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

Krister Dackland, Bo Kågström
1996 Lecture Notes in Computer Science  
An algorithm for 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].  ... 
doi:10.1007/3-540-60902-4_15 fatcat:owos5z2w6radrlgrlyp75lw52q

Blocked algorithms for the reduction to Hessenberg-triangular form revisited

B. Kågström, D. Kressner, E.S. Quintana-Ortí, G. Quintana-Ortí
2008 BIT Numerical Mathematics  
We present two variants 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.  ... 
doi:10.1007/s10543-008-0180-1 fatcat:yz2c6qv3mrga3onniq5zpgmtwy

Parallel Two-Stage Reduction of a Regular Matrix Pair to Hessenberg-Triangular Form [chapter]

Björn Adlerborn, Krister Dackland, Bo Kågström
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).  ... 
doi:10.1007/3-540-70734-4_13 fatcat:2qsj3orx2ncvnkxq5je7c64224

A QZ-method based on semiseparable matrices

Yvette Vanberghen, Raf Vandebril, Marc Van Barel
2008 Journal of Computational and Applied Mathematics  
An effective 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  ... 
doi:10.1016/ fatcat:ewqojoz2nzhxlfrgfk7arq7zda

A rational QZ method [article]

Daan Camps, Karl Meerbergen, Raf Vandebril
2018 arXiv   pre-print
This generalization 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.  ... 
arXiv:1802.04094v2 fatcat:4xk5f4ntjfhyhpfakvs7hodbx4

Efficient algorithm for simultaneous reduction to the $$m$$ m -Hessenberg-triangular-triangular form

Nela Bosner
2014 BIT Numerical Mathematics  
The algorithm is 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.  ... 
doi:10.1007/s10543-014-0516-y fatcat:rbkkewv6irh7tgju4yjstt77wy

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

Bo Kågström, Daniel Kressner
2007 SIAM Journal on Matrix Analysis and Applications  
The QZ algorithm is 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  ... 
doi:10.1137/05064521x fatcat:rqquly7osnd67pg6bsh5xpbvuq

A multishift, multipole rational QZ method with aggressive early deflation [article]

Thijs Steel, Daan Camps, Karl Meerbergen, Raf Vandebril
2020 arXiv   pre-print
The result is 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  ... 
arXiv:1902.10954v2 fatcat:7brqqaqn7vbxlg6wv3fip2svra

Anti-triangular and anti-m-Hessenberg forms for Hermitian matrices and pencils

Christian Mehl
2000 Linear Algebra and its Applications  
We derive conditions from which anti-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.  ... 
doi:10.1016/s0024-3795(00)00156-7 fatcat:cmmaijcshzb3beruzokvbouhy4

A subspace forward iteration method for solving the quadratic eigenproblem associated with the FDE formulation

Thomas E. C. Gmuer
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 − λB.  ...  Acknowledgements The authors are grateful to the anonymous referee for his/her comments which substantially improved the quality of this paper.  ... 
doi:10.1002/nme.1620290503 fatcat:emvmaz2p5nfj7dedax5lbtdf5e

Reduction of Matrix Polynomials to Simpler Forms

Yuji Nakatsukasa, Leo Taslaman, Françoise Tisseur, Ion Zaballa
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.  ... 
doi:10.1137/17m1125182 fatcat:q73djzalend4tf3ba74u3q2yqy

Numerical Methods for Linear Control Systems [chapter]

D. Boley, B. N. Datta
1997 Systems and Control in the Twenty-First Century  
For example, an algorithm for solving Ax = b is backward stable, if the computed solutionx satisfies (A + E)x = b + δb, where the matrix E and the vector δb 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  ... 
doi:10.1007/978-1-4612-4120-1_4 fatcat:gr5dlxsg5zc55awzxco2klhh7q

Pole-zero representation of descriptor systems

Pradeep Misra, Paul Van Dooren, Vassilis Syrmos
1995 Automatica  
Note that the first step 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)  ... 
doi:10.1016/0005-1098(94)00176-j fatcat:at2bdznnjrf6lgr625ceaskdly

A Householder-based algorithm for Hessenberg-triangular reduction [article]

Zvonimir Bujanović, Lars Karlsson, Daniel Kressner
2018 arXiv   pre-print
The QZ algorithm for computing eigenvalues and eigenvectors 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.  ... 
arXiv:1710.08538v2 fatcat:qjekci3ebjgjtkus2nne2h7fqe
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