Block Factorization of Hankel Matrices and Euclidean Algorithm

S. Belhaj

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 7, page 48-54
  • ISSN: 0973-5348

Abstract

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It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, whith m < n

How to cite

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Belhaj, S.. Taik, A., ed. "Block Factorization of Hankel Matrices and Euclidean Algorithm." Mathematical Modelling of Natural Phenomena 5.7 (2010): 48-54. <http://eudml.org/doc/197675>.

@article{Belhaj2010,
abstract = {It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, whith m < n},
author = {Belhaj, S.},
editor = {Taik, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {block factorization; Hankel matrices; Toeplitz matrices; Euclidean algorithm},
language = {eng},
month = {8},
number = {7},
pages = {48-54},
publisher = {EDP Sciences},
title = {Block Factorization of Hankel Matrices and Euclidean Algorithm},
url = {http://eudml.org/doc/197675},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Belhaj, S.
AU - Taik, A.
TI - Block Factorization of Hankel Matrices and Euclidean Algorithm
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/8//
PB - EDP Sciences
VL - 5
IS - 7
SP - 48
EP - 54
AB - It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, whith m < n
LA - eng
KW - block factorization; Hankel matrices; Toeplitz matrices; Euclidean algorithm
UR - http://eudml.org/doc/197675
ER -

References

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  1. S. Belhaj. A fast method to block-diagonalize a Hankel matrix. Numer Algor, 47 (2008), 15-34. Zbl1141.65020
  2. S. Belhaj. Block diagonalization of Hankel and Bézout matrices : connection with the Euclidean algorithm, submitted.  
  3. N. Ben Atti, G.M. Diaz-Toca. Block diagonalization and LU-equivalence of Hankel matrices. Linear Algebra and its Applications, 412 (2006), 247-269. Zbl1083.65041
  4. D. Bini, L. Gemignani. On the Euclidean scheme for polynomials having interlaced real zeros. Proc. 2nd ann. ACM symp. on parallel algorithms and architectures, Crete, (1990), 254-258.  
  5. D. Bini, L. Gemignani. Fast parallel computation of the polynomial remainder sequence via Bézout and Hankel matrices. SIAM J. Comput., 24 (1995), 63–77. Zbl0818.68092
  6. A. Borodin, J. VonZurGathen, J. Hopcroft. Fast parallel matrix and gcd computation. Information and Control, 52 (1982), 241-256. Zbl0507.68020
  7. G. Diaz-Toca, N. Ben Atti. Block LU factorization of Hankel and Bezout matrices and Euclidean algorithm. Int. J. Comput. Math., 86 (2009), 135-149. Zbl1158.65316
  8. W. B. Gragg, A. Lindquist. On partial realization problem. Linear Algebra Appl., 50 (1983), 277-319. Zbl0519.93024
  9. G. Heining, K. Rost. Algebraic methods for Toeplitz-like matrices and operators. Birkhäuser Verlag, Basel, 1984.  

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