Fast algorithms for perturbed Toeplitz-plus-Hankel system based on discrete cosine transform and their applications

A class of perturbed Toeplitz-plus-Hankel matrices are studied in this paper. Firstly, we present two fast algorithms for computing the eigenvalues of a Toeplitz-plus-Hankel matrix which can be diagonalized by discrete cosine transform. Based on the diagonalization of the Toeplitz-plus-Hankel matrix, algorithms for fast Toeplitz-plus-Hankel matrix-vector multiplication and solving the Toeplitz-plus-Hankel system are given. Secondly, we propose two new algorithms with less computational time to solve the perturbed Toeplitz-plus-Hankel linear system. Thirdly, image encryption and decryption utilizing the proposed algorithms are shown. Finally, the effectiveness of our proposed algorithms is verified by numerical experiments.