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

Chapter

## Abstract

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.

## Mathematics Subject Classification (2000)

65F18 15A23 15A03## Keywords

Displacement structures QR-factorization lower semiseparable plus diagonal matrices chasing procedure## Preview

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## References

- [1]A.W. Bojanczyk, R.P. Brent, and F.R. de Hoog. QR factorization of Toeplitz matrices.
*Numerische Mathematik*, 49:81–94, 1986.zbMATHCrossRefMathSciNetGoogle Scholar - [2]S. Delvaux, L. Gemignani, and M. Van Barel. Fast QR-factorization of Cauchy-like matrices.
*Linear algebra and its Applications*, 428, 2008.Google Scholar - [3]S. Delvaux and M. Van Barel. A Givens-weight representation for rank structured matrices.
*SIAM Journal on Matrix Analysis and its Applications*, 29(4):1147–1170, 2007.CrossRefGoogle Scholar - [4]S. Delvaux and M. Van Barel. A QR-based solver for rank structured matrices.
*SIAM Journal on Matrix Analysis and its Applications*, 2008. To appear.Google Scholar - [5]S. Delvaux and M. Van Barel. Unitary rank structured matrices.
*Journal of Computational and Applied Mathematics*, 215(1):49–78, 2008.zbMATHCrossRefMathSciNetGoogle Scholar - [6]P. Dewilde and A.-J. van der Veen.
*Time-varying systems and computations*. Kluwer Academic Publishers, Boston, June 1998.zbMATHGoogle Scholar - [7]D. Fasino and L. Gemignani. A Lanczos type algorithm for the
*QR*-factorization of regular Cauchy matrices.*Numerical Linear Algebra with Applications*, 9:305–319, 2002.zbMATHCrossRefMathSciNetGoogle Scholar - [8]L. Gemignani, D.A. Bini, Y. Eidelman, and I.C. Gohberg. Fast QR eigenvalue algorithms for Hessenberg matrices which are rank-one perturbations of unitary matrices.
*SIAM Journal on Matrix Analysis and its Applications*, 29, 2007.Google Scholar - [9]G. Heinig and A.W. Bojanczyk. Transformation techniques for Toeplitz and Toeplitzplus-Hankel matrices I. Transformations.
*Linear Algebra and its Applications*, 254:193–226, 1997.zbMATHCrossRefMathSciNetGoogle Scholar - [10]G. Heinig and A.W. Bojanczyk. Transformation techniques for Toeplitz and Toeplitzplus-Hankel matrices II. Algorithms.
*Linear Algebra and its Applications*, 278:11–36, 1998.zbMATHCrossRefMathSciNetGoogle Scholar - [11]G. Heinig and K. Rost. Representations of Toeplitz-plus-Hankel matrices using the trigonometric transformations with application to fast matrix-vector multiplication.
*Linear Algebra and its Applications*, 275–276:225–248, 1998.CrossRefMathSciNetGoogle Scholar - [12]T. Kailath and A.H. Sayed, editors.
*Fast reliable algorithms for matrices with structure*. SIAM, Philadelphia, PA, USA, May 1999.zbMATHGoogle Scholar - [13]V.Y. Pan.
*Structured matrices and polynomials*. Birkhäuser Springer, 2001.Google Scholar - [14]X.W. Chang and C.C Paige and G.W. Stewart. Perturbation analyses for the
*QR*factorization.*SIAM Journal on Matrix Analysis and its Applications*, 18(3):775–791, 1997.zbMATHCrossRefMathSciNetGoogle Scholar - [15]M. Van Barel, D. Fasino, L. Gemignani, and N. Mastronardi. Orthogonal rational functions and structured matrices.
*SIAM Journal on Matrix Analysis and its Applications*, 26(3):810–829, 2005.zbMATHCrossRefGoogle Scholar - [16]C.F. Van Loan.
*Computational Frameworks for the Fast Fourier Transform*. Frontiers in Applied Mathematics. SIAM, 1992.Google Scholar

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