DFT representations of Toeplitz-plus-Hankel Bezoutians with application to fast matrixvector multiplication

Abstract Representations for inverses of Toeplitz-plus-Hankel matrices and more general T + H-Bezoutians with discrete Fourier transformations are presented, which can be used for fast matrix-vector multiplication. With the help of some of the formulas multiplication by a Toeplitz-plus-Hankel matrix inverse can be carried out with six DFT's plus eight DFT's for preprocessing, which improves the so far best known result of seven DFT's plus 10 DFT's for preprocessing in [E. Bozzo, Linear Algebra Appl. 230 (1995) 127–150].

[1]  Thomas Huckle,et al.  Cauchy matrices and iterative methods for Toeplitz matrices , 1995, Optics & Photonics.

[2]  M. Naimark,et al.  The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations , 1981 .

[3]  G. Heinig,et al.  Matrix representations of Toeplitz-plus-Hankel matrix inverses , 1989 .

[4]  G. Heinig,et al.  On the inverses of Toeplitz-plus-Hankel matrices , 1988 .

[5]  Georg Heinig,et al.  Fast inversion algorithms of Toeplitz-plus-Hankel matrices , 1988 .

[6]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[7]  M. Tismenetsky,et al.  Generalized Bezoutian and the inversion problem for block matrices, I. General scheme , 1986 .

[8]  Georg Heinig,et al.  Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication , 1998 .

[9]  Karla Rost,et al.  Generalized companion matrices and matrix representations for generalized Bezoutians , 1993 .

[10]  E. Bozzo.  ALGEBRAS OF HIGHER DIMENSION FOR DISPLACEMENT DECOMPOSITIONS AND COMPUTATIONS WITH TOEPLITZ PLUS HANKEL MATRICES , 1995 .

[11]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[12]  Paul D. Gader,et al.  A variant of the Gohberg-Semencul formula involving circulant matrices , 1991 .

[13]  Adam W. Bojanczyk,et al.  Transformation Techniques for Toeplitz and Toeplitz-plus-Hankel Matrices Part I. Transformations , 1996 .

[14]  G. Strang.  A proposal for toeplitz matrix calculations , 1986 .

[15]  Enrico Bozzo,et al.  On the Use of Certain Matrix Algebras Associated with Discrete Trigonometric Transforms in Matrix Displacement Decomposition , 1995, SIAM J. Matrix Anal. Appl..

[16]  T. Chan.  An Optimal Circulant Preconditioner for Toeplitz Systems , 1988 .

[17]  Victor Y. Pan,et al.  Improved parallel computations with Toeplitz-like and Hankel-like matrices☆☆☆ , 1993 .

[18]  C. Loan.  Computational Frameworks for the Fast Fourier Transform , 1992 .

[19]  Paolo Zellini,et al.  Matrix Decompositions Using Displacement Rank and Classes of Commutative Matrix Algebras , 1995 .