The Inverses of Block Hankel and Block Toeplitz Matrices

Labahn, George; Chio, Dong Koo; Cabay, Stan

doi:10.1137/0219006

Cited by 35 publications

(28 citation statements)

References 28 publications

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“…One can compute the off-diagonal inverse formula of H using [18,Theorem 3.1] with the computation of • two left sigma bases of [H(z) T | I] T of order 2m − 1 and 2m + 1, and • two right sigma bases of [H(z) | I] of order 2m − 1 and 2m + 1. This computation can be done with O˜(s 3 m) field operation with the fast algorithm PM-Basis of [13].…”

Section: Optimizationsmentioning

confidence: 99%

Solving sparse rational linear systems

Eberly

Giesbrecht

Giorgi

et al. 2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

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We propose a new algorithm to find a rational solution to a sparse system of linear equations over the integers. This algorithm is based on a p-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox -based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art.

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Section: Optimizationsmentioning

confidence: 99%

Solving sparse rational linear systems

Eberly

Giesbrecht

Giorgi

et al. 2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

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“…Although Theorem 1 applies to any such system, a deterministic cost of O(α ω−1 M(n) log(n)) can be obtained in the particular case where the matrix is invertible. As described for example in [13], this cost follows from combining an inversion formula of [27] with σ-basis computations as in [18].…”

Section: Introductionmentioning

confidence: 99%

Solving toeplitz- and vandermonde-like linear systems with large displacement rank

Bostan

Jeannerod

Schost

2007

Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation

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Linear systems with structures such as Toeplitz-, Vandermonde-or Cauchy-likeness can be solved in O˜(α 2 n) operations, where n is the matrix size, α is its displacement rank, and O˜denotes the omission of logarithmic factors. We show that for Toeplitz-like and Vandermonde-like matrices, this cost can be reduced to O˜(α ω−1 n), where ω is a feasible exponent for matrix multiplication over the base field. The best known estimate for ω is ω < 2.38, resulting in costs of order O˜(α 1.38 n). We also present consequences for Hermite-Padé approximation and bivariate interpolation.

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“…Note that formula (57) has a close resemblance with some of the formulae that have been derived for inverses of Hankel matrices, see e.g. [23]. Also note that X c is computed more efficiently by making use of (49), (52) and the recurrence (53) rather than expression (57).…”

Section: Closed Form Inverse Of the Controllability Gramianmentioning

confidence: 95%

“…An alternative presentation of the formula relates it to the Gohberg-Semencul formulas as well as a number of related formulations of inverses of Hankel matrices, see e.g. [22] and [23].…”

Section: Introductionmentioning

confidence: 99%

The continuous closed form controllability Gramian and its inverse

Hauksdottir

Sigurdsson

2009

2009 American Control Conference

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The continuous controllability Gramian is the solution of an input Lyapunov equation in the controller (companion) form or equivalently the infinite integral of an outer product of a vector containing the impulse response and its derivatives corresponding to a unity numerator transfer function. In this paper we make use of both these viewpoints in order to derive the simple zero plaid structure of this Gramian and present the interesting links that the entries of the Gramian have to the entries of the Routh table. Moreover, an expression for the inverse of the Gramian is derived as a simple function of the coefficients of the characteristic polynomial from the fact that it is the solution of a Riccati equation.We show how the controllability Gramian forms the core part of closed form expressions of Gramians of more general MIMO systems as well as solutions of general Sylvester equations. The controllability Gramian also appears in certain zero optimization problems, either in a PID like controller setting or in a model reduction setting. The inverse of the controllability Gramian is a key element in such zero optimization.While much of the results presented can be found in closely related forms in published papers, we believe that they deserve more attention as an effective tool in numerical computations of small to mid-size systems.

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The Inverses of Block Hankel and Block Toeplitz Matrices

Cited by 35 publications

References 28 publications

Solving sparse rational linear systems

Solving sparse rational linear systems

Solving toeplitz- and vandermonde-like linear systems with large displacement rank

The continuous closed form controllability Gramian and its inverse

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