Orthogonal polyanalytic polynomials and normal matrices
Mathematics of Computation
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We present an algorithm for iteratively solving a linear system Nx=b, with a normal $N\in \mathbb{C}^{n \times n}$ and $b \in \mathbb{C}^n$, with an optimal 3-term recurrence by extending the Hermitian Lanczos method to normal matrices. This is achieved by considering the Toeplitz decomposition N=H+iK of N with Hermitian H and K. Generically, the inverse of a normal matrix is a polynomial in its Hermitian part. Using this and the fact that N and H commute, we obtain a minimization problem $$\min_{p_{j-1}\in {\cal P}_{j-1}}\left|\left| Np_{j-1}(H)b-b \right|\right|= \min_{p_{j-1}\in {\cal P}_{j-1}} \left|\left| p_{j-1}(H)Nb-b\right|\right|,$$ where ${\cal P}_{j-1}$ denotes the set of polynomials of degree j-1 at most. Thus, at the jth step, the best approximation to b needs to be found from the Krylov subspace ${\cal K}_j(H;Nb)$. Since this involves the Hermitian matrix H, this is realizable with a 3-term recurrence.