A look-ahead Bareiss algorithm for general Toeplitz matrices

Freund, Robert M.

doi:10.1007/s002110050047

Cited by 22 publications

(11 citation statements)

References 35 publications

(60 reference statements)

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“…We do not know any other methods which have been proved to be stable or weakly stable and have worst-case time bound O(n 2 ). Algorithms which involve pivoting and/or look-ahead [18,27,41,43,76] may work well in practice, but seem to require worst-case overhead O(n 3 ) to ensure stability.…”

Section: Resultsmentioning

confidence: 99%

Stability analysis of a general toeplitz systems solver

1995

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We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R T R is close to A T A. Thus, when the algorithm is used to solve the semi-normal equations R T Rx = A T b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min Ax − b 2 .

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Section: Resultsmentioning

confidence: 99%

Stability analysis of a general toeplitz systems solver

1995

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“…To cure these breakdowns a number of look-ahead variants of the classical algorithms have been introduced (e.g. [7,9]). In a new approach [14,11], T is transformed into a non-Hermitian matrix N = FT ΩF H with the Fourier matrix F and the diagonal matrix Ω with entries exp(−πij /n), j = 0, 1, ..., n − 1.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Gaussian elimination for cauchy-type matrices with application to positive definite Toeplitz matrices

Huckle¹

1998

Numerische Mathematik

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The problem of solving linear equations with a Toeplitz matrix T appears in many applications. Often T is positive definite but ill-conditioned with many small eigenvalues. In this case fast and superfast algorithms may show a very poor behavior or even break down. In recent papers the transformation of a Toeplitz matrix into a Cauchy-type matrix is proposed. The resulting new linear equations can be solved in O(n 2 ) operations using standard pivoting strategies which leads to very stable fast methods also for ill-conditioned systems. The basic tool is the formulation of Gaussian elimination for matrices with low displacement rank. In this paper, we will transform a Hermitian Toeplitz matrix T into a Cauchy-type matrix B = FTF H by applying the Fourier transform. We will prove some useful properties of B and formulate a symmetric Gaussian elimination algorithm for positive definite B . Using the symmetry and persymmetry of T we can reduce the total costs of this algorithm compared with unsymmetric Gaussian elimination. For complex Hermitian T , the complexity of the new algorithm is then nearly the same as for the Schur algorithm. Furthermore, it is possible to include some strategies for ill-conditioned positive definite matrices that are well-known in optimization. Numerical examples show that this new algorithm is fast and reliable.

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“…This proposition can be found, e.g., in [31], where it is proposed and studied an algorithm variant supported on lookahead-type techniques. Also, for Toeplitz matrices, lookahead techniques have widely studied with the aim at improving accuracy, further to ensure stability of Levinson-and Schur-type algorithms which can even break down for well-conditioned matrices [13,7]. But, in general, look-ahead algorithms for Toeplitz matrices are based on heuristics with variable results (depend on the given matrix) and they are difficult to apply in concurrent environments where we try to keep the sough-after performance of the multicore system.…”

Section: Introductionmentioning

confidence: 99%

Block pivoting implementation of a symmetric Toeplitz solver

Alonso

Dolz

Vidal

2014

Journal of Parallel and Distributed Computing

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h i g h l i g h t s• We improve the solution of symmetric Toeplitz linear systems in multicore systems.• We transform the Toeplitz matrix into a Cauchy-like one to obtain some benefits.• The problem is partitioned into two half-sized independent problems.• We use partial local pivoting to improve the accuracy of the solution.• We propose a special scheme to store data in memory that accelerates the algorithm. a r t i c l e i n f o a b s t r a c tToeplitz matrices are characterized by a special structure that can be exploited in order to obtain fast linear system solvers. These solvers are difficult to parallelize due to their low computational cost and their closely coupled data operations. We propose to transform the Toeplitz system matrix into a Cauchy-like matrix since the latter can be divided into two independent matrices of half the size of the system matrix and each one of these smaller arising matrices can be factorized efficiently in multicore computers. We use OpenMP and store data in memory by blocks in consecutive positions yielding a simple and efficient algorithm. In addition, by exploiting the fact that diagonal pivoting does not destroy the special structure of Cauchy-like matrices, we introduce a local diagonal pivoting technique which improves the accuracy of the solution and the stability of the algorithm.

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A look-ahead Bareiss algorithm for general Toeplitz matrices

Cited by 22 publications

References 35 publications

Stability analysis of a general toeplitz systems solver

Stability analysis of a general toeplitz systems solver

Symmetric Gaussian elimination for cauchy-type matrices with application to positive definite Toeplitz matrices

Block pivoting implementation of a symmetric Toeplitz solver

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