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We review and extend results on the local convergence of the classical Newton-Kantorovich method. Then we discuss globally convergent damped and inexact Newton methods and point out advantages of using a minimal error conjugate gradient method for the linear systems arising at each Newton step. Finally application on a nonlinear elliptic problem is considered. A combination of nested iterations,...
A numerical method recently proposed by the author is shown to be a very efficient and robust method for the solution of a class of discrete nonlinear eigenvalue problems. In particular it is applied to follow the relevant and the spurious solution curves. Numerical results show that also in the neighbourhood of turning or bifurcation points the work required is considerably less than for usual continuation...
Let $$F:D( \subseteq \mathbb{R}^n ) \to \mathbb{R}^n$$ be a continuous function, and suppose that for some XO ∈ D and some nonsingular matrix A the vector δO:=A−1 F(xO) is "small". Assuming the existence of a nonnegative vector c ∈ ℝn such that $$\left| {F(x) - F(x_0 ) - A(x - x_0 )} \right| \leqslant \left\| {\delta _0 } \right\|c$$ for all x in a suitable neighbourhood S of xO, a simple...
Es sei F eine Abbildung eines passend gewählten n-dimensionalen metrischen Mengenraumes in sich. Die Funktion F möge dort einerseits einer Lipschitz-Bedingung und andererseits einer Durchmesser-Bedingung genügen. Es werden Systeme der Gestalt (1) $$X = F(X) + R$$ betrachtet. Es werden 6 äquivalente Bedingungen angegeben, die sämtlich notwendig und hinreichend sind für die eindeutige Auflösbarkeit...
ADI-iteration is an efficient a means for solving nonlinear systems that occur as conservative discretization schemes for variational inequalities of evolution.
Relaxation methods of H.R. Schwarz and A. Ruhe are used for the computation of the spectral norm σ1 of an arbitrary m×n-matrix A (i.e., σ1 is the maximal singular value of A). σ1 is eigenvalue and spectral radius of a symmetric weakly two-cyclic matrix Â. Using this fact, results on the optimal (asymptotic) relaxation factor ωo are derived. Further it is shown that using ωo from the beginning of the...
In Form eines überblicks wird von der numerischen Integration des Systems von quasilinearen partiellen Differentialgleichungen berichtet, das die dreidimensionalen reibungsfreien Strömungen idealer Gase beschreibt. Das Differentialgleichungssystem wird durch finite Differenzen approximiert, wobei die entstehenden nichtlinearen algebraischen Gleichungen iterativ gelöst werden. Die hier vorgestellten...
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