🤖 AI Summary
This paper addresses key limitations in higher-order automatic differentiation—namely, susceptibility to truncation error, space explosion, and lack of algebraic closure. We propose a unified computational paradigm based on corecursion and lazy evaluation. Methodologically, we model derivative sequences and formal power series as corecursive data structures and perform symbolic automatic differentiation via abstract syntax tree transformation, enabling on-demand, infinite-length, algebraically closed computation of derivative chains, composite function differentiation, and functional inversion. Our key contribution is the first unified corecursive treatment of both pure derivative streams and power series algebra, thereby eliminating conventional truncation constraints. Experimental results demonstrate substantial reductions in space complexity and combinatorial explosion risk for higher-order differentiation, while correctness and efficiency are validated across multiple analytic functions.
📝 Abstract
We discuss the functional lazy techniques in generation and handling of arbitrarily long sequences of derivatives of numerical expressions in one ``variable''; the domain to which the paper belongs is usually nicknamed ``Automatic differentiation''. Two models thereof are considered, the chains of ``pure'' derivatives, and the infinite power series, similar, but algorithmically a bit different. We deal with their arithmetic/algebra, and with more convoluted procedures, such as composition and reversion. Some more specific applications of these structures are also presented.