D_{i j}$ is the distance between $i$ and $j$ . Let $G$ be an arbitrary connected weighted graph with $n$ vertices, where each weight is a positive definite matrix of order...

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let $A$ , $B$ be $n \times m$ matrices. The concept of matrix majorization means the $j$ th column of $A$ is majorized by the $j$ th column of $B$ and this is done for all $j$ by a doubly stochastic matrix $D$ . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

$(0, 1)$ -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

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Let $m$ and $n$ be positive integers, and let $R = (r_{1}, ..., r_{m})$ and $S = (s_{1}, ..., s_{n})$ be nonnegative integral vectors. Let $A (R, S)$ be the set of all $m \times n$ $(0, 1)$ -matrices with row sum vector $R$ and column vector $S$ . Let $R$ and $S$ be nonincreasing, and let $F (R)$ be the $m \times n$ $(0, 1)$ -matrix, where for each $i$ , the $i$ th row of $F (R, S)$ consists of $r_{i}$ 1’s followed by $(n - r_{i})$ 0’s. Let $A \in A (R, S)$ . The discrepancy of A, $disc (A)$ , is the number of positions in which $F (R)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m \times n$ matrices over...

The real symmetric matrices of odd order with a P-set of maximum size

Zhibin Du, Carlos Martins da Fonseca (2016)

Czechoslovak Mathematical Journal

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Suppose that $A$ is a real symmetric matrix of order $n$ . Denote by $m_{A} (0)$ the nullity of $A$ . For a nonempty subset $α$ of ${1, 2, ..., n}$ , let $A (α)$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $α$ . When $m_{A (α)} (0) = m_{A} (0) + | α |$ , we call $α$ a P-set of $A$ . It is known that every P-set of $A$ contains at most $⌊ n / 2 ⌋$ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As...

On the combinatorial structure of $0 / 1$ -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A $0 / 1$ -simplex is the convex hull of $n + 1$ affinely independent vertices of the unit $n$ -cube $I^{n}$ . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0 / 1$ -simplices in $I^{n}$ can be represented by $0 / 1$ -matrices $P$ of size $n \times n$ whose Gramians $G = P^{⊤} P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{- ⊤}$ of $P$ is doubly stochastic and has the...

On linear preservers of two-sided gut-majorization on $𝐌_{n, m}$

Asma Ilkhanizadeh Manesh, Ahmad Mohammadhasani (2018)

Czechoslovak Mathematical Journal

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For $X, Y \in 𝐌_{n, m}$ it is said that $X$ is gut-majorized by $Y$ , and we write $X ≺_{gut} Y$ , if there exists an $n$ -by- $n$ upper triangular g-row stochastic matrix $R$ such that $X = R Y$ . Define the relation $\sim_{gut}$ as follows. $X \sim_{gut} Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$ . The (strong) linear preservers of $≺_{gut}$ on $ℝ^{n}$ and strong linear preservers of this relation on $𝐌_{n, m}$ have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of $\sim_{gut}$ on $ℝ^{n}$ and $𝐌_{n, m}$ .

On row-sum majorization

Farzaneh Akbarzadeh, Ali Armandnejad (2019)

Czechoslovak Mathematical Journal

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Let $𝕄_{n, m}$ be the set of all $n \times m$ real or complex matrices. For $A, B \in 𝕄_{n, m}$ , we say that $A$ is row-sum majorized by $B$ (written as $A ≺^{rs} B$ ) if $R (A) ≺ R (B)$ , where $R (A)$ is the row sum vector of $A$ and $≺$ is the classical majorization on $ℝ^{n}$ . In the present paper, the structure of all linear operators $T : 𝕄_{n, m} \to 𝕄_{n, m}$ preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on $ℝ^{n}$ and then find the linear preservers of row-sum majorization of these relations on $𝕄_{n, m}$ . ...

Theoretical analysis for $ℓ_{1}$ - $ℓ_{2}$ minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

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We investigate the recovery of $k$ -sparse signals using the $ℓ_{1}$ - $ℓ_{2}$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$ -sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...