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A Matrix Approach to the Newton Formula and Divided Differences

  • J. M. Carnicer
  • Y. Khiar
  • J. M. Peña
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 8)

Abstract

The Crout factorization of a Vandermonde matrix is related with the Newton polynomial interpolation formula expressed in terms of divided differences. Another triangular factorization, which can be related with the Newton formula in terms of finite differences, is provided by the Doolittle factorization. The influence of the order of the nodes on the conditioning of the corresponding linear system is analyzed, considering the three cases of increasing order, Leja order and increasing distances to the origin. The lower triangular systems for the computation of divided and finite differences are analyzed and the conditioning of the corresponding lower triangular matrices is studied. Numerical examples are included.

Notes

Acknowledgements

This work has been partially supported by the Spanish Research Grant MTM2015-65433, by Gobierno the Aragón and Fondo Social Europeo.

References

  1. 1.
    de Boor, C., Pinkus, A.: Backward error analysis for totally positive linear systems. Numer. Math. 27, 485–490 (1976/1977)Google Scholar
  2. 2.
    Bos, L., de Marchi, S., Sommariva, A., Vianello, M.: Computing multivariate Fekete and Leja points by numerical linear algebra. SIAM J. Numer. Anal. 48 (5), 1984–1999 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carnicer, J., Khiar, Y., Peña, J.M.: Factorization of Vandermonde matrix and the Newton formula. In: Thirteenth International Conference Zaragoza-Pau on Mathematics and its Applications, Ahusborde, É., Amrouche, C., et al. (eds.) Monografías Matemáticas “García Galdeano”, vol. 40, pp. 53–60. Universidad de Zaragoza, Zaragoza (2015)Google Scholar
  4. 4.
    Demmel, J., Koev, P.: The accurate and efficient solution of a totally positive generalized Vandermonde linear system. SIAM J. Matrix Anal. Appl. 27, 142–152 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Demmel, J., Gu, M., Eisenstat, S., Slapnin̆ar, I., Veselić, K., Drmac̆, Z.: Computing the singular value decomposition with high relative accuracy. Lineal Algebra Appl. 299, 21–80 (1999)Google Scholar
  6. 6.
    Higham, N.J.: Stability analysis of algorithms for solving confluent Vandermonde-like systems. SIAM J. Matrix Anal. Appl. 1, 23–41 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics, Philadelphia (1996)zbMATHGoogle Scholar
  8. 8.
    Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29, 731–751 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Peña, J.M.: Pivoting strategies leading to small bounds of the errors for certain linear systems. IMA J. Numer. Anal. 16, 141–153 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Reichel, L.: Newton interpolation at Leja points. BIT 30 (2), 332–346 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Departamento de Matemática AplicadaUniversidad de ZaragozaZaragozaSpain

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