An iterative approach to barycentric rational Hermite interpolation

• Published in: Numerische Mathematik Vol. 140; no. 4; pp. 939 - 962
• Main Authors: Cirillo, Emiliano, Hormann, Kai
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-12-2018; Springer Nature B.V
• Subjects: Blending; Convergence; Interpolation; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Polynomials; Simulation; Theoretical; 65D05; 41A25; 41A05; 41A20
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-018-0986-y

In this paper we study an iterative approach to the Hermite interpolation problem, which first constructs an interpolant of the function values at
n
+
1
nodes and then successively adds
m
correction terms to fit the data up to the
m
th derivatives. In the case of polynomial interpolation, this simply reproduces the classical Hermite interpolant, but the approach is general enough to be used in other settings. In particular, we focus on the family of rational Floater–Hormann interpolants, which are based on blending local polynomial interpolants of degree
d
with rational blending functions. For this family, the proposed method results in rational Hermite interpolants, which depend linearly on the data, with numerator and denominator of degree at most
(
m
+
1
)
(
n
+
1
)
-
1
and
(
m
+
1
)
(
n
-
d
)
, respectively. They converge at the rate of
O
(
h
(
m
+
1
)
(
d
+
1
)
)
as the mesh size
h
converges to 0. After deriving the barycentric form of these interpolants, we prove the convergence rate for
m
=
1
and
m
=
2
, and show that the approximation results compare favourably with other constructions.