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Kai Brandhorst1, Martin Head-Gordon1

  • 1Department of Chemistry, University of California, Berkeley, California 94720, United States, and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, United States.

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Summary
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This study introduces an efficient algorithm for computing Cholesky factors of sparse symmetric positive definite matrices. The method separates algebraic and numeric tasks, improving performance without numerical thresholding.

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Area of Science:

  • Numerical Analysis
  • Computational Linear Algebra
  • Computer Science

Background:

  • Sparse symmetric positive definite matrices are crucial in scientific computing.
  • Existing algorithms for Cholesky factorization can be computationally intensive.
  • Efficient computation of matrix inverses is also a significant challenge.

Purpose of the Study:

  • To develop an efficient algorithm for computing Cholesky factors and inverses of sparse symmetric positive definite matrices.
  • To improve computational performance by separating algebraic and numerical tasks.
  • To provide a robust method that avoids numerical thresholding.

Main Methods:

  • A novel algorithm separating the computation into algebraic and numerical phases.
  • Utilizing graph theory for reordering and determining the exact nonzero structure.
  • Employing highly optimized dense linear algebra kernels for the numerical factorization.

Main Results:

  • The proposed algorithm demonstrates efficient, nonlinear scaling for Cholesky factorization and inverse computation.
  • The separation of tasks and graph-based approach preserve sparsity effectively.
  • Performance comparisons show advantages over standard library and thresholding-based sparse implementations.

Conclusions:

  • The presented algorithm offers a significant performance improvement for sparse Cholesky factorization and inverse computation.
  • The method's robustness and efficiency make it suitable for large-scale problems.
  • Further considerations for handling positive semidefinite matrices are discussed.