On Solving Block Toeplitz Systems Using a Block Schur Algorithm

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

doi:10.1109/icpp.1994.136

Cited by 7 publications

(6 citation statements)

References 10 publications

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“…The effect of the communications increases if we take into account the small computational cost of this method when applied to structured matrices. The behaviour of this algorithm has been experimentally confirmed by its authors on distributed memory multicomputers [16].…”

Section: The Parallel Algorithmmentioning

confidence: 60%

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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: 60%

“…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%

An Efficient Parallel Algorithm to Solve Block?Toeplitz Systems

2005

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“…Although this algorithm is efficient in one processor [37], the communication cost limits the performance with more processors, mainly because of the point-to-point messages. We have implemented two parallel versions of this algorithm using different block distributions [33].…”

Section: Parallel Algorithmmentioning

confidence: 99%

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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“…For instance, it can be found parallel algorithms to solve Toeplitz systems using systolic arrays [1] or dealing only with positive definite matrices or with symmetric matrices [2]. There also exist parallel algorithms for shared memory computers [3,4,5] and, more recently, several parallel algorithms for distributed architectures have been proposed [6].…”

Section: Introductionmentioning

confidence: 99%

The Symmetric–Toeplitz Linear System Problem in Parallel

Alonso

Vidal

2005

Lecture Notes in Computer Science

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Many algorithms exist that exploit the special structure of Toeplitz matrices for solving linear systems. Nevertheless, these algorithms are difficult to parallelize due to its lower computational cost and the great dependency of the operations involved that produces a great communication cost. The foundation of the parallel algorithm presented in this paper consists of transforming the Toeplitz matrix into a another structured matrix called Cauchy-like. The particular properties of Cauchy-like matrices are exploited in order to obtain two levels of parallelism that makes possible to highly reduce the execution time. The experimental results were obtained in a cluster of PC's.

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On Solving Block Toeplitz Systems Using a Block Schur Algorithm

Cited by 7 publications

References 10 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

The Symmetric–Toeplitz Linear System Problem in Parallel

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