Displacement ranks of matrices and linear equations

Kailath, T.; Kung, Sun Yuan; Morf, M.

doi:10.1016/0022-247x(79)90124-0

Cited by 429 publications

(216 citation statements)

References 11 publications

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“…The algorithm from [6] for such matrices with q terms can be easily modified ( [11]) to an algorithm requiring 5q−4 2 n 2 multiplications and additions. In essence, it is the algorithm already proposed in [7] but within the confines of a block method.…”

Section: Tools Of Researchmentioning

confidence: 99%

“…The smallest q is called the Toeplitz rank (displacement rank in the terminology of [7]) of T . Toeplitz ranks were introduced in [7].…”

Section: Tools Of Researchmentioning

confidence: 99%

“…Toeplitz ranks were introduced in [7]. In [11] it was proved that the Toeplitz rank of a product of two matrices is not greater than the sum of the factors' ranks increased by 1.…”

Section: Tools Of Researchmentioning

confidence: 99%

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Which circulant preconditioner is better?

Strela

Тыртышников

1996

Math. Comp.

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Abstract. The eigenvalue clustering of matrices S −1 n An and C −1 n An is experimentally studied, where An, Sn and Cn respectively are Toeplitz matrices, Strang, and optimal circulant preconditioners generated by the Fourier expansion of a function f (x). Some illustrations are given to show how the clustering depends on the smoothness of f (x) and which preconditioner is preferable. An original technique for experimental exploration of the clustering rate is presented. This technique is based on the bisection idea and on the Toeplitz decomposition of a three-matrix product CAC, where A is a Toeplitz matrix and C is a circulant. In particular, it is proved that the Toeplitz (displacement) rank of CAC is not greater than 4, provided that C and A are symmetric.

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Section: Tools Of Researchmentioning

confidence: 99%

“…The smallest q is called the Toeplitz rank (displacement rank in the terminology of [7]) of T . Toeplitz ranks were introduced in [7].…”

Section: Tools Of Researchmentioning

confidence: 99%

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Which circulant preconditioner is better?

Strela

Тыртышников

1996

Math. Comp.

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“…Also, (JH) T (JH) = H T H. Thus, our results apply equally to Hankel and Toeplitz matrices. Our results might also be extended to more general classes of matrices with a displacement structure [50,51,52,53]. For simplicity we restrict our attention to the Toeplitz case.…”

Section: Introductionmentioning

confidence: 94%

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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“…In all this paper, we closely follow the existing formalism of structured matrix computations developed in previous work by Morf [34], Bitmead-Anderson [3], Kailath and co-authors [28,29], Pan [37,38], Kaltofen [30], Cardinal [14], etc.…”

mentioning

confidence: 99%

Algorithms for Structured Linear Systems Solving and Their Implementation

Hyun

Lebreton

Schost

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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ABSTRACTere exists a vast literature dedicated to algorithms for structured matrices, but relatively few descriptions of actual implementations and their practical performance. In this paper, we consider the problem of solving Cauchy-like systems, and its application to mosaic Toeplitz systems, in two contexts: rst in the unit cost model (which is a good model for computations over nite elds), then over Q. We introduce new variants of previous algorithms and describe an implementation of these techniques and its practical behavior. We pay a special a ention to particular cases such as the computation of algebraic approximants.

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Displacement ranks of matrices and linear equations

Cited by 429 publications

References 11 publications

Which circulant preconditioner is better?

Which circulant preconditioner is better?

Stability analysis of a general toeplitz systems solver

Algorithms for Structured Linear Systems Solving and Their Implementation

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