On the similarity to nonnegative and Metzler Hessenberg forms

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.

Acceleration of Hessenberg reduction for 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.

Acceleration of Hessenberg reduction for 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.

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

We 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.

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

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.

Algorithms for Solving Nonhomogeneous Generalized 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.

An Input/Output Efficient Algorithm for Hessenberg Reduction

Reduction of an [Formula: see text] nonsymmetric matrix to Hessenberg form which takes [Formula: see text] flops and [Formula: see text] I/Os is a major performance bottleneck in the computing of its eigenvalues. Usually to improve the performance, this Hessenberg reduction is computed in two steps: the first one reduces the matrix to a banded Hessenberg form, and the second one further reduces it to Hessenberg form by incorporating more matrix-matrix operations in the computation. We analyse the two steps of the Hessenberg reduction problem on the external memory model (of Aggarwal and Vitter) for their I/O complexities. We propose and analyse a tile based algorithm for the first step of the reduction and show that it takes [Formula: see text] I/Os. For the reduction of a banded Hessenberg matrix of bandwidth [Formula: see text] to Hessenberg form, we propose an algorithm, which uses tight packing of bulges, and requires only [Formula: see text] I/Os. Combining the results of the two steps of the reduction, we show that the Hessenberg reduction can be performed in [Formula: see text] I/Os, when [Formula: see text] is sufficiently large. To the best of our knowledge, the proposed algorithm is the first I/O optimal algorithm for Hessenberg reduction; the worst case I/O complexity matches the known lower bound of the problem.