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The usual usual "implementation" "implementation" ofreal numbers as
floating point numbers on exist- iing ng computers computers has
the well-known disadvantage that most of the real numbers are not
exactly representable in floating point. Also the four basic
arithmetic operations can usually not be performed exactly. For
numerical algorithms there are frequently error bounds for the
computed approximation available. Traditionally a bound for the
infinity norm is estima- ted using ttheoretical heoretical
ccoonncceeppttss llike ike the the condition condition number
number of of a a matrix matrix for for example. example. Therefore
Therefore the error bounds are not really available in practice
since their com- putation requires more or less the exact solution
of the original problem. During the last years research in
different areas has been intensified in or- der to overcome these
problems. As a result applications to different concrete problems
were obtained. The LEDA-library (K. Mehlhorn et al.) offers a
collection of data types for combinatorical problems. In a series
of applications, where floating point arith- metic fails, reliable
results are delivered. Interesting examples can be found in
classical geometric problems. At the Imperial College in London was
introduced a simple principle for "exact arithmetic with real
numbers" (A. Edalat et al.), which uses certain nonlinear
transformations. Among others a library for the effective
computation of the elementary functions already has been
implemented.
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