Norm Equalities and Inequalities for Tridiagonal Perturbed Toeplitz Operator Matrices

An inexact Krylov subspace method for large generalized Hankel eigenproblems

“…Firstly, we need to solve two large Hankel systems, but the cost becomes prohibitive if the desired accuracy is too high. To overcome this 12 1.8834 × 10 −7 8.3084 × 10 −8 0.1829 0.2851 3.0463 2 14 1.8834 × 10 −7 3.4707 × 10 −7 0.3205 1.1403 219.0806 2 16 1.8834 × 10 −7 2.4584 × 10 −8 0.8528 11.9347 -2 18 1.8834 × 10 −7 2.2137 × 10 −9 1.6465 -- 12 3.7268 × 10 −8 1.4190 × 10 −9 0.3370 870.5256 3.7431 2 14 9.8939 × 10 −10 5.4106 × 10 −12 0.6775 -181.6075 2 16 3.7268 × 10 −8 8.0073 × 10 −10 2.3588 --2 18 3.7268 × 10 −8 9.4170 × 10 −10 13.5398 -difficulty, we establish a relationship between the error of the Hankel systems and the residual of the Hankel eigenproblem, and we provide a cheap stopping criterion for solving the Hankel systems inexactly. Numerical results show that our "inexact" strategy outperform solving the Hankel systems "exactly", especially when the Hankel systems are large.…”

“…Recently, the shift-and-invert Arnoldi or Lanczos method has been used in designing fast algorithms for the generalized Toeplitz eigenproblem [9], and Toeplitz matrix exponential [10,11]. Toeplitz matrices have various applications [12][13][14][15], due to the special structure of Toeplitz matrices, there are many fast algorithms for solving Toeplitz matrix problems [16][17][18][19] and various formula for the inversion of Toeplitz matrix [20][21][22], the products of the inverse of a Toeplitz matrix and a vector can be implemented using several FFTs [10,11,21]. For a Hankel matrix, the inverse can be obtained by solving two large Hankel linear systems, and the matrix-vector products in the shift-and-invert Arnoldi method can also be realized efficiently by using FFTs.…”