Eigenvalue computation of totally nonnegative upper Hessenberg matrices based on a variant of the discrete hungry Toda equation

2015 ◽  
Author(s):  
Ryo Sumikura ◽  
Akiko Fukuda ◽  
Emiko Ishiwata ◽  
Yusaku Yamamoto ◽  
Masashi Iwasaki ◽  
...  
Keyword(s):  
Toda Equation ◽  
Eigenvalue Computation ◽  
Hessenberg Matrices ◽  
Discrete Hungry Toda Equation

Periodic convergence in the discrete hungry Toda equation

2018 ◽  
Vol 51 (34) ◽  
pp. 344001
Author(s):  
You Takahashi ◽  
Masashi Iwasaki ◽  
Akiko Fukuda ◽  
Emiko Ishiwata ◽  
Yoshimasa Nakamura
Keyword(s):  
Toda Equation ◽  
Discrete Hungry Toda Equation

On a shifted LR transformation derived from the discrete hungry Toda equation

2012 ◽  
Vol 170 (1) ◽  
pp. 11-26 ◽  
Author(s):  
Akiko Fukuda ◽  
Yusaku Yamamoto ◽  
Masashi Iwasaki ◽  
Emiko Ishiwata ◽  
Yoshimasa Nakamura
Keyword(s):  
Toda Equation ◽  
Discrete Hungry Toda Equation

Error analysis for matrix eigenvalue algorithm based on the discrete hungry Toda equation

2012 ◽  
Vol 61 (2) ◽  
pp. 243-260 ◽  
Author(s):  
Akiko Fukuda ◽  
Yusaku Yamamoto ◽  
Masashi Iwasaki ◽  
Emiko Ishiwata ◽  
Yoshimasa Nakamura
Keyword(s):  
Error Analysis ◽  
Toda Equation ◽  
Matrix Eigenvalue ◽  
Discrete Hungry Toda Equation

scholarly journals On the similarity to nonnegative and Metzler Hessenberg forms

2021 ◽  
Vol 10 (1) ◽  
pp. 1-8
Author(s):  
Christian Grussler ◽  
Anders Rantzer
Keyword(s):  
Standard Form ◽  
Complete Characterization ◽  
Nonnegative Matrices ◽  
Positive Systems ◽  
Third Order ◽  
Hessenberg Form ◽  
Hessenberg Matrices ◽  
Open Question ◽  
Order Continuous

Abstract We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions n 3, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of n = 3 also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if n 4. In particular, this provides the first standard form for controllable third order continuous-time positive systems via a positive controller-Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions n 5 remain similar to Metzler Hessenberg matrices.

On the Full-Discrete Extended Generalised q-Difference Toda System

2017 ◽  
Vol 72 (8) ◽  
pp. 703-709
Author(s):  
Chuanzhong Li ◽  
Anni Meng
Keyword(s):  
Difference Equation ◽  
Hamiltonian Structure ◽  
Soliton Solutions ◽  
Toda Equation ◽  
Lax Pairs ◽  
Toda System ◽  
Toda Hierarchy

AbstractIn this paper, we construct a full-discrete integrable difference equation which is a full-discretisation of the generalised q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of an extended generalised full-discrete q-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the extended full-discrete generalised q-Toda hierarchy are given.

scholarly journals Representation of doubly infinite matrices as non-commutative Laurent series

2017 ◽  
Vol 5 (1) ◽  
pp. 250-257 ◽  
Author(s):  
María Ivonne Arenas-Herrera ◽  
Luis Verde-Star
Keyword(s):  
Laurent Series ◽  
Infinite Matrices ◽  
General Properties ◽  
Hessenberg Matrices ◽  
Diagonal Representation

Abstract We present a new way to deal with doubly infinite lower Hessenberg matrices based on the representation of the matrices as the sum of their diagonal submatrices. We show that such representation is a simple and useful tool for computation purposes and also to obtain general properties of the matrices related with inversion, similarity, commutativity, and Pincherle derivatives. The diagonal representation allows us to consider the ring of doubly infinite lower Hessenberg matrices over a ring R as a ring of Laurent series in one indeterminate, with coefficients in the ring of R-valued sequences that don’t commute with the indeterminate.

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