A Givens-Weight Representation for Rank Structured Matrices

Delvaux, Steven; Barel, Marc Van

doi:10.1137/060654967

Cited by 23 publications

(64 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the literature, various efficient representations for rank structured matrices have been proposed, and efficient and accurate algorithms have been developed using these representations [1,3,7,8,9,12,21,16,19,20,17,26,27]. In particular, several efficient algorithms have been developed for approximating a symmetric matrix A by a symmetric semi-separable matrix, accurate to a constant multiple of any given tolerance τ > 0 [9,12,21].…”

Section: Matrixmentioning

confidence: 99%

Direction-Preserving and Schur-Monotonic Semiseparable Approximations of Symmetric Positive Definite Matrices

Gu¹,

Li²,

Vassilevski³

2010

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract. For a given symmetric positive definite matrix A ∈ R N×N , we develop a fast and backward stable algorithm to approximate A by a symmetric positive-definite semi-separable matrix, accurate to a constant multiple of any prescribed tolerance. In addition, this algorithm preserves the product, AZ, for a given matrix Z ∈ R N×d , where d ≪ N. Our algorithm guarantees the positive-definiteness of the semi-separable matrix by embedding an approximation strategy inside a Cholesky factorization procedure to ensure that the Schur complements during the Cholesky factorization all remain positive definite after approximation. It uses a robust direction-preserving approximation scheme to ensure the preservation of AZ. We present numerical experiments and discuss potential implications of our work.

show abstract

Section: Matrixmentioning

confidence: 99%

Direction-Preserving and Schur-Monotonic Semiseparable Approximations of Symmetric Positive Definite Matrices

Gu¹,

Li²,

Vassilevski³

2010

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

“…This will allow the use of efficient representations to represent the matrix S during the chasing scheme, such as sequentially semiseparable, quasiseparable, uv, or Givens-weight representations, see e.g. [6,3]. The best choice of representation depends on many factors.…”

Section: Qr-factorization Of Displacement Structured Matricesmentioning

confidence: 99%

QR-factorization of Displacement Structured Matrices Using a Rank Structured Matrix Approach

Delvaux

Gemignani

Barel

2010

Numerical Methods for Structured Matrices and Applications

Self Cite

View full text Add to dashboard Cite

A general scheme is proposed for computing the QR-factorization of certain displacement structured matrices, including Cauchy-like, Vandermonde-like, Toeplitz-like and Hankel-like matrices, hereby extending some earlier work for the QR-factorization of the Cauchy matrix. The algorithm employs a chasing scheme for the recursive construction of a diagonal plus semiseparable matrix of semiseparability rank r, where r is equal to the given displacement rank. The complexity is O(r 2 n 2 ) operations in the general case, and O(rn 2 ) operations in the Toeplitz-and Hankel-like case, where n denotes the matrix size. Numerical experiments are provided.

show abstract

“…We refer to (Vandebril et al, 2005) and (Vandebril et al, 2007) for a broad bibliographic overview on the topic. Note also the alternative approach of Givens and unitary weights in Delvaux and Van Barel (2007).…”

Section: Introductionmentioning

confidence: 99%

Time and space efficient generators for quasiseparable matrices

Pernet

Storjohann

2018

Journal of Symbolic Computation

View full text Add to dashboard Cite

The class of quasiseparable matrices is defined by the property that any submatrix entirely below or above the main diagonal has small rank, namely below a bound called the order of quasiseparability. These matrices arise naturally in solving PDE's for particle interaction with the Fast Multi-pole Method (FMM), or computing generalized eigenvalues. From these application fields, structured representations and algorithms have been designed in numerical linear algebra to compute with these matrices in time linear in the matrix dimension and either quadratic or cubic in the quasiseparability order. Motivated by the design of the general purpose exact linear algebra library LinBox, and by algorithmic applications in algebraic computing, we adapt existing techniques introduce novel ones to use quasiseparable matrices in exact linear algebra, where sub-cubic matrix arithmetic is available. In particular, we will show, the connection between the notion of quasiseparability and the rank profile matrix invariant, that we have introduced in 2015. It results in two new structured representations, one being a simpler variation on the hierarchically semiseparable storage, and the second one exploiting the generalized Bruhat decomposition. As a consequence, most basic operations, such as computing the quasiseparability orders, applying a vector, a block vector, multiplying two quasiseparable matrices together, inverting a quasiseparable matrix, can be at least as fast and often faster than previous existing algorithms.

show abstract

A Givens-Weight Representation for Rank Structured Matrices

Cited by 23 publications

References 14 publications

Direction-Preserving and Schur-Monotonic Semiseparable Approximations of Symmetric Positive Definite Matrices

Direction-Preserving and Schur-Monotonic Semiseparable Approximations of Symmetric Positive Definite Matrices

QR-factorization of Displacement Structured Matrices Using a Rank Structured Matrix Approach

Time and space efficient generators for quasiseparable matrices

Contact Info

Product

Resources

About