Fast and accurate evaluation of dual Bernstein polynomials
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Fast and accurate evaluation of dual Bernstein polynomials Filip Chudy1 · Paweł Wo´zny1 Received: 24 April 2020 / Accepted: 3 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Dual Bernstein polynomials find many applications in approximation theory, computational mathematics, numerical analysis, and computer-aided geometric design. In this context, one of the main problems is fast and accurate evaluation both of these polynomials and their linear combinations. New simple recurrence relations of low order satisfied by dual Bernstein polynomials are given. In particular, a firstorder non-homogeneous recurrence relation linking dual Bernstein and shifted Jacobi orthogonal polynomials has been obtained. When used properly, it allows to propose fast and numerically efficient algorithms for evaluating all n + 1 dual Bernstein polynomials of degree n with O(n) computational complexity. Keywords Recurrence relations · Bernstein basis polynomials · Dual Bernstein polynomials · Jacobi polynomials
1 Introduction Let us introduce the inner product ·, ·α,β by f, gα,β :=
1
(1 − x)α x β f (x)g(x) dx,
0
where α, β > −1.
Paweł Wo´zny
[email protected] Filip Chudy [email protected] 1
Institute of Computer Science, University of Wrocław, ul. F. Joliot-Curie 15, 50-383 Wrocław, Poland
(1.1)
Numerical Algorithms
Let Πn (n ∈ N) denote the set of polynomials of degree at most n. Recall that the (α,β) shifted Jacobi polynomial of degree n, Rn ∈ Πn , is defined by Rn(α,β) (x) :=
n (α + 1)n (−n)k (n + α + β + 1)k (1 − x)k n! k!(α + 1)k
(n = 0, 1, . . .).
k=0
(1.2) Here (c) (c ∈ C; ∈ N) denotes the Pochhammer symbol, (c)0 := 1,
(c) := c(c + 1) . . . (c + − 1) ( ≥ 1).
These polynomials satisfy the second-order recurrence relation of the form (α,β)
(α,β)
ξ0 (n)Rn(α,β) (x) + ξ1 (n)Rn+1 (x) + ξ2 (n)Rn+2 (x) = 0
(n = 0, 1, . . .), (1.3)
where (1.4) ξ0 (n) := −2(n + α + 1)(n + β + 1)(2n + σ + 3), 2 2 ξ1 (n) := (2n + σ + 2) (2n+σ +1)(2n + σ + 3)(2x −1) + α − β , (1.5) ξ2 (n) := −2(n + 2)(n + σ + 1)(2n + σ + 1),
(1.6)
and σ := α + β + 1 (cf., e.g., [16, §1.8]). Remark 1 Notice that the recurrence relation (1.3) can be used, for example, in fast (α,β) and accurate methods for evaluating the values Rn (x) for a given x, α, β and all 0 ≤ n ≤ N, where N is a fixed natural number, with O(N) computational complexity. For more details about performing computations with recurrence relations properly, see [23]. Shifted Jacobi polynomials are orthogonal with respect to the inner product (1.1), i.e., (α,β) (α,β) Rk , R = δk hk (k, ∈ N), α,β
where δk is the Kronecker delta (δk = 0 for k = and δkk = 1) and hk := K
(α + 1)k (β + 1)k k!(2k/σ + 1)(σ )k
(k = 0, 1, . . .)
with K := Γ (α + 1)Γ (β + 1)/Γ (σ + 1). For more properties and applications of (α,β) polynomials Rn , see, e.g., [1, 16]. n n Let B0 , B1 , . . . , Bnn ∈ Πn be Bernstein basis polynomials given by n i Bin (x) := (i = 0, 1, . . . , n; n ∈ N). (1.7) x (1 − x)n−i i Definition 2
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