A Hermitian Lanczos method for normal matrices (Q2784399)

An algorithm is presented for iteratively solving a linear system \(Nx= b\), with a normal matrix \(N\), with an optional 3-term recurrence by extending the Hermitian Lanczos method to normal matrices. To this end the Toeplitz decomposition of \(N\), defined via \(N= H+iK\) with Hermitian \(H\) and \(K\), is employed. The key is to notice that \(N^{-1}\) is generically a polynomial in \(H\). Consequently, the fact that \(N\) and \(H\) commute is employed to extend the Hermitian Lanczos method to normal matrices without losing the optimality or increasing the length of the recurrence. In addition to the basic algorithm, its restarted and rotated implementation, aimed at nongeneric or nearly nongeneric normal matrices, are considered too.