On the Exponentiation of Interval Matrices

Alexandre Goldsztejn (1) , Arnold Neumaier (2)

1 LINA - Laboratoire d'Informatique de Nantes Atlantique
2 Fakultät für Mathematik [Wien]

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.

Keywords: Interval analysis; matrix exponentiation

Domains: Numerical Analysis [cs.NA]; Numerical Analysis [math.NA]; Computer Arithmetic; Computational Complexity [cs.CC]

Alexandre Goldsztejn, Arnold Neumaier. On the Exponentiation of Interval Matrices. 2009. ⟨hal-00411330v1⟩