A recurrence formula for generalized divided differences and some applications

Mühlbach, G.

doi:10.1016/0021-9045(73)90104-4

Cited by 56 publications

(38 citation statements)

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“…tp being analytic inside the simply closed contour W. From (5.1) it can be'seen that most of the present" paper's assertions apply also to the divided difference case, thus recovering some of the results in [Mühlbach 1973[Mühlbach , 1978 on classieal and generalized divided differences.…”

Section: Final Remarksmentioning

confidence: 56%

A unified approach to B-spline recursions and knot insertion, with application to new recursion formulas

Walz

1995

Adv Comput Math

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Abstract.We present a uni:f1edapproach to and a generalization of almost all known recursion schemes concerning B-spline functions. This indudes formulas for the computation of a B-spline's values, its derivatives (ordinary and partial), and for a knot insertion method for B-spline curves. Fu~thermore, our generalization allows us to derive interesting new relations for these purposes. Keywords.B In this paper we would like to present a unified approach to these formulas; we will do this by proving generalized relations for each of the above-mentioned situations: Bspline value recursions in Seetion 2, formulas for a B-spline's derivatives in Seetion 3, and knot insertion in Section 4. Our generalizations do not only coveralmost all formulas. mentioned in the first paragraph as special cases, but they allow also to derive quite .easily some interesting new relations.The approach we are going to present is in all cases based on thevery nice contour integral representation of B-splines (1.2),due to G. Meinardus [Meinardus 1974].

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Section: Final Remarksmentioning

confidence: 56%

A unified approach to B-spline recursions and knot insertion, with application to new recursion formulas

Walz

1995

Adv Comput Math

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“…a relation which allows us to compute a generalized divided difference of order, say, m + j ,directly from those of order m for arbitrary j EIN. This generalizes results of Mühlbach [10,11] and enables us to compute generalized divided differences recursively, even if the underlying function space is spanned by a non-complete Chebyshev system. Furthermore, we give a contour integral representation of the generalized divided difference operator.…”

Section: O Introductionmentioning

confidence: 72%

“…';;'" (fm) = 1. This is precisely the "usual" recurrence formula for genera.lized divided differences , which is a very fundamental property and was found by Mühlbach in [10] with other methods.…”

Section: ß~~M+i(j)mentioning

confidence: 99%

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Generalized divided differences, with applications to generalized B-splines

Walz

1992

Calcolo

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“…Both the divided and forward difference algorithms are discussed in standard numerical analysis textbooks [10]. We refer the readers to [32,21] for a generalization of divided differences to the multivariate setting. We now describe how a bivariate version of the divided difference and forward difference algorithms can be obtained to evaluate a bivariate polynomial along a grid {(a j , b k )}.…”

Section: Lagrange Evaluation Algorithmmentioning

confidence: 99%

A unified approach to evaluation algorithms for multivariate polynomials

Lodha

Goldman

1997

Math. Comp.

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Abstract. We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the Bernstein-Bézier, multinomial (or Taylor), Lagrange and Newton bases. This unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of Lbases, a class of bases that include the Bernstein-Bézier, multinomial, and a rich subclass of Lagrange and Newton bases. All of the known evaluation algorithms can be generated either by considering up recursive evaluation algorithms for L-bases or by examining change of basis algorithms for L-bases. For polynomials of degree n in s variables, the class of up recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity O(n s+1 ), a nested multiplication algorithm with computational complexity O(n s log n) and a ladder recurrence algorithm with computational complexity O(n s ). These algorithms also generate a new generalization of the Aitken-Neville algorithm for evaluation of multivariate polynomials expressed in terms of Lagrange L-bases. The second class of algorithms, based on certain change of basis algorithms between L-bases, include a nested multiplication algorithm with computational complexity O(n s ), a divided difference algorithm, a forward difference algorithm, and a Lagrange evaluation algorithm with computational complexities O(n s ), O(n s ) and O(n) per point respectively for the evaluation of multivariate polynomials at several points.

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A recurrence formula for generalized divided differences and some applications

Cited by 56 publications

References 0 publications

A unified approach to B-spline recursions and knot insertion, with application to new recursion formulas

A unified approach to B-spline recursions and knot insertion, with application to new recursion formulas

Generalized divided differences, with applications to generalized B-splines

A unified approach to evaluation algorithms for multivariate polynomials

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