On effective criterion of stability of partial indices for matrix polynomials

Adukova, N. V.; Adukov, V. M.

doi:10.1098/rspa.2020.0012

Cited by 3 publications

(6 citation statements)

References 16 publications

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“…s=1, r=0. Then, by the above-mentioned theorem 3.1 from [23], the left (right) partial indices of scriptP1false(zfalse) are stable if and only if rank T−s=false(ϰ+1false)p−ϰfalse(rank Ts=false(ϰ+1false)p−ϰfalse), that is: rank T−1=4 (rank T1=4). Thus, we can conclude that τ-indices are λ1τ=λ2τ=1 and ρ1τ=ρ2τ=1.…”

Section: On Numerical Realization Of the Explicit Algorithmmentioning

confidence: 99%

“…The Laurent coefficients of −1 − (z) are found directly from the recurrence formulae (2.4) In order to verify the stability criterion for matrix polynomials (theorem 3.1 in [23]), we need to compute the τ -ranks of matrices…”

Section: On Numerical Realization Of the Explicit Algorithmmentioning

confidence: 99%

“…In order to verify the stability criterion for matrix polynomials (theorem 3.1 in [23]), we need to compute the τ-ranks of matrices T−1=(c−1c−2c0c−1c1c0c2c1)1emand1emT1=(c1c0c−1c−2c2c1c0c−1).…”

Section: On Numerical Realization Of the Explicit Algorithmmentioning

confidence: 99%

“…Hence, all subsequent steps for the explicit factorization scriptP1false(zfalse) may be unstable. To apply the effective criterion of stability for partial indices [23], we have to verify that a certain matrix with float point entries, built using the matrix scriptP1false(zfalse), has the full rank. In turn, this is an ill-condition problem itself if the condition number of the matrix is large enough ([24], ch.…”

Section: Introductionmentioning

confidence: 99%

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An explicit Wiener–Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package

2022

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We discuss an explicit algorithm for solving the Wiener–Hopf factorization problem for matrix polynomials. By an exact solution of the problem, we understand the one constructed by a symbolic computation. Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material.

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Section: On Numerical Realization Of the Explicit Algorithmmentioning

confidence: 99%

Section: On Numerical Realization Of the Explicit Algorithmmentioning

confidence: 99%

Section: On Numerical Realization Of the Explicit Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An explicit Wiener–Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package

2022

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“…At present, in the general case, it is not known even when a matrix function admits a canonical factorization, that is, a factorization with the zero partial indices. The effective stability criteria for the indices are known for triangular second order matrix functions [12] and for Laurent matrix polynomials [13].…”

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confidence: 99%

Normalization of Wiener - Hopf factorization for $2\times 2$ matrix functions and its application

Adukov

2022

Ufimsk. Mat. Zh.

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In this work we cover a gap existing in the general Wiener-Hopf factorization theory of matrix functions. It is known that factors in such factorization are not determined uniquely and in the general case, there are no known ways of normalizing the factorization ensuring its uniqueness. In the work we introduce the notion of 𝑃 -normalized factorization. Such normalization ensures the uniqueness of the Wiener-Hopf factorization and gives an opportunity to find the Birkhoff factorization. For the second order matrix function we show that the factorization of each matrix function can be reduced to the 𝑃 -normalized factorization. We describe all possible types of such factorizations, obtain the conditions ensuring the existence of such normalization and find the form of the factors for such type of the normalization. We study the stability of 𝑃 -normalization under a small perturbation of the initial matrix function. The results are applied for specifying the Shubin theorem on the continuity of the factors and for obtaining the explicit estimates of the absolute errors of the factors for an approximate factorization.

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Stability of Factorization Factors of the Canonical Factorization of Wiener–hopf Matrix Functions

Адукова¹,

Дильман²

2021

MMPh

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The problem of Wiener–Hopf factorization of matrix functions is one of the most demanded problems of mathematical analysis. However, its application is constrained by the fact that at the present time, there are no methods for constructively constructing factorization in the general case. In addition, the problem is far and by unstable, that is, a small perturbation of the original matrix function can lead to a change in the integer invariants of the problem (partial indices), and the factorization factors of the original and perturbed matrix functions may not be close. This means that the dependence of factors on perturbation is not continuous. The situation is complicated by the fact that factorization factors are found in a non-unique way, and therefore, before comparing factorizations, they need to be normalized. This problem is also not solved in the general case. In the well-known theorem of M.A. Shubin, the normalization problem is bypassed in the following way: it is proved that if the original and perturbed matrix functions have the same sets of partial indices, then their factorizations with close factorization factors exist. It is clear that in this case it is impossible to assess the degree of their proximity efficiently. In the proposed work, the theorem of M.A. Shubin is refined for the case when the original matrix function admits a canonical factorization. In this case, it is indicated how the canonical factorizations of two sufficiently close matrix functions should be normalized so that their factorization factors are also close enough. The main result of the work is to obtain explicit estimates, in terms of factorization of the original matrix function, for the absolute error in the approximate calculation of factors. The estimates are obtained by using the Toeplitz operator technique.

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On effective criterion of stability of partial indices for matrix polynomials

Cited by 3 publications

References 16 publications

An explicit Wiener–Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package

An explicit Wiener–Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package

Normalization of Wiener - Hopf factorization for $2\times 2$ matrix functions and its application

Stability of Factorization Factors of the Canonical Factorization of Wiener–hopf Matrix Functions

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