A fast algorithm for evaluating the matrix polynomial I+A+. . .+A/sup N-1/

1992 ◽  
Vol 39 (4) ◽  
pp. 299-300 ◽  
Author(s):  
L. Lei ◽  
T. Nakamura
Keyword(s):  
Fast Algorithm ◽  
Matrix Polynomial ◽  
The Matrix

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals Linearization schemes for Hermite matrix polynomials

2015 ◽  
Author(s):  
Nikta Shayanfar ◽  
Heike Fassbender
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Matrix Polynomial ◽  
Multiple Input Multiple Output ◽  
Matrix Pencil ◽  
Least Square ◽  
Coupled Systems ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
The Matrix

The polynomial eigenvalue problem is to find the eigenpair of $(\lambda,x) \in \mathbb{C}\bigcup \{\infty\} \times \mathbb{C}^n \backslash \{0\}$ that satisfies $P(\lambda)x=0$, where $P(\lambda)=\sum_{i=0}^s P_i \lambda ^i$ is an $n\times n$ so-called matrix polynomial of degree $s$, where the coefficients $P_i, i=0,\cdots,s$, are $n\times n$ constant matrices, and $P_s$ is supposed to be nonzero. These eigenvalue problems arise from a variety of physical applications including acoustic structural coupled systems, fluid mechanics, multiple input multiple output systems in control theory, signal processing, and constrained least square problems. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Such methods convert the eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploit and preserve the structure and properties of the original eigenvalue problem. The linearizations have been extensively studied with respect to the basis that the matrix polynomial is expressed in. If the matrix polynomial is expressed in a special basis, then it is desirable that its linearization be also expressed in the same basis. The reason is due to the fact that changing the given basis ought to be avoided \cite{H1}. The authors in \cite{ACL} have constructed linearization for different bases such as degree-graded ones (including monomial, Newton and Pochhammer basis), Bernstein and Lagrange basis. This contribution is concerned with polynomial eigenvalue problems in which the matrix polynomial is expressed in Hermite basis. In fact, Hermite basis is used for presenting matrix polynomials designed for matching a series of points and function derivatives at the prescribed nodes. In the literature, the linearizations of matrix polynomials of degree $s$, expressed in Hermite basis, consist of matrix pencils with $s+2$ blocks of size $n \times n$. In other words, additional eigenvalues at infinity had to be introduced, see e.g. \cite{CSAG}. In this research, we try to overcome this difficulty by reducing the size of linearization. The reduction scheme presented will gradually reduce the linearization to its minimal size making use of ideas from \cite{VMM1}. More precisely, for $n \times n$ matrix polynomials of degree $s$, we present linearizations of smaller size, consisting of $s+1$ and $s$ blocks of $n \times n$ matrices. The structure of the eigenvectors is also discussed.

scholarly journals Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

2019 ◽  
Vol 35 ◽  
pp. 116-155
Author(s):  
Biswajit Das ◽  
Shreemayee Bora
Keyword(s):  
Eigenvalue Problem ◽  
Matrix Polynomial ◽  
Vector Spaces ◽  
Backward Error ◽  
Rectangular Matrix ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
Stable Manner ◽  
The Matrix ◽  
Matrix Pencils

The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

scholarly journals Using Graph Partitioning for Scalable Distributed Quantum Molecular Dynamics

Algorithms ◽  
2019 ◽  
Vol 12 (9) ◽  
pp. 187 ◽  
Author(s):  
Hristo N. Djidjev ◽  
Georg Hahn ◽  
Susan M. Mniszewski ◽  
Christian F. A. Negre ◽  
Anders M. N. Niklasson
Keyword(s):  
Graph Partitioning ◽  
Matrix Polynomial ◽  
Quantum Molecular Dynamics ◽  
Matrix Polynomials ◽  
Quantum Mechanical Description ◽  
The Matrix ◽  
Partitioning Problem ◽  
The Individual ◽  
Multi Body ◽  
Mathematical Justification

The simulation of the physical movement of multi-body systems at an atomistic level, with forces calculated from a quantum mechanical description of the electrons, motivates a graph partitioning problem studied in this article. Several advanced algorithms relying on evaluations of matrix polynomials have been published in the literature for such simulations. We aim to use a special type of graph partitioning to efficiently parallelize these computations. For this, we create a graph representing the zero–nonzero structure of a thresholded density matrix, and partition that graph into several components. Each separate submatrix (corresponding to each subgraph) is then substituted into the matrix polynomial, and the result for the full matrix polynomial is reassembled at the end from the individual polynomials. This paper starts by introducing a rigorous definition as well as a mathematical justification of this partitioning problem. We assess the performance of several methods to compute graph partitions with respect to both the quality of the partitioning and their runtime.

SYNTHESIS OF LINEAR MULTICHANNEL REGULATORS USING STRUCTURAL TRANSFORMATIONS

2017 ◽  
pp. 7-20 ◽  
Author(s):  
Kurbonmurod Mullomirakovich Bobobekov ◽  
Alexander Aleksandrovich Voevoda
Keyword(s):  
Dynamical System ◽  
Matrix Polynomial ◽  
Polynomial Matrix ◽  
Structural Transformations ◽  
Polynomial Expansion ◽  
Mass System ◽  
Channel System ◽  
The Matrix ◽  
Left And Right ◽  
Modal Method

The paper considers the problem of using left and right matrix polynomial decompositions in the synthesis of regulators using modal method. The left matrix polynomial expansion is assumed. In the synthesis of regulators, the complexity of computations essentially increases when the inequality of the column/row degrees of the polynomial matrix of the "denominator" of the object. After transition from the left matrix polynomial expansion tothe right one, the authors propose to accomplish a number of unimodal transformations of the matrix of the object’s "denominator", which will result in bringing unimodal matrices into different sections of a structure chart. In this case, equivalent transformations are performed. They "equalize" the column/row degrees of the polynomial matrix of the "denominator" of the object (in order to retain equivalence there are introduced at once two unimodal matrices, opposite to each other). The proposed technique is illustrated by the example of a two-channel system corresponding to a dynamical system, which, in its turn is corresponding to a three-mass system with springs.

Standard pairs and existence of symmetric multiscaling functions

2019 ◽  
Vol 18 (02) ◽  
pp. 1950057
Author(s):  
A. T. Mithun ◽  
M. C. Lineesh
Keyword(s):  
Matrix Polynomial ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Standard Pair ◽  
Compactly Supported ◽  
Matrix Polynomials ◽  
The Matrix ◽  
Symbol Function ◽  
Necessary And Sufficient ◽  
Multiscaling Function

Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed.

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