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 \(B=FTF^{\rm H}\) 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.