Generalized Rouche's theorem and its application to multivariate autoregressions

Monden, Yoshimi; Arimoto, Suguru

doi:10.1109/tassp.1980.1163469

Cited by 23 publications

(10 citation statements)

References 9 publications

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“…4 with different attenuation filters. Instead of the delay proportional design in (41), all one-pole filters have the same target frequency response corresponding to an average delay length. As a consequence, the decay time of the neighboring modes are largely different, while the overall shape still follows the target reverberation time.…”

Section: A Attenutationmentioning

confidence: 99%

Modal Decomposition of Feedback Delay Networks

Schlecht

Habets

2019

IEEE Trans. Signal Process.

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Feedback delay networks (FDNs) belong to a general class of recursive filters which are widely used in sound synthesis and physical modeling applications. We present a numerical technique to compute the modal decomposition of the FDN transfer function. The proposed pole finding algorithm is based on the Ehrlich-Aberth iteration for matrix polynomials and has improved computational performance of up to three orders of magnitude compared to a scalar polynomial root finder. We demonstrate how explicit knowledge of the FDN's modal behavior facilitates analysis and improvements for artificial reverberation. The statistical distribution of mode frequency and residue magnitudes demonstrate that relatively few modes contribute a large portion of impulse response energy.

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Section: A Attenutationmentioning

confidence: 99%

Modal Decomposition of Feedback Delay Networks

Schlecht

Habets

2019

IEEE Trans. Signal Process.

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“…In order to extend this criterion to the case of matrix polynomials we need a generalisation of the Rouché theorem to matrix polynomials. We report the following result of [34] which we rephrase in a simpler way better suited for our problem.…”

Section: Choosing Initial Approximationsmentioning

confidence: 99%

Solving polynomial eigenvalue problems by means of the Ehrlich–Aberth method

Бини

Noferini

2013

Linear Algebra and its Applications

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Given the n × n matrix polynomial P (x) = k i=0 P i x i , we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial det P (x), is treated in polynomial form rather than in matrix form by means of the Ehrlich-Aberth iteration. The main computational issues are discussed, namely, the choice of the starting approximations needed to start the Ehrlich-Aberth iteration, the computation of the Newton correction, the halting criterion, and the treatment of eigenvalues at infinity. We arrive at an effective implementation which provides more accurate approximations to the eigenvalues with respect to the methods based on the QZ algorithm. The case of polynomials having special structures, like palindromic, Hamiltonian, symplectic, etc., where the eigenvalues have special symmetries in the complex plane, is considered. A general way to adapt the Ehrlich-Aberth iteration to structured matrix polynomial is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.

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“…The disadvantage is that all the coefficients need to be multiplied by A −1 k , which could be costly. (2) A version of Theorem 3.3 (applied to A −1 k P ) was also obtained in [3] by using a different generalization of Rouché's theorem from [16] and [20], which limits that result to the spectral norm. Because it is based on Theorem 3.2, Theorem 3.3 holds for any matrix norm induced by a vector norm.…”

Section: Generalized Rouché and Pellet Theoremsmentioning

confidence: 99%