A Package for Rational Matrices

rationalmatrices collects classes, methods and tools for creating and manipulating rational matrices, i.e. matrices whose entries are rational functions. It provides the fundamentals for the R package RLDM (Rational Linear Dynamic Models).

Installation

remotes::install_github("bfunovits/rationalmatrices")

Classes

There exist many different representations for rational matrices.

This package in particular deals with

• left matrix fraction representations (ARMA), implemented as lmfd class,
• right matrix fraction representations, implemented as rmfd class, and
• state space representations, implemented as stsp class.

As a special case, polynomial matrices, as polm class, and Laurent polynomial matrices, as lpolm class, which allow for negative powers are implemented as well. The latter class is special in the sense that even though it is a superset of polm object, it cannot be coerced to stsp, lfmd, or rmfd objects and interaction with other classes does not make sense in many cases.

The coefficients of the power series expansion of the rational function are stored as pseries objects and a collection of values of the rational function may be stored as zvalues objects.

The package offers tools to convert one representation into another (equivalent) representation. Some of these methods are quite sophisticated, e.g. pseries2stsp is based on the Ho-Kalman Realization Algorithm.

General Matrix Methods

The package provides "standard" generic functions for these objects:

• General methods, like print, dim, plot.
• Arithmetic operations, like addition and (matrix) multiplication.
• Extract parts of the matrix, transposition, (column or row) bind two or more matrices.

Special Methods

The package also provides some specific methods for rational matrices:

• Compute poles and zeroes of rational matrices.
• Check properties of the rational matrix, like stability and inverse stability.
• Normal Forms for polynomial matrices, like the Hermite normal form, the Smith form, and the Wiener-Hopf factorization.
• Check for left primeness with is.coprime()
• Compute the derivative (with respect to $z$)
• Tools related to state space representations, e.g. computation of controllability and observability matrices.

Authors

Wolfgang Scherrer and Bernd Funovits