The subject of this work is the study of Birkhoff’s polynomial interpolation. Given a list <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ Z_n=[z_{0},...,z_{n}] $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Z</mi> <mi>n</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ (n+1) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> distinct nodes, of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \mathbb {K}=\mathbb {R} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">K</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \mathbb {C} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>, we will seek to study the questions of existence and uniqueness of a polynomial P such that P and a number of its derivatives take, in these nodes, given values. Recently, Messaoudi et al. (Numer. Algorithms 80, 253–278 2019) presented a new algorithm for computing the Hermite interpolation polynomial called the Generalized Recursive Polynomial Interpolation Algorithm (GRPIA). In this paper, we will give a new formulation of the Birkhoff polynomial interpolation problem and derive a new algorithm, called the Recursive Hermite-Birkhoff Polynomial Interpolation Algorithm (RHBPIA), to solve the Birkhoff interpolation problem and generalize the GRPIA. A new existing result will be established. The numerical stability of this algorithm will also be studied and some examples will be given.

RHBPIA: a new algorithm for computing Hermite-Birkhoff interpolation polynomials

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发表日期：

2025年6月2日

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Omar Rhouni,
Mohammed Errachid,
Mustapha Esghir

摘要

The subject of this work is the study of Birkhoff’s polynomial interpolation. Given a list $ Z_n=[z_{0},...,z_{n}] $ $Z_{n} = [z_{0}, . . ., z_{n}]$ with $ (n+1) $ $(n + 1)$ distinct nodes, of $ \mathbb {K}=\mathbb {R} $ $K = R$ or $ \mathbb {C} $ $C$ , we will seek to study the questions of existence and uniqueness of a polynomial P such that P and a number of its derivatives take, in these nodes, given values. Recently, Messaoudi et al. (Numer. Algorithms 80, 253–278 2019) presented a new algorithm for computing the Hermite interpolation polynomial called the Generalized Recursive Polynomial Interpolation Algorithm (GRPIA). In this paper, we will give a new formulation of the Birkhoff polynomial interpolation problem and derive a new algorithm, called the Recursive Hermite-Birkhoff Polynomial Interpolation Algorithm (RHBPIA), to solve the Birkhoff interpolation problem and generalize the GRPIA. A new existing result will be established. The numerical stability of this algorithm will also be studied and some examples will be given.