Abstract
Verification in Computer Algebra Systems. In this paper, we have attempted to demonstrate that the question of condition, i.e. of the sensitivity of results w.r.t. perturbations of data, may play a role in algebraic algorithms, even if they are carried out in rational arithmetic. Poor condition is traced to a near-degeneracy of the situation specified by the data. Thus in a process called verification in this context, the presence of a genuinely degenerate problem near the specified problem should be discovered before or during the execution of the algorithm; the algorithm should switch to a stable modification in this case. Such modified versions are obtained by regarding the specified problem as a perturbation of the nearby degenerate one. Finally, it is indicated how these ideas may also lead to safe implementations of algebraic algorithms in floating-point arithmetic.
These ideas are developed considering the integration of rational functions, the choice of basis in multivariate polynomial interpolation, and the computation of zeros of multivariate polynomial systems.
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Dedicated to Professor U. Kulisch on the occasion of his 60th birthday
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© 1993 Springer-Verlag
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Stetter, H.J. (1993). Verification in Computer Algebra Systems. In: Albrecht, R., Alefeld, G., Stetter, H.J. (eds) Validation Numerics. Computing Supplementum, vol 9. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6918-6_18
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DOI: https://doi.org/10.1007/978-3-7091-6918-6_18
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-82451-1
Online ISBN: 978-3-7091-6918-6
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