An Elementary Proof of the Exponential Conditioning of Real Vandermonde Matrices

Stefano Serra Capizzano

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 761-768
  • ISSN: 0392-4033

Abstract

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We provide and discuss an elementary proof of the exponential con- ditioning of real Vandermonde matrices which can be easily given in undergraduate courses: we exclusively use the definition of conditioning and the sup-norm formula on [ - 1 , 1 ] for Chebyshev polynomials of first kind. The same proof idea works virtually unchanged for the famous Hilbert matrix.

How to cite

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Serra Capizzano, Stefano. "An Elementary Proof of the Exponential Conditioning of Real Vandermonde Matrices." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 761-768. <http://eudml.org/doc/290357>.

@article{SerraCapizzano2007,
abstract = {We provide and discuss an elementary proof of the exponential con- ditioning of real Vandermonde matrices which can be easily given in undergraduate courses: we exclusively use the definition of conditioning and the sup-norm formula on $[-1, 1]$ for Chebyshev polynomials of first kind. The same proof idea works virtually unchanged for the famous Hilbert matrix.},
author = {Serra Capizzano, Stefano},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {761-768},
publisher = {Unione Matematica Italiana},
title = {An Elementary Proof of the Exponential Conditioning of Real Vandermonde Matrices},
url = {http://eudml.org/doc/290357},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Serra Capizzano, Stefano
TI - An Elementary Proof of the Exponential Conditioning of Real Vandermonde Matrices
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 761
EP - 768
AB - We provide and discuss an elementary proof of the exponential con- ditioning of real Vandermonde matrices which can be easily given in undergraduate courses: we exclusively use the definition of conditioning and the sup-norm formula on $[-1, 1]$ for Chebyshev polynomials of first kind. The same proof idea works virtually unchanged for the famous Hilbert matrix.
LA - eng
UR - http://eudml.org/doc/290357
ER -

References

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  1. BECKERMANN, B., The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numer. Math., 85-4 (2000). Zbl0965.15003MR1771780DOI10.1007/PL00005392
  2. GAUTSCHI, W. - INGLESE, G., Lower bounds for the condition number of Vandermonde matrices, Numer. Math., 52 (1988), 241-250. Zbl0646.15003MR929571DOI10.1007/BF01398878
  3. LI, R. C., Vandermonde Matrices with Chebyshev Nodes, Technical Report, 2005-02, Department of Mathematics, University of Kentucky, January 2005. Linear Algebra Appl., to appear. MR2398120DOI10.1016/j.laa.2007.10.029
  4. MASON, J. - HANDSCOMB, D., Chebyshev polynomials, Chapman and Hall, Boca Raton, 2003. MR1937591
  5. TAYLOR, J., The condition of Gram matrices and related problems, Proc. Royal Soc. Edinburgh, 80A-1/2 (1978), 45-56. MR529568DOI10.1017/S030821050001012X
  6. TODD, J., The condition of certain matrices. II, Arch. Math., 5 (1954), 249-257. Zbl0055.35502MR66033DOI10.1007/BF01898363
  7. TYRTYSHNIKOV, E., How bad are Hankel matrices?, Numer. Math., 67-2 (1994), 261-269. Zbl0797.65039MR1262784DOI10.1007/s002110050027
  8. WILF, H., Finite sections of some classical inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, 1970. Zbl0199.38301MR271762

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