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    Gaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splines

    250793.pdf (4.725Mb)
    Access Status
    Open access
    Authors
    Barton, M.
    Calo, Victor
    Date
    2016
    Type
    Journal Article
    
    Metadata
    Show full item record
    Citation
    Barton, M. and Calo, V. 2016. Gaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splines. Journal of Computational and Applied Mathematics. 296: pp. 709-723.
    Source Title
    Journal of Computational and Applied Mathematics
    DOI
    10.1016/j.cam.2015.09.036
    ISSN
    0377-0427
    Faculty
    Faculty of Science and Engineering
    School
    Department of Applied Geology
    URI
    http://hdl.handle.net/20.500.11937/51543
    Collection
    • Curtin Research Publications
    Abstract

    We introduce a new concept for generating optimal quadrature rules for splines. To generate an optimal quadrature rule in a given (target) spline space, we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one, while preserving the number of quadrature points and therefore optimality. The quadrature nodes and weights are, considered as a higher-dimensional point, a zero of a particular system of polynomial equations. As the space is continuously deformed by changing the source knot vector, the quadrature rule gets updated using polynomial homotopy continuation. For example, starting with C1 cubic splines with uniform knot sequences, we demonstrate the methodology by deriving the optimal rules for uniform C2 cubic spline spaces where the rule was only conjectured to date. We validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen.

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