Squinting at Power Series

Data streams are an ideal vehicle for handling power series. Stream implementations can be read off directly from simple recursive equations that define operations such as multiplication, substitution, exponentiation, and reversion of series. The bookkeeping that bedevils these algorithms when they are expressed in traditional languages is completely hidden when they are expressed in stream terms. Communicating processes are the key to the simplicity of the algorithms. Working versions are presented in newsqueak, the language of Pike’s ‘‘squint’’ system; their effectiveness depends critically on the stream protocol. Series and streams Power series are natural objects for stream processing. Pertinent computations are neatly describable by recursive equations. CSP (communicating sequential process) languages1, 2 are good vehicles for implementation. This paper develops the algorithmic ideas and reduces them to practice in a working CSP language,3 not coincidentally illustrating the utility of concurrent programming notions in untangling the logic of some intricate computations on sequences. This elegant approach to power series computation, first demonstrated but never published by Kahn and MacQueen as an application of their data stream system, is still little known.4 Power series are represented as streams of coefficients, the exponents being given implicitly by ordinal position. (This is an infinite analogue of the familiar representation of a polynomial as an array of coefficients.) Thus the power series for the exponential function e x = n = 0 Σ ∞ x n / n! = 1 + x + 2 1 _ _ x 2 + 6 1 _ _ x 3 + 24 1 _ __ x 4 + . . .