Interval arithmetic has been extensively applied to systems of linear equations by the interval matrix arithmetic community. This paper demonstrates through simple examples that some of this work can be viewed as particular instantiations of an abstract “design operation,” the RANGE operation of the Labeled Interval Calculus formalism for inference about sets of possibilities in design. These particular operations promise to solve a variety of design problems that lay beyond the reach of the original Labeled Interval Calculus. However, the abstract view also leads to a new operation, apparently overlooked by the matrix mathematics community, that should also be useful in design; the paper provides an algorithm for computing it.

This paper defines, develops algorithms for, and illustrates the utility in design of a class of mathematical operations. These accept as inputs a system of linear constraint equations, Ax = b, an interval matrix of values for the coefficients, A, and an interval vector of values for either x or b. They return a set of values for the “domain” of the other vector, in the sense that all combinations of the output vector values set and values for A, when inserted into the constraint equation, correspond to values for the input vector that lie within the input interval. These operations have been mostly overlooked by the interval matrix arithmetic community, but are mathematically interesting and useful in the design, for example, of structures.

This paper defines, develops algorithms for, and illustrates the design use of a class of mathematical operations. These operations accept as inputs a system of linear constraint equations, Ax = b, an interval matrix of values for the coefficients A, and an interval vector of values for either x or b. They return a set of values for the other variable that is “sufficient” in this sense. Suppose that ◯ is an interval of input vectors, and  an interval matrix. Then, one Sufficient-Points operation returns a set of vectors ~ such that for each b in ~, the set of x values that can be produced by inserting all the values of  into Ax = b is a superset of the input vector x. These operations have been partly overlooked by the interval matrix mathematics community, but are mathematically interesting and useful in the design, for example, of circuits.