Dr. Zlatko Drmac Profile

Dr. Zlatko Drmac

Assistant Professor at Univ of Zagreb

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Proceedings Article | 20 November 2001 Paper

Efficient and accurate computation of generalized singular-value decompositions

Zlatko Drmac

Proceedings Volume 4474, (2001) https://doi.org/10.1117/12.448660

KEYWORDS: Lithium, Neptunium, Matrices, Signal processing, Current controlled current source, Fiber reinforced polymers, Algorithms

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We present a new family of algorithms for accurate floating--point computation of the singular value decomposition (SVD) of various forms of products (quotients) of two or three matrices. The main goal of such an algorithm is to compute all singular values to high relative accuracy. This means that we are seeking guaranteed number of accurate digits even in the smallest singular values. We also want to achieve computational efficiency, while maintaining high accuracy. To illustrate, consider the SVD of the product A=B^TSC. The new algorithm uses certain preconditioning (based on diagonal scalings, the LU and QR factorizations) to replace A with A'=(B')^TS'C', where A and A' have the same singular values and the matrix A' is computed explicitly. Theoretical analysis and numerical evidence show that, in the case of full rank B, C, S, the accuracy of the new algorithm is unaffected by replacing B, S, C with, respectively, D₁B, D₂SD₃, D₄C, where D_i, i=1,...,4 are arbitrary diagonal matrices. As an application, the paper proposes new accurate algorithms for computing the (H,K)-SVD and (H₁,K)-SVD of S.

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