A superfast structured solver for Toeplitz linear systems via randomized sampling

We propose a superfast solver for Toeplitz linear systems based on rank structured matrix methods and randomized sampling. The solver uses displacement equations to transform a Toeplitz matrix T into a Cauchy-like matrix C, which is known to have low-numerical-rank offdiagonal blocks. Thus, we design a fast scheme for constructing a hierarchically semiseparable (HSS) matrix approximation to C, where the HSS generators have internal structures. Unlike classical HSS methods, our solver employs randomized sampling techniques together with fast Toeplitz matrixvector multiplications, and thus converts the direct compression of the off-diagonal blocks of C into the compression of much smaller blocks. A strong rank-revealing QR factorization method is used to generate/preserve certain special structures, and also to ensure stability. A fast ULV HSS factorization scheme is provided to take advantage of the special structures. We also propose a precomputation procedure for the HSS construction so as to further improve the efficiency. The complexity of these methods is significantly lower than some similar Toeplitz solvers for large matrix size n. Detailed flop counts are given, with the aid of a rank relaxation technique. The total cost of our methods includes O(n) flops for HSS operations and O(n log ² n) flops for matrix multiplications via FFTs, where n is the order of T. Various numerical tests on classical examples, including ill-conditioned ones, demonstrate the efficiency, and also indicate that the methods are stable in practice. This work shows a practical way of using randomized sampling in the development of fast rank structured methods.