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Direct Methods for Linear Systems with Inexact Input Data

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 2009年第2-3期26卷 279-296页

作者： Mayer, Guenter Univ Rostock Inst Math D-18051 Rostock Germany

We give a survey on direct methods for interval linear systems. We also consider various kinds of solution sets and show how the interval hull can be computed.

关键词： interval linear system solution set interval hull interval gaussian algorithm interval Cholesky method interval Trench algorithm interval Bareiss algorithm

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NEW CRITERIA FOR THE FEASIBILITY OF THE CHOLESKY METHOD WITH interval DATA

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2008年第4期30卷 1392-1405页

作者： Alefeld, G. Mayer, G. Univ Karlsruhe Inst Angew & Numer Math KIT D-76128 Karlsruhe Germany Univ Rostock Inst Math D-18051 Rostock Germany

We present some new criteria for the feasibility of the interval Cholesky method. In particular, we relate this feasibility to that of the interval gaussian algorithm.

关键词： linear interval equations Cholesky method interval Cholesky algorithm gaussian algorithm interval gaussian algorithm linear system of equations criteria of feasibility

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On regular and singular interval systems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2007年第2期199卷 220-228页

作者： Mayer, Guenter Univ Rostock Inst Math D-18051 Rostock Germany

We give a survey on interval linear systems discussing problems for regular systems as well as for singular ones. We consider several solution sets and direct methods to enclose them. Moreover we study iterative methods, particularly the total step method as the basis for other ones. We also use this method for enclosing solutions of singular linear systems. (c) 2006 Elsevier B.V. All rights reserved.

关键词： interval linear system solution set interval gaussian algorithm interval Cholesky method total step method singular interval linear system

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A NEW CRITERION TO GUARANTEE THE FEASIBILITY OF THE interval gaussian algorithm

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1993年第2期14卷 408-419页

作者： FROMMER, A MAYER, G BERG UNIV GESAMTHSCH WUPPERTAL GH WUPPERTAL FACHBEREICH MATH GAUSSSTR 20 W-5600 WUPPERTAL GERMANY

The main subject of this paper is the interval gaussian algorithm, which produces an interval vector [x]G analogously to its well-known counterpart in classical numerical analysis. Criteria are derived to guarantee the existence of [x]G if the n X n inter-val matrix [A] is degenerate to a point matrix or if its comparison matrix [[A]] is irreducible and diagonally dominant. While in the first case all classical criteria of feasibility apply, the second case yields to a criterion which seems to be new. A way to construct matrices such that [X]G does not exist, although it does for each matrix A is-an-element-of [A], is also indicated.

关键词： LINEAR interval EQUATIONS gaussian algorithm interval gaussian algorithm LINEAR SYSTEMS OF EQUATIONS CRITERIA OF FEASIBILITY

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SOME REMARKS ON 2 interval - ARITHMETIC MODIFICATIONS OF THE NEWTON METHOD

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COMPUTING 1992年第1期48卷 125-128页

作者： MAYER, G 1. Institut für Angewandte Mathematik Universit?t Karlsruhe (TH) Kaiserstra?e 12 D-W-7500 Karlsruhe 1 Federal Republic of Germany

In this paper we consider the interval Newton method and its simplified version to find tight bounds for the solutions of systems of nonlinear equations. Referring to a paper of Alefeld on this subject, we derive a connection between the two matrices used there to formulate sufficient criteria for the convergence of these methods. We also answer an open question concerning the break down of the Newton method after a finite number of steps.

关键词： NONLINEAR EQUATIONS SYSTEMS OF NONLINEAR EQUATIONS interval NEWTON METHOD interval gaussian algorithm

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COMPARISON-THEOREMS FOR ITERATIVE METHODS BASED ON STRONG SPLITTINGS

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SIAM JOURNAL ON NUMERICAL ANALYSIS 1987年第1期24卷 215-227页

作者： MAYER, G Univ Karlsruhe Karlsruhe West Ger Univ Karlsruhe Karlsruhe West Ger

By means of strong splittings of an $n \times n$ interval M-matrix $\mathcal {A}$ one can construct iterative methods to enclose the solution set $\{ x\} : = \{ x|Ax = b,A \in A,b \in \mathfrak{b}\} $, where $\mathfrak{b}$ is an interval vector and $x \in \mathbb{R}^n $. We derive comparison theorems for an upper bound of the asymptotic convergence factor and for the fixpoints resulting from those methods.

关键词： 65F10 65G10 65H10 interval arithmetic iterative methods systems of linear equations interval gaussian algorithm asymptotic convergence factor

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