Computations with infinite Toeplitz matrices and polynomials

Бини, Дарио Андреа; Gemignani, Luca; Meini, Beatrice

doi:10.1016/s0024-3795(01)00341-x

Cited by 74 publications

(51 citation statements)

References 62 publications

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“…The presentation of Algorithm 2 here is slightly different from that in [3]. We would have [3] exactly.…”

Section: Solving Hyperbolic Qepsmentioning

confidence: 97%

“…We would have [3] exactly. However, this difference has no effect on the sequences {X i } and {B i }.…”

Section: Solving Hyperbolic Qepsmentioning

confidence: 99%

“…If they do not satisfy (3), then the QEP is not hyperbolic. If they satisfy (3) and Q((λ n +λ n+1 )/2) < 0, then the QEP is hyperbolic; otherwise it is not hyperbolic.…”

Section: Algorithmmentioning

confidence: 99%

“…The two solvents can be found efficiently by applying an algorithm based on cyclic reduction (see [3], for example). The algorithm can be introduced as in [19].…”

Section: Solving Hyperbolic Qepsmentioning

confidence: 99%

“…Most of the analysis of this algorithm can be found in [3]. For general matrices A, B, C, the algorithm may break down (a singular B i is encountered).…”

Section: Solving Hyperbolic Qepsmentioning

confidence: 99%

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Algorithms for hyperbolic quadratic eigenvalue problems

Guo

Lancaster

2005

Math. Comp.

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Abstract. We consider the quadratic eigenvalue problem (or the QEP) (λ 2 A + λB + C)x = 0, where A, B, and C are Hermitian with A positive definite. The QEP is called hyperbolic if (x * Bx) 2 > 4(x * Ax)(x * Cx) for all nonzero x ∈ C n . We show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the QEP. A hyperbolic QEP is overdamped if B is positive definite and C is positive semidefinite. We show that a hyperbolic QEP (whose eigenvalues are necessarily real) is overdamped if and only if its largest eigenvalue is nonpositive. For overdamped QEPs, we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction. We also present a new measure for the degree of hyperbolicity of a hyperbolic QEP.

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“…The presentation of Algorithm 2 here is slightly different from that in [3]. We would have [3] exactly.…”

Section: Solving Hyperbolic Qepsmentioning

confidence: 97%

“…We would have [3] exactly. However, this difference has no effect on the sequences {X i } and {B i }.…”

Section: Solving Hyperbolic Qepsmentioning

confidence: 99%

“…If they do not satisfy (3), then the QEP is not hyperbolic. If they satisfy (3) and Q((λ n +λ n+1 )/2) < 0, then the QEP is hyperbolic; otherwise it is not hyperbolic.…”

Section: Algorithmmentioning

confidence: 99%

“…The two solvents can be found efficiently by applying an algorithm based on cyclic reduction (see [3], for example). The algorithm can be introduced as in [19].…”

Section: Solving Hyperbolic Qepsmentioning

confidence: 99%

“…Most of the analysis of this algorithm can be found in [3]. For general matrices A, B, C, the algorithm may break down (a singular B i is encountered).…”

Section: Solving Hyperbolic Qepsmentioning

confidence: 99%

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Algorithms for hyperbolic quadratic eigenvalue problems

Guo

Lancaster

2005

Math. Comp.

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Iterative and doubling algorithms for Riccati‐type matrix equations: A comparative introduction

Poloni

2020

GAMM-Mitteilungen

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We review a family of algorithms for Lyapunov-and Riccati-type equations which are all related to each other by the idea of doubling: they construct the iterate Q k = X 2 k of another naturally-arising fixed-point iteration (X h) via a sort of repeated squaring. The equations we consider are Stein equations X − A * X A = Q, Lyapunov equations A * X + X A + Q = 0, discrete-time algebraic Riccati equations X = Q + A * X(I + G X) −1 A, continuous-time algebraic Riccati equations Q + A * X + X A − X G X = 0, palindromic quadratic matrix equations A + Q Y + A * Y 2 = 0, and nonlinear matrix equations X + A * X −1 A = Q. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.

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A parallel radix-4 block cyclic reduction algorithm

Myllykoski

Rossi

2013

Numer. Linear Algebra Appl.

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A conventional block cyclic reduction algorithm operates by halving the size of the linear system at each reduction step, i.e., the algorithm is a radix-2 method. An algorithm analogous to the block cyclic reduction known as the radix-q PSCR method allows the use of higher radix numbers and is thus more suitable for parallel architectures as it requires fever reduction steps. This paper presents an alternative and more intuitive way of deriving a radix-4 block cyclic reduction method for systems with a coefficient matrix of the form tridiag{−I, D, −I}. This is done by modifying an existing radix-2 block cyclic reduction method. The resulting algorithm is then parallelized by using the partial fraction technique. The parallel variant is demonstrated to be less computationally expensive when compared to the radix-2 block cyclic reduction method in the sense that the total number of emerging sub-problems is reduced. The method is also shown to be numerically stable and equivalent to the radix-4 PSCR method. The numerical results archived correspond to the theoretical expectations.

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Computations with infinite Toeplitz matrices and polynomials

Cited by 74 publications

References 62 publications

Algorithms for hyperbolic quadratic eigenvalue problems

Algorithms for hyperbolic quadratic eigenvalue problems

Iterative and doubling algorithms for Riccati‐type matrix equations: A comparative introduction

A parallel radix-4 block cyclic reduction algorithm

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