Batch matrix exponentiation

Abstract

Matrix-matrix multiplication can be considered a linchpin of applied numerical dense linear algebra as the performance of many common dense linear algebra packages is closely tied to the performance of matrix-matrix multiplication. Batch matrix-matrix multiplication, the matrix-matrix multiplication of a large number of relatively small matrices, is a developing area within dense linear algebra and is relevant to various application areas such as phylogenetics, finite element modeling, image processing, fluid dynamics, and hydrodynamics. Using batch matrix-matrix multiplication as the foundation, we have developed an optimized batch matrix exponentiation algorithm in CUDA that outperforms cublasXgemmBatched for small square matrices. After introducing the original motivation for our problem, matrix exponentiation from the phylogenetics domain, we discuss our algorithm in the context of both cublasXgemmBatched, and two alternative GPU methods for the numerical computation of matrix exponentiation: Lagrange interpolation, and Newton interpolation. All comparisons are done on both the Fermi and the Kepler architectures.