A global analysis of Newton iterations for determining turning points

Vladimír Janovský; Viktor Seige

Displaying similar documents to “A global analysis of Newton iterations for determining turning points”

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...

Bifurcation of periodic solutions to differential inequalities in $ℝ^{3}$

Miroslav Bosák, Milan Kučera (1992)

Czechoslovak Mathematical Journal

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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.

Exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems

Shao-Yuan Huang, Ping-Han Hsieh (2023)

Czechoslovak Mathematical Journal

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We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems $\{\begin{matrix} - {[φ (u^{'})]}^{'} = λ u^{p} (1 - \frac{u}{N}) & in (- L, L), \\ u (- L) = u (L) = 0, \end{matrix}$ where $p > 1$ , $N > 0$ , $λ > 0$ is a bifurcation parameter, $L > 0$ is an evolution parameter, and $φ (u)$ is either $φ (u) = u$ or $φ (u) = u / \sqrt{1 - u^{2}}$ . We prove that the corresponding bifurcation curve is $\subset$ -shape. Thus, the exact multiplicity of positive solutions can be obtained.

Bifurcation from a saddle connection in functional differential equations: An approach with inclination lemmas

Hans-Otto Walther

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CONTENTSIntroduction...........................................................................................................................................5 I.........................................................................................................................................................151. Preliminaries...................................................................................................................................152. Solutions of a family...

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...

A two-stage Newton-like method for computing simple bifurcation points of nonlinear equations depending on two parameters

Gerd Pönisch (1990)

Banach Center Publications

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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...

On the application of Newton's and chord methods of bifurcation problems.

Elgindi, M.B.M. (1994)

International Journal of Mathematics and Mathematical Sciences

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Bifurcation theorems for nonlinear problems with lack of compactness

Francesca Faraci, Roberto Livrea (2003)

Annales Polonici Mathematici

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We deal with a bifurcation result for the Dirichlet problem ⎧ $- Δ_{p} u = μ / {| x |}^{p} {| u |}^{p - 2} u + λ f (x, u)$ a.e. in Ω, ⎨ ⎩ $u_{| \partial Ω} = 0$ . Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number $λ *_{μ}$ such that for every $λ \in] 0, λ *_{μ} [$ the above problem admits a nonzero weak solution $u_{λ}$ in $W ₀^{1, p} (Ω)$ satisfying $l i m_{λ \to 0 ⁺} | | u_{λ} | | = 0$ .