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Fast and stable contour integration for high order divided differences via elliptic functions


Authors: M. Lopez-Fernandez and S. Sauter
Journal: Math. Comp. 84 (2015), 1291-1315
MSC (2010): Primary 65D30, 30E20, 33B99, 39A70, 65R20
DOI: https://doi.org/10.1090/S0025-5718-2014-02890-1
Published electronically: August 28, 2014
MathSciNet review: 3315509
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Abstract: In this paper, we will present a new method for evaluating high order divided differences for certain classes of analytic, possibly, operator-valued functions. This is a classical problem in numerical mathematics but also arises in new applications such as the use of generalized convolution quadrature to solve retarded potential integral equations. The functions which we will consider are allowed to grow exponentially to the left complex half-plane, polynomially to the right half-plane and have an oscillatory behaviour with increasing imaginary part. The interpolation points are scattered in a large real interval. Our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule.


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Additional Information

M. Lopez-Fernandez
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Email: maria.lopez@math.uzh.ch

S. Sauter
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Email: stas@math.uzh.ch

DOI: https://doi.org/10.1090/S0025-5718-2014-02890-1
Keywords: Divided differences, numerical approximation of contour integrals, Jacobi elliptic functions, convolution quadrature
Received by editor(s): August 2, 2012
Received by editor(s) in revised form: August 7, 2013
Published electronically: August 28, 2014
Additional Notes: The first author was partially supported by the Spanish grant MTM 2010-19510 and the Swiss grant SNSF 200021_140685
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.