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I/O efficient QR and QZ algorithms

Sraban Kumar Mohanty, Sajith Gopalan
2012 2012 19th International Conference on High Performance Computing  
We propose a tile based small-bulge multishift QR algorithm which is scalable and more amenable for multicore architecture than the traditional panel based algorithms and that under certain conditions  ...  We analyse the multishift QR algorithm [1]; this algorithm chases × bulges, where is the number of shifts, using both matrix-vector and matrix-matrix operations.  ...  A QR step applied on a Hessenberg matrix produces a Hessenberg matrix. ( 3 / √ ) I/Os The explicit (single shift) QR algorithm introduces shifts in the basic QR algorithm to accelerate its convergence:  ... 
doi:10.1109/hipc.2012.6507492 dblp:conf/hipc/MohantyG12 fatcat:n3eiibo4uneyza7tbgnvcctfea

On Aggressive Early Deflation in Parallel Variants of the QR Algorithm [chapter]

Bo Kågström, Daniel Kressner, Meiyue Shao
2012 Lecture Notes in Computer Science  
Multishift and aggressive early deflation (AED) techniques have led to significantly more efficient sequential implementations of the QR algorithm during the last decade.  ...  In this paper, we discuss a two-level approach for performing AED in a parallel environment, where the lower level consists of a novel combination of AED with the pipelined QR algorithm implemented in  ...  implementations of the parallel QR algorithm.  ... 
doi:10.1007/978-3-642-28151-8_1 fatcat:rcgk3q263rfbpjpeptpv5ie434

A multishift QR iteration without computation of the shifts

Augustin A. Dubrulle, Gene H. Golub
1994 Numerical Algorithms  
matrix computations, eigenvalues, QR algorithm Each iteration of the multishift QR algorithm of Bai and Demmel requires the computation of a "shift vector" defined by m shifts of the origin of the spectrum  ...  ABSTRACT Each iteration of the multishift QR algorithm of Bai and Demmel requires the computation of a "shift vector" defined by m shifts of the origin of the spectrum, which control the convergence of  ...  In such cases, the most efficient implementation of the QR iteration is likely to consist of a sequence of m]k multishift iterations, each using k of the shifts, k ::; 16 .  ... 
doi:10.1007/bf02140681 fatcat:ubbje2phyjcg5nn7ko234pu53y

The Multishift QR Algorithm. Part I: Maintaining Well-Focused Shifts and Level 3 Performance

Karen Braman, Ralph Byers, Roy Mathias
2002 SIAM Journal on Matrix Analysis and Applications  
It replaces the large diagonal bulge in the multishift QR sweep with a chain of many small bulges.  ...  The small-bulge multishift QR sweep admits nearly any number of simultaneous shifts-even hundreds-without adverse effects on the convergence rate.  ...  Familiarity with the multishift QR algorithm [4] as implemented in LAPACK version 2 [1] is helpful. We will refer to the iterative step from bulge-introduction to bulge-disappearance as a QR sweep  ... 
doi:10.1137/s0895479801384573 fatcat:nzqfkdo2gfe7rmxlenvkxgia6m

A multishift Hessenberg method for pole assignment of single-input systems

A. Varga
1996 IEEE Transactions on Automatic Control  
The proposed algorithm belongs to the family of Hessenberg methods and is based on an implicit multishift QR-like technique.  ...  A new algorithm is proposed for the pole assignment of single-input linear time-invariant systems.  ...  A particular one-step multishift variant of the proposed method is very well suited for computer implementation.  ... 
doi:10.1109/9.545716 fatcat:3w6zcnllbjb37jdavkzthvgnsy

A multiple shift QR-step for structured rank matrices

Raf Vandebril, Marc Van Barel, Nicola Mastronardi
2010 Journal of Computational and Applied Mathematics  
This paper will explain a general multishift implementation for Hessenberg-like matrices (semiseparable matrices are a special case and hence also admit this approach).  ...  A first attempt for computing the eigenvalues of nonsymmetric matrices via intermediate Hessenberg-like matrices (i.e. a matrix having all subblocks in the lower triangular part of rank at most one) was  ...  Conclusions In this manuscript a multishift implementation of the QR-algorithm for Hessenberg-like matrices was developed.  ... 
doi:10.1016/ fatcat:sc2ivgomo5fuhco5v3cuzb4t3y

Multishift Variants of the QZ Algorithm with Aggressive Early Deflation

Bo Kågström, Daniel Kressner
2007 SIAM Journal on Matrix Analysis and Applications  
An extension of the small-bulge multishift QR algorithm is developed, which chases chains of many small bulges instead of only one bulge in each QZ iteration.  ...  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  ...  Inspired by the works of Braman, Byers, and Mathias [7] and Lang [36] for the QR algorithm, we propose multishift QZ iterations that chase a tightly coupled chain of bulge pairs instead of only one  ... 
doi:10.1137/05064521x fatcat:rqquly7osnd67pg6bsh5xpbvuq

The Multishift QR Algorithm. Part II: Aggressive Early Deflation

Karen Braman, Ralph Byers, Roy Mathias
2002 SIAM Journal on Matrix Analysis and Applications  
Aggressive early deflation is a QR algorithm deflation strategy that takes advantage of matrix perturbations outside of the subdiagonal entries of the Hessenberg QR iterate.  ...  The small-bulge multishift QR sweep with aggressive early deflation maintains a high rate of execution of floating point operations while significantly reducing the number of operations required.  ...  The implementation of the large-bulge multishift QR algorithm in LAPACK selects its shifts to be the eigenvalues of a trailing principal submatrix.  ... 
doi:10.1137/s0895479801384585 fatcat:jpmvfrssdnb4xoirldnrhl5p4y

QR-like algorithms for eigenvalue problems

David S. Watkins
2000 Journal of Computational and Applied Mathematics  
In the year 2000 the dominant method for solving matrix eigenvalue problems is still the QR algorithm. This paper discusses the family of GR algorithms, with emphasis on the QR algorithm.  ...  Included are historical remarks, an outline of what GR algorithms are and why they work, and descriptions of the latest, highly parallelizable, versions of the QR algorithm.  ...  It also has a double-shift implicit QR code, which is used on matrices (or portions of matrices) under 50 × 50. For larger matrices a multishift QR code is used.  ... 
doi:10.1016/s0377-0427(00)00402-7 fatcat:neysnxgrnjdutouvrscwzkmc4a

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

An Implicit Multishift $QR$-Algorithm for Hermitian Plus Low Rank Matrices

Raf Vandebril, Gianna M. Del Corso
2010 SIAM Journal on Scientific Computing  
a QR-step on a Hessenberg matrix.  ...  a QR-step on a Hessenberg matrix.  ...  This representation allows performing implicit multishift QR-steps involving only O(n) operations for each iteration.  ... 
doi:10.1137/090754522 fatcat:fgoe3rlmwresjpqy3toy7dxi4m

Optimally packed chains of bulges in multishift QR algorithms

Lars Karlsson, Daniel Kressner, Bruno Lang
2014 ACM Transactions on Mathematical Software  
After an initial reduction to Hessenberg form, a QR iteration can be viewed as chasing a small bulge from the top left to the bottom right corner along the subdiagonal of A.  ...  This directly translates into a reduced chain length by 33% compared to the state-of-the-art LAPACK implementation of the QR algorithm.  ...  Acknowledgments We thank Bo Kågström and Meiyue Shao for inspiring discussions on the subject of this article.  ... 
doi:10.1145/2559986 fatcat:u25ozktwozhdpe3r5o644bm2pq

Performance Modeling and Optimal Block Size Selection for the Small-Bulge Multishift QR Algorithm [chapter]

Yusaku Yamamoto
2006 Lecture Notes in Computer Science  
The small-bulge multishift QR algorithm proposed by Braman, Byers and Mathias is one of the most e cient algorithms for computing the eigenvalues of nonsymmetric matrices on processors with hierarchical  ...  Thus our model can beused to automatically select the optimal value of m for a given matrix size on a given architecture.  ...  This work is partially supported by the Ministry of Education, Science, Sports and 13 Culture, Grant in Aid for Scienti c Research on Priority Areas, "i-explosion" (No. 18049014), Grant-in-aid for Scienti  ... 
doi:10.1007/11946441_44 fatcat:zqiajcj2bnavllouyydttfc7bq

On Deflations in Extended QR Algorithms

Thomas Mach, Raf Vandebril
2014 SIAM Journal on Matrix Analysis and Applications  
Further, we present a generalization of aggressive early deflation to the extended QR algorithms.  ...  Further, we present a generalization of aggressive early deflation to the extended QR algorithms.  ...  Further implementations might improve this by packing the bulges tighter and using blocked updates, as is done in recent implementations of Francis's QR algorithm.  ... 
doi:10.1137/130935665 fatcat:6mus4lvkv5fq5gcbf46w4rfjxu

A Novel Parallel QR Algorithm for Hybrid Distributed Memory HPC Systems

Robert Granat, Bo Kågström, Daniel Kressner
2010 SIAM Journal on Scientific Computing  
A novel variant of the parallel QR algorithm for solving dense nonsymmetric eigenvalue problems on hybrid distributed high performance computing (HPC) systems is presented.  ...  Numerous numerical experiments confirm the superior performance of our parallel QR algorithm in comparison with the existing ScaLAPACK code, leading to an implementation that is one to two orders of magnitude  ...  The authors are grateful to Björn Adlerborn and Lars Karlsson for helpful discussions on the subject and for constructive comments on earlier versions of this paper, and toÅke Sandgren for valuable support  ... 
doi:10.1137/090756934 fatcat:pscnkbwqxnfkbmo6hoyrxnasga
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