Geodesic
Quasilinear Structures in Stochastic Arithmetic and their Application
Serdica Journal of Computing, Tome 10 (2016) no. 1, pp. 001-012.

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Stochastic arithmetic has been developed as a model for computing with imprecise numbers. In this model, numbers are represented by independent Gaussian variables with known mean value and standard deviation and are called stochastic numbers. The algebraic properties of stochastic numbers have already been studied by several authors. Anyhow, in most life problems the variables are not independent and a direct application of the model to estimate the standard deviation on the result of a numerical computation may lead to some overestimation of the correct value. In this work “quasilinear” algebraic structures based on standard stochastic arithmetic are studied and, from pure abstract algebraic considerations, new arithmetic operations called “inner stochastic addition and subtraction” are introduced. They appear to be stochastic analogues to the inner interval addition and subtraction used in interval arithmetic. The algebraic properties of these operations and the involved algebraic structures are then studied. Finally, the connection of these inner operations to the correlation coefficient of the variables is developed and it is shown that they allow the computation with non-independent variables. The corresponding methodology for the practical application of the new structures in relation to problems analogous to “dependency problems” in interval arithmetic is given and some numerical experiments showing the interest of these new operations are presented. ACM Computing Classification System (1998): D.2.4, G.3, G.4.
Keywords: Validation of Numerical Software, Stochastic Arithmetic, Linear Algebra, Imprecise Data, CESTAC Method, Scientific Computing 
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Markov, Svetoslav; Alt, René; Lamotte, Jean-Luc. Quasilinear Structures in Stochastic Arithmetic and their Application. Serdica Journal of Computing, Tome 10 (2016) no. 1, pp. 001-012. http://geodesic.mathdoc.fr/item/SJC_2016_10_1_a0/