Simplification of Symbolic-Numerical Interval Expressions∗

Although interval arithmetic is increasingly used in combination with computer algebra and other methods, both approaches — symbolic-algebraic and interval-arithmetic — are used separately. Implementing symbolic interval arithmetic seems not suitable due to the exponential growth in the “size” of the end-points. In this paper we propose a methodology for “true” symbolic-algebraic manipulations on symbolic-numerical interval expressions involving interval variables instead of symbolic intervals. Due to the better algebraic properties, resembling to classical analysis, and the containment of classical interval arithmetic as a special case, we consider the algebraic extension of conventional interval arithmetic as an appropriate environment for solving interval algebraic problems. Based on the distributivity relations, a general framework for simplification of symbolicnumerical expressions involving intervals is given and some of the wider implications of the theory pertaining to interval algebraic problems are discussed.