Meiyue Shao

In block eigenvalue algorithms, such as the subspace iteration algorithm and the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm, a large block size is often employed to achieve robustness and rapid convergence. However, using a large block size also increases the computational cost. Traditionally, the block size is typically reduced after convergence of some eigenpairs, known as deflation. In this work, we propose a nondeflation-based, more aggressive technique, where the block size is adjusted dynamically during the algorithm. This technique can be applied to a wide range of block eigensolvers, reducing computational cost without compromising convergence speed. We present three adaptive strategies for adjusting the block size, and apply them to four well-known eigensolvers as examples. Theoretical analysis and numerical experiments are provided to illustrate the efficiency of the proposed technique. In practice, an overall acceleration of 20% to 30% is observed. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/IzayoiYuuki/Shrink-and-expand_public and in the supplementary materials ( Shrink-and-expand_public-master.zip [70.2MB]). [Formula: see text]

Recent development on mixed precision techniques has largely enhanced the performance of various linear algebra solvers, one of which is the solver for the least squares problem. By transforming least squares problems into augmented linear systems, mixed precision techniques are capable of refining the lower precision solution to the working precision. In this paper, we propose mixed precision iterative refinement algorithms for two variants of the least squares problems—the least squares problem with linear equality constraints (LSE) and the generalized least squares problem (GLS). Both classical and GMRES‐based iterative refinement can be applied to augmented systems of these two problems to improve the accuracy of the solution. For reasonably well‐conditioned problems, our algorithms reduce the execution time by a factor of 40% on average compared to the fixed precision ones from LAPACK on the x86‐64 architecture.

In this work, the infinite GMRES algorithm, recently proposed by [Jarlebring and Correnty, SIAM J. Matrix Anal. Appl., 43 (2022), pp. 1382–1405] is employed in contour integral-based nonlinear eigensolvers, avoiding the computation of costly factorizations at each quadrature node to solve the linear systems efficiently. Several techniques are applied to make the infinite GMRES memory-friendly, computationally efficient, and numerically stable in practice. More specifically, we analyze the relationship between polynomial eigenvalue problems and their scaled linearizations, and provide a novel weighting strategy which can significantly accelerate the convergence of infinite GMRES in this particular context. We also adopt the technique of TOAR to infinite GMRES to reduce the memory footprint. Theoretical analysis and numerical experiments are provided to illustrate the efficiency of the proposed algorithm.

Householder orthogonalization plays an important role in numerical linear algebra. It attains perfect orthogonality regardless of the conditioning of the input. However, in the context of a nonstandard inner product, it becomes difficult to apply Householder orthogonalization, partly due to the lack of an initial orthonormal basis. We propose strategies to overcome this obstacle and discuss algorithms and variants of Householder orthogonalization with a nonstandard inner product. Theoretical analysis and numerical experiments demonstrate that our approach is numerically stable under mild assumptions.

The locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm is a popular approach for computing a few smallest eigenvalues and the corresponding eigenvectors of a large Hermitian positivedefinite matrix A. In this work, we propose a mixed precision variant of LOBPCGthat uses a (sparse) Cholesky factorization of A computed in reduced precision as the preconditioner. To further enhance performance, a mixed precision orthogonalization strategyis proposed. To analyze the impact of reducing precision in the preconditioner on performance, we carry out a rounding error and convergence analysis of PINVIT, a simplified variant of LOBPCG. Our theoretical results predict and our numerical experiments confirm that the impact on convergence remains marginal.In practice, our mixed precision LOBPCG algorithm typically reduces thecomputation time by a factor of 1.4--2.0 on both CPUs and GPUs. MSC Classification: c