Inexact Newton-type method for solving large-scale absolute value equation $Ax-|x| = b$

Jingyong Tang

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Displaying similar documents to “Inexact Newton-type method for solving large-scale absolute value equation $A x - | x | = b$ ”

New quasi-Newton method for solving systems of nonlinear equations

Ladislav Lukšan, Jan Vlček (2017)

Applications of Mathematics

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We propose a new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR or LU decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires $O (n^{2})$ arithmetic operations per iteration in contrast with the Newton method, which requires $O (n^{3})$ operations per iteration. Computational experiments confirm the...

Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations

Monnanda Erappa Shobha, Ioannis K. Argyros, Santhosh George (2014)

Applicationes Mathematicae

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We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations KF(x) = y. It is assumed that the available data is $y^{δ}$ with $| | y - y^{δ} | | \leq δ$ , K: Z → Y is a bounded linear operator and F: X → Z is a nonlinear operator where X,Y,Z are Hilbert spaces. Two cases of F are considered: where $F^{'} {(x ₀)}^{- 1}$ exists (F’(x₀) is the Fréchet derivative of F at an initial guess x₀) and where F is a monotone operator....

Newton’s method over global height fields

Xander Faber, Adam Towsley (2014)

Journal de Théorie des Nombres de Bordeaux

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For any field $K$ equipped with a set of pairwise inequivalent absolute values satisfying a product formula, we completely describe the conditions under which Newton’s method applied to a squarefree polynomial $f \in K [x]$ will succeed in finding some root of $f$ in the $v$ -adic topology for infinitely many places $v$ of $K$ . Furthermore, we show that if $K$ is a finite extension of the rationals or of the rational function field over a finite field, then the Newton approximation sequence fails to converge...

A hybrid method for nonlinear least squares that uses quasi-Newton updates applied to an approximation of the Jacobian matrix

Lukšan, Ladislav, Vlček, Jan

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In this contribution, we propose a new hybrid method for minimization of nonlinear least squares. This method is based on quasi-Newton updates, applied to an approximation $A$ of the Jacobian matrix $J$ , such that $A^{T} f = J^{T} f$ . This property allows us to solve a linear least squares problem, minimizing $∥ A d + f ∥$ instead of solving the normal equation $A^{T} A d + J^{T} f = 0$ , where $d \in R^{n}$ is the required direction vector. Computational experiments confirm the efficiency of the new method.

On the Newton partially flat minimal resistance body type problems

M. Comte, Jesus Ildefonso Díaz (2005)

Journal of the European Mathematical Society

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We study the flat region of stationary points of the functional $\int_{Ω} F (| \nabla u (x) |) d x$ under the constraint $u \leq M$ , where $Ω$ is a bounded domain in $ℝ^{2}$ . Here $F (s)$ is a function which is concave for $s$ small and convex for $s$ large, and $M > 0$ is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when $Ω$ is a ball. We also analyze some other qualitative properties. Moreover,...

Convergence acceleration of shifted $L R$ transformations for totally nonnegative Hessenberg matrices

Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura (2020)

Applications of Mathematics

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We design shifted $L R$ transformations based on the integrable discrete hungry Toda equation to compute eigenvalues of totally nonnegative matrices of the banded Hessenberg form. The shifted $L R$ transformation can be regarded as an extension of the extension employed in the well-known dqds algorithm for the symmetric tridiagonal eigenvalue problem. In this paper, we propose a new and effective shift strategy for the sequence of shifted $L R$ transformations by considering the concept of the Newton...

Finite-dimensional Pullback Attractors for Non-autonomous Newton-Boussinesq Equations in Some Two-dimensional Unbounded Domains

Cung The Anh, Dang Thanh Son (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study the existence and long-time behavior of weak solutions to Newton-Boussinesq equations in two-dimensional domains satisfying the Poincaré inequality. We prove the existence of a unique minimal finite-dimensional pullback $D_{σ}$ -attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms.

Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of $C^{2}$

Arkadiusz Płoski (1990)

Annales Polonici Mathematici

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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 adaptive $s$ -step conjugate gradient algorithm with dynamic basis updating

Erin Claire Carson (2020)

Applications of Mathematics

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The adaptive $s$ -step CG algorithm is a solver for sparse symmetric positive definite linear systems designed to reduce the synchronization cost per iteration while still achieving a user-specified accuracy requirement. In this work, we improve the adaptive $s$ -step conjugate gradient algorithm by the use of iteratively updated estimates of the largest and smallest Ritz values, which give approximations of the largest and smallest eigenvalues of $A$ , using a technique due to G. Meurant and...

Prime ideal factorization in a number field via Newton polygons

Lhoussain El Fadil (2021)

Czechoslovak Mathematical Journal

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Let $K$ be a number field defined by an irreducible polynomial $F (X) \in ℤ [X]$ and $ℤ_{K}$ its ring of integers. For every prime integer $p$ , we give sufficient and necessary conditions on $F (X)$ that guarantee the existence of exactly $r$ prime ideals of $ℤ_{K}$ lying above $p$ , where $\bar{F} (X)$ factors into powers of $r$ monic irreducible polynomials in $𝔽_{p} [X]$ . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of $ℤ_{K}$ lying above $p$ ....

Integral equalities for functions of unbounded spectral operators in Banach spaces

Benedetto Silvestri

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The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form $g (R_{F}) \int f_{x} (R_{F}) d μ (x) = h (R_{F})$ . (1) They involve functions of the kind $X ∋ x \mapsto f_{x} (R_{F}) \in B (F)$ , where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

Hukuhara's differentiable iteration semigroups of linear set-valued functions

Andrzej Smajdor (2004)

Annales Polonici Mathematici

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Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. A family $F^{t} : t \geq 0$ of continuous linear set-valued functions $F^{t} : K \to c c (K)$ is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function $Φ (t, x) = F^{t} (x)$ is a solution of the problem $D_{t} Φ (t, x) = Φ (t, G (x)) : = ⋃ Φ (t, y) : y \in G (x)$ , Φ(0,x) = x, for x ∈ K and t ≥ 0, where $D_{t} Φ (t, x)$ denotes the Hukuhara derivative of Φ(t,x) with respect to t and $G (x) : = l i m_{s \to 0 +} (F^{s} (x) - x) / s$ for x ∈ K.

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...

Convergence in the dual of certain $K M_{p}$ -spaces

Larry Kitchens, Charles Swartz (1974)

Colloquium Mathematicae

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Zero points of quadratic matrix polynomials

Opfer, Gerhard, Janovská, Drahoslava

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Our aim is to classify and compute zeros of the quadratic two sided matrix polynomials, i.e. quadratic polynomials whose matrix coefficients are located at both sides of the powers of the matrix variable. We suppose that there are no multiple terms of the same degree in the polynomial $𝐩$ , i.e., the terms have the form $𝐀_{j} 𝐗^{j} 𝐁_{j}$ , where all quantities $𝐗, 𝐀_{j}, 𝐁_{j}, j = 0, 1, ..., N,$ are square matrices of the same size. Both for classification and computation, the essential tool is the description of the polynomial $𝐩$ by a matrix...

Iterating along a Prikry sequence

Spencer Unger (2016)

Fundamenta Mathematicae

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We introduce a new method which combines Prikry forcing with an iteration between the Prikry points. Using our method we prove from large cardinals that it is consistent that the tree property holds at ℵₙ for n ≥ 2, $ℵ_{ω}$ is strong limit and $2^{ℵ_{ω}} = ℵ_{ω + 2}$ .

Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

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We show that the dimer model on a bipartite graph $Γ$ on a torus gives rise to a quantum integrable system of special type, which we call a. The phase space of the classical system contains, as an open dense subset, the moduli space $Ł_{Γ}$ of line bundles with connections on the graph $Γ$ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs $Γ_{1}$ and $Γ_{2}$ areif the Newton polygons of the corresponding partition functions coincide up to translation....

Displaying similar documents to “Inexact Newton-type method for solving large-scale absolute value equation A x - | x | = b ”

Displaying similar documents to “Inexact Newton-type method for solving large-scale absolute value equation $A x - | x | = b$ ”