Workshop: Structured Numerical Linear Algebra Problems: Algorithms and Applications Cortona, Italy,

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WorkshopStructured Numerical Linear Algebra
ProblemsAlgorithms and ApplicationsCortona,
Italy, September 19-24, 2004

• Interpolation and Approximation on Chebyshev
  extrema nodes

Alfredo Eisinberg Giuseppe Fedele DEIS
University of Calabria - Italy
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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
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A property on elementary symmetric functions
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A property on elementary symmetric functions
Example
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A property on elementary symmetric functions
Example (continued)
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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symmetric functions Unpublished
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A property on elementary symmetric functions
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A property on elementary symmetric functions
Inverse of the Vandermonde matrix
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A property on elementary symmetric functions
Inverse of the Vandermonde matrix
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A property on elementary symmetric functions
Inverse of the Vandermonde matrix
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A property on elementary symmetric functions
Factorization
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Chebyshev nodes
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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.
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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.
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Why Gauss-Lobatto nodes?
L. Brutman, A Note on Polynomial Interpolation
at the Chebyshev Extrema Nodes, Journal of
Approx. Theory 42, 283-292 (1984).
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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).
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Notes Chebyshev nodes
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Chebyshev nodes
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Interpolation on Gauss-Lobatto nodes
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Gauss-Lobatto Chebyshev nodes (extrema)
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Gauss-Lobatto Chebyshev nodes (extrema)
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Gauss-Lobatto Chebyshev nodes (extrema)
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Factorization
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Factorization
A. Eisinberg, G. Fedele Vandermonde systems on
Gauss-Lobatto Chebyshev nodes Unpublished
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Properties of Q
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Properties of H
n9
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Algorithm details
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Algorithm details
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Algorithm details
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Algorithm details
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Numerical experiments
Primal system
Dual system
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Numerical experiments
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Numerical experiments
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Computational cost
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Frobenius norms
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Frobenius norms
A. Eisinberg, G. Fedele Vandermonde systems on
Gauss-Lobatto Chebyshev nodes Unpublished
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Determinant
A. Eisinberg, G. Fedele Vandermonde systems on
Gauss-Lobatto Chebyshev nodes Unpublished
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Discrete orthogonal polynomials on Gauss-Lobatto
nodes
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Discrete orthogonal polynomials on G-L nodes
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Discrete orthogonal polynomials on G-L nodes
A. Eisinberg, G. Fedele Discrete orthogonal
polynomials on Gauss-Lobatto Chebyshev
nodes Unpublished
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Discrete orthogonal polynomials on G-L nodes
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Discrete orthogonal polynomials on G-L nodes
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Discrete orthogonal polynomials on G-L nodes
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(No Transcript)
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Discrete orthogonal polynomials on G-L nodes
Inner products
Three-terms recurrence relation
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Numerical results
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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.
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Numerical results
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Numerical results
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Numerical results
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Applications
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Eigensystems
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Eigensystems
A. Eisinberg, G. Fedele A property on the
elementary symmetric functions Unpublished
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Differentiation matrices
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The process of obtaining approximations to the
values of a function at the collocation points
can be expressed as a matrix-vector
multiplication the matrices involved are called
spectral differentiation matrices.
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Differentiation matrices
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Differentiation matrices
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Differentiation matrices
B. D. Welfert Generation of pseudospectral
differentiation matrices I SIAM J. Numer. Anal.,
34, 1640-1657 (1997).
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LMS filter
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LMS filter
S/N 0.1
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Unpublished papers
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Papers
A. Eisinberg, G. Fedele Accurate floating point
summation a new approach A. Eisinberg, G.
Fedele A property on the elementary symmetric
functions. A. Eisinberg, G. Fedele On the
inversion of the Vandermonde matrix. A.
Eisinberg, G. Fedele, C. Imbrogno Vandermonde
systems on equidistant nodes in 0,1. A.
Eisinberg, G. Fedele, G. Franzè On the Lebesgue
constant for Lagrange interpolation on
equidistant nodes. A. Eisinberg, G.
Fedele Discrete orthogonal polynomials on
equidistant nodes. A. Eisinberg, G.
Fedele Vandermonde systems on Gauss-Lobatto
Chebyshev nodes. A. Eisinberg, G.
Fedele Discrete orthogonal polynomials on
Gauss-Lobatto Chebyshev nodes.
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Tank you