The QR-algorithm is a renowned method for computing all eigenvalues of an arbitrary matrix. A preliminary unitary similarity transformation to Hessenberg form is indispensable for keeping the computational complexity of the subsequent QR-steps under control. In this paper, a whole new family of matrices, sharing the major qualities of Hessenberg matrices, will be put forward. This gives rise to...

A new framework for transforming arbitrary matrices to compressed representations is presented. The framework provides a generic way of transforming a matrix via unitary similarity transformations to e.g. Hessenberg, Hessenberg-like and combinations of both. The new algorithms are deduced, based on the QR-factorization of the original matrix. Based on manipulations with Givens transformations, ...

The purpose of this paper is to develop a spectral analysis of the Hessenberg matrix obtained by the GMRES algorithm used for solving a linear system with a singular matrix. We prove that the singularity of the Hessenberg matrix depends on the nature of A and some others criteria like the zero eigenvalue multiplicity and the projection of the initial residual on particular subspaces. We also in...

The Rational Krylov Sequence (RKS) method can be seen as a generalisation of Arnoldi's method. It projects a matrix pencil onto a smaller subspace; this projection results in a small upper Hessenberg pencil. As for the Arnoldi method, RKS can be restarted implicitly, using the QR decomposition of a Hessenberg matrix. This restart comes with a projection of the subspace using a rational function...

0. Introduction. The QR algorithm was developed by Francis (1960) to find the eigenvalues (or roots) of real or complex matrices. We shall consider it here in the context of exact arithmetic. Sufficient conditions for convergence, listed in order of increasing generality have been given by Francis [1], Kublanovskaja [3], Parlett [4], and Wilkinson [8]. It seems that necessary and sufficient con...

0. Introduction. The QR algorithm was developed by Francis (1960) to find the eigenvalues (or roots) of real or complex matrices. We shall consider it here in the context of exact arithmetic. Sufficient conditions for convergence, listed in order of increasing generality have been given by Francis [1], Kublanovskaja [3], Parlett [4], and Wilkinson [8]. It seems that necessary and sufficient con...

Let P (m,n) denote the maximum permanent of an n-by-n lower Hessenberg (0, 1)-matrix with m entries equal to 1. A “staircased” structure for some matrices achieving this maximum is obtained, and recursive formulas for computing P (m,n) are given. This structure and results about permanents are used to determine the exact values of P (m,n) for n ≤ m ≤ 8n/3 and for all nnz(Hn) − nnz(Hbn/2c) ≤ m ≤...

This contribution considers the problem of transforming a regular matrix pair (A;B) to generalized Schur form. The focus is on blocked algorithms for the reduction process that typically includes two major steps. The rst is a two-stage reduction of a regular matrix pair (A;B) to condensed form (H;T ) using orthogonal transformations Q and Z such that H = QAZ is upper Hessenberg and T = QBZ is u...

In this paper, a new method based on the generalized Purcell method is proposed to solve the usual least-squares problem arising in the GMRES method. The theoretical aspects and computational results of the method are provided. For the popular iterative method GMRES, the decomposition matrices of the Hessenberg matrix is obtained by using a simple recursive relation instead of Givens rotations....

We show that every n × n complex nonderogatory matrix is similar to a unique unit upper Hessenberg Toeplitz matrix. The proof is constructive, and can be adapted to nonderogatory matrices with entries in any field of characteristic zero or characteristic greater than n. We also prove that every n× n complex matrix with n ≤ 4 is similar to a Toeplitz matrix.