A Kogbetliantz-type algorithm for the hyperbolic SVD

@article{Novakovic2020AKA,
  title={A Kogbetliantz-type algorithm for the hyperbolic SVD},
  author={Vedran Novakovi'c and Sanja Singer},
  journal={Numerical Algorithms},
  year={2020},
  volume={90},
  pages={523 - 561}
}
In this paper, a two-sided, parallel Kogbetliantz-type algorithm for the hyperbolic singular value decomposition (HSVD) of real and complex square matrices is developed, with a single assumption that the input matrix, of order n, admits such a decomposition into the product of a unitary, a non-negative diagonal, and a J-unitary matrix, where J is a given diagonal matrix of positive and negative signs. When J = ±I, the proposed algorithm computes the ordinary SVD. The paper’s most important… 

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References

SHOWING 1-10 OF 38 REFERENCES

The LAPW method with eigendecomposition based on the Hari-Zimmermann generalized hyperbolic SVD

An accurate, highly parallel algorithm for the generalized eigendecomposition of a matrix pair, given in a factored form, of complex and Hermitian matrices, considered the golden standard in Condensed Matter Physics.

Batched computation of the singular value decompositions of order two by the AVX-512 vectorization

The vectorized approach is shown to be about three times faster than processing each matrix in isolation, while slightly improving accuracy over the straightforward method for the $2\times 2$ SVD.

A Jacobi eigenreduction algorithm for definite matrix pairs

SummaryWe propose a Jacobi eigenreduction algorithm for symmetric definite matrix pairsA, J of small to medium-size real symmetric matrices withJ2=I,J diagonal (neitherJ norA itself need be

The hyberbolic singular value decomposition and applications

The hyperbolic SVD accurately and efficiently finds the eigenstructure of any matrix that is expressed as the difference of two matrix outer products and applies in problems where the conventional SVD cannot be employed.

Dynamic ordering for a parallel block-Jacobi SVD algorithm

Parallelizing the Kogbetliantz Method: A First Attempt

The paper investigates a way how can the two-sided Jacobi-type method for computing the singular value decomposition of triangular matrices, known as Kogbetliantz method, be adapted for use with

Full block J-Jacobi method for Hermitian matrices

Solution of linear equations by diagonalization of coefficients matrix

This form is very convenient since the multiplication of matrices is performed by an electronic computer in almost no time. The unitary matrices U and T are known to exist (for example modal matrices

Block-oriented J-Jacobi methods for Hermitian matrices

Structured Factorizations in Scalar Product Spaces

A key feature of this analysis is the identification of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations.