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About the Qualitative Computation of Jordan Forms
Author(s) -
Chatelin F.,
Braconnier Th.
Publication year - 1994
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19940740211
Subject(s) - citation , library science , mathematics , physics , computer science
The central question in Scientiic Computing today is to determine the level of conndence that one can have in the results massively produced by workstations and supercomputers. This question is even more acute that the problems to be solved are more and more diicult mathematically (nonlinear, ill-posed and of large size). Besides problems of arithmetic reliability, the quality of an approximate solution depends crucially on the numerical method at use. Arithmetic reliability and numerical quality are both related to the amount of stability of the exact mathematical problem to be solved, which models the physical phenomenon under study. It is thanks to the computers that in the seventies, the scientists could become aware of the theoretical role of chaos in phenomena ruled by nonlinear equations. The conndence that industry engineers can put in the results produced by scientiic computing is subordinated today to the development of reliable tools for analysis and/or control of computations from the arithmetical and/or numerical point of view. It is important to distinguish in a computer result the amount due to the arithmetic instability (curable) from the amount due to a possible mathematical or physical instability (unavoidable) but which can be studied by theoretical means. It is because industry uses more and more reliable models of physical reality at the edge of instability that new types of numerical instability appear, which were previously known from only a handful of mathematicians. To that end we have developed a novel theory of nite precision computation, called Qualitative Computing. It consists of two main objectives : 1. for computations where the innuence of nite precision arithmetic remains moderate, to control the global error on the computed result, 2. for computations dominated by round-oo (such as chaotic computations), to use the errors to reveal mathematical properties which are out of reach of any nite precision computation. In such cases, computations are said to be \impossible" (because all signiicant digits are wrong") and the quantiication {exact computing, for example{ becomes meaningless. However some information, of a more qualitative type, can be extracted from results which are \wrong" in a classical sense.