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In a recent article, the class of functions from the integers to the integerscomputable in polynomial time has been characterized using discrete ordinarydifferential equations (ODE), also known as finite differences. Doing so, wepointed out the fundamental role of linear (discrete) ODEs and classical ODEtools such as changes of variables to capture computability and complexitymeasures, or as a tool for programming. In this article, we extend the approachto a characterization of functions from the integers to the reals computable inpolynomial time in the sense of computable analysis. In particular, we providea characterization of such functions in terms of the smallest class offunctions that contains some basic functions, and that is closed bycomposition, linear length ODEs, and a natural effective limit schema.

A characterization of polynomial time computable functions from the integers to the reals using discrete ordinary differential equations