On block Householder algorithms for the reduction of a matrix to Hessenberg form

2003 ◽  
Author(s):  
A.A. Dubrulle
Keyword(s):  
Hessenberg Form

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.

scholarly journals The probabilities of extinction in a branching random walk on a strip

2020 ◽  
Vol 57 (3) ◽  
pp. 811-831
Author(s):  
Peter Braunsteins ◽  
Sophie Hautphenne
Keyword(s):  
Random Walk ◽  
Iterative Method ◽  
Fixed Points ◽  
Branching Processes ◽  
Extinction Probability ◽  
Branching Random Walk ◽  
Infinite Type ◽  
Hessenberg Form ◽  
Global Extinction ◽  
Countably Infinite

AbstractWe consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

scholarly journals Counter Example: The Algorithm of Determinant of Centrosymmteric Matrix based on Lower Hessenberg Form

2020 ◽  
Vol 6 (1) ◽  
pp. 20-29
Author(s):  
Nur Khasanah ◽  
Farikhin Farikhin
Keyword(s):  
Block Matrix ◽  
The Other ◽  
Numerical Examples ◽  
Block Matrices ◽  
Hessenberg Form ◽  
Counter Example ◽  
Centrosymmetric Matrix

The algorithm for computing determinant of centrosymmetric matrix has been evaluated before. This algorithm shows the efficient computational determinant process on centrosymmetric matrix by working on block matrix only. One of block matrix at centrosymmetric matrix appearing on this algorithm is lower Hessenberg form. However, the other block matrices may possibly appear as block matrix for centrosymmetric matrix’s determinant. Therefore, this study is aimed to show the possible block matrices at centrosymmetric matrix and how the algorithm solve the centrosymmetric matrix’s determinant. Some numerical examples for different cases of block matrices on determinant of centrosymmetric matrix are given also. These examples are useful for more understanding for applying the algorithm with different cases.

scholarly journals Acceleration of Hessenberg reduction for nonsymmetric matrix.

2021 ◽  
Author(s):  
Hesamaldin Nekouei
Keyword(s):  
Hybrid Algorithm ◽  
Eigenvalue Problems ◽  
Symmetric Matrices ◽  
Real Matrix ◽  
Great Efficiency ◽  
Matrix Size ◽  
Hessenberg Form ◽  
Large Matrix ◽  
Speed Up ◽  
Nonsymmetric Matrix

The worth of finding a general solution for nonsymmetric eigenvalue problems is specified in many areas of engineering and science computations, such as reducing noise to have a quiet ride in automotive industrial engineering or calculating the natural frequency of a bridge in civil engineering. The main objective of this thesis is to design a hybrid algorithm (based on CPU-GPU) in order to reduce general non-symmetric matrices to Hessenberg form. A new blocks method is used to achieve great efficiency in solving eigenvalue problems and to reduce the execution time compared with the most recent related works. The GPU part of proposed algorithm is thread based with asynchrony structure (based on FFT techniques) that is able to maximize the memory usage in GPU. On a system with an Intel Core i5 CPU and NVIDA GeForce GT 635M GPU, this approach achieved 239.74 times speed up over the CPU-only case when computing the Hessenberg form of a 256 * 256 real matrix. Minimum matrix order (n), which the proposed algorithm supports, is sixteen. Therefore, supporting this matrix size is led to have the large matrix order range.

scholarly journals Algorithms for Solving Nonhomogeneous Generalized Sylvester Matrix Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Ehab A. El-Sayed ◽  
Eid E. El Behady
Keyword(s):  
Explicit Solution ◽  
Second Order ◽  
New Method ◽  
Matrix Equations ◽  
Numerical Examples ◽  
First Order ◽  
Hessenberg Form ◽  
Sylvester Matrix ◽  
The Matrix ◽  
Sylvester Matrix Equations

This paper considers a new method to solve the first-order and second-order nonhomogeneous generalized Sylvester matrix equations AV+BW= EVF+R and MVF2+DV F+KV=BW+R, respectively, where A,E,M,D,K,B, and F are the arbitrary real known matrices and V and W are the matrices to be determined. An explicit solution for these equations is proposed, based on the orthogonal reduction of the matrix F to an upper Hessenberg form H. The technique is very simple and does not require the eigenvalues of matrix F to be known. The proposed method is illustrated by numerical examples.

On the reduction of matrix polynomials to Hessenberg form

2016 ◽  
Vol 31 ◽  
pp. 321-334 ◽  
Author(s):  
Thomas Cameron
Keyword(s):  
Matrix Polynomial ◽  
Rational Functions ◽  
Constructive Proof ◽  
Hessenberg Matrix ◽  
Original Matrix ◽  
Matrix Polynomials ◽  
Hessenberg Form ◽  
Special Cases ◽  
Square Matrix

It is well known that every real or complex square matrix is unitarily similar to an upper Hessenberg matrix. The purpose of this paper is to provide a constructive proof of the result that every square matrix polynomial can be reduced to an upper Hessenberg matrix, whose entries are rational functions and in special cases polynomials. It will be shown that the determinant is preserved under this transformation, and both the finite and infinite eigenvalues of the original matrix polynomial can be obtained from the upper Hessenberg matrix.

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