Symmetric Gaussian elimination for Cauchy-type matrices with application to positive definite Toeplitz matrices (Q1392764)

New methods for solving systems of linear equations with Hermitian positive definite Toeplitz matrices are given. The given matrix is first transformed into a Cauchy-type matrix, then symmetric Gaussian elimination is applied. For moderately ill-conditioned matrices total pivoting is proposed. For very ill-conditioned matrices an alternative method is described. This involves the replacement of small diagonal entries by a small number. The total computational cost is \(O(n^2)\). It is shown by computational experiments that the proposed algorithms are stable and reliable. Relative advantages of the author's methods over other known methods are carefully documented.