New quasi-Newton method for solving systems of nonlinear equations

Ladislav Lukšan; Jan Vlček

Displaying similar documents to “New quasi-Newton method for solving systems of nonlinear equations”

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

Jingyong Tang (2024)

Applications of Mathematics

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Newton-type methods have been successfully applied to solve the absolute value equation $A x - | x | = b$ (denoted by AVE). This class of methods usually solves a system of linear equations exactly in each iteration. However, for large-scale AVEs, solving the corresponding system exactly may be expensive. In this paper, we propose an inexact Newton-type method for solving the AVE. In each iteration, the proposed method solves the corresponding system only approximately. Moreover, it adopts a new line...

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....

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.

A modified limited-memory BNS method for unconstrained minimization derived from the conjugate directions idea

Vlček, Jan, Lukšan, Ladislav

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A modification of the limited-memory variable metric BNS method for large scale unconstrained optimization of the differentiable function $f : ℛ^{N} \to ℛ$ is considered, which consists in corrections (based on the idea of conjugate directions) of difference vectors for better satisfaction of the previous quasi-Newton conditions. In comparison with [11], more previous iterations can be utilized here. For quadratic objective functions, the improvement of convergence is the best one in some sense, all...

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...

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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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...

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,...

New hybrid conjugate gradient method for nonlinear optimization with application to image restoration problems

Youcef Elhamam Hemici, Samia Khelladi, Djamel Benterki (2024)

Kybernetika

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The conjugate gradient method is one of the most effective algorithm for unconstrained nonlinear optimization problems. This is due to the fact that it does not need a lot of storage memory and its simple structure properties, which motivate us to propose a new hybrid conjugate gradient method through a convex combination of $β_{k}^{R M I L}$ and $β_{k}^{H S}$ . We compute the convex parameter $θ_{k}$ using the Newton direction. Global convergence is established through the strong Wolfe conditions. Numerical experiments...

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...

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...

A dual-parameter double-step splitting iteration method for solving complex symmetric linear equations

Beibei Li, Jingjing Cui, Zhengge Huang, Xiaofeng Xie (2024)

Applications of Mathematics

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We multiply both sides of the complex symmetric linear system $A x = b$ by $1 - i ω$ to obtain a new equivalent linear system, then a dual-parameter double-step splitting (DDSS) method is established for solving the new linear system. In addition, we present an upper bound for the spectral radius of iteration matrix of the DDSS method and obtain its quasi-optimal parameter. Theoretical analyses demonstrate that the new method is convergent when some conditions are satisfied. Some tested examples are...