Preconditioned Jacobi SVD Algorithm Outperforms PDGESVD

Bečka, Martin; Okša, Gabriel

doi:10.1007/978-3-030-43229-4_47

Cited by 2 publications

(1 citation statement)

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“…The Jacobi method is a very old method for diagonalizing a real symmetric matrix [31]. Recently, the Jacobi method has received much attention due to its natural suitability for parallel computation [7] and high accuracy in finite precision [15]. Another interesting aspect of Jacobi's method with proper procedure ordering shows sweep-quadratic convergence rate after sufficient iterations [38,43].…”

Section: Introductionmentioning

confidence: 99%

A mixed precision Jacobi method for the symmetric eigenvalue problem

Zhang¹,

Bai²

2022

Preprint

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The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we propose a mixed precision Jacobi method for the symmetric eigenvalue problem. We first compute the eigenvalue decomposition of a real symmetric matrix by an eigensolver at low precision and we obtain a low-precision matrix of eigenvectors. Then by using the modified Gram-Schmidt orthogonalization process to the low-precision eigenvector matrix in high precision, a high-precision orthogonal matrix is obtained, which is used as an initial guess for the Jacobi method. We give the rounding error analysis for the proposed method and the quadratic convergence of the proposed method is established under some sufficient conditions. We also present a mixed precision one-side Jacobi method for the singular value problem and the corresponding rounding error analysis and quadratic convergence are discussed. Numerical experiments on CPUs and GPUs are conducted to illustrate the efficiency of the proposed mixed precision Jacobi method over the original Jacobi method.

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