An SVD analysis of equispaced polynomial interpolation

Runge showed more than a century ago that polynomial interpolation of degree N on a uniform, evenly-spaced grid may diverge as N → ∞ . Here, we present a new angle on polynomial interpolation and the Runge Phenomenon by performing an SVD analysis of the interpolation matrix. We prove that the observ...

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Bibliographic Details
Published in:Applied numerical mathematics Vol. 59; no. 10; pp. 2534 - 2547
Main Authors: Boyd, John P., Sousa, Alan M.
Format: Journal Article
Language:English
Published: Kidlington Elsevier B.V 01.10.2009
Elsevier
Subjects:
Approximations and expansions
Exact sciences and technology
Interpolation
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Polynomial interpolation
Runge Phenomenon
Sciences and techniques of general use
SVD analysis
Interpolation
Runge Phenomenon
Polynomial interpolation
SVD analysis
Polynomial
Matrix analysis
Filtering
Grid pattern
Error estimation
Singular value
Numerical approximation
Numerical analysis
Applied mathematics
Condition number
Growth of error
ISSN:0168-9274, 1873-5460
Online Access:Get full text
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Summary:Runge showed more than a century ago that polynomial interpolation of degree N on a uniform, evenly-spaced grid may diverge as N → ∞ . Here, we present a new angle on polynomial interpolation and the Runge Phenomenon by performing an SVD analysis of the interpolation matrix. We prove that the observed exponential-with- N error growth of the Runge Phenomenon requires that at least the smallest singular value of the interpolation matrix must decay to zero exponentially fast with N, and the matrix condition number κ must grow exponentially with N. Experimentally, four-fifths of the singular values are O ( 1 ) ; the remaining one-fifth of singular values decay exponentially fast with increasing mode number j. The corresponding U modes of the SVD factorization are well approximated by Hermite functions. SVD filtering, which is a common strategy for coping with ill-conditioned matrices by omitting modes of tiny singular value, gave errors no smaller than 7% for Runge's exemplary f ( x ) . Any similar anti-Runge strategy that is based on treating the “good” modes (of O ( 1 ) singular values) differently from the “bad” modes of small singular value is likely to fail because these two groups of modes are strongly coupled. A better strategy for exploiting the SVD factorization remains elusive.
ISSN:0168-9274
1873-5460
DOI:10.1016/j.apnum.2009.05.012