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  • interpolation  (2)
  • rational interpolation  (2)
  • barycentric formula  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Rational interpolation through the optimal attachment of poles to the interpolating polynomial (2000)
    Springer
    Numerical algorithms 23 (2000), S. 315-328 
    Details
    ISSN: 1572-9265
    Keywords: interpolation ; rational interpolation ; optimal interpolation ; 65D05 ; 41A05 ; 41A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markov's inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed, written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen norm of the error.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019168504808
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A matrix for determining lower complexity barycentric representations of rational interpolants (2000)
    Springer
    Numerical algorithms 24 (2000), S. 17-29 
    Details
    ISSN: 1572-9265
    Keywords: interpolation ; rational interpolation ; barycentric representation ; barycentric weights ; complexity ; 65D05 ; 41A05 ; 41A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Among the representations of rational interpolants, the barycentric form has several advantages, for example, with respect to stability of interpolation, location of unattainable points and poles, and differentiation. But it also has some drawbacks, in particular the more costly evaluation than the canonical representation. In the present work we address this difficulty by diminishing the number of interpolation nodes embedded in the barycentric form. This leads to a structured matrix, made of two (modified) Vandermonde and one Löwner, whose kernel is the set of weights of the interpolant (if the latter exists). We accordingly modify the algorithm presented in former work for computing the barycentric weights and discuss its efficiency with several examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019180807534
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    A closed formula for the čebyšev barycentric weights of optimal approximation in H2 (1993)
    Springer
    Numerical algorithms 5 (1993), S. 155-163 
    Details
    ISSN: 1572-9265
    Keywords: Optimal approximation ; barycentric formula ; čebyšev points ; AMS(MOS) 65D05 ; 65D15 ; 41A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The barycentric formula has several advantages over other means of evaluating the polynomial interpolating a function betweenn points in an interval. In particular, it is much more stable for sets of points clustered at the extremities of the interval, as are all the sets guaranteeing a good approximation forn sufficiently large. Also, it requires onlyO(n) operations for every function to be interpolated, once some weights, which depend only on the points, have been computed. Computing those weights usually requiresO(n2) operations; for čebyšev points, however,O(n) operations suffice. We show here that all the above is also true for the optimal evaluation of functionals in H2 by giving a closed formula for the corresponding weights.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02215678
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    Paper (German National Licenses)
    Fulltext
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