From polynomial interpolation to Chebfunjm/wiki/uploads/LMS/Trefthen-chebfun.pdfBurden & Faires...
Transcript of From polynomial interpolation to Chebfunjm/wiki/uploads/LMS/Trefthen-chebfun.pdfBurden & Faires...
From polynomial interpolation to ChebfunNick Trefethen, University of Oxford
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What's the integral of exp(-x) cos(6x)5 sin(5x)6 from –1 to 1?
Maple or Mathematica can figure out the answer exactly:
To combat the combinatorial explosion we use floating-point arithmetic: as we calculate, we round each number to 16 digits.
CHEBFUN VISION: floating-point arithmetic for functions, not just numbers.Represent functions by (piecewise) polynomial interpolants.
“Computing with symbolic feel and numerical speed”
Through any n+1 data values,there is a unique polynomial interpolant of degree n.
As Chebfun calculates, it chooses each polynomial degree to achieve 16 digits.
• Extend floating-point arithmetic from numbers to functions.• Overload Matlab’s vectors and matrices to functions and operators.
WHAT THE BOOKS SAY ABOUT POLYNOMIAL INTERPOLATION
Dahlquist & Björck (1974): “But there are many functions which are not at all suited for approximation by a single polynomial in the entire interval which is of interest.”
Forsythe, Malcolm & Moler (1977): “Polynomials of high degree can lead to difficult problems associated with roundoff error.”
Fröberg (1985): “Although Lagrangian interpolation is sometimes useful in theoretical investigations, it is rarely used in practical computations.”
Kahaner, Moler & Nash (1989): “Polynomial interpolation is a bad idea.”
Cormen, Leiserson & Rivest (1990): “Interpolation is a notoriously tricky problem from the point of view of numerical stability.”
Stoer & Bulirsch (1993): “While theoretically important, Lagrange's formula is, in general, not as suitable for actual calculations as some other methods to be described below.”
Stewart (1996): “Unfortunately, there are functions for which interpolation at the Chebyshevpoints fails to converge.”
Kincaid & Cheney (2002): “For most continuous functions, the quantity || f −pn ||∞ willnot converge to 0.”
Burden & Faires (2005): “The oscillatory nature of high degree polynomials, and the property that a fluctuation over a small portion of the interval can induce large fluctuations over the entire range, restricts their use.”
Parlett, SIAM Review book review (2010): “You do not want to meet a polynomial of degree 1000 on a dark night.”
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All this is untrue.
Interpolants in equispaced points are bad, but Chebyshev points are fine.
Some algorithms are numerically unstable,but the barycentric interpolation formula is bulletproof.
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CHEBYSHEV POINTS IN [−1,1]
xj = cos( j/n), 0 j n . Clustered near the boundaries.
Chebyshev… Bernstein… Lanczos… Clenshaw… Fox… Elliott… Mason… Rivlin… Good… Salzer… Orszag… Geddes…
f = continuous function on [–1,1]
p* = best max-norm degree n polynomial approximation of f
p = degree n interpolant of f in the Chebyshev pts.
|| f – p|| : error in Chebyshev interpolation
|| f – p*|| : smallest possible error among all polynomials
NOTATION FOR FIVE THEOREMSABOUT POLYNOMIAL INTERPOLATION
IN CHEBYSHEV POINTS
Theorem 1. || f − p || [ 2 + (2/) log n ] || f − p*|| .
Theorem 2. If f , f ’ ,…, f (k−1) are absolutely continuous and f (k) has bounded variation, then || f − p|| = O( n−k ) .
Theorem 3. If f is analytic in the closed ellipse in the complex planewith foci 1 and semiaxis lengths summing to ρ , then
|| f − p|| = O( ρ−n ) .
Theorem 4. Barycentric interpolation formula:
p(x) = .
Theorem 5. The barycentric formula is numerically stable.
" (−1) j f (x j ) / ( x − x j )
" (−1) j / ( x − x j )
Ehlich & Zeller 1966
Mastroianni & Szabados 1995
follows from Bernstein 1912
N. J. Higham 2004
M. Riesz 1916Salzer 1972
"NEAR-
BEST"
"SPE
CTRAL A
CCURACY"
BARYCENTRIC I
NTERP.
SIAM 2013
“Chebyshev technology”
Textbook Software
Textbook 2
Exploring ODESTrefethen, Birkisson, and Driscoll
To be published by SIAM in 2018and also freely available online
Demonstration