Implicit Cholesky algorithms for singular values and vectors of triangular matrices

Fernando, Kasun; Parlett, Beresford Ν.

doi:10.1002/nla.1680020604

Cited by 9 publications

(9 citation statements)

References 18 publications

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“…The implicit Cholesky algorithm from [4] can easily be extended to skew-symmetric matrices in several ways. One way is to use a one-sided factorization, which is exploited in generic GR algorithms (see [9]).…”

Section: Implicit Cholesky-like Algorithmmentioning

confidence: 99%

“…This algorithm is a skew-symmetric generalization of the algorithm constructed in [4]. Application of this algorithm to the Bunch factor of a skew symmetric matrix produces the orthogonal skew-symmetric differential qd (osdqd) algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by belief that eigenvalue algorithms for symmetric matrices can be extended to algorithms for skewsymmetric matrices, we construct an implicit Cholesky-like algorithm for full skew-symmetric matrices. This algorithm is a skew-symmetric generalization of the algorithm constructed in [4]. Application of this algorithm to the Bunch factor of a skew symmetric matrix produces the orthogonal skew-symmetric differential qd (osdqd) algorithm.…”

Section: Introductionmentioning

confidence: 99%

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Skew-Symmetric Differential qd Algorithm

Singer

2005

Appl. Num. Anal. Comp. Math.

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Differential qd (dqd) algorithm with shifts is probably the fastest known algorithm which computes eigenvalues of symmetric tridiagonal matrices with high relative accuracy.In this paper we will construct a similar algorithm for computing eigenvalues of skew-symmetric matrices, which is based on implicit usage of both the QR and the symplectic QR factorizations. If we apply this algorithm to tridiagonal skew-symmetric matrices, we obtain the skew-symmetric dqd algorithm. This algorithm also enjoys high relative stability. However, incorporation of shifts is much harder then in the symmetric case, and yet to be implemented.Finally, the standard algorithm for computing the eigenvalues of tridiagonal skew-symmetric matrices can also be interpreted in the context of the skew-symmetric dqd algorithm.

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Section: Implicit Cholesky-like Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Skew-Symmetric Differential qd Algorithm

Singer

2005

Appl. Num. Anal. Comp. Math.

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“…Originally, the differential qd (dqd) algorithm was proposed by Rutishauser [28,29]. Parlett and Fernando [14,15,22] discovered the shifted variant (dqds) as an improvement to Demmel and Kahan's zero shift QR [7] for computing singular values of a bidiagonal matrix to high relative accuracy, see also [27].…”

mentioning

confidence: 99%

“…First, consider the dqds algorithm for computing the singular values of a bidiago- nal matrix [14,15,22,27]. There an invalid operation signals that stability has been lost.…”

mentioning

confidence: 99%

Benefits of IEEE‐754 Features in Modern Symmetric Tridiagonal Eigensolvers

Marques¹,

Riedy²,

̈mel³

2006

SIAM J. Sci. Comput.

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Abstract. Bisection is one of the most common methods used to compute the eigenvalues of symmetric tridiagonal matrices. Bisection relies on the Sturm count: for a given shift σ, the number of negative pivots in the factorization T − σI = LDL T equals the number of eigenvalues of T that are smaller than σ. In IEEE-754 arithmetic, the value ∞ permits the computation to continue past a zero pivot, producing a correct Sturm count when T is unreduced. Demmel and Li showed in the 90s that using ∞ rather than testing for zero pivots within the loop could improve performance significantly on certain architectures.When eigenvalues are to be computed to high relative accuracy, it is often preferable to work with LDL T factorizations instead of the original tridiagonal T . One important example is the MRRR algorithm. When bisection is applied to the factored matrix, the Sturm count is computed from LDL T which makes differential stationary and progressive qds algorithms the methods of choice.While it seems trivial to replace T by LDL T , in reality these algorithms are more complicated: in IEEE-754 arithmetic, a zero pivot produces an overflow followed by an invalid exception (NaN) that renders the Sturm count incorrect.We present alternative, safe formulations that are guaranteed to produce the correct result. Benchmarking these algorithms on a variety of platforms shows that the original formulation without tests is always faster provided no exception occurs. The transforms see speed-ups of up to 2.6× over the careful formulations.Tests on industrial matrices show that encountering exceptions in practice is rare. This leads to the following design: First, compute the Sturm count by the fast but unsafe algorithm. Then, if an exception occurred, recompute the count by a safe, slower alternative.The new Sturm count algorithms improve the speed of bisection by up to 2× on our test matrices. Furthermore, unlike the traditional tiny-pivot substitution, proper use of IEEE-754 features provides a careful formulation that imposes no input range restrictions.

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A note on generating finer‐grain parallelism in a representation tree

Vömel

2011

Numerical Linear Algebra App

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SUMMARY The representation tree lies at the heart of the algorithm of Multiple Relatively Robust Representations for computing orthogonal eigenvectors of a symmetric tridiagonal matrix without Gram–Schmidt. A representation tree describes the incremental shift relations between relatively robust representations of eigenvalue clusters of an unreduced tridiagonal matrix, which are needed to strongly separate close eigenvalues in the relative sense. At the bottom of the representation tree, each leaf defines a relatively isolated eigenvalue to high relative accuracy. The shape of the representation tree plays a pivotal role for complexity and available parallelism: a deeper tree consisting of multiple levels of nodes involves tasks associated to more work (i.e., eigenvalue refinement to resolve eigenvalue clusters) and less parallelism (i.e., a longer critical path as well as potential data movement and synchronization). An embarrassingly parallel, ideal tree on the other hand consists of a root and leaves only. As highly parallel hybrid graphics processing unit/multicore platforms with large memory now become available as commodity platforms, exploiting parallelism in traditional algorithms becomes key to modernizing the components of standard software libraries such as LAPACK. This paper focuses on LAPACK's Multiple Relatively Robust Representations algorithm and investigates the critical case where a representation tree contains a long sequential chain of large (fat) nodes that hamper parallelism. This key problem needs to be addressed as it concerns all sorts of computing environments, distributed computing, symmetric multiprocessor, as well as hybrid graphics processing unit/multicore architectures. We present an improved representation tree that often offers a significantly shorter critical path and finer computational granularity of smaller tasks that are easier to schedule. In a study of selected synthetic and application matrices, we show that an average 75% reduction in the length of the critical path and 82% reduction in task granularity can be achieved. Copyright © 2011 John Wiley & Sons, Ltd.

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Implicit Cholesky algorithms for singular values and vectors of triangular matrices

Cited by 9 publications

References 18 publications

Skew-Symmetric Differential qd Algorithm

Skew-Symmetric Differential qd Algorithm

Benefits of IEEE‐754 Features in Modern Symmetric Tridiagonal Eigensolvers

A note on generating finer‐grain parallelism in a representation tree

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