Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations

Authors:
Dario A. Bini;Luca Gemignani;Victor Y. Pan
Affiliations:
Dipartimento di Matematica, Universit‘a di Pisa, Via F. Buonarroti 2, 56127, Pisa, Italy;Dipartimento di Matematica, Universit‘a di Pisa, Via F. Buonarroti 2, 56127, Pisa, Italy;Department of Mathematics and Computer Science, Lehman College of CUNY, Via F. Buonarroti 2, 10468, Bronx, NY, USA
Venue:
Numerische Mathematik
Year:
2005

Citing 0
Cited 23

Can the TPRI structure help us to solve the algebraic eigenproblem?

SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
A unitary Hessenberg QR-based algorithm via semiseparable matrices

Journal of Computational and Applied Mathematics
Computing the zeros of a Fourier series or a Chebyshev series or general orthogonal polynomial series with parity symmetries

Computers & Mathematics with Applications
Geometric applications of the Bezout matrix in the Lagrange basis

Proceedings of the 2007 international workshop on Symbolic-numeric computation
Null space and eigenspace computations with additive preprocessing

Proceedings of the 2007 international workshop on Symbolic-numeric computation
Real root-finding

Proceedings of the 2007 international workshop on Symbolic-numeric computation
Companion matrix pencils for hermite interpolants

Proceedings of the 2007 international workshop on Symbolic-numeric computation
Structured matrix methods for polynomial root-finding

Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Additive preconditioning and aggregation in matrix computations

Computers & Mathematics with Applications
Eigen-solving via reduction to DPR1 matrices

Computers & Mathematics with Applications
Computations with quasiseparable polynomials and matrices

Theoretical Computer Science
Preconditioning, randomization, solving linear systems, eigen-solving, and root-finding

Proceedings of the 2009 conference on Symbolic numeric computation
A cutting plane method for solving convex optimization problems over the cone of nonnegative polynomials

WSEAS Transactions on Mathematics
A multiple shift QR-step for structured rank matrices

Journal of Computational and Applied Mathematics
A unitary Hessenberg QR-based algorithm via semiseparable matrices

Journal of Computational and Applied Mathematics
Real and complex polynomial root-finding with eigen-solving and preprocessing

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Algebraic and numerical algorithms

Algorithms and theory of computation handbook
New progress in real and complex polynomial root-finding

Computers & Mathematics with Applications
An Implicit Multishift $QR$-Algorithm for Hermitian Plus Low Rank Matrices

SIAM Journal on Scientific Computing
Polynomial algebra for Birkhoff interpolants

Numerical Algorithms
Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices

SIAM Journal on Matrix Analysis and Applications
Efficient polynomial root-refiners: A survey and new record efficiency estimates

Computers & Mathematics with Applications
Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

Computer Aided Geometric Design

Quantified Score

Hi-index	0.00

Visualization

Abstract

We introduce a class ** of n×n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of **, we show that if A ∈ ** then A′=RQ ∈ **, where A=QR is the QR decomposition of A. This allows one to implement the QR iteration for computing the eigenvalues and the eigenvectors of any A ∈ ** with O(n) arithmetic operations per iteration and with O(n) memory storage. This iteration, applied to generalized companion matrices, provides new O(n2) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations. Numerical experiments confirm the effectiveness and the robustness of our approach.