High performance algorithms for Toeplitz and block Toeplitz matrices

Gallivan, Kyle A.; Thirumalai, S.; Dooren, Paul Van; Vermaut, Vincent

doi:10.1016/0024-3795(95)00649-4

Cited by 41 publications

(29 citation statements)

References 17 publications

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“…In [16,17,32], S. Thirumalai presents an efficient block algorithm to perform the previous decomposition. This algorithm uses a method that is similar to the one in LAPACK [2] to compute and apply Householder transformations [3,28] following a notation called W Y .…”

Section: The Parallel Algorithmmentioning

confidence: 99%

“…The sequential version of the method is very efficient, as shown in [17,32]. However, the parallel implementation has a large communication cost.…”

Section: The Parallel Algorithmmentioning

confidence: 99%

“…However, if we use this technique, we cannot apply the transformations in a blocked way, as proposed in [17]. Instead, we have to perform a displacement after transforming each row of the generator to the proper form.…”

Section: Reducing the Communication Costmentioning

confidence: 99%

“…Although our algorithm does not exploit the blocked application of the transformations shown in [17], the hyperbolic transformations used on each iteration produce more accurate results. In [29], it is shown that if the hyperbolic transformations are applied directly as in [17], the GSA may be unstable, while if they are applied in a factorized way, the algorithm is stable.…”

Section: Implementation Of the Unitary Transformationsmentioning

confidence: 99%

“…In [29], it is shown that if the hyperbolic transformations are applied directly as in [17], the GSA may be unstable, while if they are applied in a factorized way, the algorithm is stable. More specifically, our algorithm uses the transformation method presented in [13].…”

Section: Implementation Of the Unitary Transformationsmentioning

confidence: 99%

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An Efficient Parallel Algorithm to Solve Block?Toeplitz Systems

2005

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Abstract. In this paper, we present an efficient parallel algorithm to solve Toeplitz-block and block-Toeplitz systems in distributed memory multicomputers. This algorithm parallelizes the Generalized Schur Algorithm to obtain the semi-normal equations. Our parallel implementation reduces the communication cost and optimizes the memory access. The experimental analysis on a cluster of personal computers shows the scalability of the implementation. The algorithm is portable because it is based on standard tools and libraries, such as ScaLAPACK and MPI.

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Section: The Parallel Algorithmmentioning

confidence: 99%

“…The sequential version of the method is very efficient, as shown in [17,32]. However, the parallel implementation has a large communication cost.…”

Section: The Parallel Algorithmmentioning

confidence: 99%

Section: Reducing the Communication Costmentioning

confidence: 99%

Section: Implementation Of the Unitary Transformationsmentioning

confidence: 99%

Section: Implementation Of the Unitary Transformationsmentioning

confidence: 99%

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An Efficient Parallel Algorithm to Solve Block?Toeplitz Systems

2005

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Solving the block-Toeplitz least-squares problem in parallel

Alonso¹,

Badía²,

Vidal³

2004

Concurrency Computat.: Pract. Exper.

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SUMMARYIn this paper we present two versions of a parallel algorithm to solve the block-Toeplitz least-squares problem on distributed-memory architectures. We derive a parallel algorithm based on the seminormal equations arising from the triangular decomposition of the product T T T . Our parallel algorithm exploits the displacement structure of the Toeplitz-like matrices using the Generalized Schur Algorithm to obtain the solution in O(mn) flops instead of O(mn 2 ) flops of the algorithms for non-structured matrices. The strong regularity of the previous product of matrices and an appropriate computation of the hyperbolic rotations improve the stability of the algorithms. We have reduced the communication cost of previous versions, and have also reduced the memory access cost by appropriately arranging the elements of the matrices. Furthermore, the second version of the algorithm has a very low spatial cost, because it does not store the triangular factor of the decomposition. The experimental results show a good scalability of the parallel algorithm on two different clusters of personal computers.

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A fast algorithm for computing the smallest eigenvalue of a symmetric positive‐definite Toeplitz matrix

Mastronardi

Barel

Vandebril

2008

Numerical Linear Algebra App

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SUMMARYRecent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive-definite (SPD) Toeplitz matrix.An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016 DOI: 10. /j.laa.2007. The algorithm relies on the computation of the R factor of the Q R factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R −1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed.

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High performance algorithms for Toeplitz and block Toeplitz matrices

Cited by 41 publications

References 17 publications

An Efficient Parallel Algorithm to Solve Block?Toeplitz Systems

An Efficient Parallel Algorithm to Solve Block?Toeplitz Systems

Solving the block-Toeplitz least-squares problem in parallel

A fast algorithm for computing the smallest eigenvalue of a symmetric positive‐definite Toeplitz matrix

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