On the Complexity of Composition and Generalized Composition of Power Series

Title:

On the Complexity of Composition and Generalized Composition of Power Series

Author(s):

Brent, R. P.
Traub, Joseph F.

Date:

1985

Type:

Technical reports

Department:

Computer Science

Permanent URL:

http://hdl.handle.net/10022/AC:P:11681

Series:

Columbia University Computer Science Technical Reports

Part Number:

CUCS16285

Publisher:

Department of Computer Science, Columbia University

Publisher Location:

New York

Abstract:

Let F(x) = f1x + f2(x)(x) + . . . be a formal power series over a field Delta. Let F superscript 0(x) = x and for q = 1,2 . . . , define F superscript q(x) = F superscript (q1) (F(x)). The obvious algorithm for computing the first n terms of F superscript q(x) is by the composition position analogue of repeated squaring. This algorithm has complexity about log 2 q times that of a single composition. The factor log 2 q can be eliminated in the computation of the first n terms of (F(x)) to the q power by a change of representation, using the logarithm and exponential functions. We show the factor log 2 q can also be eliminated for the composition problem. F superscript 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 superscript q(x) whenever it is defined.

Subject(s):

Computer science
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