On the algebraic Properties of Stochastic Arithmetic. Comparison to Interval Arithmetic

On the numerical solution to linear problems using stochastic arithmetic

“…Stochastic numbers can be defined as elements of the direct sum V ⊕ S of a vector space V (of mean values) and a s-space S (of standard deviations) both of same dimension k. Namely, let V = V k be a k-dimensional vector space with a basis (v (1) , ..., v (k) ) and let S = S k be a k-dimensional s-space having a basis (s (1) , ..., s (k) ). Then we say that (v (1) , ..., v (k) ; s (1) , ..., s (k) ) is a basis of the kdimensional space V k ⊕ S k .…”

“…Then we say that (v (1) , ..., v (k) ; s (1) , ..., s (k) ) is a basis of the kdimensional space V k ⊕ S k . Such a setting allows us to consider numerical problems involving vectors and matrices, wherein the numeric variables have been substituted by stochastic ones.…”

“…As regard to addition the system of standard deviations is an abelian monoid with cancellation law. This monoid can be embedded in an additive group and after a suitable extension of multiplication by scalars one obtains a so-called s-space, which is closely related to a vector space [1], [4]. This allows us to introduce in s-spaces concepts like linear combination, basis, dimension etc.…”

On the solution to numerical problems using stochastic arithmetic

“…The proof in the general case (s real) is similar to the one for the proper case (s nonnegative) see [1].…”

“…This work is part of a more general one, which aims,to study the abstract algebraic structures induced by the operations on stochastic numbers and to compare the relevant theory with the performance of the CESTAC method [1], [2], [6], [7]. In the paper we restrict ourselves to the arithmetic operations addition, negation and multiplication by scalars and the relation inclusion.…”