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

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• 作者：Michael Bartoň^a ; ^{Michael.Barton@kaust.edu.sa" class="auth_mail" title="E-mail the corresponding author} ; Victor Manuel Calo^a ; ^b ; ^{Victor.Calo@kaust.edu.sa" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Gaussian quadrature ; B-splines ; Well-constrained polynomial system ; Polynomial homotopy continuation
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：296
• 期：Complete
• 页码：709-723
• 全文大小：2124 K

文摘

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$C^1$ cubic splines with uniform knot sequences, we demonstrate the methodology by deriving the optimal rules for uniform C²$C^2$ 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.