Fast algorithms for the solution of perturbed symmetric Toeplitz linear system and its applications

We consider solving the perturbed symmetric Toeplitz linear system and the multiplication of the perturbed symmetric Toeplitz matrix and the vector. In terms of solving the perturbed symmetric Toeplitz linear system, we present two new approaches for the solution. Based on the decomposing, embedding and trigonometric transformation methods for symmetric Toeplitz matrix–vector multiplication, and order-reduction algorithms, five novel methods for the product of the perturbed symmetric Toeplitz matrix and the vector are given. The proposed fast algorithms are applied to image encryption and decryption and numerical results are reported to show the proposed algorithms are promising.