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  • 1
    Electronic Resource
    Electronic Resource
    Inclusion of Zeros of Nowhere Differentiable n-Dimensional Functions (1997)
    Springer
    Reliable computing 3 (1997), S. 5-16 
    Details
    ISSN: 1573-1340
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A method is described for calculating verified error bounds for zeros of a system of nonlinear equations f(x) = 0 with continuous f : R" → R". We do not require existence of partial derivatives of f, and the function f may even be known only up to some finite precision ε 〉 0. An inclusion may contain infinitely many zeros of f. An example of a continuous but nowhere differentiable function is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1009967901122
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Intervals and the Origins of Calculus (1998)
    Springer
    Reliable computing 4 (1998), S. 191-197 
    Details
    ISSN: 1573-1340
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Interval (two-sided) estimates developed by John Wallis in the 17 century can be considered as an important step towards development of Newton-Leibniz's calculus.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1009989211143
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  • 3
    Electronic Resource
    Electronic Resource
    A Note on Epsilon-Inflation (1998)
    Springer
    Reliable computing 4 (1998), S. 371-375 
    Details
    ISSN: 1573-1340
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The epsilon-inflation proved to be useful and necessary in many verification algorithms. Different definitions of an epsilon-inflation are possible, depending on the context. Recently, certain theoretical justifications and optimality results were proved for an epsilon-inflation without absolute term. In this note we show that in currently used interval iterations the epsilon-inflation without absolute term does not serve the purpose it is defined for. A new epsilon-inflation is proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1024419816707
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  • 4
    Electronic Resource
    Electronic Resource
    Fast and Parallel Interval Arithmetic (1999)
    Springer
    BIT 39 (1999), S. 534-554 
    Details
    ISSN: 1572-9125
    Keywords: Interval arithmetic ; parallel computer ; BLAS ; midpoint-radius ; infimum-supremum
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Infimum-supremum interval arithmetic is widely used because of ease of implementation and narrow results. In this note we show that the overestimation of midpoint-radius interval arithmetic compared to power set operations is uniformly bounded by a factor 1.5 in radius. This is true for the four basic operations as well as for vector and matrix operations, over real and over complex numbers. Moreover, we describe an implementation of midpoint-radius interval arithmetic entirely using BLAS. Therefore, in particular, matrix operations are very fast on almost any computer, with minimal effort for the implementation. Especially, with the new definition it is seemingly the first time that full advantage can be taken of the speed of vector and parallel architectures. The algorithms have been implemented in the Matlab interval toolbox INTLAB.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022374804152
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  • 5
    Electronic Resource
    Electronic Resource
    Ill-Conditionedness Need not be Componentwise Near to Ill-Posedness for Least Squares Problems (1999)
    Springer
    BIT 39 (1999), S. 143-151 
    Details
    ISSN: 1572-9125
    Keywords: Componentwise distance ; condition number ; ill-posed ; least squares ; pseudoinverse ; underdetermined linear systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The condition number of a problem measures the sensitivity of the answer to small changes in the input, where “small” refers to some distance measure. A problem is called ill-conditioned if the condition number is large, and it is called ill-posed if the condition number is infinity. It is known that for many problems the (normwise) distance to the nearest ill-posed problem is proportional to the reciprocal of the condition number. Recently it has been shown that for linear systems and matrix inversion this is also true for componentwise distances. In this note we show that this is no longer true for least squares problems and other problems involving rectangular matrices. Problems are identified which are arbitrarily ill-conditioned (in a componentwise sense) whereas any componentwise relative perturbation less than 1 cannot produce an ill-posed problem. Bounds are given using additional information on the matrix.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022377410087
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  • 6
    Book
    Book
    Software Engineering im Scientific Computing; Beiträge eines Workshops in Hamburg 6.-8. Juni 1995 (1996)
    Braunschweig u.a. :Vieweg,
    Details
    Title: Software Engineering im Scientific Computing; Beiträge eines Workshops in Hamburg 6.-8. Juni 1995
    Contributer: Mackens, Wolfgang , Rump, Siegfried M.
    Publisher: Braunschweig u.a. :Vieweg,
    Year of publication: 1996
    Pages: 315 S.
    Series Statement: Angewandte Mathematik/Informatik
    Type of Medium: Book
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