0

0

0

4

# PolynomialMatrices

The PolynomialMatrices Julia package provides a support for calculations with univariate polynomial matrices, that is, matrices whose entries are univariate polynomials, such as

[ 3s²+2s+1     1 ]
P(s) =  |                |
[    2s        s ].

Tighly related to systems of linear ordinary differential or difference equations, these mathematical objects are useful in disciplines such as automatic control and signal processing.

## Polynomial matrix in Julia as an Array of polynomials

A matrix of univariate polynomials can be created in Julia by combining the functionality of the standard Julia Array (or Matrix) type and the Poly type provided by the Polynomials package. Our example polynomial matrix can be entered in Julia as

julia> using Polynomials

julia> M = [Poly([1, 2, 3]) Poly([1]); Poly([0,2]) Poly([0,1])]
2×2 Array{Polynomials.Poly{Int64,2}:
Poly(1 + 2⋅x + 3⋅x^2)  Poly(1)
Poly(2⋅x)              Poly(x)

The trouble with the resulting matrix is that the number of operations we can do over such objest is quite limited. If, for example, we want to check column-reducedness or perform triangularization, plain Julia has no functionality for that. And now PolynomialMatrices package enters the stage...

## Polynomial matrix in Julia as PolyMatrix

Using PolynomialMatrices package, we convert our array of polynomials into a PolyMatrix type as in

julia> using PolynomialMatrices

julia> P = PolyMatrix(M)
2×2 PolyArray{Int64,2}:
Poly(1 + 2⋅x + 3⋅x^2)  Poly(1)
Poly(2⋅x)              Poly(x)

for which many operations are now defined. For example, the matrix can be evaluated at a given value of its independent variable

julia> P(1)
2×2 Array{Int64,2}:
6  1
2  1

Or, a polynomial matrix P can be transformed into an upper triangular R by premultplication by unimodular U using

julia> R,U = rtriang(P)
(Poly{Float64}[Poly(-0.5454545454545455) Poly(-0.5454545454545454 + 0.6666666666666669*x + 0.5757575757575755*x^2 - 0.36363636363636365*x^3); Poly(0.0) Poly(-0.2357022603955159*x + 0.4714045207910319*x^2 + 0.7071067811865472*x^3)], Poly{Float64}[Poly(-0.5454545454545454 + 0.24242424242424268*x) Poly(0.4242424242424243 + 0.5757575757575756*x - 0.36363636363636365*x^2); Poly(-0.4714045207910317*x) Poly(0.23570226039551567 + 0.47140452079103196*x + 0.7071067811865472*x^2)])

julia> R
2×2 PolyMatrix{Float64,Array{Float64,2},Val{:x},2}:
Poly(-0.545455)  Poly(-0.545455 + 0.666667*x + 0.575758*x^2 - 0.363636*x^3)
Poly(0.0)        Poly(-0.235702*x + 0.471405*x^2 + 0.707107*x^3)

And some more computation with polynomial matrices is offered by the package.

## Polynomial matrix viewed (and entered) as a matrix polynomial

A very useful interpretation of a polynomial matrix is that of a matrix polynomial. That is, a polynomial whose coefficients are not just numbers but matrices. Our original example can thus be written as

[ 1  1 ]   [ 2  0 ]     [ 3  0 ]
P(s) =  |      | + |      | s + |      | s²
[ 0  0 ]   [ 2  1 ]     [ 0  0 ].

Hence, a natural way to enter a polynomial matrix in Julia is by entering a 3D array of coefficients matrices (of the corresponding matrix polynomial).

julia> A = zeros(Int,2,2,3);
julia> A[:,:,1] = [1 1;0 0];
julia> A[:,:,2] = [2 0;2 1];
julia> A[:,:,3] = [3 0;0 0];

julia> P = PolyMatrix(A,:s)
2×2 PolyArray{Int64,2}:
Poly(1 + 2⋅s + 3⋅s^2)  Poly(1)
Poly(2⋅s)              Poly(s)

## Polynomial matrix stored internally (and entered) as a dictionary

A PolyMatrix is implemented and stored as a dict, mapping from the powers (of the variables) to the coefficient matrices. It can thus also be constructed from a dict as in

julia> d = Dict(0=>[1 1;0 0], 1=>[2 0;2 1], 2=>[3 0;0 0]);
julia> P = PolyMatrix(d,:s)
2×2 PolyArray{Int64,2}:
Poly(1 + 2⋅s + 3⋅s^2)  Poly(1)
Poly(2⋅s)              Poly(s)

Individual coefficient matrices can be accessed accordingly --- the coefficient dictionary is obtained using coeffs function and the individual coefficient matrices are accessed using keys. For example, the coefficient matrix with the 1st power of the variable can be obtained using

julia> C = coeffs(P)
DataStructures.SortedDict{Int64,Array{Int64,2},Base.Order.ForwardOrdering} with 3 entries:
0 => [1 1; 0 0]
1 => [2 0; 2 1]
2 => [3 0; 0 0]

julia> C[1]
2×2 Array{Int64,2}:
2  0
2  1

## List of functions for PolyMatrix objects

The functions for polynomial matrices implemented in PolynomialMatrices package are:

### Inquiry about parameters of the polynomial matrix

• coeffs: dictionary of coefficient matrices, keys are the powers
• degree, col_degree, row_degree: degree, column and row degrees
• variable, vartype: polynomial corresponding to the variable, symbol of the variable
• high_col_deg_matrix, high_row_deg_matrix: coefficient matrices corresponding to leading column and row degrees, respectively.

### Analysis

• is_col_proper, is_row_proper: checking if the matrix is column- and row-proper (also column- and row-reduced)

### Reductions, conversions

• colred, rowred: column and row degree reduction of a polynomial matrix
• ltriang, rtriang: conversion to a lower left and uppper right triangular polynomial matrix
• hermite: conversion to hermite form.

## Future plans

• det for computing the determinant of a polynomial matrix
• roots for computing the roots (or zeros) of a polynomial matrix
• some state space realization from a fraction of two polynomial matrices
• ...
• separate (but related) packages PolynomialMatrixEquations and PolynomialMatrixFactorizations are planned.

## PolynomialMatrices package is restricted to univariate polynomials only, for multivariate polynomials look elsewhere

As it is implemented now, PolyMatrix objects do not allow for mixing different variables --- a PolyMatrix object can only operate together with PolyMatrix objects with the same variable. For multivariate polynomials, you may want to check MultivariatePolynomials package.

10/01/2019

9 months ago

131 commits