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Roel Van Beeumen, Wim Michiels, Karl Meerbergen, Linearization of Lagrange and Hermite interpolating matrix polynomials, IMA Journal of Numerical Analysis, Volume 35, Issue 2, April 2015, Pages 909–930, https://doi.org/10.1093/imanum/dru019
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Abstract
This paper considers interpolating matrix polynomials P(λ) in Lagrange and Hermite bases. A classical approach to investigating the polynomial eigenvalue problem P(λ) x = 0 is linearization, by which the polynomial is converted into a larger matrix pencil with the same eigenvalues. Since the current linearizations of degree n Lagrange polynomials consist of matrix pencils with n+2 blocks, they introduce additional eigenvalues at infinity. Therefore, we introduce new linearizations which overcome this. Initially, we restrict to Lagrange and barycentric Lagrange matrix polynomials and give two new and more compact linearizations, resulting in matrix pencils of n+1 and n blocks for polynomials of degree n. For the latter, there is a one-to-one correspondence between the eigenpairs of P(λ) and the eigenpairs of the pencil. We also prove that these linearizations are strong. Moreover, we show how to exploit the structure of the proposed matrix pencils in Krylov-type methods, so that in this case we only have to deal with linear system solves of matrices of the original matrix polynomial dimension. Finally, we generalize for Hermite interpolation and introduce new linearizations for Hermite Lagrange and barycentric Hermite matrix polynomials. Again, we can show that the linearizations are strong and that there is a one-to-one correspondence of the eigenpairs.
