When Structure is Silent: Opportunities for Algorithmic Dispatch in Linear Algebra

Abstract

Algorithmic dispatch is essential for performance in linear-algebra-intensive systems. A persistent challenge lies in the treatment of structured matrices. Although such matrices are often described as sparse, the term structured is more precise, as it highlights exploitable properties - such as bandedness or triangularity - whose algorithmic advantages extend beyond sparsity alone. When the dispatch strategy leaves these structures unrecognized, valuable opportunities for optimization are lost. Recent advances in generative AI offer the promise of linking these silent structures to more effective algorithmic and architectural choices, supplying much of the missing connective tissue in computational linear algebra. However, AI-synthesized dispatch strategies also raise important questions about their theoretical soundness. This work introduces analytical criteria - grounded in time-complexity analysis - to determine when structure-aware dispatch delivers tangible gains. We examine the overheads of structure detection and data-format conversion, characterizing their impact on speedup and slowdown. We illustrate these concepts through a case study on LU factorization applied to banded matrices stored in a dense format, demonstrating results that align with theoretical bounds and reveal substantial gains in both performance and memory usage. These analyses underscore the need for more intelligent dispatch strategies to recognize and exploit silent structures - an underused path to high-performance linear algebra.