RMPIA: a new algorithm for computing the Lagrange matrix interpolation polynomials

Let σ 0 , σ 1 ,⋯, σ n be a set of n + 1 distinct real numbers (i.e., σ i ≠ σ j , for i ≠ j ) and F 0 , F 1 ,⋯ , F n , be given real s × r matrices, we know that there exists a unique s × r matrix polynomial P n ( λ ) of degree n such that P n ( σ i ) = F i , for i = 0,1,⋯ , n , P n is the matrix int...

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Vydáno v:Numerical algorithms Ročník 92; číslo 1; s. 849 - 867
Hlavní autoři: Messaoudi, Abderrahim, Sadok, Hassane
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.01.2023
Springer Nature B.V
Springer Verlag
Edice:Numerical Methods and Scientific Computing CIRM, Luminy, France 8-12 November 2021
Témata:
Algebra
Algorithms
Computation
Computer Science
Eigenvalues
Interpolation
Mathematics
Numeric Computing
Numerical Analysis
Original Paper
Polynomials
Real numbers
Theory of Computation
Recursive polynomial interpolation algorithm
Sylvester’s identity
MSC 65F
Matrix polynomial
MSC 15A
Schur complement
Lagrange matrix polynomial interpolation problem
ISSN:1017-1398, 1572-9265
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Shrnutí:Let σ 0 , σ 1 ,⋯, σ n be a set of n + 1 distinct real numbers (i.e., σ i ≠ σ j , for i ≠ j ) and F 0 , F 1 ,⋯ , F n , be given real s × r matrices, we know that there exists a unique s × r matrix polynomial P n ( λ ) of degree n such that P n ( σ i ) = F i , for i = 0,1,⋯ , n , P n is the matrix interpolation polynomial for the set {( σ i , F i ), i = 0,1,⋯ , n }. The matrix polynomial P n ( λ ) can be computed by using the Lagrange formula or the barycentric method. This paper presents a new method for computing matrix interpolation polynomials. We will reformulate the Lagrange matrix interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Recursive Matrix Polynomial Interpolation Algorithm (RMPIA) in full and simplified versions, and some properties of this algorithm will be studied. Cost and storage of this algorithm with the classical formulas will be studied and some examples will also be given.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-022-01357-0