A Package for Generating, Manipulating, and Testing Convolution Algorithms

This report describes a Maple package created for exploring the techniques of Winograd, Nussbaumer, and others for computing “fast” convolution algorithms. After codifying known convolution techniques into a common framework of bilinear algorithms built from parameterized matrices and algebraic operators, Maple’s symbolic and algebraic computation facilities are used to derive and manipulate these algorithms. The package provides an infrastructure for generating, manipulating, testing, and combining various convolution convolution algorithms within an interactive environment. The algorithms generated by the package can be exported to a domain-specific language called SPL (Signal Processing Language) and then translated into efficient C or FORTRAN code by the SPL compiler. By combining the strengths of Maple and the SPL compiler the benefits of existing algebraic computation tools are obtained without the need to embed high-performance compiler technology into a computer algebra system. The resulting enviroment allows for the systematic application of the algebraic theory developed over the years to produce correct and efficient programs. Numerous algorithmic choices can be tried, allowing for rapid testing of various optimizations to find the best combinations of algorithms for a particular size convolution on a particular computer. Furthermore, automatic code generation and algebraic verification provides the ability to construct non-trivial examples with confidence that the resulting code is correct.