On the Exponentiation of Interval Matrices
For researchers working on interval analysis and numerical computation, this work addresses the fundamental challenge of dependency loss in interval matrix exponentiation, though the NP-hardness result is theoretical and the method is an adaptation.
The paper proves that computing a sharp enclosure of the interval matrix exponential is NP-hard and adapts the scaling and squaring method to interval matrices, which significantly reduces dependency loss compared to direct interval evaluation of the Taylor series.
The numerical computation of the exponentiation of a real matrix has been intensively studied. The main objective of a good numerical method is to deal with round-off errors and computational cost. The situation is more complicated when dealing with interval matrices exponentiation: Indeed, the main problem will now be the dependency loss of the different occurrences of the variables due to interval evaluation, which may lead to so wide enclosures that they are useless. In this paper, the problem of computing a sharp enclosure of the interval matrix exponential is proved to be NP-hard. Then the scaling and squaring method is adapted to interval matrices and shown to drastically reduce the dependency loss w.r.t. the interval evaluation of the Taylor series.