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Chapter
ApCoA = Embedding Commutative Algebra into Analysis
I take a philosophical look at Approximate Commutative Algebra and present my view of computational algebra over \(\mathbb{C}\) , highligh...
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Chapter and Conference Paper
Proposal for the Algorithmic Use of the BKK-Number in the Algebraic Reduction of a O-dimensional Polynomial System
For a regular 0-dimensional system P of polynomials with numerical coefficients, its BKK-number m equals the number of its zeros, counting multiplicities. In this paper, I analyze how the knowledge of m may be us...
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Chapter and Conference Paper
Algebraic Predicates for Empirical Data
It is widely assumed that the assignment of truth values to non-trivial algebraic predicates containing numerical data is possible only if the data are exact and if exact computation is employed. But in many appl...
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Chapter and Conference Paper
Condition Analysis of Overdetermined Algebraic Problems
In analogy to numerical linear algebra, the evolving numerical polynomial algebra studies the modifications of classical polynomial algebra necessary to accomodate inaccurate data and inexact computation. A stand...
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Chapter and Conference Paper
Polynomials with Coefficients of Limited Accuracy
In Scientific Computing, data often have a limited accuracy. With polynomial modelling functions, this affects the meaningful accuracy of their zeros. We derive constructive criteria for judging the validity o...
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Article
Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems
The eigenproblem method calculates the solutions of systems of polynomial equations \( f_1(x_1, \ldots , x_s)=0,\ldots,f_m(x_1, \ldots , x_s)=0\) . It consists in fixing a suit...
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Article
Professor W. Knödel, 60 years
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Article
Professor R. Albrecht, 60 years
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Chapter and Conference Paper
Sequential defect correction for high-accuracy floating-point algorithms
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Article
Professor Rudolf Inzinger
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Chapter and Conference Paper
Modular analysis of numerical software
We have attempted to demonstrate how the design of control modules in numerical software may be based on rigorous analysis to a larger extent. The relative merits of different designs may more reliably be eval...
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Article
The defect correction principle and discretization methods
Recently, a number of closely related techniques for error estimation and iterative improvement in discretization algorithms have been proposed. In this article, we expose the common structural principle of al...
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Chapter and Conference Paper
Considerations concerning a theory for ode-solvers
In today’s general purpose software packages for initial value problems in ODE’s, the course of the computation is normally determined by a tolerance parameter δ: The particular integration procedure and the s...
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Chapter and Conference Paper
Towards a theory for discretizations of stiff differential systems
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Article
Buchbesprechungen
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Chapter and Conference Paper
Cyclic finite-difference methods for ordinary differential equations
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Chapter
Economical Global Error Estimation
This paper has no immediate relation to stiff differential equations; in fact, modifications which may be necessary to adapt some of the following techniques to badly stiff systems have not been elaborated. Th...
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Book
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Chapter
General Discretization Methods
In this introductory chapter we consider general aspects of discretization methods. Much of the theory is applicable not only to standard discretization methods for ordinary differential equations (both initia...
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Chapter
Runge-Kutta Methods
One-step methods (see Def. 2.1.8) form a particularly simple class of f. s. m. for IVP 1. Among these, a certain class of methods has commonly been associated with the names of C. Runge and W. Kutta and is wid...
