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Register a new userComparison Theorems for Splittings of M-matrices in (block) Hessenberg Form
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Added by
Federico G. Poloni
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Some variants of the (block) Gauss-Seidel iteration for the solution of linear systems with $M$-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix $rho(P_{GS})geq rho(P_S)geq rho(P_{AGS})$, where $P_{GS}, P_S, P_{AGS}$ are the iteration matrices of the Gauss-Seidel, staircase, and anti-Gauss-Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation.
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We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix $A=G+U V^H$, where $Gin mathbb C^{ntimes n}$ is a unitary matrix represented in some compressed format using $O(nk)$ parameters and $U$ and $V$ are $ntimes k$ matrices with $k< n$. At the core of these methods is a certain structured decomposition, referred to as a LFR decomposition, of $A$ as product of three possibly perturbed unitary $k$ Hessenberg matrices of size $n$. It is shown that in most interesting cases an initial LFR decomposition of $A$ can be computed very cheaply. Then we prove structural properties of LFR decompositions by giving conditions under which the LFR decomposition of $A$ implies its Hessenberg shape. Finally, we describe a bulge chasing scheme for converting the initial LFR decomposition of $A$ into the LFR decomposition of a Hessenberg matrix by means of unitary transformations. The reduction can be performed at the overall computational cost of $O(n^2 k)$ arithmetic operations using $O(nk)$ storage. The computed LFR decomposition of the Hessenberg reduction of $A$ can be processed by the fast QR algorithm presented in [8] in order to compute the eigenvalues of $A$ within the same costs.
We develop two fast algorithms for Hessenberg reduction of a structured matrix $A = D + UV^H$ where $D$ is a real or unitary $n times n$ diagonal matrix and $U, V inmathbb{C}^{n times k}$. The proposed algorithm for the real case exploits a two--stage approach by first reducing the matrix to a generalized Hessenberg form and then completing the reduction by annihilation of the unwanted sub-diagonals. It is shown that the novel method requires $O(n^2k)$ arithmetic operations and it is significantly faster than other reduction algorithms for rank structured matrices. The method is then extended to the unitary plus low rank case by using a block analogue of the CMV form of unitary matrices. It is shown that a block Lanczos-type procedure for the block tridiagonalization of $Re(D)$ induces a structured reduction on $A$ in a block staircase CMV--type shape. Then, we present a numerically stable method for performing this reduction using unitary transformations and we show how to generalize the sub-diagonal elimination to this shape, while still being able to provide a condensed representation for the reduced matrix. In this way the complexity still remains linear in $k$ and, moreover, the resulting algorithm can be adapted to deal efficiently with block companion matrices.
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