Michel Goze, Nicolas Goze, Elisabeth Remm: Algebraic computations on the set of real intervals, p.393-402

Abstract:

The set of closed intervals of $R$ is provided with a semigroup structure. We complete this semigroup to obtain a $2$-dimensional vector space. But neither associative nor nonassociative algebra structure satisfying a natural property on the product of intervals can be defined on this vector space. We define an embedding of this vector space into a $4$-dimensional associative algebra and we compute all the arithmetic operations on intervals in this algebra. As applications, we study polynomial functions and the problem of diagonalization of matrices whose elements are intervals.

Key Words: Intervals analysis, associative algebras, matrices of intervals, polynomial equations.

2000 Mathematics Subject Classification: Primary: 65G40;
Secondary: 08A02.

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