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The development of hierarchical matrices as a tool to accurately approximate dense matrices of large dimensions allows addressing problems of scales that would be difficult to address using the conventional representations. This opens up an important range of opportunities in various domains of application. However, the value of this mathematical tool could be diminished if it cannot make an efficient use of modern hardware architectures. In this work, we implement a recently proposed algorithm to solve Sylvester equations involving two types of hierarchical matrices, HODLR and HALR. We also perform an experimental evaluation on a multicore CPU system to detect the principal performance bottlenecks, and discuss the main challenges to develop high-performance implementation of this type of algorithms.

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Implementation of the Sylvester Equation for HODLR and HALR \(\mathcal {H}\) -Matrices

  • 会议论文

  • 发表日期：

    2026年

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  • Gonzalo Berger,
  • Florencia Uslenghi,
  • Ernesto Dufrechou,
  • Pablo Ezzatti

摘要

The development of hierarchical matrices as a tool to accurately approximate dense matrices of large dimensions allows addressing problems of scales that would be difficult to address using the conventional representations. This opens up an important range of opportunities in various domains of application. However, the value of this mathematical tool could be diminished if it cannot make an efficient use of modern hardware architectures. In this work, we implement a recently proposed algorithm to solve Sylvester equations involving two types of hierarchical matrices, HODLR and HALR. We also perform an experimental evaluation on a multicore CPU system to detect the principal performance bottlenecks, and discuss the main challenges to develop high-performance implementation of this type of algorithms.