ON THE COMPLEXITY OF COMPOSITION AND GENERALIZED COMPOSITION OF POWER SERIES

R. P. BRENT AND J. F. TRAUB

Abstract

Let F (x) = f_{1}x + f_{2}x^{2} + ⋯ be a formal power series over a field Δ. Let F^{(0)}(x) = x and, for q = 1, 2, …, define F^{(q)}(x) = F^{(q−1)}(F (x)). The obvious algorithm for computing the first n terms of F^{(q)}(x) is by the composition analogue of repeated squaring. This algorithm has complexity about log_{2} q times that of a single composition. Brent [1] showed that the factor log_{2} q can be eliminated in the computation of the first n terms of (F (x))^{q} by a change of representation, using the logarithm and exponential functions. We show here that the factor log_{2} q can also be eliminated for the composition problem, unless the complexity of composition is quasi-linear.

F^{(q)}(x) can often, but not always, be defined for more general q. We give algorithms and complexity bounds for computing the first n terms of F^{(q)}(x) whenever it is defined.

We conclude the paper with some open problems.

Comments

Only the Abstract is given here. The full paper appeared as [3]. For related work, see [2].

References

1. R. P. Brent, “Multiple-precision zero-finding methods and the complexity of elementary function evaluation”, in Analytic Computational Complexity (edited by J. F. Traub), Academic Press, New York, 1975, 151–176. MR 52#15938, 54#11843; Zbl 342.65031. rpb028.

2. R. P. Brent and H. T. Kung, “Fast algorithms for manipulating formal power series”, Journal of the ACM 25 (1978), 581–595. CR 20#34535, MR 58#25090, Zbl 388.68052. rpb045.

3. R. P. Brent and J. F. Traub, “On the complexity of composition and generalized composition of power series”, SIAM J. on Computing 9 (1980), 54–66. MR 81b:68042. Also appeared as Report TR CMU-CS-78-128, Department of Computer Science, Carnegie-Mellon University (May 1978), 34 pp. Abstract appeared in SIGSAM Bulletin 12, 2 (May 1978), 9. rpb050.

(Brent) Australian National University, Canberra, Australia

(Traub) Carnegie-Mellon University, Pittsburgh, Pennsylvania, USA

1991 Mathematics Subject Classification. Primary 68Q25; Secondary 30D05, 65Y20, 68Q40. Key words and phrases. Composition, fast algorithms, formal power series, symbolic computation,

generalized composition, functional equations, Schroeder function, iteration, similarity transformations. Received by the editors May 23, 1978, and in revised form December 18, 1978. This research was supported in part by the National Science Foundation under Grant MCS75-222-55

and the Office of Naval Research under Contract N00014-76-C-0370, NR 044-422. The work of the first author was also supported in part by the National Science Foundation under Grant 1-442427-21164-2 at the University of California at Berkeley.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1980, SIAM. Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb050a typeset using AMS-L^{A}TEX