## MULTISHIFT VARIANTS OF THE QZ ALGORITHM WITH Aggressive Early Deflation

Citations: | 20 - 14 self |

### BibTeX

@MISC{Kågström_multishiftvariants,

author = {Bo Kågström and Daniel Kressner},

title = {MULTISHIFT VARIANTS OF THE QZ ALGORITHM WITH Aggressive Early Deflation},

year = {}

}

### OpenURL

### Abstract

New variants of the QZ algorithm for solving the generalized eigenvalue problem are proposed. 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. This allows the effective use of level 3 BLAS operations, which in turn can provide efficient utilization of high performance computing systems with deep memory hierarchies. Moreover, an extension of the aggressive early deflation strategy is proposed, which can identify and deflate converged eigenvalues long before classic deflation strategies would. Consequently, the number of overall QZ iterations needed until convergence is considerably reduced. As a third ingredient, we reconsider the deflation of infinite eigenvalues and present a new deflation algorithm, which is particularly effective in the presence of a large number of infinite eigenvalues. Combining all these developments, our implementation significantly improves existing implementations of the QZ algorithm. This is demonstrated by numerical experiments with random matrix pairs as well as with matrix pairs arising from various applications.

### Citations

847 |
Accuracy and Stability of Numerical Algorithms
- Higham
- 2002
(Show Context)
Citation Context ...yielding a computed solution ˆx. This implies that ˆx is the exact solution of a slightly perturbed system (2.5) (T + F)ˆx = e1, ‖F ‖2 ≤ cT ‖T ‖2, where cT is not much larger than the unit roundoff u =-=[22]-=-. Now, consider the Householder matrix H1(ˆx) = I − ˜ β˜v˜v T , where ˜ β ∈ R, ˜v ∈ R n , such that (I − ˜ β˜v˜v T )ˆx = ˜γe1 for some scalar ˜γ. The computation of the quantities ˜ β, ˜v defining H1(... |

742 | A set of level 3 basic linear algebra subprograms - DONGARRA, DUCROZ, et al. - 1990 |

612 | Matrix Perturbation Theory
- Stewart, Sun
- 1990
(Show Context)
Citation Context ...T, although some care must be taken to implement this computation in a safe manner, see [37, 45]. Moreover, the leading k columns of the orthogonal matrices Z and Q span a pair of deflating subspaces =-=[41]-=- if the (k+1, k) subdiagonal entry of the matrix S vanishes. A reordering of the diagonal blocks of S and T can be used to compute other deflating subspaces, see [23, 27, 44]. The eigenvalues of (A, B... |

572 |
The theory of matrices
- Gantmacher
- 1959
(Show Context)
Citation Context ...gorithm may utterly fail to correctly identify infinite eigenvalues, especially if the index of the matrix pair, defined as the size of the largest Jordan block associated with an infinite eigenvalue =-=[18]-=-, is larger than one [37]. In the context of differential-algebraic equations (DAE), the index of (A, B) corresponds to the index of the DAE B ˙x = Ax + f. Many applications, such as multibody systems... |

234 | Users’ guide for the harwell-boeing sparse matrix collection - Duff, Grimes, et al. - 1992 |

139 |
An algorithm for generalized matrix eigenvalue problems
- Moler, Stewart
- 1973
(Show Context)
Citation Context ...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 . It goes back to Moler and Stewart in 1973 =-=[37]-=- and has undergone only a few modifications during the next 25 years, notably through works by Ward [47, 48], Kaufman [29], Dackland and K˚agström [12]. Non-orthogonal variants of the QZ algorithm inc... |

136 | Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations - Brenan, Campbell, et al. - 1996 |

86 | GEMM-Based Level 3 BLAS: HighPerformance Model Implementations and Performance Evaluation Benchmark
- Kagstrom, Ling, et al.
- 1995
(Show Context)
Citation Context ...algorithm, we propose multishift QZ iterations that chase a tightly coupled chain of bulge pairs instead of only one bulge pair per iteration. This allows the effective use of level 3 BLAS operations =-=[15, 25, 26]-=- during the bulge chasing process, which in turn can provide efficient utilization of today’s high performance computing systems with deep memory hierarchies. Tightly coupled bulge chasing has also su... |

67 | Differential-Algebraic Equations and Their Numerical Treatment, Teubner - Griepentrog, März - 1986 |

61 |
A Test Matrix Collection for Non-Hermitian Eigenvalue Problems (Release 1.0
- Bai, Day, et al.
- 1997
(Show Context)
Citation Context ...e of the multishift QZ algorithm with aggressive early deflation for matrix pairs that arise from practically relevant applications. We have selected 16 matrix pairs from the Matrix Market collection =-=[3]-=-, 6 matrix pairs from model reduction benchmark collections [11, 30], and 4 matrix pairs arising from the computation of corner singularities of elliptic PDEs [38]. A more detailed description of the ... |

58 | A collection of benchmark examples for model reduction of linear time invariant dynamical systems. SLICOT Working
- Chahlaoui, Dooren
(Show Context)
Citation Context ...n for matrix pairs that arise from practically relevant applications. We have selected 16 matrix pairs from the Matrix Market collection [3], 6 matrix pairs from model reduction benchmark collections =-=[11, 30]-=-, and 4 matrix pairs arising from the computation of corner singularities of elliptic PDEs [38]. A more detailed description of the selected matrix pairs along with individual performance results can ... |

51 |
The generalized Schur decomposition of an arbitrary pencil A − λB: Robust software with error bounds and application
- Demmel, K˚agström
- 1993
(Show Context)
Citation Context ...d extra caution, and so called staircase-type algorithms can be used for identifying singular cases by computing a generalized upper triangular (GUPTRI) form of (A, B) (e.g., see Demmel and K˚agström =-=[13, 14]-=-). Three ingredients make the QZ algorithm work effectively. First, the matrix pair (A, B) is reduced to Hessenberg-triangular form, i.e., orthogonal matrices Q and Z are computed so that H = Q T AZ i... |

50 |
On a block implementation of the Hessenberg multishift QR iteration
- Bai, Demmel
- 1989
(Show Context)
Citation Context ...ix pair (H(j), T (j)), see [52], Theorem 2.4 can be extended to the case j > n − m − 1. Early attempts to improve the performance of the QR algorithm focused on using shift polynomials of high degree =-=[4]-=-, leading to medium-order Householder matrices during the QR iteration and enabling the efficient use of WY representations. This approach, however, has proved disappointing due to the fact that the c... |

39 |
de Geijn. High-performance implementation of the level-3 BLAS
- Goto, van
- 2008
(Show Context)
Citation Context ...me depends on the performance of TRMM relative to GEMM, which may vary depending on BLAS implementations used for the target architecture and actual matrix sizes (e.g., see [25, 26]). A recent report =-=[20]-=- has identified computing environments for which TRMM performs significantly worse than GEMM, especially for the matrix dimensions arising in our application. In such a setting, it is more favorable t... |

30 | Differential-algebraic equations: Analysis and numerical solution - Kunkel, Mehrmann - 2006 |

28 | A framework for symmetric band reduction
- Bischof, Lang, et al.
(Show Context)
Citation Context ... be deflated, this degrades the overall performance of the multishift QZ algorithm. A higher computation/communication ratio can be attained by using windowing techniques similar to those proposed in =-=[5, 12, 32]-=-. In the following, we illustrate such an algorithm, conceptually close to a recently presented block algorithm for reordering standard and generalized Schur forms [32]. Consider a matrix pair (H, T) ... |

26 | Computing eigenspaces with specified eigenvalues of a regular matrix pair (A, B) and condition estimation: theory, algorithms and software. Numer. Algorithms
- K˚agström, Poromaa
- 1996
(Show Context)
Citation Context ...pan a pair of deflating subspaces [41] if the (k+1, k) subdiagonal entry of the matrix S vanishes. A reordering of the diagonal blocks of S and T can be used to compute other deflating subspaces, see =-=[23, 27, 44]-=-. The eigenvalues of (A, B) are read off from (S, T) as follows. The 2 × 2 diagonal blocks correspond to pairs of complex conjugate eigenvalues. The real eigenvalues are given in pairs (sii, tii) corr... |

26 |
Numerical Methods and Software for General and Structured Eigenvalue Problems
- Kressner
- 2004
(Show Context)
Citation Context ...that the matrix pair under consideration, which will be denoted by (H, T), is already in Hessenberg-triangular form. Efficient algorithms for reducing a given matrix pair to this form can be found in =-=[12, 31]-=-. For the moment, we also assume that (H, T) is an unreduced matrix pair, i.e., all subdiagonal entries of H as well as all diagonal entries of T are different from zero. The latter condition implies ... |

21 |
The multishift QR algorithm. ii. aggressive early deflation
- Braman, Byers, et al.
(Show Context)
Citation Context ...y been used in the reduction of a matrix pair (Hr, T) in block Hessenberg-triangular form, where Hr has r subdiagonals, to Hessenberg-triangular form (H, T) [12]. Recently, Braman, Byers, and Mathias =-=[7]-=- also presented a new, advanced deflation strategy, the so called aggressive early deflation. Combining this deflation strategy with multishift QR iterations leads to a variant of the QR algorithm, wh... |

21 |
Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint
- Rabier, Rheinboldt
- 2000
(Show Context)
Citation Context ...ns (DAE), the index of (A, B) corresponds to the index of the DAE B ˙x = Ax + f. Many applications, such as multibody systems and electrical circuits, lead to DAEs with index at least two, see, e.g., =-=[8, 39, 43]-=-. If the matrix pair (A, B) has an infinite eigenvalue then the matrix B is singular. This implies that at least one of the diagonal entries in the upper triangular matrix T in the Hessenberg-triangul... |

21 |
Shifting strategies for the parallel QR algorithm
- Watkins
- 1994
(Show Context)
Citation Context ...flop reduction offered by the block triangular structure into an actual decrease of execution time. As for the tiny-bulge multishift QR algorithm, we have to be aware of so called vigilant deflations =-=[6, 49]-=-, i.e., zero or tiny subdiagonal elements in H that arise during the chasing process. In order to preserve the information contained in the bulge pairs, the chain of bulge pairs must be reintroduced i... |

20 |
Blocked algorithms and software for reduction of a regular matrix pair to generalized Schur form
- Dackland, K˚agström
- 1999
(Show Context)
Citation Context ... n×n . It goes back to Moler and Stewart in 1973 [37] and has undergone only a few modifications during the next 25 years, notably through works by Ward [47, 48], Kaufman [29], Dackland and K˚agström =-=[12]-=-. Non-orthogonal variants of the QZ algorithm include the LZ algorithm by Kaufman [28] and the AB algorithm for pencils by Kublanovskaya [34]. The purpose of the QZ algorithm is to compute a generaliz... |

20 | The transmission of shifts and shift blurring in the QR algorithm
- Watkins
- 1996
(Show Context)
Citation Context ...This effect is caused by shift blurring: with increasing m the eigenvalues of the bulge pairs, which should represent the shifts in exact arithmetic, often become extremely sensitive to perturbations =-=[50, 51, 33]-=-. Already for moderate m, say m ≥ 15, the shifts may be completely contaminated by round-off errors during the bulge chasing process. Not surprisingly, we made similar observations in numerical experi... |

18 |
The multishift QR algorithm. i. maintaining well-focused shifts and level 3 performance
- Braman, Byers
(Show Context)
Citation Context ...generalized Schur form (S, T) into smaller subproblems. This paper describes improvements for the latter two ingredients, QZ iterations and deflations. Inspired by the works of Braman, Byers, Mathias =-=[6]-=-, 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 bulge pair per iteration. This allows the effective use ... |

15 |
Oberwolfach Benchmark Collection
- Korvink, Rudyni
(Show Context)
Citation Context ...n for matrix pairs that arise from practically relevant applications. We have selected 16 matrix pairs from the Matrix Market collection [3], 6 matrix pairs from model reduction benchmark collections =-=[11, 30]-=-, and 4 matrix pairs arising from the computation of corner singularities of elliptic PDEs [38]. A more detailed description of the selected matrix pairs along with individual performance results can ... |

15 |
Forward stability and transmission of shifts in the QR algorithm
- Watkins
- 1995
(Show Context)
Citation Context ...This effect is caused by shift blurring: with increasing m the eigenvalues of the bulge pairs, which should represent the shifts in exact arithmetic, often become extremely sensitive to perturbations =-=[50, 51, 33]-=-. Already for moderate m, say m ≥ 15, the shifts may be completely contaminated by round-off errors during the bulge chasing process. Not surprisingly, we made similar observations in numerical experi... |

14 |
A direct method for reordering eigenvalues in the generalized real Schur form of a regular matrix pair
- K˚agström
- 1992
(Show Context)
Citation Context ...pan a pair of deflating subspaces [41] if the (k+1, k) subdiagonal entry of the matrix S vanishes. A reordering of the diagonal blocks of S and T can be used to compute other deflating subspaces, see =-=[23, 27, 44]-=-. The eigenvalues of (A, B) are read off from (S, T) as follows. The 2 × 2 diagonal blocks correspond to pairs of complex conjugate eigenvalues. The real eigenvalues are given in pairs (sii, tii) corr... |

13 |
AB-algorithm and its modifications for the spectral problems of linear pencils of matrices
- Kublanovskaya
- 1984
(Show Context)
Citation Context ...rks by Ward [47, 48], Kaufman [29], Dackland and K˚agström [12]. Non-orthogonal variants of the QZ algorithm include the LZ algorithm by Kaufman [28] and the AB algorithm for pencils by Kublanovskaya =-=[34]-=-. 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 ... |

13 |
Effiziente Orthogonaltransformationen bei der Eigen- und Singulärwertzerlegung
- Lang
- 1997
(Show Context)
Citation Context ...hur form (S, T) into smaller subproblems. This paper describes improvements for the latter two ingredients, QZ iterations and deflations. Inspired by the works of Braman, Byers, Mathias [6], 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 bulge pair per iteration. This allows the effective use of level 3 BLAS... |

13 |
Balanced truncation model reduction for semidiscretized Stokes equation
- Stykel
(Show Context)
Citation Context ...ngular form has one or more diagonal entries close to zero. Each of these diagonal entries admits the deflation of an infinite eigenvalue. Some applications, such as semi-discretized Stokes equations =-=[42]-=-, lead to matrix pairs that have a large number of infinite eigenvalues. Consequently, a substantial amount of computational work in the QZ algorithm is spent for deflating these eigenvalues. We will ... |

13 |
Balancing the generalized eigenvalue problem
- Ward
- 1981
(Show Context)
Citation Context ...sized regular matrix pairs (A, B) with A, B ∈ R n×n . It goes back to Moler and Stewart in 1973 [37] and has undergone only a few modifications during the next 25 years, notably through works by Ward =-=[47, 48]-=-, Kaufman [29], Dackland and K˚agström [12]. Non-orthogonal variants of the QZ algorithm include the LZ algorithm by Kaufman [28] and the AB algorithm for pencils by Kublanovskaya [34]. The purpose of... |

12 |
Parallel and blocked algorithms for reduction of a regular matrix pair to Hessenberg-triangular and generalized Schur forms
- Adlerborn, Dackland, et al.
- 2002
(Show Context)
Citation Context ...ions can be gained from using perturbations more general than those of Lemma 6.3. To illustrate the effectiveness of Lemma 6.3, let us consider the following matrix pair, which has been considered in =-=[1]-=- as an extension of the motivating example in [7]: (6.5) ⎛⎡ (H, T) = ⎜⎢ ⎝⎣ 6 5 4 3 2 1 0.001 1 0 0 0 0 0.001 2 0 0 0 0.001 3 0 0 0.001 4 0 0.001 5 ⎤ ⎥ ⎦ , ⎡ ⎢ ⎣ ⎦, ⎤ ⎦. 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1... |

11 |
The multishift QR algorithm: is it worth the trouble
- Dubrulle
- 1991
(Show Context)
Citation Context ... use of WY representations. This approach, however, has proved disappointing due to the fact that the convergence of such a large-bulge multishift QR algorithm is severely affected by roundoff errors =-=[16]-=-. This effect is caused by shift blurring: with increasing m the eigenvalues of the bulge pairs, which should represent the shifts in exact arithmetic, often become extremely sensitive to perturbation... |

10 | Block algorithms for reordering standard and generalized Schur forms
- KRESSNER
(Show Context)
Citation Context ...ent algorithms for deflating infinite eigenvalues within the QZ algorithm. This approach is conceptually close to blocked algorithms for reordering eigenvalues in standard and generalized Schur forms =-=[32]-=-. The rest of this paper is organized as follows. In Section 2, we review and extend conventional multishift QZ iterations, and provide some new insight into their numerical backward stability. Multis... |

10 |
The combination shift QZ algorithm
- Ward
- 1975
(Show Context)
Citation Context ...sized regular matrix pairs (A, B) with A, B ∈ R n×n . It goes back to Moler and Stewart in 1973 [37] and has undergone only a few modifications during the next 25 years, notably through works by Ward =-=[47, 48]-=-, Kaufman [29], Dackland and K˚agström [12]. Non-orthogonal variants of the QZ algorithm include the LZ algorithm by Kaufman [28] and the AB algorithm for pencils by Kublanovskaya [34]. The purpose of... |

10 | Performance of the QZ algorithm in the presence of infinite eigenvalues
- Watkins
(Show Context)
Citation Context ...d as chasing a bulge pair from the top left corner to the bottom right corner of (H, T), the question arises how the information contained in the shifts is passed during this chasing process. Watkins =-=[52]-=- discovered a surprisingly simple relationship; the intended shifts are the finite eigenvalues of the bulge pairs. To explain this in more detail, suppose that the implicit shifted QZ iteration with m... |

9 |
CoCoS – computation of corner singularities
- Pester
- 2005
(Show Context)
Citation Context ...irs from the Matrix Market collection [3], 6 matrix pairs from model reduction benchmark collections [11, 30], and 4 matrix pairs arising from the computation of corner singularities of elliptic PDEs =-=[38]-=-. A more detailed description of the selected matrix pairs along with individual performance results can be found in Appendix A. In the following, we summarize these results and sort the matrix pairs ... |

9 |
Algorithm 590: DSUBSP and EXCHQZ: FORTRAN subroutines for computing deflating subspaces with specified spectrum
- Dooren
- 1982
(Show Context)
Citation Context ...pan a pair of deflating subspaces [41] if the (k+1, k) subdiagonal entry of the matrix S vanishes. A reordering of the diagonal blocks of S and T can be used to compute other deflating subspaces, see =-=[23, 27, 44]-=-. The eigenvalues of (A, B) are read off from (S, T) as follows. The 2 × 2 diagonal blocks correspond to pairs of complex conjugate eigenvalues. The real eigenvalues are given in pairs (sii, tii) corr... |

9 |
Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem
- Watkins, Elsner
- 1994
(Show Context)
Citation Context ...or the following implementations of the QZ algorithm: DHGEQZ: LAPACK version 3.0 implementation as described in the original paper by Moler and Stewart [37] with some of the modifications proposed in =-=[29, 47, 53]-=-, see also Section 2.1. KDHGEQZ: Blocked variant of DHGEQZ, developed by Dackland and K˚agström [12]. MULTIQZ: Multishift QZ algorithm based on tightly coupled chains of tightly bulges as described in... |

8 |
Generalized Singular Values with Algorithms and Applications
- Loan
- 1973
(Show Context)
Citation Context ...airs (α, β) of the bivariate polynomial det(βA −αB), can be directly computed from the diagonal blocks of S and T, although some care must be taken to implement this computation in a safe manner, see =-=[37, 45]-=-. Moreover, the leading k columns of the orthogonal matrices Z and Q span a pair of deflating subspaces [41] if the (k+1, k) subdiagonal entry of the matrix S vanishes. A reordering of the diagonal bl... |

6 |
GEMM-based level 3 BLAS: Algorithms for the model implementations
- K˚agström, Ling, et al.
- 1999
(Show Context)
Citation Context ...algorithm, we propose multishift QZ iterations that chase a tightly coupled chain of bulge pairs instead of only one bulge pair per iteration. This allows the effective use of level 3 BLAS operations =-=[15, 25, 26]-=- during the bulge chasing process, which in turn can provide efficient utilization of today’s high performance computing systems with deep memory hierarchies. Tightly coupled bulge chasing has also su... |

4 | The descriptor controllability radius
- Byers
- 1994
(Show Context)
Citation Context ...of being effectively computed and tested. Finding the minimum among all reducing perturbations of the form (6.4) is closely related to finding the distance to uncontrollability of a descriptor system =-=[9]-=-. This connection along with numerical methods for computing the distance to uncontrollability will be studied in a forthcoming paper. However, in preliminary numerical experiments with the multishift... |

4 |
Some thoughts on the QZ algorithm for solving the generalized eigenvalue problem
- Kaufman
- 1977
(Show Context)
Citation Context ...ix pairs (A, B) with A, B ∈ R n×n . It goes back to Moler and Stewart in 1973 [37] and has undergone only a few modifications during the next 25 years, notably through works by Ward [47, 48], Kaufman =-=[29]-=-, Dackland and K˚agström [12]. Non-orthogonal variants of the QZ algorithm include the LZ algorithm by Kaufman [28] and the AB algorithm for pencils by Kublanovskaya [34]. The purpose of the QZ algori... |

4 |
Kronecker’s canonical form and the QZ-algorithm, Linear Algebra and Its Applications 28: 285–303. scientific computing
- Wilkinson
- 1979
(Show Context)
Citation Context ...ove-mentioned singularities reliably and otherwise well-conditioned eigenvalues can change drastically, meaning that the values of the computed pairs (sii, tii) cannot be trusted (e.g., see Wilkinson =-=[54]-=- for several illustrative examples.) Moreover, it is impossible to decide just by inspection whether sii = ǫ1 and tii = ǫ2, with ǫ1 and ǫ2 tiny, correspond to a finite eigenvalue ǫ1/ǫ2 or to a true si... |

3 |
Singular matrix pencils
- K˚agström
(Show Context)
Citation Context ...e eigenvalue, in principle, the only reliable and robust way to identify all infinite eigenvalues is to apply a preprocessing step with a staircase-type of algorithm. By applying the GUPTRI algorithm =-=[13, 14, 24]-=- to a regular pair (A, B) with infinite eigenvalues, we get (4.1) U T (A, B)V = ([ A11 A12 0 Ainf ] [ B11 B12 , 0 Binf where U and V are orthogonal transformation matrices, (Ainf, Binf) reveals the Jo... |

3 |
The LZ-algorithm to solve the generalized eigenvalue problem
- Kaufman
- 1974
(Show Context)
Citation Context ...ifications during the next 25 years, notably through works by Ward [47, 48], Kaufman [29], Dackland and K˚agström [12]. Non-orthogonal variants of the QZ algorithm include the LZ algorithm by Kaufman =-=[28]-=- and the AB algorithm for pencils by Kublanovskaya [34]. 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... |

3 | On the Eigensystems of Graded Matrices
- Stewart
(Show Context)
Citation Context ...hk,k| + |hk+1,k+1|). It is known for standard eigenvalue problems that, especially in the presence of graded matrices, the use of the criterion (3.3) gives higher accuracy in the computed eigenvalues =-=[40]-=-. We have observed similar accuracy improvements for the QZ algorithm when using (3.3) in favour of (3.2). We have also encountered examples where both criteria give similar accuracy but with slightly... |

3 |
Where is the nearest non-regular pencil? Linear Algebra and its Applications
- Byers, He, et al.
- 1998
(Show Context)
Citation Context ...pond to a finite eigenvalue ǫ1/ǫ2 or to a true singular pencil. Anyhow, with this information we know that βA − αB is close to a singular pencil. (Note that the converse of this statement is not true =-=[10, 24]-=-.) Although the QZ algorithm delivers erratic results for singular or almost singular cases, the computed results are still exact for small perturbations of the original matrix pair (A, B). To robustl... |

2 | On the use of larger bulges in the QR algorithm
- Kressner
(Show Context)
Citation Context ...This effect is caused by shift blurring: with increasing m the eigenvalues of the bulge pairs, which should represent the shifts in exact arithmetic, often become extremely sensitive to perturbations =-=[50, 51, 33]-=-. Already for moderate m, say m ≥ 15, the shifts may be completely contaminated by round-off errors during the bulge chasing process. Not surprisingly, we made similar observations in numerical experi... |

2 |
Solution of index-2-DAEs and its application in circuit simulation. Dissertation, Humboldt-Univ. zu
- Tischendorf
- 1996
(Show Context)
Citation Context ...ns (DAE), the index of (A, B) corresponds to the index of the DAE B ˙x = Ax + f. Many applications, such as multibody systems and electrical circuits, lead to DAEs with index at least two, see, e.g., =-=[8, 39, 43]-=-. If the matrix pair (A, B) has an infinite eigenvalue then the matrix B is singular. This implies that at least one of the diagonal entries in the upper triangular matrix T in the Hessenberg-triangul... |