NDLI: Computing ECT-B-splines recursively

Content Provider	Springer Nature Link
Author	Mühlbach, Günter W. Tang, Yuehong
Copyright Year	2005
Abstract	ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as Neville–Aitken weights of certain generalized interpolation problems. For multiple knots they are limits of Neville–Aitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval [a,b] connected at inner knots all of multiplicities zero by full connection matrices A $^{[i]}$ that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECT-spline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves.
Ending Page	78
Page Count	44
Starting Page	35
File Format	PDF
ISSN	10171398
e-ISSN	15729265
Journal	Numerical Algorithms
Issue Number	1
Volume Number	41
Language	English
Publisher	Springer US
Publisher Date	2005-12-13
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	ECT-spline curves Interpolation Algebra Algorithms Numeric Computing de-Boor algorithm ECT-systems ECT-B-splines Theory of Computation Mathematics Spline approximation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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