Title: Workshop: Structured Numerical Linear Algebra Problems: Algorithms and Applications Cortona, Italy,
1WorkshopStructured 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
2Outline
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
3A property on elementary symmetric functions
4A property on elementary symmetric functions
Example
5A property on elementary symmetric functions
Example (continued)
6A 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symmetric functions Unpublished
7A property on elementary symmetric functions
8A property on elementary symmetric functions
Inverse of the Vandermonde matrix
9A property on elementary symmetric functions
Inverse of the Vandermonde matrix
10A property on elementary symmetric functions
Inverse of the Vandermonde matrix
11A property on elementary symmetric functions
Factorization
12Chebyshev nodes
13Chebyshev nodes Standard
A. Eisinberg, G. Fedele Polynomial interpolation
and related algorithms Twelfth International
Colloquium on Num. Anal. And Computer Science
with Appl. Plovdiv, 2003.
14Chebyshev nodes Extended
A. Eisinberg, G. Fedele Polynomial interpolation
and related algorithms Twelfth International
Colloquium on Num. Anal. And Computer Science
with Appl. Plovdiv, 2003.
15Why Gauss-Lobatto nodes?
L. Brutman, A Note on Polynomial Interpolation
at the Chebyshev Extrema Nodes, Journal of
Approx. Theory 42, 283-292 (1984).
16Why 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).
17Notes Chebyshev nodes
18Chebyshev nodes
19Interpolation on Gauss-Lobatto nodes
20Gauss-Lobatto Chebyshev nodes (extrema)
21Gauss-Lobatto Chebyshev nodes (extrema)
22Gauss-Lobatto Chebyshev nodes (extrema)
23Factorization
24Factorization
A. Eisinberg, G. Fedele Vandermonde systems on
Gauss-Lobatto Chebyshev nodes Unpublished
25Properties of Q
26Properties of H
n9
27Algorithm details
28Algorithm details
29Algorithm details
30Algorithm details
31Numerical experiments
Primal system
Dual system
32Numerical experiments
33Numerical experiments
34Computational cost
35Frobenius norms
36Frobenius norms
A. Eisinberg, G. Fedele Vandermonde systems on
Gauss-Lobatto Chebyshev nodes Unpublished
37Determinant
A. Eisinberg, G. Fedele Vandermonde systems on
Gauss-Lobatto Chebyshev nodes Unpublished
38Discrete orthogonal polynomials on Gauss-Lobatto
nodes
39Discrete orthogonal polynomials on G-L nodes
40Discrete orthogonal polynomials on G-L nodes
A. Eisinberg, G. Fedele Discrete orthogonal
polynomials on Gauss-Lobatto Chebyshev
nodes Unpublished
41Discrete orthogonal polynomials on G-L nodes
42Discrete orthogonal polynomials on G-L nodes
43Discrete orthogonal polynomials on G-L nodes
44(No Transcript)
45Discrete orthogonal polynomials on G-L nodes
Inner products
Three-terms recurrence relation
46Numerical results
47Numerical 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.
48Numerical results
49Numerical results
50Numerical results
51Applications
52Eigensystems
53Eigensystems
A. Eisinberg, G. Fedele A property on the
elementary symmetric functions Unpublished
54Differentiation matrices
55The 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.
56Differentiation matrices
57Differentiation matrices
58Differentiation matrices
B. D. Welfert Generation of pseudospectral
differentiation matrices I SIAM J. Numer. Anal.,
34, 1640-1657 (1997).
59LMS filter
60LMS filter
S/N 0.1
61Unpublished papers
62Papers
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.
63Tank you