Box math and KSM: Extending Sherman-Morrison to functions of interval matrices

Certain advantages of midpoint/radius notation for boxes are well known. Box notation, a concise midpoint/radius scheme employing a `box operator', π, significantly simplifies box calculations. The box math approach to interval analysis, emphasizing `image-centered' representations and assessment of quality of approximation, demonstrates the power of mid-point/ radius methods. A prime example is the KSM method, extending complex analysis type expansions to functions of matrix boxes. This works particularly well for the inverse of a matrix box, leading to simple formulas for the hull of the solution of a linear box equation (interval matrix equation). An outline of key ideas of box math and proofs of some basic KSM theorems is presented below.