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Workshop: Structured Numerical Linear Algebra Problems: Algorithms and Applications Cortona, Italy, September 19-24, 200

Workshop: Structured Numerical Linear Algebra Problems: Algorithms and Applications Cortona, Italy, September 19-24, 2004. Interpolation and Approximation on Chebyshev extrema nodes. Alfredo Eisinberg Giuseppe Fedele DEIS – University of Calabria - Italy. Outline.

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Workshop: Structured Numerical Linear Algebra Problems: Algorithms and Applications Cortona, Italy, September 19-24, 200

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  1. Workshop:Structured Numerical Linear Algebra Problems:Algorithms and ApplicationsCortona, Italy, September 19-24, 2004 Interpolation and Approximation on Chebyshev extrema nodes Alfredo Eisinberg Giuseppe Fedele DEIS – University of Calabria - Italy

  2. Outline A property on the elementary symmetric functions Explicit factorization of the inverse of the Vandermonde matrix Symmetric functions for G-L nodes Vandermonde systems on G-L nodes Discrete orthogonal polynomials on G-L nodes Applications

  3. A property on elementary symmetric functions

  4. A property on elementary symmetric functions Example

  5. A property on elementary symmetric functions Example (continued)

  6. A property on elementary symmetric functions A. Eisinberg, C. Picardi On the inversion of Vandermonde matrix IFAC, Kyoto, Japan, 1981. A. Eisinberg, G. Fedele A property on the elementary symmetric functions Unpublished

  7. A property on elementary symmetric functions

  8. A property on elementary symmetric functions Inverse of the Vandermonde matrix

  9. A property on elementary symmetric functions Inverse of the Vandermonde matrix

  10. A property on elementary symmetric functions Inverse of the Vandermonde matrix

  11. A property on elementary symmetric functions Factorization

  12. Chebyshev nodes

  13. Chebyshev nodes Standard A. Eisinberg, G. Fedele Polynomial interpolation and related algorithms Twelfth International Colloquium on Num. Anal. And Computer Science with Appl. Plovdiv, 2003.

  14. Chebyshev nodes Extended A. Eisinberg, G. Fedele Polynomial interpolation and related algorithms Twelfth International Colloquium on Num. Anal. And Computer Science with Appl. Plovdiv, 2003.

  15. Why Gauss-Lobatto nodes? “It has been also shown that the set of nodes coinciding with the Chebyshev extrema, failed to be a good approximation to the optimal interpolation set. Nevertheless, this set of nodes is of considerable interest since as was established by Ehlick and Zeller, the norm of corresponding interpolation operator is less than the norm of the operator (where T is the set of Chebyshev nodes) induced by interpolation at the Chebyshev root. Namely, the following relation holds: ” L. Brutman, A Note on Polynomial Interpolation at the Chebyshev Extrema Nodes, Journal of Approx. Theory 42, 283-292 (1984).

  16. Why Gauss-Lobatto nodes? “For some sets of nodes which are of special importance in the interpolation theory, such as equidistant nodes, Chebyshev roots and extrema and others, the behavior of the Lebesgue function is well investigated.” L. Brutman, Lebesgue functions for polynomial interpolation – a survey, Annals of Numerical Mathematics 4, 111-127 (1997).

  17. Notes: Chebyshev nodes

  18. Chebyshev nodes

  19. Interpolation on Gauss-Lobatto nodes

  20. Gauss-Lobatto Chebyshev nodes (extrema)

  21. Gauss-Lobatto Chebyshev nodes (extrema)

  22. Gauss-Lobatto Chebyshev nodes (extrema)

  23. Factorization A. Eisinberg, G. Franzè, N. Salerno Rectangular Vandermonde matrices on Chebyshev nodes Linear Algebra Appl., 283 (1998), 205-219.

  24. Factorization A. Eisinberg, G. Fedele Vandermonde systems on Gauss-Lobatto Chebyshev nodes Unpublished

  25. Properties of Q n=9

  26. Properties of H n=9

  27. Algorithm details

  28. Algorithm details

  29. Algorithm details

  30. Algorithm details

  31. Numerical experiments Primal system: Dual system:

  32. Numerical experiments

  33. Numerical experiments

  34. Computational cost

  35. Frobenius norms

  36. Frobenius norms A. Eisinberg, G. Fedele Vandermonde systems on Gauss-Lobatto Chebyshev nodes Unpublished

  37. Determinant A. Eisinberg, G. Fedele Vandermonde systems on Gauss-Lobatto Chebyshev nodes Unpublished

  38. Discrete orthogonal polynomials on Gauss-Lobatto nodes

  39. Discrete orthogonal polynomials on G-L nodes

  40. Discrete orthogonal polynomials on G-L nodes A. Eisinberg, G. Fedele Discrete orthogonal polynomials on Gauss-Lobatto Chebyshev nodes Unpublished

  41. Discrete orthogonal polynomials on G-L nodes

  42. Discrete orthogonal polynomials on G-L nodes

  43. Discrete orthogonal polynomials on G-L nodes

  44. Discrete orthogonal polynomials on G-L nodes Inner products Three-terms recurrence relation

  45. Numerical results

  46. Numerical results EF = Our algorithm 4mn flops CB = Conte – De Boor algorithm 10mn flops S. D. Conte, C. De Boor Elementary Numerical Analysis McGraw Hill, 2nd ed., 1972.

  47. Numerical results

  48. Numerical results

  49. Numerical results

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