A note on the complexity of algebraic differentiation

Authors:
B. F. Caviness;H. I. Epstein
Affiliations:
Rensselaer Polytechnic Institute, Troy, N.Y.;Boston College, Chestnut Hill, Massachusetts
Venue:
ACM SIGSAM Bulletin
Year:
1977

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Abstract

It is a well-known empirical result that differentiation, especially higher order differentiation, of simple expressions can lead to long and complex expressions. In this paper we give some theoretical results that help to explain this phenomenon. In particular we show that in certain representations there exist expressions whose representations require σ(n) space but whose (k+1)-th order derivatives require σ(n(n+k-1/k)) space and hence require at least σ(n(n+k-1/k)) time to compute.