Abstract
Over the years, a lot of effort has been put into the development of self-validated computational (SVC) models that will be able to overcome the “memoryless nature” of interval arithmetic, i.e., to take into account the dependencies between variables involved in a computation and/or reduce the so-called wrapping effect. This effort has resulted in several such models worth mentioning: ellipsoid calculus (Chernousko, Izv Akad Nauk SSSR, Tekh Kibern 3:3–11; 4:3–11; 5:5–11, 1980, [27]), Vályi, Ellipsoidal calculus for estimation and control, Birkhäuser, Boston, 1997, [269]), constrained interval arithmetic (Lodwick, Constrained interval arithmetic, 1999, [131]), Hansen’s generalized interval arithmetic (Hansen, A generalized interval arithmetic, Springer, Berlin, 1975, [72]), affine arithmetic (de Figueiredo, Stolfi, Self-validated numerical methods and applications, 1997, [33]), reduced affine arithmetic (Messine, New affine forms in interval branch and bound algorithms, 1999, [139]), and revised affine arithmetic (Vu, Sam-Haroud, Faltings, A generic scheme for combining multiple inclusion representations in numerical constraint propagation, 2004, [271]).
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Notes
- 1.
Hansens’s generalized interval arithmetic records only the correlations between output quantities \(z_i\) and inputs \(x_j\), but not among the inputs nor the outputs between the quantities involved in a computation [33].
- 2.
In Hansen’s generalized intervals, the joint range may be a non-convex polygon.
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Skalna, I. (2018). Alternative Arithmetic. In: Parametric Interval Algebraic Systems. Studies in Computational Intelligence, vol 766. Springer, Cham. https://doi.org/10.1007/978-3-319-75187-0_2
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DOI: https://doi.org/10.1007/978-3-319-75187-0_2
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