workshop: structured numerical linear algebra problems: algorithms and applications cortona, italy,...
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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. - PowerPoint PPT PresentationTRANSCRIPT
Workshop:Structured Numerical Linear Algebra Problems:Algorithms and ApplicationsCortona, Italy, September 19-24, 2004
Interpolation and Approximation on
Chebyshev extrema nodes
Alfredo EisinbergGiuseppe FedeleDEIS – University of Calabria - Italy
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
A property on elementary symmetric functions
A property on elementary symmetric functions
)3,3()2,3()1,3()0,3(
)()()(
)3,3()2,3()1,3(
1)0,3(,,
23
3213231212
3213
321
323121
321
3213
xxx
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xxx
xxxX
n
Example
A property on elementary symmetric functions
))(()3,3())(()2,3(
))(()1,3(,,
2313
2312
1312
3213
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xxxX
Example(continued)
A property on elementary symmetric functions
A. Eisinberg, C. PicardiOn the inversion of Vandermonde matrixIFAC, Kyoto, Japan, 1981.
A. Eisinberg, G. FedeleA property on the elementary symmetric functions Unpublished
A property on elementary symmetric functions
Inverse of theVandermonde matrix
A property on elementary symmetric functions
Inverse of theVandermonde matrix
A property on elementary symmetric functions
A property on elementary symmetric functions
Inverse of theVandermonde matrix
A property on elementary symmetric functions
Factorization
Chebyshev nodes
Chebyshev nodesStandard
A. Eisinberg, G. FedelePolynomial interpolation and related algorithmsTwelfth International Colloquium on Num. Anal. AndComputer Science with Appl.Plovdiv, 2003.
Chebyshev nodesExtended
A. Eisinberg, G. FedelePolynomial interpolation and related algorithmsTwelfth International Colloquium on Num. Anal. AndComputer Science with Appl.Plovdiv, 2003.
Why Gauss-Lobatto nodes?“It has been also shown that the set of nodescoinciding with the Chebyshev extrema, failed to be a goodapproximation to the optimal interpolation set. Nevertheless, thisset of nodes is of considerable interest since as was established byEhlick and Zeller, the norm of corresponding interpolation operator is less than the norm of the operator (where T is theset of Chebyshev nodes) induced by interpolation at the Chebyshevroot. Namely, the following relation holds:
”
mkmkU 0)/cos(
)(UPn )(TPn
,...6,4,2 ,10 ,)(
,...5,3,1 ,)()(
21
1
mm
TP
mTPUP
mmm
mm
L. Brutman, A Note on Polynomial Interpolation at the Chebyshev Extrema Nodes, Journal of Approx. Theory 42, 283-292 (1984).
Why Gauss-Lobatto nodes?
“For some sets of nodes which are of special importance in theinterpolation theory, such as equidistant nodes, Chebyshev rootsand extrema and others, the behavior of the Lebesgue functionis well investigated.”
L. Brutman, Lebesgue functions for polynomial interpolation – a survey,Annals of Numerical Mathematics 4, 111-127 (1997).
Notes: Chebyshev nodes
Chebyshev nodes
Interpolationon Gauss-Lobatto
nodes
Gauss-Lobatto Chebyshev nodes (extrema)
Gauss-Lobatto Chebyshev nodes (extrema)
Gauss-Lobatto Chebyshev nodes (extrema)
Factorization
A. Eisinberg, G. Franzè, N. SalernoRectangular Vandermonde matrices on Chebyshev nodesLinear Algebra Appl., 283 (1998), 205-219.
Factorization
A. Eisinberg, G. FedeleVandermonde systems on Gauss-Lobatto Chebyshev nodesUnpublished
FHUDSWn
Properties of Qn=9
Properties of Hn=9
2,...,2,1 ,11,,1 nkkHkH
niinHniH
njjnH j
,...,2,1 ,,12
12
,
2,...,2,1 ,)1(12,1
21
2,...,1, ;
2,...,3,2
,,ij,H
,1
22),1)(1(modcos,
niijni
jiHn
njijiH
2...,2,1 ;
2,...,2,1
,,12
)1(,12
1
nknj
jknHjknH j
2...,2,1 ;,...,2,1
,12
,)1(12
, 1
nkni
kniHkniH i
Algorithm details
nf
fff
f3
2
1
2/)1(
2/
3
2
1
1
nn f
fff
b
fFb 1
Algorithm details
1b12 bHb
Algorithm details
23 bQb
)9()8()7()6()5()4()3()2()1(
2
2
2
2
2
2
2
2
2
bbbbbbbbb
)1(8 2b)3(3b )3(9 2b )5(4 2b )7(2b
Algorithm details
311 bK
nsolef
Numerical experiments
Primal system:
Dual system:
Numerical experiments
Numerical experiments
Computational cost
Frobenius norms
Frobenius norms
A. Eisinberg, G. FedeleVandermonde systems on Gauss-Lobatto Chebyshev nodesUnpublished
Determinant
A. Eisinberg, G. FedeleVandermonde systems on Gauss-Lobatto Chebyshev nodesUnpublished
Discrete orthogonalpolynomials on
Gauss-Lobatto nodes
Discrete orthogonal polynomials on G-L nodes
Discrete orthogonal polynomials on G-L nodes
A. Eisinberg, G. FedeleDiscrete orthogonal polynomials on Gauss-Lobatto Chebyshev nodesUnpublished
Discrete orthogonal polynomials on G-L nodes
Discrete orthogonal polynomials on G-L nodes
Discrete orthogonal polynomials on G-L nodes
Inner products
Three-terms recurrencerelation
Discrete orthogonal polynomials on G-L nodes
Numerical results
Numerical resultsEF = Our algorithm
4mn flops
CB = Conte – De Boor algorithm10mn flops
S. D. Conte, C. De BoorElementary Numerical AnalysisMcGraw Hill, 2nd ed., 1972.
Numerical results
Numerical results
Numerical results
Applications
Eigensystems
Eigensystems
A. Eisinberg, G. FedeleA property on the elementary symmetric functions Unpublished
Differentiationmatrices
The process of obtaining approximations to the valuesof a function at the collocation points can be expressedas a matrix-vector multiplication; the matrices involvedare called spectral differentiation matrices.
Differentiation matrices
Differentiation matrices
Differentiation matrices8)( xxf
B. D. WelfertGeneration of pseudospectral differentiation matrices ISIAM J. Numer. Anal., 34, 1640-1657 (1997).
LMS filter
LMS filter
398429.00128.30sin02995.2
425389.00645.10sin96023.1 22025.181587.3)( 72685.0
tt
etg t
7/30sin27/10sin24)( ttetf t
S/N = 0.1
Unpublished papers
PapersA. Eisinberg, G. FedeleAccurate floating point summation: a new approach
A. Eisinberg, G. FedeleA property on the elementary symmetric functions.
A. Eisinberg, G. FedeleOn the inversion of the Vandermonde matrix.
A. Eisinberg, G. Fedele, C. ImbrognoVandermonde systems on equidistant nodes in [0,1].
A. Eisinberg, G. Fedele, G. FranzèOn the Lebesgue constant for Lagrange interpolation on equidistant nodes.
A. Eisinberg, G. FedeleDiscrete orthogonal polynomials on equidistant nodes.
A. Eisinberg, G. FedeleVandermonde systems on Gauss-Lobatto Chebyshev nodes.
A. Eisinberg, G. FedeleDiscrete orthogonal polynomials on Gauss-Lobatto Chebyshev nodes.
Tank you