Linearization of Lagrange and Hermite interpolating matrix polynomials

نویسندگان

  • Roel Van Beeumen
  • Wim Michiels
  • Karl Meerbergen
چکیده

This paper considers interpolating matrix polynomials P (λ) in Lagrange and Hermite bases. A classical approach to investigate 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 multiple 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.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

gH-differentiable of the 2th-order functions interpolating

Fuzzy Hermite interpolation of 5th degree generalizes Lagrange interpolation by fitting a polynomial to a function f that not only interpolates f at each knot but also interpolates two number of consecutive Generalized Hukuhara derivatives of f at each knot. The provided solution for the 5th degree fuzzy Hermite interpolation problem in this paper is based on cardinal basis functions linear com...

متن کامل

COMPOSITE INTERPOLATION METHOD AND THE CORRESPONDING DIFFERENTIATION MATRIX

Properties of the hybrid of block-pulse functions and Lagrange polynomials based on the Legendre-Gauss-type points are investigated and utilized to define the composite interpolation operator as an extension of the well-known Legendre interpolation operator. The uniqueness and interpolating properties are discussed and the corresponding differentiation matrix is also introduced. The appl...

متن کامل

On an Interpolation Process of Lagrange–hermite Type

Abstract. We consider a Lagrange–Hermite polynomial, interpolating a function at the Jacobi zeros and, with its first (r−1) derivatives, at the points ±1. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator in certain suitable weighted L-spaces, 1 < p < ∞, proving a Marcinkiewicz inequality involving the derivative of the polynomial at ...

متن کامل

Operational matrices with respect to Hermite polynomials and their applications in solving linear differential equations with variable coefficients

In this paper, a new and efficient approach is applied for numerical approximation of the linear differential equations with variable coeffcients based on operational matrices with respect to Hermite polynomials. Explicit formulae which express the Hermite expansion coeffcients for the moments of derivatives of any differentiable function in terms of the original expansion coefficients of the f...

متن کامل

Solution of Non Linear Singular Perturbation Equation Using Hermite Collocation Method

A numerical method for solving second order, transient, parabolic partial differential equation is presented. The spatial discretization is based on Hermite collocation method (HCM). It is a combination of orthogonal collocation method and piecewise cubic Hermite interpolating polynomials. The solution is obtained in terms of cubic Hermite interpolating basis. Numerical results have been plotte...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2013