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Batch Matrix Exponentiation

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Numerical Computations with GPUs

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.

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Notes

  1. 1.

    In this work, we refer to general matrix–matrix multiplication as GEMM, in adherence with the Basic Linear Algebra Subroutines (BLAS) standard [5].

  2. 2.

    Here, M is the dimension of the probability matrix and number of sites in the model. For example, M = 4 for the nucleotide model.

  3. 3.

    We use the following flop count throughout this work, regardless of the algorithm, implementation, or architecture:

    $$\displaystyle{ flops = n {\ast} (3m^{3} + 2m) }$$
    (3.9)

    where n is the number of branch lengths, and m is the dimension of the matrix E from Eq. (3.8). This count comes from Ln. 24 and 32 of Cd. 3.

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  1. Georgia Institute of Technology, Atlanta, GA, 30332, USA

    M. Graham Lopez & Mitchel D. Horton

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  1. M. Graham Lopez
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  2. Mitchel D. Horton
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Correspondence to Mitchel D. Horton .

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  1. National Center for Supercomputing Applications, University of Illinois, Urbana, Illinois, USA

    Volodymyr Kindratenko

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Lopez, M.G., Horton, M.D. (2014). Batch Matrix Exponentiation. In: Kindratenko, V. (eds) Numerical Computations with GPUs. Springer, Cham. https://doi.org/10.1007/978-3-319-06548-9_3

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  • DOI: https://doi.org/10.1007/978-3-319-06548-9_3

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