A fast algorithm for block Toeplitz systems with tensor structure

Abstract

We study the solution of block Toeplitz system Tx = b by the preconditioned conjugate gradient (PCG) method where T = T (1) ⊗ T (2) ⊗ … ⊗ T ( m) and T ( i) , i = 1, 2, …, m, are n × n Toeplitz matrices. The preconditioner C ̃ is a matrix that preserves the tensor structure of T and is close to T in Frobenius norm. With a fast algorithm, we show that C ̃ is a good preconditioner for solving block Toeplitz systems with tensor structure. Only O( mn m log n) operations are required for the solution of the preconditioned systems. An application is given here.