Factoring Block Fiedler Companion Matrices

Corso, Gianna M. Del; Poloni, Federico; Robol, Leonardo; Vandebril, Raf

doi:10.1007/978-3-030-04088-8_7

Cited by 2 publications

(6 citation statements)

References 35 publications

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“…Thus, the experiments reveal that the bound k ≤ ⌈ n 2 ⌉ is tight. However, the computed unitary termQ does not coincide with the one obtained theoretically in (8). This fact should not be surprising, because the representation is not necessarily unique.…”

Section: The Fiedler Pentadiagonal Linearizationmentioning

confidence: 75%

“…Given a monic polynomial of degree n ,

p false(x false) = x^{n} - \sum_{i = 0}^{n - 1} p_{i} x^{i}

, define

F_{0} : = [\begin{array}{cc} p_{0} \\ I_{n - 1} \end{array}],

F_{i} : = [\begin{array}{ccc} I_{i - 1} \\ G false(p_{i} false) \\ I_{n - i - 1} \end{array}], G (p_{i}) : = [\begin{array}{cc} 0 & 1 \\ 1 & p_{i} \end{array}], i = 1, …, n - 1 .

Then, the matrix

F = F_{1} F_{3} … F_{2 ⌊ \frac{n}{2} ⌋ - 1} F_{0} F_{2} … F_{2 ⌊ \frac{n - 1}{2} ⌋}

is a linearization of p ( x ) and has the pentadiagonal structure depicted in Figure . From the other works, we know that these matrices are unitary‐plus‐rank‐ k with k at most

⌈ \frac{n}{2} ⌉

. In particular, one can observe that

F = Q + G<...>

…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…These matrices are the subject of our study: Our aim is to provide a theoretical characterization of these matrices in terms of their singular‐ or eigenvalues. It has been noted that several linearizations of (matrix) polynomials are low‐rank perturbations of unitary or Hermitian matrices (see, e.g., other works and the references therein). Computing the roots of these polynomials hence coincides with computing the eigenvalues of the associated matrices.…”

Section: Introductionmentioning

confidence: 99%

“…It has been noted that several linearizations of (matrix) polynomials are low rank perturbations of unitary or Hermitian matrices (see, e.g., [12,18,19,32,33] and the references therein). Computing the roots of these polynomials hence coincides with computing the eigenvalues of the associated matrices.…”

Section: Introductionmentioning

confidence: 99%

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When is a matrix unitary or Hermitian plus low rank?

Corso

Poloni

Robol

et al. 2019

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Summary Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing complexity once the matrix has been reduced, for instance, to tridiagonal or Hessenberg form. Recently, fast and reliable eigensolvers dealing with low‐rank perturbations of unitary and Hermitian matrices have been proposed. These structured eigenvalue problems appear naturally when computing roots, via confederate linearizations, of polynomials expressed in, for example, the monomial or Chebyshev basis. Often, however, it is not known beforehand whether or not a matrix can be written as the sum of a Hermitian or unitary matrix plus a low‐rank perturbation. In this paper, we give necessary and sufficient conditions characterizing the class of Hermitian or unitary plus low‐rank matrices. The number of singular values deviating from 1 determines the rank of a perturbation to bring a matrix to unitary form. A similar condition holds for Hermitian matrices; the eigenvalues of the skew‐Hermitian part differing from 0 dictate the rank of the perturbation. We prove that these relations are linked via the Cayley transform. Then, based on these conditions, we identify the closest Hermitian or unitary plus rank k matrix to a given matrix A, in Frobenius and spectral norm, and give a formula for their distance from A. Finally, we present a practical iteration to detect the low‐rank perturbation. Numerical tests prove that this straightforward algorithm is effective.

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Section: The Fiedler Pentadiagonal Linearizationmentioning

confidence: 75%

“…Given a monic polynomial of degree n ,

p false(x false) = x^{n} - \sum_{i = 0}^{n - 1} p_{i} x^{i}

, define

F_{0} : = [\begin{array}{cc} p_{0} \\ I_{n - 1} \end{array}],

F_{i} : = [\begin{array}{ccc} I_{i - 1} \\ G false(p_{i} false) \\ I_{n - i - 1} \end{array}], G (p_{i}) : = [\begin{array}{cc} 0 & 1 \\ 1 & p_{i} \end{array}], i = 1, …, n - 1 .

Then, the matrix

F = F_{1} F_{3} … F_{2 ⌊ \frac{n}{2} ⌋ - 1} F_{0} F_{2} … F_{2 ⌊ \frac{n - 1}{2} ⌋}

is a linearization of p ( x ) and has the pentadiagonal structure depicted in Figure . From the other works, we know that these matrices are unitary‐plus‐rank‐ k with k at most

⌈ \frac{n}{2} ⌉

. In particular, one can observe that

F = Q + G<...>

…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

When is a matrix unitary or Hermitian plus low rank?

Corso

Poloni

Robol

et al. 2019

Numerical Linear Algebra App

Self Cite

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show abstract

“…Note that when the actual eigenval- Table 7: Results on Fiedler pentadiagonal matrices [38] associated to scalar polynomials. As proved in [20,18] the rank-correction for dense polynomials is in general k = n/2 but it can be lower in the case the polynomial is sparse.…”

Section: Numerical Experimentsmentioning

confidence: 93%

Fast QR iterations for unitary plus low rank matrices

2019

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Some fast algorithms for computing the eigenvalues of a (block) companion matrix have recently appeared in the literature. In this paper we generalize the approach to encompass unitary plus low rank matrices of the form A = U + XY H where U is a general unitary matrix. Three important cases for applications are U unitary diagonal, U unitary block Hessenberg and U unitary in block CMV form. Our extension exploits the properties of a larger matrixÂ obtained by a certain embedding of the Hessenberg reduction of A suitable to maintain its structural properties. We show thatÂ can be factored as product of lower and upper unitary Hessenberg matrices possibly perturbed in the first k rows, and, moreover, such a data-sparse representation is well suited for the design of fast eigensolvers based on the QR iteration. The resulting algorithm is fast and backward stable.We first recall some basic properties of unitary matrices which play an important role in the derivation of our methods.Lemma 1 Let U be a unitary matrix of size n. Then rank (U (α, β)) = rank (U (J\α, J\β)) + |α| + |β| − n where J = {1, 2, . . . , n} and α and β are subsets of J. If α = {1, . . . , h} and β = J\α, then we have rank (U (1 : h, h+1 : n)) = rank (U (h+1 : n, 1 : h)), for all h = 1, . . . , n−1.

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Factoring Block Fiedler Companion Matrices

Cited by 2 publications

References 35 publications

When is a matrix unitary or Hermitian plus low rank?

When is a matrix unitary or Hermitian plus low rank?

Fast QR iterations for unitary plus low rank matrices

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