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Reduction of a regular matrix pair (A, B) to block Hessenberg-triangular form
[chapter]

1996
*
Lecture Notes in Computer Science
*

An algorithm for

doi:10.1007/3-540-60902-4_15
fatcat:owos5z2w6radrlgrlyp75lw52q
*reduction**of**a**regular**matrix**pair*(*A*;*B*)*to**block**Hessenberg*-*triangular**form*is presented. ... This condensed*form*Q T (*A*;*B*)Z = (H; T), where H and T are*block*upper*Hessenberg*and upper*triangular*, respectively, and Q and Z orthogonal, may serve as*a*rst step in the solution*of*the generalized ... Elementwise*Reduction**to**Hessenberg*-*Triangular**Form*The elementwise algorithm reduces*a**regular**matrix**pair*(*A*;*B*)*to*upper*Hessenberg*-*triangular**form*5]. ...##
###
Blocked algorithms for the reduction to Hessenberg-triangular form revisited

2008
*
BIT Numerical Mathematics
*

We present two variants

doi:10.1007/s10543-008-0180-1
fatcat:yz2c6qv3mrga3onniq5zpgmtwy
*of*Moler and Stewart's algorithm for reducing*a**matrix**pair**to**Hessenberg*-*triangular*(HT)*form*with increased data locality in the access*to*the matrices. ... algorithm in subroutine DGGHRD from LAPACK, Dackland and Kågström's two-stage algorithm for the HT*form*, and*a*modified version*of*the latter which requires considerably less flops. (2000) : 65F15, 65Y20 ... This so-called*Hessenberg*-*triangular*(HT)*form**of*the*matrix**pair*(*A*,*B*) yields*a*significant*reduction*in the computational cost during the iterative part*of*the QZ algorithm. ...##
###
Parallel Two-Stage Reduction of a Regular Matrix Pair to Hessenberg-Triangular Form
[chapter]

2001
*
Lecture Notes in Computer Science
*

*A*parallel two-stage ScaLAPACK-style algorithm for

*reduction*

*of*

*a*

*regular*

*matrix*

*pair*(

*A*;

*B*)

*to*

*Hessenberg*-

*triangular*

*form*is presented. ... In this contribution we present

*a*parallel two-stage algorithm for

*reduction*

*of*

*a*

*regular*

*matrix*

*pair*(

*A*;

*B*)

*to*

*Hessenberg*-

*triangular*

*form*(H; T). ... for

*reduction*

*of*

*a*

*regular*

*matrix*

*pair*(

*A*;

*B*)

*to*

*Hessenberg*-

*triangular*

*form*(H; T). ...

##
###
A QZ-method based on semiseparable matrices

2008
*
Journal of Computational and Applied Mathematics
*

An effective

doi:10.1016/j.cam.2007.07.032
fatcat:ewqojoz2nzhxlfrgfk7arq7zda
*reduction**of**a**matrix**pair**to*lower semiseparable, upper*triangular**form*will be presented as well as*a*QZ-iteration for this*matrix**pair*. ... It will also be shown, that the QZ-iteration for*a*semiseparable-*triangular**matrix**pair*is closely related*to*the QZ-iteration for*a**Hessenberg*-*triangular**matrix**pair*. ... The*reduction**to*lower semiseparable-*triangular**form*The*reduction**of**a**pair**of*matrices*A*and*B*,*to**Hessenberg*-*triangular**form*, via unitary transformations Q and Z is well known, and can be found for ...##
###
A rational QZ method
[article]

2018
*
arXiv
*
pre-print

This generalization

arXiv:1802.04094v2
fatcat:4xk5f4ntjfhyhpfakvs7hodbx4
*of*the classical QZ method operates implicitly on*a**Hessenberg*,*Hessenberg*pencil instead*of*on*a**Hessenberg*,*triangular*pencil. ... In this article we study*Hessenberg*,*Hessenberg*pencils, link them*to*rational Krylov subspaces, propose*a*direct*reduction*method*to*such*a*pencil, and introduce the implicit rational QZ step. ... The authors thank Jared Aurentz and Thomas Mach for the fruitful discussions and suggestions related*to*this project, and the referees for their valuable feedback. ...##
###
Efficient algorithm for simultaneous reduction to the $$m$$ m -Hessenberg-triangular-triangular form

2014
*
BIT Numerical Mathematics
*

The algorithm is

doi:10.1007/s10543-014-0516-y
fatcat:rbkkewv6irh7tgju4yjstt77wy
*a**blocked*version*of*the algorithm described by Miminis and Page (1982) which reduces*A**to*the m-*Hessenberg**form*, and*B*and E*to*the*triangular**form*. ... The proposed*blocked*algorithm for the m-Hessenbergtriangular-*triangular**reduction*is based on the aggregated Givens rotations, which are*a*generalization*of*the*blocked*algorithm for the*Hessenberg*-*triangular*... ALGORITHM 1: The*blocked*algorithm for*reduction**of*matrices*A*,*B*, and E*to*the m-*Hessenberg*-*triangular*-*triangular**form*. ...##
###
Page 8850 of Mathematical Reviews Vol. , Issue 2000m
[page]

2000
*
Mathematical Reviews
*

Summary: “

*A*two-stage*blocked*algorithm for*reduction**of**a**regular**matrix**pair*(*A*,*B*)*to*upper*Hessenberg*-*triangular**form*is presented. ... for*reduction**of**a**regular**matrix**pair**to*generalized Schur*form*. ...##
###
Multishift Variants of the QZ Algorithm with Aggressive Early Deflation

2007
*
SIAM Journal on Matrix Analysis and Applications
*

The QZ algorithm is

doi:10.1137/05064521x
fatcat:rqquly7osnd67pg6bsh5xpbvuq
*a*numerically backward stable method for computing generalized eigenvalues and deflating subspaces*of*small-*to*mediumsized*regular**matrix**pairs*(*A*,*B*) with*A*,*B*∈ R n×n . ... The purpose*of*the QZ algorithm is*to*compute*a*generalized Schur decomposition*of*(*A*,*B*), i.e., orthogonal matrices Q and Z so that S = Q T AZ is quasi-upper*triangular*with 1 × 1 and 2 × 2*blocks*on ... Tightly coupled bulge chasing has also successfully been used in the*reduction**of**a**matrix**pair*(H r , T ) in*block**Hessenberg*-*triangular**form*, where H r has r subdiagonals,*to**Hessenberg*-*triangular**form*...##
###
A multishift, multipole rational QZ method with aggressive early deflation
[article]

2020
*
arXiv
*
pre-print

The result is

arXiv:1902.10954v2
fatcat:7brqqaqn7vbxlg6wv3fip2svra
*a*multishift, multipole iteration on*block**Hessenberg*pencils which allows one*to*stick*to*real arithmetic for*a*real input pencil. ... In this paper we extend the rational QZ method by introducing shifts and poles*of*higher multiplicity in the*Hessenberg*pencil, which is*a*pencil consisting*of*two*Hessenberg*matrices. ... The authors are grateful*to*Paul Van Dooren and Nicola Mastronardi for their help with the iterative refinement procedure for 2×2 with 2×2 swaps [9] which was essential for handling 2×2*blocks*accurately ...##
###
Anti-triangular and anti-m-Hessenberg forms for Hermitian matrices and pencils

2000
*
Linear Algebra and its Applications
*

We derive conditions from which anti-

doi:10.1016/s0024-3795(00)00156-7
fatcat:cmmaijcshzb3beruzokvbouhy4
*triangular*and anti-m-*Hessenberg**forms*for general (including singular) Hermitian pencils can be obtained under unitary equivalence transformations. ... .,*pairs**of*Hermitian matrices, arise in many applications, such as linear quadratic optimal control or quadratic eigenvalue problems. ... Rodman for his support during my research visit at the College*of*William and Mary. ...##
###
A subspace forward iteration method for solving the quadratic eigenproblem associated with the FDE formulation

1990
*
International Journal for Numerical Methods in Engineering
*

*A*Generalized Francis-QR based method is also developed

*to*compute the generalized Schur decomposition

*of*the

*matrix*

*pair*(

*A*,

*B*). ... The main purpose

*of*this work is

*to*give

*a*generalization

*of*the Subspace Iteration Method

*to*compute the largest eigenvalues and their corresponding eigenvectors

*of*the

*matrix*pencil

*A*− λ

*B*. ... Acknowledgements The authors are grateful

*to*the anonymous referee for his/her comments which substantially improved the quality

*of*this paper. ...

##
###
Reduction of Matrix Polynomials to Simpler Forms

2018
*
SIAM Journal on Matrix Analysis and Applications
*

*A*square

*matrix*can be reduced

*to*simpler

*form*via similarity transformations. Here "simpler

*form*" may refer

*to*diagonal (when possible),

*triangular*(Schur), or

*Hessenberg*

*form*. ... In this work we introduce

*a*practical means

*to*reduce

*a*

*matrix*polynomial with nonsingular leading coefficient

*to*

*a*simpler (diagonal,

*triangular*,

*Hessenberg*)

*form*while preserving the degree and the eigenstructure ...

*Reduction*

*to*other simple

*forms*like

*block*-diagonal,

*block*-

*triangular*or

*Hessenberg*

*forms*was also considered. ...

##
###
Numerical Methods for Linear Control Systems
[chapter]

1997
*
Systems and Control in the Twenty-First Century
*

For example, an algorithm for solving Ax =

doi:10.1007/978-1-4612-4120-1_4
fatcat:gr5dlxsg5zc55awzxco2klhh7q
*b*is backward stable, if the computed solutionx satisfies (*A*+ E)x =*b*+ δ*b*, where the*matrix*E and the vector δ*b*are small in some measure. ... For example, the condition number*of*the linear systems problem Ax =*b*is Cond(*A*) =*A**A*−1 . ... It is stable. 3.5*Reduction**to*Controller-*Hessenberg**Form*Given the*pair**of*matrices (*A*,*B*), there always exists an orthogonal*matrix*P such that P*B*= R =*B*0 , P AP T = H, where H is in*block**Hessenberg*...##
###
Pole-zero representation of descriptor systems

1995
*
Automatica
*

Note that the first step

doi:10.1016/0005-1098(94)00176-j
fatcat:at2bdznnjrf6lgr625ceaskdly
*of*the QZ algorithm is the*reduction**of*the*pair*(E.*A*)*to**a**triangular*-*Hessenberg**form*. ...*A*.*b*, c. d. n), The total computational cost can be broken down as follows: (*a*) (*b*) (c) Obfaining observer*Hessenberg**form*. The initial*reduction**of*E*to*an upper-*triangular**matrix*requires z 3 ? ...*Hessenberg*-*triangular**reduction*and transfer function matrices*of*singular systems. /EEE Trans. Circuits sysl.. CAS-36, 907-912. Misra. P. and R. V. Pate1 (1987) ...##
###
A Householder-based algorithm for Hessenberg-triangular reduction
[article]

2018
*
arXiv
*
pre-print

The QZ algorithm for computing eigenvalues and eigenvectors

arXiv:1710.08538v2
fatcat:qjekci3ebjgjtkus2nne2h7fqe
*of**a**matrix*pencil*A*- λ*B*requires that the matrices first be reduced*to**Hessenberg*-*triangular*(HT)*form*. ... The current method*of*choice for HT*reduction*relies entirely on Givens rotations regrouped and accumulated into small dense matrices which are subsequently applied using*matrix*multiplication routines ... Part*of*this work was done while the first author was*a*postdoctoral researcher atÉcole polytechnique fédérale de Lausanne, Switzerland. ...
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