Computing the Smallest Eigenvalue of a Symmetric Toeplitz Matrix
نویسنده
چکیده
In this note we discuss a method of order 1 + √ 3 for computing the smallest eigenvalue λ1 of a symmetric and positive definite Toeplitz matrix. It generalizes and improves a method introduced in [7] which is based on rational Hermitean interpolation of the secular equation. Taking advantage of a further rational approximation of the secular equation which is essentially for free and which yields lower bounds of λ1 we obtain an improved stopping criterion.
منابع مشابه
A hybrid method for computing the smallest eigenvalue of a symmetric and positive definite Toeplitz matrix
In this paper we suggest a hybrid method for computing the smallest eigenvalue of a symmetric and positive definite Toeplitz matrix which takes advantage of two types of methods, Newton’s method for the characteristic polynomial and projection methods based on rational interpolation of the secular equation.
متن کاملA Schur–based algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz matrix
Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. Many of them compute the smallest eigenvalue in an iterative fashion, relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exp...
متن کاملSymmetric Schemes for Computing the Minimum Eigenvalue of a Symmetric Toeplitz Matrix
In 8] and 9] W. Mackens and the present author presented two generalizations of a method of Cybenko and Van Loan 4] for computing the smallest eigenvalue of a symmetric, positive deenite Toeplitz matrix. Taking advantage of the symmetry or skew symmetry of the corresponding eigenvector both methods are improved considerably.
متن کاملA fast algorithm for computing the smallest eigenvalue of a symmetric positive-definite Toeplitz matrix
Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive definite Toeplitz matrix. Several algorithms have been proposed in the literature. Many of them compute the smallest eigenvalue in an iterative fashion, relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exp...
متن کاملThe Minimum Eigenvalue of a Symmetric Positive De nite Toeplitz Matrix and Rational Hermitian Interpolation
A novel method for computing the minimal eigenvalue of a symmetric positive deenite Toeplitz matrix is presented. Similarly to the algorithm of Cybenko and Van Loan it is a combination of bisection and a root nding method. Both phases of the method are accelerated considerably by rational Hermite interpolation of the secular equation. For randomly generated test problems of dimension 800 the av...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002