Testing algorithms for linear matrix polynomial equations

We reduce the problem of computing the rank and a null-space basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree, d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension, n and degree, d. If the latter multiplication is done in MM(n,d)= O~(nωd operations, with ω the exponent of matrix multiplication over K, then the algorithm uses O~MM(n,d) operations in, K. For m x n matrices of rank r and degree d, the cost expression is O(nmr ω-2d). The soft-O notation O~ indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module.

Polynomial Control Toolbox: A New Aid for Teaching Linear Systems in MATLAB

This paper gives an algorithm for computing the inverses of polynomial matrices. The Gauss-Jordan inversion method, which is commonly used for the numerical inversion of constant matrices, can be applied to the inversion of polynomial matrices. It requires, however, operations with polynomials. Moreover, in the division of the operations, if the common factors in the divisor polynomials and the dividend polynomials are not eliminated, the resultant inverse contains polynomials of high degree in the numerators and the denominators. An algorithm given in this paper requires only operations with constant matrices. This algorithm gives an inverse in the minimal degree form if a polynomial matrix to be inverted is not a special form. It is shown that this is faster than existing ones. Also, it is an extension of the Souriau-Frame-Faddeev one. Some examples are illustrated to show the feasibility of the algorithm.

A persistent problem for hedge fund researchers presents itself in the form of inconsistent and diverse style classifications within and across database providers. In this article, single-manager hedge funds from the Hedge Fund Research (HFR) and Hedgefund.Net (HFN) databases are classified on the basis of a common factor extracted using the factor axis methodology. It is assumed that the returns of all sample hedge funds are attributable to a common factor that is shared across hedge funds within one classification, and a specific factor that is unique to a particular hedge fund. In contrast to earlier research and the application of principal component analysis (Fung and Hsieh [1997, 2001]), factor axis seeks to determine how much of the covariance in the dataset is attributable to common factors (commonality). Factor axis largely ignores the diagonal elements of the covariance matrix, and orthogonal factor rotation maximizes the covariance between hedge fund return series. In an iterative approach, common factors are extracted until all return series are described by one common and one specific factor. Prior to factor extraction, the series are tested for autoregressive moving-average processes, and the residuals of such models are used in further analysis to improve upon squared correlations as initial factor estimates. The methodology is applied to the July 1995 to June 2010 timeframe. The results indicate that the number of distinct style classifications is reduced in comparison with the arbitrary self-select classifications of the databases.

A CAD Package for Multivariable Control Systems Presented by Polynomial Matrices

Design of steady-state minimum variance controllers

In this paper the rate of stability of solutions of matrix polynomial equations of the typeA0+A1X+A2X2+...+A m X m =0 is studied. Particular attention is given to the case where the matrix polynomialL(λ):=A0+A1λ+A2λ2+...+A m λ m is weakly hyperbolic, i.e., for every non-zero vectorx the scalar polynomial 〈L(λ)x, x〉 has only real roots. Also the rate of stability of solutions of matrix quadratic equations of the typeXBX+XA-DX-C=0 is studied. Here the special case that is of interest to continuous-time optimal control theory, that is, the case whereB=B* is positive semidefinite andC=C*,A=−D*, is discussed in detail. The analogous theory for the discrete-time optimal control leads to the equation $$X = A^* XA + Q - (B^* XA)^* (R + B^* XB)^{ - 1} B^* XA,$$ and the rate of stability of solutions of this equation is also studied. Most of the problems are discussed in both real and complex settings.

Metamodel-based approaches are very often used for nonlinear optimization. The Successive Response Surface Method (SRSM) (Roux et al. in Int J Numer Methods Eng 42:517–534, 1998; Stander and Craig in Eng Comput 19:431–450, 2002) is one example for the application of such an approach. SRSM computes a smoothing approximation for each system response in each iteration based on a set of sample points generated in a defined domain around the current optimum. This domain is shifted and contracted in each iteration according to the resulting current best point. By default, a linear polynomial is used as the approximation function. A new Linear Interpolation Approach (LInA) has been developed to reduce the number of nonlinear analyses required during the optimization. In contrast to SRSM, LInA uses interpolating linear polynomials rather than regressions. Linear approximation is used to exclude artificial local minima between sample points. Due to the fact that no oversampling is required anymore, interpolation results in a lower number of nonlinear analyses without losing accuracy. A second reduction of the required analyses is achieved by avoiding a complete resampling in each iteration. It is assumed that the vectors defined from the current best point to the remaining sample points should be as orthogonal as possible. Therefore, LInA starts with initial sample points from a Koshal design of experiments (DOE) locally around the initial design point. After having computed the new best point based on the linear approximation, only a few existing sample points are deleted in each iteration and replaced by new ones. Complete QR decomposition is used to generate the new sample points in an orthogonal complementary subspace. First studies with two nonlinear structural optimization problems have shown that the LInA approach leads to a noteworthy reduction of the number of required nonlinear analyses in comparison to SRSM. Furthermore, LInA is compared to two kriging approaches using a subdomain similar as LInA does. LInA shows results comparable to kriging with respect to the number of analysis runs.

The features of generating the n th sequence of refined Fibonacci polynomial matrices of the m th order, the elements of which are Fibonacci polynomials of no higher than (m+n–2)th number, are considered. The obtained Fibonacci matrices allow finding both their determinants and inverse matrices suitable for matrix encryption of block data. It was found that even over the last decade a significant number of publications have been published, each of which substantiates different approaches to generating sequences of Fibonacci polynomial matrices and proves the feasibility of their use for encrypting block data. It was found that the use of such matrices as a separate procedure for protecting block data in the theory and practice of cryptography is extremely rare. The Fibonacci polynomial matrix sequences from the 2nd to the 5th order are analyzed, according to which shortcomings in the traditional approach to forming the structure of the elements of such matrices are revealed, first of all, the number (k) of its different elements, which are Fibonacci polynomials of the not higher than (n–1) th power. It is established that for Fibonacci matrices of any order (m) of such different Fibonacci polynomials there will be only k=3 , the structure of which will depend only on the number (n) of the sequence of the Fibonacci polynomial matrix. Such a small number is not only uninformative and easily analyzable by a cryptanalyst but also vulnerable to cryptanalysis. The structure of the elements of Fibonacci polynomial matrices is specified, the number of which already depends on the order of the matrix(m) and is k=m+1 . A method for generating a sequence of refined Fibonacci polynomial matrices has been developed, which consists in using a recurrent matrix relation, according to which the next polynomial matrix is formed by multiplying the variable x sequentially by the elements of the current matrix, adding the elements of the newly generated matrix to those of the previous matrix, after which all similar terms are grouped in the formed elements. The mechanisms for forming a sequence of 8 polynomial refined Fibonacci matrices from the 2nd to the 4th order, the elements of which are Fibonacci polynomials of no higher than (m+n–2) th power, have been presented, which made it possible to analyze not only the features of their construction, but also to understand the corresponding procedures for finding their determinants and inverse matrices. It was found that the proposed structure of the elements of the nth sequence of Fibonacci polynomial matrices of the mth order has an interesting property, according to which it is possible to avoid the use of the recurrent matrix relation, and to generate the corresponding Fibonacci polynomial matrices only by the numbers (m+n–2–j) th sequence of Fibonacci polynomials, the specific values of which depend on their location in the matrix and the number of its column, namely ∀j∈[0÷(m−1)] . Software has been developed that allows generating both sequences of refined Fibonacci polynomial matrices of the mth order, and finding their determinants and inverse polynomial matrices of a similar order. An example of the application of the matrix method of encrypting block data with a refined Fibonacci polynomial matrix is given, which allows the interested reader to understand the basic principle of encrypting both the initial message and decrypting the encrypted message.

The polynomial matrix using the block coefficient matrix representation auto-regressive moving average (referred to as the PM-ARMA) model is constructed in this paper for actively controlled multi-degree-of-freedom (MDOF) structures with time-delay through equivalently transforming the preliminary state space realization into the new state space realization. The PM-ARMA model is a more general formulation with respect to the polynomial using the coefficient representation auto-regressive moving average (ARMA) model due to its capability to cope with actively controlled structures with any given structural degrees of freedom and any chosen number of sensors and actuators. (The sensors and actuators are required to maintain the identical number.) under any dimensional stationary stochastic excitation.

A matrix Euclidean algorithm induced by state space realization

Linear Algebra and Polynomial Matrices

A polynomial matrix method for computing stable rational doubly coprime factorizations

An algorithm for calculating vapour-liquid equilibrium in multi-component systems is presented. The algorithm uses the Newton-Raphson method to solve the non-linear algebraic equations defining equilibrium. Iterative sequences have been organised to trace the progress of a calculation to ensure an efficient and progress of a calculation to ensure an efficient and successful calculation. Initialisation of the calculation procedure is automatic and relies only on parameters procedure is automatic and relies only on parameters characterising the components in a system. Important features discussed in this paper include a generalised flash calculation defined as a calculation of equilibrium conditions specified by a feed and any valid combination of two thermodynamic variables, and the incorporation of any suitable equations of state for evaluating liquid and vapour properties. An example problem is studied to illustrate the efficiency of the algorithm and a comparison is made between the Newton-Raphson method and the successive substitution method which is commonly used in phase-equilibrium calculations. Introduction Phase-equilibrium calculations are often required in the solution of problems involving simultaneous flow of liquid and vapour. Retrograde condensation in gas-condensate reservoirs and gas liberation in volatile oil reservoirs are examples of two-phase flow problems currently attracting interest. In simulating such flows, knowledge of the mutual influences between the flow of the reservoir fluid and the thermodynamics of equilibrium is essential. Some insight into the complex nature of these influences can be gained using compositional reservoir simulators. The basic function of these simulators is to determine numerical solutions of the partial differential equations governing the conservation partial differential equations governing the conservation of mass, momentum etc of each component in a system coupled with the algebraic equations defining vapour-liquid equilibrium throughout the system. During the calculation procedure, a solution of the equations defining equilibrium is required at every time wad space coordinate to evaluate, for example, local flow coefficients etc. The solution of these equations can be either computed internally in a simulator or supplied as multi-dimensional tables, for example, equilibrium ratios as functions of pressure, temperature and composition. In Both cases, a routine is required to determine vapour-liquid equilibrium, hereinafter abbreviated to VLE. The equations defining VLE are non-linear and in general can only be solved by iterative methods. Most commonly used is the method of successive substitution which is based on that proposed by Prausnitz and Chueh. However, this method exhibits slow or no convergence at conditions where the gradients of the thermodynamic variables are large, for example, in the vicinity of dew-, bubble- and critical point. To overcome convergence problems, Fussell and Yanosik developed an iterative sequence based on the Newton-Raphson method. A close examination of their paper revealed that their analysis can be extended. This prompted further investigations, the results of which are described in this paper. The contents of this paper are arranged as follows. The equations defining VLE are first presented. On inspection, it became apparent that a flash calculation can be generalised to a VLE calculation specified by a feed and any valid combination of two thermodynamic variables, An algorithm based on the Newton-Raphson method for generalised flash calculations is then described, followed by a description of an automatic starring procedure which relies only on parameters characterizing the components in a system. A system studied by Yarborough was selected as a basis on which the efficiency of the algorithm was assessed, and a comparison between the Newton-Raphson method and the successive substitution method was made. Phase-equilibrium calculations require equations of state to evaluate relevant properties pertaining to liquid and vapour. With several equations of state, for example ref. 4–7, available to an investigator, an algorithm which incorporates any suitable equations of state for liquid and vapour is desirable. The analysis has therefore been kept general throughout this investigation to allow for the insertion of any suitable equations of state. The experimental results of Yarborough were also used to assess the numerical results computed from the equation of Soave and that of Peng and Robinson.

This paper presents new delay-dependent robust linear matrix inequality (LMI) conditions for a robust H_∞ filter for discrete time systems affected by time-varying state delays and missing measurements. Our attention is focused on the analysis and design of robust H_∞ filters. LMI relaxations based on homogeneous polynomial matrices of an arbitrary degree are used to determine the state-space realization of the filter. The missing measurements are described by a binary switching sequence satisfying a Bernoulli distribution. Numerical examples are presented to illustrate the effectiveness and applicability of the proposed method.