We illustrate how the liberal use of high-order procedural abstractions and infinite streams helps us to express some of the vocabulary and methods of numerical analysis. We develop a software toolbox encapsulating the technique of Richardson extrapolation, and we apply these tools to the problems of numerical integration and differentiation. By separating the idea of Richardson extrapolation from its use in particular circumstances, we indicate how numerical programs can be written that exhibit the structure of the ideas from which they are formed.

Many programs are executed according to the conventional, eager execution order, for which verification of execution costs is well-understood. However, there are other execution orders in use. One such order in common use is lazy execution or lazy evaluation, which is mostly demand-driven. Laziness supports better decompositions of algorithms, e.g., into modular producers and consumers, which enables compositional reasoning of answer correctness, but then timing correctness is more elusive. This thesis gives a formal method for verifying lazy timing, compositional with respect to program structure; it is an extension of a predicative programming theory. Predicative programming theories are formal methods that unify both specifications and programs as predicates or boolean-typed expressions over memory state and other quantities of interest. Their strengths are mathematical simplicity and support of program development and verification by incremental refinements. Among these theories, Hehner's a Practical Theory of Programming has the further strength of leaving termination and timing open rather than a built-in, and therefore is a flexible substrate for various timing schemes corresponding to various execution strategies. We use this substrate for our method for lazy timing. This thesis also proves soundness of the eager timing scheme in Hehner's work with respect to an eager operational semantics, and our lazy timing scheme with respect to a lazy operational semantics. Thus, if refinements promise an upper time bound, then execution actually stops within that time. Lastly, this thesis outlines a space of more operational semantics. It is possible ground for more execution strategies.

Many programs are executed according to the conventional, eager execution order, for which verification of execution costs is well-understood. However, there are other execution orders in use. One such order in common use is lazy execution or lazy evaluation, which is mostly demand-driven. Laziness supports better decompositions of algorithms, e.g., into modular producers and consumers, which enables compositional reasoning of answer correctness, but then timing correctness is more elusive. This thesis gives a formal method for verifying lazy timing, compositional with respect to program structure; it is an extension of a predicative programming theory. Predicative programming theories are formal methods that unify both specifications and programs as predicates or boolean-typed expressions over memory state and other quantities of interest. Their strengths are mathematical simplicity and support of program development and verification by incremental refinements. Among these theories, Hehner's a Practical Theory of Programming has the further strength of leaving termination and timing open rather than a built-in, and therefore is a flexible substrate for various timing schemes corresponding to various execution strategies. We use this substrate for our method for lazy timing. This thesis also proves soundness of the eager timing scheme in Hehner's work with respect to an eager operational semantics, and our lazy timing scheme with respect to a lazy operational semantics. Thus, if refinements promise an upper time bound, then execution actually stops within that time. Lastly, this thesis outlines a space of more operational semantics. It is possible ground for more execution strategies.