On the exact modelling of linear systems

2019 ◽  
Vol 37 (3) ◽  
pp. 730-751
Author(s):  
Georgia G Pechlivanidou ◽  
Nicholas P Karampetakis
Keyword(s):  
Inverse Problem ◽  
Differential Operator ◽  
Programming Language ◽  
Computer Algebra ◽  
Continuous Time ◽  
Computer Algebra System ◽  
Polynomial Matrix ◽  
Main Theme ◽  
Polynomial Matrices ◽  
System A

Abstract It is well known that given the continuous-time AutoRegressive representation $A\left ( \rho \right ) \beta \left ( t\right ) =0,$ where $\rho $ denotes the differential operator and $A\left ( \rho \right ) $ a regular polynomial matrix, we can always construct the smooth behaviour of this system, by using the finite zero structure of $A\left ( \rho \right ) $. The main theme of this work is to study the following inverse problem: given a specific smooth behaviour, find a family of regular polynomial matrices $A\left ( \rho \right ) $, such that the system $A\left ( \rho \right ) \beta \left ( t\right ) =0$ has exactly the prescribed behaviour. Following an idea coming from Antoulas & Willems (1993) and Willems (1986, 1991) we present an algorithm which solve this problem and can be easily implemented either in a computer programming language like C++ or in a computer algebra system like Mathematica.

The Maple computer algebra system: A review

1995 ◽  
Vol 10 (3) ◽  
pp. 329-337 ◽  
Author(s):  
John Hutton ◽  
James Hutton
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
System A

An Algorithm for the Inverse Solution of Geodesic Sailing without Auxiliary Sphere

2014 ◽  
Vol 67 (5) ◽  
pp. 825-844 ◽  
Author(s):  
Wei-Kuo Tseng
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Computational Cost ◽  
Inverse Solution ◽  
Series Expansions ◽  
Trigonometric Functions ◽  
Innovative Method

An innovative algorithm to determine the inverse solution of a geodesic with the vertex or Clairaut constant located between two points on a spheroid is presented. This solution to the inverse problem will be useful for solving problems in navigation as well as geodesy. The algorithm to be described derives from a series expansion that replaces integrals for distance and longitude, while avoiding reliance on trigonometric functions. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A finite difference method together with an innovative method of iteration that approximates Newton's method is presented which overcomes these shortcomings encountered for nearly antipodal regions. The method provided here, which does not involve an auxiliary sphere, was aided by the Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the users accuracy objectives and which are limited only by machine precisions.

Algebraic Methods for the Construction of Algebraic-Difference Equations With Desired Behavior

2018 ◽  
Vol 34 ◽  
pp. 1-17 ◽  
Author(s):  
Lazaros Moysis ◽  
Nicholas Karampetakis
Keyword(s):  
Difference Equations ◽  
Polynomial Matrix ◽  
Elementary Divisor ◽  
System Of Equations ◽  
Polynomial Matrices ◽  
Linear System Of Equations ◽  
System A ◽  
Auto Regressive ◽  
Algebraic Difference ◽  
Forward Operator

For a given system of algebraic and difference equations, written as an Auto-Regressive (AR) representation $A(\sigma)\beta(k)=0$, where $\sigma $ denotes the shift forward operator and $A\left( \sigma \right) $ a regular polynomial matrix, the forward-backward behavior of this system can be constructed by using the finite and infinite elementary divisor structure of $A\left( \sigma \right) $. This work studies the inverse problem: Given a specific forward-backward behavior, find a family of regular or non-regular polynomial matrices $A\left( \sigma \right) $, such that the constructed system $A\left( \sigma \right) \beta \left( k\right) =0$ has exactly the prescribed behavior. It is proved that this problem can be reduced either to a linear system of equations problem or to an interpolation problem and an algorithm is proposed for constructing a system satisfying a given forward and/or backward behavior.

scholarly journals Some useful MATHEMATICA teaching examples

2005 ◽  
Vol 18 (2) ◽  
pp. 329-344
Author(s):  
Milan Tasic ◽  
Predrag Stanimirovic ◽  
Ivan Stanimirovic ◽  
Marko Petkovic ◽  
Nebojsa Stojkovic
Keyword(s):  
Programming Language ◽  
Computer Algebra ◽  
Computer Algebra System

We show how a computer algebra system in MATHEMATICA can be used in several elementary courses in mathematics for students. We have also developed an application in programming language DELPHI for testing students in MATHEMATICA.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

Sign in / Sign up

Export Citation Format

Share Document