Symmetric Gaussian elimination for cauchy-type matrices with application to positive definite Toeplitz matrices

Abstract

The problem of solving linear equations with a Toeplitz matrix T appears in many applications. Often T is positive definite but ill-conditioned with many small eigenvalues. In this case fast and superfast algorithms may show a very poor behavior or even break down. In recent papers the transformation of a Toeplitz matrix into a Cauchy-type matrix is proposed. The resulting new linear equations can be solved in O(n 2 ) operations using standard pivoting strategies which leads to very stable fast methods also for ill-conditioned systems. The basic tool is the formulation of Gaussian elimination for matrices with low displacement rank. In this paper, we will transform a Hermitian Toeplitz matrix T into a Cauchy-type matrix B = FTF H by applying the Fourier transform. We will prove some useful properties of B and formulate a symmetric Gaussian elimination algorithm for positive definite B . Using the symmetry and persymmetry of T we can reduce the total costs of this algorithm compared with unsymmetric Gaussian elimination. For complex Hermitian T , the complexity of the new algorithm is then nearly the same as for the Schur algorithm. Furthermore, it is possible to include some strategies for ill-conditioned positive definite matrices that are well-known in optimization. Numerical examples show that this new algorithm is fast and reliable.