# Analytical aspects of matrix interpolation problems and its applications

Dharm Prakash Singh<EMAIL_ADDRESS>Amit Ujlayan  Department of
Applied Mathematics, School of Vocational Studies and Applied Sciences,
Gautam Buddha University, Gr. Noida - 201312, India<EMAIL_ADDRESS>
###### Abstract

In this paper, the $mn$-dimensional space of tensor-product polynomials of two
variables, of degree at most $(m-1)+(n-1)$, is considered. A theory of two-
variate polynomials is developed by establishing the algebra and basic
algebraic properties with respect to the usual addition, scalar
multiplication, and a newly defined algebraic operation in the considered
space. Further, the existence of the considered space is established with
respect to the matrix interpolation problem (MIP), $P(i,j)=a_{ij}$ for all
$1\leq i\leq m$, $1\leq j\leq n$, corresponds to a given matrix
$(a_{ij})_{m\times n}$ in the space of $m\times n$ order real matrices. The
poisedness of the MIP is proved and three formulae are presented to construct
the respective polynomial in the considered space. After that, using
construction formulae, a polynomial map from the space of $m\times n$ order
real matrices to the considered space is defined. Some properties of the
polynomial map are investigated and some isomorphic structures between the
spaces are installed. It is proved that the considered space is isomorphic to
the space of $m\times n$ order real matrices with respect to the algebra
structure. The polynomials in the considered space with respect to the MIP’s
for the given matrices also preserve the geometric properties of the matrices
such as transpose, symmetry, and skew-symmetry. Some examples are included to
demonstrate and verify the results.

###### keywords:

Two-variate polynomial interpolation , polynomial algebra , algebraic
structures Mathematics Subject Classification (2010): 41A05 , 41A10 , 41A63 ,
65F99 , 15A99

##  1 Introduction

Let $\mathbb{R}^{m\times n}$ denotes the space of $m\times n$ order real
matrices and $(a_{ij})_{m\times n}$ denotes the matrix $A=(a_{ij})_{m\times
n}$, where $m,n\in\mathbb{N}$, [1, 2]. Again, let $\Pi^{d}$ be the space of
all $d$-variate polynomials and $\Pi_{N}^{d}$ be
${{d+N}\choose{N}}$-dimensional subspace of $\Pi^{d}$, of degree at most $N$,
over the field $\mathbb{F}$, where $d,N\in\mathbb{N}$, [3, 4]. For a given
finite linearly independent set $\Theta$ of functionals and an associated
vector $Y=(y_{\theta}:\theta\in\Theta)$ of prescribed values, find a
polynomial $p$ in $\Pi^{d}$ such that

$\Theta p=Y,\quad i.e.,\quad\theta p=y_{\theta},\quad\theta\in\Theta,$ (1)

represents general form of the polynomial interpolation problem, [5, 3].

Let $\mathcal{S}$ be a subspace of $\Pi^{d}$, the polynomial interpolation
problem (1) with respect to the set $\Theta$ is said to be _poised_ or
_correct_ for $\mathcal{S}$, if for any $Y\in\mathbb{F}^{\Theta}$ there exists
a unique $p\in\mathcal{S}$ such that $\Theta p=Y$, [5]. In other words, the
interpolation problem (1) with respect to a finite set $\Theta$ of the
cardinality $q$, $q\in\mathbb{N}$ is poised in $\mathcal{S}$, if and only if,
$q=dim(\mathcal{S})$ and the determinant of the sample matrix, [6, 7],

$\Theta\mathcal{B}=\begin{bmatrix}\theta
p:\theta\in\Theta,p\in\mathcal{B}\end{bmatrix}\in\mathbb{F}^{{\Theta}\times\mathcal{B}}$
(2)

is non-zero for any choice of basis $\mathcal{B}$ of $\mathcal{S}$. The
problem is said to be _singular_ if $det(\Theta\mathcal{B})=0$ for every
choice of the set $\Theta$, _regular_ if $det(\Theta\mathcal{B})=0$ for no
choice of the set $\Theta$, and almost regular if $det(\Theta\mathcal{B})=0$
only on a subset of $\mathbb{R}^{d}$ of measure zero, [5, 6, 7, 8, 9].

###  1.1 Motivation and the considered problem

Nowadays, a matrix is the majorly used data structure in computer-oriented
models and algorithms of science and engineering. Particularly, in computer
graphics [10], signal processing [11], image processing [10, 11], computer-
aided designs in control systems [12, 13], thermal engineering [14] to name a
few. The generation of the respective three-dimensional smooth interpolating
surface is one of the important tools for analysis, comparison, and
interpretation. In medical imaging, the surface interpolation plays a very
important role in computer-assisted surgical planning and medical diagnosis,
[15].

The matrices and matrix algebra are used in a very large-scale to solve the
computational problems as their solutions constitute an important and major
section of scientific, engineering, and numerical computations on the modern
computers. The computational problems build a challenging and application-
oriented segment of effectual research. In last decades, an intense research
has been witnessed to develop efficient numerical algorithms for the matrix
computations and a rapid progress have been noticed in the formation of
efficient and effective algorithms for the polynomials in support of
algebraic, symbolic, numerical, and scientific computations, [16, 17, 18].

In this regard, considering the matrix interpolation problem [19], defined as
follows:

###### Definition 1.1

Let $(a_{ij})\in\mathbb{R}^{m\times n}$ be a given matrix. For a given
subspace $\mathcal{P}$ of $\Pi^{2}$ and the set
$\mathcal{D}=\left\\{(i,j):i=1,2,\ldots,m,j=1,2,\ldots,n\right\\}$ of pairwise
distinct interpolation points or nodes, find a polynomial $P\in\mathcal{P}$,
such that

$P(i,j)=a_{ij}\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,n$ (*)

is defined as _Matrix Interpolation Problem (MIP)_.

For the given matrix $(a_{ij})\in\mathbb{R}^{m\times n}$, the MIP (* ‣ 1.1)
can not be poised in $\Pi_{N}^{2}$ for almost every $N\in\mathbb{N}$, in view
of the fact that,

  1. (i)

if $mn=dim\Pi_{N}^{2}$, the determinant of the respective sample matrix is
zero.

  2. (ii)

if $mn\neq dim\Pi_{N}^{2}$, the necessary condition of $\frac{(N+1)(N+2)}{2}$
interpolation points does not hold (see section 1.1 in [19]).

###  1.2 Brief review of literature

Let $k$ be a positive integer and
$\Theta=\left\\{x_{1},x_{2},\ldots,x_{k}\right\\}\subset\mathbb{R}^{d}$ be the
set of pairwise disjoint points. Then, for a given subspace
$\mathcal{H}\subset\Pi^{d}$ and some given constants
$f_{1},f_{2},\ldots,f_{k}$ in $\mathbb{R}$, the problem to find a polynomial
$p\in\mathcal{H}$ with respect to $\Theta$, such that

$\displaystyle p(y_{i})=f_{i}\textrm{ for all }i=1,2,...,k$ (3)

can be defined as Lagrange interpolation problem. The interpolation points
$x_{i}$, $i=1,2,...,k$ are called interpolation sites or nodes and
$\mathcal{H}$ is known as the interpolation space, [3, 8].

For the given set $\Theta$ and a subspace $\mathcal{H}$, if the problem (3) is
poised in $\mathcal{H}$, then the required polynomial $p\in\mathcal{H}$ can be
constructed using the interpolation formulae discussed in [4, 6].

In general, the problem (3) need not be _poised_ in a given subspace
$\mathcal{H}$ of $\Pi_{N}^{d}$, with respect to the given set $\Theta$ of
dimension ${{d+N}\choose{N}}$. As in the case of $d\geq 2$, for a given
${{d+N}\choose{N}}$-dimensional subspace, of a certain total degree, the
poisedness of the problem does not independent of the position and geometry
(or configuration) of the data points. Also, the basic structure and nature of
the interpolation changes utterly and the existence of _Haar Spaces_ of
dimension higher than one vanishes, [3, 4, 5, 8].

However, there always exist at least one set $\Theta$ of dimension
${{d+N}\choose{N}}$ in $\mathbb{R}^{d}$ such that the interpolation problem
(3) can poised in $\Pi_{N}^{d}$, [4, 5, 8]. Again, the Kergin interpolation
[20] insures that, there always exist at least one subspace of $\Pi_{r-1}^{d}$
in which the interpolation problem (3) can be poised for some arbitrarily
known set $\Theta$ of dimension $r$, $r\in\mathbb{N}$. Hence, the Lagrange
interpolation problem (3) can never be singular, [8]. The work in this
manuscript focuses on a special case of two-variate interpolation problem.
Therefore, we restrict our self to $d=2$.

For $d=2$, the MIP (* ‣ 1.1) is a class of the Lagrange interpolation
problems. Therefore, due to Kergin, there must exist at at least one
$mn$-dimensional subspace of $\Pi^{2}$ in which the MIP (* ‣ 1.1) can always
be poised with respect to the set $\mathcal{D}$ for a given matrix
$a_{ij}\in\mathbb{R}^{m\times n}$. In [19], the constructive existence of two
$mn$-dimensional subspaces ${}_{n}^{1}\Pi_{m,n}^{2}$ and
${}_{1}^{m}\Pi_{m,n}^{2}$ has been established, by projecting two-dimensional
interpolating points into one-dimension such that all projection are different
using a bijective transformation approach and univariate polynomial
interpolation, in which the MIP (* ‣ 1.1) can always be poised, where
${}_{\alpha}^{\beta}\Pi_{m,n}^{2}$ is the the space of all two variable
polynomials in $x$ and $y$, of degree at most $mn-1$, in two parameters
$\alpha$ and $\beta$, with real coefficients, such that, if
$P\in{{}_{\alpha}^{\beta}\Pi_{m,n}^{2}}$, then

$\displaystyle P(x,y)=\sum_{k=0}^{mn-1}{{\lambda_{k}}(\alpha x+\beta y)^{k}}.$
(4)

Further, it is concluded that the MIP (* ‣ 1.1) can always be poised in
${}_{\alpha}^{\beta}\Pi_{m,n}^{2}$ for unaccountably infinite number of
choices of the parameters $\alpha$ and $\beta$, provided the condition $\alpha
i_{1}+\beta j_{1}\neq\alpha i_{2}+\beta j_{2}$ for all $i_{1}\neq i_{2}$ and
$j_{1}\neq j_{2}$ holds, where
$i_{1},i_{2}\in\left\\{1,2,\ldots,m\right\\}\textrm{ and
}j_{1},j_{2}\in\left\\{1,2,\ldots,n\right\\}$.

From the computational and algebraic point of view, we have identified some
flaws of the solutions and the established subspaces which are listed as
follows:

  1. 1.

The degree of the interpolating polynomials is up to $mn-1$, which is somehow
high from the computational point of view.

  2. 2.

The interpolating polynomials in these subspaces, corresponds to the MIP (* ‣
1.1), do not satisfy the geometric structures of the given matrix
$a_{ij}\in\mathbb{R}^{m\times n}$ such as transpose, symmetry, and skew-
symmetry.

  3. 3.

The suggested subspaces are isomorphic to $\mathbb{R}^{m\times n}$ with
respect to the vector algebra structure, but they do not preserve
multiplication structure as defined in the space of matrices
$\mathbb{R}^{m\times n}$.

The identified demerits in the solutions and the interpolation spaces is
motivating us to identify at least one $mn$-dimensional space of two variable
polynomials, of least possible degree, in which the MIP(* ‣ 1.1) can always be
poised and which can preserve almost every algebraic as well as geometric
structures of the space $\mathbb{R}^{m\times n}$.

In [21], the author, Seimatsu Narumi suggested a $(m+1)(n+1)$-dimensional
space of two variable polynomials, of total degree $mn$ (degree of the
variables not greater than $m$ and $n$ respectively). The suggested space is
known as the space of tensor-product polynomials in two variable. Narumi
claimed this space as a correct interpolation space for a two-variate
interpolation problem $P(a_{i},b_{j})=f(a_{i},b_{j})$, corresponds to a two-
variate real-valued function $f(x,y)$, by sampling the data with respect to
equally spaced interpolation points of the form $(a_{i},b_{j})$, $0\leq i\leq
m$, $0\leq j\leq n$. A method of bivariate divided difference is discussed to
evaluate the coefficients of the required polynomial in the suggested space,
but the existence of the space or the approach of used interpolation method is
not mentioned.

For a given matrix $a_{ij}\in\mathbb{R}^{m\times n}$, the MIP (* ‣ 1.1)
becomes a special case of the Narumi interpolation problem, where
$f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ and the interpolation points takes the
form $(i,j)$, $1\leq i\leq m$, $1\leq j\leq n$. Therefore, in this part of
work, the $mn$-dimensional space of tensor-product polynomials in two
variable, of degree at most $(m-1)+(n-1)$, is considered.

###  1.3 Objectives and arrangement of the article

The main objective of this paper is to develop a theory of two-variate
polynomials in the considered space with respect to the algebraic operations
and geometric properties of the space $\mathbb{R}^{m\times n}$. The second
objective of this paper is to establish the existence of the considered space
in which the MIP(* ‣ 1.1) can be poised for all $a_{ij}\in\mathbb{R}^{m\times
n}$ and present some formulae to construct the respective polynomial in the
established space. Defining a polynomial map from $\mathbb{R}^{m\times n}$ to
the considered space, investigation of some algebraic properties of the
polynomial map, and establish some isomorphic structures between the spaces
are the other prime objectives.

The rest of the paper is consolidated as follows. In section 2, the
$mn$-dimensional space of tensor-product polynomials in two variables, of
degree at most $(m-1)+(n-1)$, will be defined. The basic algebra and some
algebraic properties of the defined space will be discussed with respect to
usual addition, scalar multiplication, and a newly defined algebraic
operation. In section 3, the existence of the defined space will be
established and the poisedness of the MIP (* ‣ 1.1) in it will be proved. Some
formulae will be presented to construct the two-variate polynomial in the
space corresponds to the MIP (* ‣ 1.1) for a given matrix
$(a_{ij})\in\mathbb{R}^{m\times n}$. In section 4, the polynomial map from
$\mathbb{R}^{m\times n}$ to the established subspace will be defined. Some
properties of the additive and multiplicative structures will be discussed. In
section 5, some examples will be included to validate the theoretical findings
and to observe the geometrical prospects. Finally, section 6 will conclude the
paper.

##  2 A theory of two-variate polynomials

In this section, the $mn$-dimensional space of tensor-product polynomial in
two variables, of degree at most $(m-1)+(n-1)$, will be defined. Some
properties of the defined space will be investigated and a theory of two-
variate polynomials will be developed in the defined space with respect to
some algebraic operations and geometric properties of the space
$\mathbb{R}^{m\times n}$. Some definitions, examples, counterexamples,
remarks, and theorems will be included to identify the properties of the
defined space.

###### Definition 2.1

Let $\mathcal{P}_{m}^{n}$ denotes the space of two variable polynomials in $x$
and $y$, of degree at most $(m-1)+(n-1)$, with real coefficients such that, if
$P\in\mathcal{P}_{m}^{n}$, then

$P(x,y)={\sum_{k_{1}=0}^{m-1}{\sum_{k_{2}=0}^{n-1}}}{\lambda_{{k_{1},}{k_{2}}}}{x^{k_{1}}}{y^{k_{2}}}.$
(**)

Here, dim $\mathcal{P}_{m}^{n}=mn$.

###  2.1 Equality, addition, and scalar multiplication of the polynomials in
$\mathcal{P}_{m}^{n}$

The equality, addition and scalar multiplication of the polynomials in
$\mathcal{P}_{m}^{n}$ is defined as the standard definitions of the two-
variate polynomials.

###### Theorem 2.1

The space $\mathcal{P}_{m}^{n}$ is an $mn$-dimensional subspace of
$\Pi_{m+n-2}^{2}$.

Proof: The space $\mathcal{P}_{m}^{n}$ is an $mn$-dimensional non-empty subset
of the vector space $\Pi_{m+n-2}^{2}$. Suppose $P,Q\in\mathcal{P}_{m}^{n}$ be
two polynomials, given as

$\displaystyle
P(x,y)={\sum_{k_{1}=0}^{m-1}{\sum_{k_{2}=0}^{n-1}}}{\lambda_{{k_{1},}{k_{2}}}}{x^{k_{1}}}{y^{k_{2}}},\quad
Q(x,y)={\sum_{k_{1}=0}^{m-1}{\sum_{k_{2}=0}^{n-1}}}{\mu_{{k_{1},}{k_{2}}}}{x^{k_{1}}}{y^{k_{2}}},$

and $\alpha$ be a real scalar. Then,

$\displaystyle\qquad P(x,y)+Q(x,y)$
$\displaystyle={\sum_{k_{1}=0}^{m-1}{\sum_{k_{2}=0}^{n-1}}}{\lambda_{{k_{1},}{k_{2}}}}{x^{k_{1}}}{y^{k_{2}}}+{\sum_{k_{1}=0}^{m-1}{\sum_{k_{2}=0}^{n-1}}}{\mu_{{k_{1},}{k_{2}}}}{x^{k_{1}}}{y^{k_{2}}}$
$\displaystyle={\sum_{k_{1}=0}^{m-1}{\sum_{k_{2}=0}^{n-1}}}({\lambda_{{k_{1},}{k_{2}}}+\mu_{{k_{1},}{k_{2}}}}){x^{k_{1}}}{y^{k_{2}}},$
$\displaystyle\textrm{and }\;\;\alpha
P(x,y)=\alpha{\sum_{k_{1}=0}^{m-1}{\sum_{k_{2}=0}^{n-1}}}(\lambda_{{k_{1},}{k_{2}}}){x^{k_{1}}}{y^{k_{2}}}={\sum_{k_{1}=0}^{m-1}{\sum_{k_{2}=0}^{n-1}}}({\alpha}\lambda_{{k_{1},}{k_{2}}}){x^{k_{1}}}{y^{k_{2}}}.$

i.e., $P(x,y)+Q(x,y),\alpha P(x,y)\in\mathcal{P}_{m}^{n}$ for some scalars
$\delta_{{k_{1},}{k_{2}}}={\lambda_{{k_{1},}{k_{2}}}+\mu_{{k_{1},}{k_{2}}}}$
and $\sigma_{{k_{1},}{k_{2}}}={\alpha}\lambda_{{k_{1},}{k_{2}}}$. This
completes the proof.

###### Corollary 2.1

The spaces $\mathbb{R}^{m\times n}$ and $\mathcal{P}_{m}^{n}$ are isomorphic
with respect to the vector space structure.

###### Remark 2.1

The set of standard basis of the vector space $\mathcal{P}_{m}^{n}$, is

$\left\\{x^{k_{1}-1}y^{k_{2}-1}:1\leq k_{1}\leq m,1\leq k_{2}\leq n,\textrm{
where }k_{1},k_{2},m,n\in\mathbb{N}\right\\}.$ (5)

###  2.2 Product of the polynomials in $\mathcal{P}_{m}^{n}$

###### Definition 2.2

Let $P\in\mathcal{P}_{m}^{n}$ and $Q\in\mathcal{P}_{n}^{q}$ be two
polynomials, then the operation ‘$\otimes$’ given by

$P(x,y)\otimes Q(x,y)={\sum_{k=1}^{n}(P(x,k)\times Q(k,y))}\textrm{ in
}\mathcal{P}_{m}^{q},$ (6)

is defined as DP product.

###### Example 2.1

Let $P\in\mathcal{P}_{2}^{2}$ and $Q\in\mathcal{P}_{2}^{3}$ be two polynomials
given as $P(x,y)=14x-10xy+13y-18$ and
$Q(x,y)=\frac{1}{2}(14y^{2}-11xy^{2}+41xy-34x-52y+44)$ respectively. Then, the
definition (2.2) implies that $P(x,y)\otimes
Q(x,y)=\frac{1}{2}(60xy^{2}-224xy-79y^{2}+184x+295y-242)$ in
$\mathcal{P}_{2}^{3}$. But, $Q(x,y)\otimes P(x,y)$ is not defined.

###### Remark 2.2

If $P(x,y)\otimes Q(x,y)$ is defined, it is not necessary that $Q(x,y)\otimes
P(x,y)$ is also defined.

###### Example 2.2

Let $P\in\mathcal{P}_{1}^{2}$ and $Q\in\mathcal{P}_{2}^{1}$ be two polynomials
given as $P(x,y)=2y-3$ and $Q(x,y)=-2x+3$ respectively. Then, the definition
(2.2) implies that $P(x,y)\otimes Q(x,y)=-2\in\mathcal{P}_{1}^{1}$ and
$Q(x,y)\otimes P(x,y)=-4xy+6x+6y-9\in\mathcal{P}_{2}^{2}$.

###### Remark 2.3

If $P\in\mathcal{P}_{m}^{n}$ and $Q\in\mathcal{P}_{p}^{q}$, then
$P(x,y)\otimes Q(x,y)$ and $Q(x,y)\otimes P(x,y)$ both are defined, if and
only if $n=p$ and $m=q$.

###### Example 2.3

Let $P\in\mathcal{P}_{2}^{2}$ and $Q\in\mathcal{P}_{2}^{2}$ be two polynomials
given as $P(x,y)=-x-y+3$ and $Q(x,y)=-2xy+3i+3j-4$ respectively. Then, the
definition (2.2) implies that $P(x,y)\otimes Q(x,y)=y-x\in\mathcal{P}_{2}^{2}$
and $Q(x,y)\otimes P(x,y)=x-y\in\mathcal{P}_{2}^{2}$, Here, $P(x,y)\otimes
Q(x,y)\neq Q(x,y)\otimes P(x,y)$.

###### Remark 2.4

Let $P,Q\in\mathcal{P}_{n}^{n}$, then $P(x,y)\otimes Q(x,y)$ and
$Q(x,y)\otimes P(x,y)$ both are defined, but it is not necessary that
$P(x,y)\otimes Q(x,y)=Q(x,y)\otimes P(x,y)$.

###### Example 2.4

Let $P\in\mathcal{P}_{2}^{2}$ and $Q\in\mathcal{P}_{2}^{2}$ be two polynomials
given as $P(x,y)=3xy-3x-4y+4$ and $Q(x,y)=2xy-5x-4y+10$ respectively. Then,
the definition (2.2) implies that $P(x,y)\otimes
Q(x,y)=0\in\mathcal{P}_{2}^{2}$ and $Q(x,y)\otimes
P(x,y)=xy-x-2y\in\mathcal{P}_{2}^{2}$.

###### Remark 2.5

$P(x,y)\otimes Q(x,y)=0$ need not implies that either $P(x,y)=0$ or $Q(x,y)=0$
or $Q(x,y)\otimes P(x,y)=0$, i.e., the set $\mathcal{P}_{n}^{n}$ possess zero-
divisor property.

####  2.2.1 Some properties of the DP product

######  Theorem 2.2 (DP product closure)

The set $\mathcal{P}_{m}^{n}$ satisfies closure property under the operation
$\otimes$, if and only if $m=n$.

Proof: Suppose $\mathcal{P}_{m}^{n}$ satisfies closure property under the
operation $`\otimes^{\prime}$, then it has to prove that $m=n$ for all
$m,n\in\mathbb{N}$. Let $P_{1},P_{2}\in\mathcal{P}_{m}^{n}$ and if $m\neq n$,
then by the definition of DP product (2.2), neither $P_{1}(x,y)\otimes
P_{2}(x,y)$ nor $P_{2}(x,y)\otimes P_{1}(x,y)$ is defined. The contra-positive
approach of proof, completes the proof of first part.

Conversely, suppose that $m=n$ for all $m,n\in\mathbb{N}$. Let us consider
that $P_{1},P_{2}\in\mathcal{P}_{n}^{n}$, then
$P_{1}(x,k)\in\mathcal{P}_{n}^{1}$ and $P_{2}(k,y)\in\mathcal{P}_{1}^{n}$, for
some fixed $k$, where $1\leq k\leq n$, $n\in\mathbb{N}$. Thus,

$\displaystyle\quad P_{1}(x,k)\times P_{2}(k,y)\in\mathcal{P}_{n}^{n}\textrm{
for all }1\leq k\leq n\quad\quad\textrm{by definition of DP product}$
$\displaystyle\Rightarrow{\sum_{k=1}^{n}(P_{1}(x,k)\times
P_{2}(k,y))}\in\mathcal{P}_{n}^{n}\quad\quad\quad\quad\quad\quad\quad\textrm{by
additive closure}$ $\displaystyle\Rightarrow P_{1}(x,y)\otimes
P_{2}(x,y))\in\mathcal{P}_{n}^{n}\quad\quad\quad\quad\quad\quad\quad\quad\quad\textrm{by
definition of DP product}$

This completes the proof.

######  Theorem 2.3 (DP product and zero polynomial)

Suppose $P\in\mathcal{P}_{m}^{n}$, then

  1. (i)

$P(x,y)\otimes O(x,y)=0\in\mathcal{P}_{m}^{p}$ for all
$O(x,y)=0\in\mathcal{P}_{n}^{p}$,

  2. (ii)

$O(x,y)\otimes P(x,y)=0\in\mathcal{P}_{p}^{n}$ for all
$O(x,y)=0\in\mathcal{P}_{p}^{m}$.

Proof: Suppose $P\in\mathcal{P}_{m}^{n}$, then
(i) For all $O(x,y)=0\in\mathcal{P}_{n}^{p}$, we get

$P(x,y)\otimes O(x,y)={\sum_{k=1}^{n}(P(x,k)\times
O(k,y))}={\sum_{k=1}^{n}(P(x,k)\times 0}=0\in\mathcal{P}_{m}^{p}.$

(ii) For all $O(x,y)=0\in\mathcal{P}_{p}^{m}$, we get

$O(x,y)\otimes P(x,y)={}{\sum_{k=1}^{n}(O(x,k)\times
P(k,y))}={\sum_{k=1}^{n}(0\times P(x,k))}=0\in\mathcal{P}_{p}^{n}.$

So every entry of the product is the scalar zero, i.e., the result is a zero
polynomial. This completes the proof.

######  Theorem 2.4 (DP product and scalar multiplication)

Suppose $P\in\mathcal{P}_{m}^{n}$, $Q\in\mathcal{P}_{n}^{p}$, and $\alpha$ be
a real scalar. Then,

$\alpha(P(x,y)\otimes Q(x,y))={\left(\alpha P(x,y)\right)}\otimes
Q(x,y)=P(x,y)\otimes{\left(\alpha Q(x,y)\right)}\textrm{ in
}\mathcal{P}_{m}^{p}.$

Proof: Let $P\in\mathcal{P}_{m}^{n}$, $Q\in\mathcal{P}_{n}^{p}$ and $\alpha$
be a real scalar. Then,

$\displaystyle\alpha(P(x,y)\otimes Q(x,y))$
$\displaystyle=\alpha\left({\sum_{k=1}^{n}{\left(P(x,k)\times
Q(k,y)\right)}}\right)\quad\textrm{by definition of DP product}$
$\displaystyle=\alpha{\sum_{k=1}^{n}{\left(P(x,k)\times
Q(k,y)\right)}}\quad\textrm{by scalar multiplication}$
$\displaystyle={\sum_{k=1}^{n}{\alpha\left(P(x,k)\times
Q(k,y)\right)}}\quad\textrm{by scalars distribution}$
$\displaystyle={\sum_{k=1}^{n}{\left(\left(\alpha P(x,k)\right)\times
Q(k,y)\right)}}\quad\textrm{by scalars multiplication}$
$\displaystyle={\left(\alpha P(x,y)\right)}\otimes Q(x,y)\quad\textrm{by
definition of DP product}$

so the polynomials $\alpha(P(x,y)\otimes Q(x,y))$ and ${\left(\alpha
P(x,y)\right)}\otimes Q(x,y)$ are equal, entry-by-entry, and by the definition
of polynomial equality, they are equal polynomials in $\mathcal{P}_{m}^{p}$.
Similarly, the polynomials $\alpha(P(x,y)\otimes Q(x,y))$ and
$P(x,y)\otimes(\alpha Q(x,y))$ are equal in $\mathcal{P}_{m}^{p}$. This
completes the proof.

######  Theorem 2.5 (DP product associativity)

Suppose $P\in\mathcal{P}_{m}^{n}$, $Q\in\mathcal{P}_{n}^{p}$ and
$R\in\mathcal{P}_{p}^{q}$, then

$P(x,y)\otimes\left(Q(x,y)\otimes R(x,y)\right)=\left(P(x,y)\otimes
Q(x,y)\right)\otimes R(x,y)\textrm{ in }\mathcal{P}_{m}^{q}.$

Proof: Let $P\in\mathcal{P}_{m}^{n}$, $Q\in\mathcal{P}_{n}^{p}$ and
$R\in\mathcal{P}_{p}^{q}$. Then, $P(x,y)\otimes\left(Q(x,y)\otimes
R(x,y)\right)$

$\displaystyle=P(x,y)\otimes\left({\sum_{l=1}^{p}{\left(Q(x,l)\times
R(l,y)\right)}}\right)\quad\textrm{by definition of DP product}$
$\displaystyle={\sum_{k=1}^{n}{\left(P(x,k)\times\left({\sum_{l=1}^{p}{\left(Q(k,l)\times
R(l,y)\right)}}\right)\right)}}\quad\textrm{by definition of DP product}$
$\displaystyle={\sum_{k=1}^{n}{\sum_{l=1}^{p}}{P(x,k)\times Q(k,l)\times
R(l,y)}}\quad\textrm{by distributivity in $\mathbb{R}$}$
$\displaystyle={\sum_{l=1}^{p}{\sum_{k=1}^{n}}{P(x,k)\times Q(k,l)\times
R(l,y)}}\quad\textrm{by switching the order of finite sums}$
$\displaystyle={\sum_{l=1}^{p}{\sum_{k=1}^{n}}{R(l,y)\times P(x,k)\times
Q(k,l)}}\quad\textrm{by commutativity in $\mathbb{R}$}$
$\displaystyle={\sum_{l=1}^{p}{\left(R(l,y)\times\left({\sum_{k=1}^{n}{\left(P(x,k)\times
Q(k,l)\right)}}\right)\right)}}\quad\textrm{since $R(l,y)$ is independent of
the index $k$}$
$\displaystyle={\sum_{l=1}^{p}{\left(R(l,y)\times\left(P(x,l)\otimes
Q(x,l)\right)\right)}}\quad\textrm{by definition of DP product}$
$\displaystyle={\sum_{l=1}^{p}{\left(\left(P(x,l)\otimes Q(x,l)\right)\times
R(l,y)\right)}}\quad\textrm{by commutativity in $\mathbb{R}$}$
$\displaystyle=\left(P(x,y)\otimes Q(x,y)\right)\otimes R(x,y)\quad\textrm{by
definition of DP product}$

So the polynomials $P(x,y)\otimes\left(Q(x,y)\otimes R(x,y)\right)$ and
$\left(P(x,y)\otimes Q(x,y)\right)$ $\otimes R(x,y)$ are equal, entry-by-
entry, and by the definition of polynomial equality, they are equal polynomial
in $\mathcal{P}_{m}^{q}$. This completes the proof.

######  Theorem 2.6 (DP product distributivity across addition)

Suppose $P,S\in\mathcal{P}_{m}^{n}$ and $Q,R\in\mathcal{P}_{n}^{p}$, then
(i) Left Distributive Law:

$P(x,y)\otimes{\left(Q(x,y)+R(x,y)\right)}=P(x,y)\otimes Q(x,y)+P(x,y)\otimes
R(x,y)\textrm{ in }\mathcal{P}_{m}^{p}.$

(ii) Right Distributive Law:

${\left(P(x,y)+S(x,y)\right)}\otimes Q(x,y)=P(x,y)\otimes Q(x,y)+S(x,y)\otimes
Q(x,y)\textrm{ in }\mathcal{P}_{m}^{p}.$

Proof: Let us consider that $P\in\mathcal{P}_{m}^{n}$,
$Q\in\mathcal{P}_{n}^{p}$, $R\in\mathcal{P}_{n}^{p}$ and
$S\in\mathcal{P}_{m}^{n}$. Then,

$\displaystyle(i)\textrm{ }P(x,y)\otimes{\left(Q(x,y)+R(x,y)\right)}$
$\displaystyle={}{\sum_{k=1}^{n}(P(x,k)\times{\left(Q(k,y)+R(k,y)\right)}})$
$\displaystyle={}{\sum_{k=1}^{n}{\left(P(x,k)\times Q(k,y)+P(x,k)\times
R(k,y)\right)}}$ $\displaystyle={}{\sum_{k=1}^{n}{\left(P(x,k)\times
Q(k,y)\right)}}+{\sum_{k=1}^{n}{\left(P(x,k)\times R(k,y)\right)}}$
$\displaystyle={}P(x,y)\otimes Q(x,y)+P(x,y)\otimes R(x,y).$

So the polynomials $P(x,y)\otimes{\left(Q(x,y)+R(x,y)\right)}$ and
$P(x,y)\otimes Q(x,y)$ \+ $P(x,y)$ $\otimes R(x,y)$ are equal, entry-by-entry,
and by the definition of polynomial equality, they are equal polynomials in
$\mathcal{P}_{m}^{p}$.

$\displaystyle(ii)\textrm{ }{\left(P(x,y)+S(x,y)\right)}\otimes Q(x,y)$
$\displaystyle={}{\sum_{k=1}^{n}({\left(P(x,k)+S(x,k)\right)\times Q(k,y))}}$
$\displaystyle={}{\sum_{k=1}^{n}{\left(P(x,k)\times Q(k,y)+S(x,k)\times
Q(k,y)\right)}}$ $\displaystyle={}{\sum_{k=1}^{n}{\left(P(x,k)\times
Q(k,y)\right)}}+{\sum_{k=1}^{n}{\left(S(x,k)\times Q(k,y)\right)}}$
$\displaystyle={}P(x,y)\otimes Q(x,y)+S(x,y)\otimes Q(x,y).$

So the polynomials ${\left(P(x,y)+S(x,y)\right)}\otimes Q(x,y)$ and
$P(x,y)\otimes Q(x,y)$+$S(x,y)$ $\otimes Q(x,y)$ are equal, entry-by-entry,
and by the definition of polynomial equality, they are equal polynomials in
$\mathcal{P}_{m}^{p}$. This completes the proof.

######  Definition 2.3 (Identity under DP product)

Let $P\in\mathcal{P}_{m}^{n}$ be any polynomial. Then, the polynomial

  1. (i)

$I_{n}\in\mathcal{P}_{n}^{n}$ is said to right identity, if $P(x,y)\otimes
I_{n}(x,y)=P(x,y)$,

  2. (ii)

$I_{m}\in\mathcal{P}_{m}^{m}$ is said to be left identity, if
$I_{m}(x,y)\otimes P(x,y)=P(x,y)$.

In general, the polynomial $I_{n}\in\mathcal{P}_{n}^{n}$ is said to be the
identity, if

$P(x,y)\otimes I_{n}(x,y)=P(x,y)=I_{n}(x,y)\otimes P(x,y)$ (7)

for all $P\in\mathcal{P}_{n}^{n}$.

###### Example 2.5

Let $P\in\mathcal{P}_{3}^{2}$, $Q\in\mathcal{P}_{2}^{2}$, and
$R\in\mathcal{P}_{3}^{3}$ and be three polynomials given by
$P(x,y)=ax^{2}y+bxy+cx^{2}+dx+ey+f$, $Q(x,y)=2xy-3x-3y+5$, and
$R(x,y)=\frac{1}{2}(3x^{2}y^{2}-12x^{2}y-12xy^{2}+10x^{2}+49xy+10y^{2}-42x-42y+38)$
respectively. Then, the definition (2.2) implies that $P(x,y)\otimes
Q(x,y)=ax^{2}y+bxy+cx^{2}+dx+ey+f$ and $R(x,y)\otimes
P(x,y)=ax^{2}y+bxy+cx^{2}+dx+ey+f$. Therefore, $Q\in\mathcal{P}_{2}^{2}$ and
$R\in\mathcal{P}_{3}^{3}$ are the right and left identity of
$P\in\mathcal{P}_{3}^{2}$ respectively.

###### Example 2.6

Let $P(x,y)=axy+bx+cy+d$ and $Q(x,y)=2xy-3x-3y+5$ be two polynomials in
$\mathcal{P}_{2}^{2}$. Then, the definition (2.2) implies that

$P(x,y)\otimes Q(x,y)=axy+bx+cy+d=Q(x,y)\otimes P(x,y).$

Thus, $P(x,y)\otimes Q(x,y)=P(x,y)=Q(x,y)\otimes P(x,y)$ holds. Hence,
$Q(x,y)=2xy-3x-3y+5$ is the identity polynomial in $\mathcal{P}_{2}^{2}$.

###### Remark 2.6

The polynomials $I_{1}(x,y)=1$, $I_{2}(x,y)=2xy-3x-3y+5$, and
$I_{3}(x,y)=\frac{1}{2}(3x^{2}y^{2}-12x^{2}y-12xy^{2}+10x^{2}+49xy+10y^{2}-42x-42y+38)$
are the identity polynomials in $\mathcal{P}_{1}^{1}$, $\mathcal{P}_{2}^{2}$
and $\mathcal{P}_{3}^{3}$ respectively.

######  Definition 2.4 (Inverse under DP product)

Let $P\in\mathcal{P}_{n}^{n}$ be any polynomial. Then, the polynomial
$Q\in\mathcal{P}_{n}^{n}$ is said to be the inverse of $P(x,y)$, if

$P(x,y)\otimes Q(x,y)=I_{n}(x,y)=Q(x,y)\otimes P(x,y),$ (8)

where $I_{n}\in\mathcal{P}_{n}^{n}$ is the identity polynomial.

###### Remark 2.7

Let $\alpha_{1},\alpha_{2},\ldots,\alpha_{n}$ be $n$ scalars. The polynomial
$P\in\mathcal{P}_{n}^{n}$ is invertible (or non-singular), if

${\sum_{i=1}^{n}{\alpha_{i}.P(i,y)}}=0\Rightarrow\alpha_{i}=0\textrm{ for all
}i=1,2,\ldots,n,$ (9)

or

${\sum_{j=1}^{n}{\alpha_{j}.P(x,j)}}=0\Rightarrow\alpha_{j}=0\textrm{ for all
}j=1,2,\ldots,m.$ (10)

###### Example 2.7

Let $P(x,y)=-10xy+14x+13y-18$ and $Q(x,y)=15xy-21x-20y+28$ be two polynomials
in $\mathcal{P}_{2}^{2}$. Suppose $\alpha_{1},\alpha_{2}$ be two scalars, then
${\sum_{i=1}^{2}{\alpha_{i}.P(i,y)}}=0$ and
${\sum_{j=1}^{2}{\alpha_{j}.P(x,j)}}=0$ implies that
$\alpha_{1}=\alpha_{2}=0$. Therefore, the polynomial $P\in\mathcal{P}_{2}^{2}$
is invertible. The, the polynomial $Q\in\mathcal{P}_{2}^{2}$ is not
invertible, since ${\sum_{i=1}^{2}{\alpha_{i}.P(i,y)}}=0$ implies that
$(\alpha_{1},\alpha_{2})=(2k,k)$, where $k$ is an arbitrary constant. Suppose
$R\in\mathcal{P}_{2}^{2}$ be the inverse of $P\in\mathcal{P}_{2}^{2}$, given
as

$R(x,y)=\beta_{1}xy+\beta_{2}x+\beta_{3}y+\beta_{4};\quad\beta_{1},\beta_{2},\beta_{3},\beta_{4}\in\mathbb{R}.$
(11)

Since $I_{2}(x,y)=2xy-3x-3y+5$ is the identity in $\mathcal{P}_{2}^{2}$,
therefore the definition (2.4) implies that $\beta_{1}=0$,
$\beta_{2}=-\frac{1}{2}$, $\beta_{3}=-1$, and $\beta_{4}=\frac{7}{2}$. Hence,
$R(x,y)=-\frac{1}{2}(x+2y-7)$ is the inverse of the given polynomial
$P\in\mathcal{P}_{2}^{2}$.

######  Definition 2.5 (Eigenvalue of the polynomial)

Let $P\in\mathcal{P}_{n}^{n}$ and $X\in\mathcal{P}_{n}^{1}$ be two
polynomials. Suppose $\lambda$ is a scalar, if

$P(x,y)\otimes X(x,1)=\lambda.X(x,1),$ (12)

then $\lambda$ is said to be “eigenvalue” of the polynomial
$P\in\mathcal{P}_{n}^{n}$ and $X\in\mathcal{P}_{n}^{1}$ is the “eigen-
polynomial” corresponding to the eigenvalue $\lambda$.

###### Example 2.8

Let $P\in\mathcal{P}_{2}^{2}$ and $X_{1},X_{2}\in\mathcal{P}_{2}^{1}$ be three
polynomials such that $P(x,y)=-xy+4x+3y-5$, $X_{1}(x,1)=5x-8$ and
$X_{2}(x,1)=x$. Then,

$P(x,y)\otimes X_{1}(x,1)=-X_{1}(x,1)\textrm{ and }P(x,y)\otimes
X_{2}(x,1)=7X_{2}(x,1).$

Hence, $\lambda=-1$ and $\lambda=7$ are the eigenvalues of the polynomial
$P(x,y)=-xy+4x+3y-5$, where $X_{1}(x,1)=5x-8$ and $X_{2}(x,1)=x$ are the
eigen-polynomials corresponding to the eigenvalues $\lambda=-1$ and
$\lambda=7$ respectively.

######  Definition 2.6 (Power of the polynomial)

Let $P\in\mathcal{P}_{n}^{n}$ and $r\geq 2$, $r\in\mathbb{N}$, then

${\left(P(x,y)\right)}^{2}=P(x,y)\otimes P(x,y)\in\mathcal{P}_{n}^{n}.$ (13)

###### Remark 2.8

The polynomial $P\in\mathcal{P}_{n}^{n}$ is said to

  1. 1.

involuntary, if ${\left(P(x,y)\right)}^{2}=I_{n}(x,y)$, where $I_{n}(x,y)$ is
the identity polynomial in $\mathcal{P}_{n}^{n}$.

  2. 2.

idempotent, if ${\left(P(x,y)\right)}^{2}=P(x,y)$.

  3. 3.

nilpotent with index $r$, if ${\left(P(x,y)\right)}^{r}=0$, where $r$ is the
least positive integer.

  4. 4.

periodic with index $r$, if ${\left(P(x,y)\right)}^{r+1}=P(x,y)$, where $r$ is
the least positive integer.

###  2.3 Transpose, symmetry, skew-symmetry, and orthogonality of the
polynomials in $\mathcal{P}_{m}^{n}$

######  Definition 2.7 (Transpose of the polynomial)

Let $P(x,y)$ be a polynomial in $\mathcal{P}_{m}^{n}$, then its transpose is
defined as

$P^{T}(x,y)=P(y,x),$ (14)

such that $P^{T}\in\mathcal{P}_{n}^{m}$.

###### Remark 2.9

The polynomial $P\in\mathcal{P}_{n}^{n}$ is said to

  1. 1.

symmetric, if $P(x,y)=P(y,x)$.

  2. 2.

skew-symmetric, if $P(x,y)=-P(y,x)$.

  3. 3.

orthogonal, if $P(x,y)\otimes P^{T}(x,y)=I_{n}(x,y)=P^{T}(x,y)\otimes P(x,y)$,
where $I_{n}(x,y)$ is the identity polynomial in $\mathcal{P}_{n}^{n}$.

##  3 Existence of the subspace $\mathcal{P}_{m}^{n}$, poisedness of the MIP,
and some polynomial construction formulae

In this section, the existence of the subspace $\mathcal{P}_{m}^{n}$ against
the solution of the MIP (* ‣ 1.1) and the poisedness of the MIP (* ‣ 1.1) in
it will be proved. Three polynomial construction formulae will be derived
using univariate Lagrange, Newton-forward difference, and Newton-backward
difference interpolation formulae [22, 23] with respect to the MIP(* ‣ 1.1)
for a given matrix $(a_{ij})\in\mathbb{R}^{m\times n}$ in the subspace
$\mathcal{P}_{m}^{n}$. Some arguments and other construction methods will be
remarked.

###### Theorem 3.1

Let $a_{ij}\in\mathbb{R}^{m\times n}$ be any given matrix. Then, there exists
a unique polynomial $P\in\mathcal{P}_{m}^{n}$ with respect to the set
$\mathcal{D}$ which satisfy the MIP (* ‣ 1.1) .

Proof: The proof consist of two parts, existence and uniqueness. Both will be
proved one by one as follows:

Existence:  Let $A=(a_{ij})\in\mathbb{R}^{m\times n}$ be a given matrix. Then,
for the given set of nodes

$\Omega_{m}=\left\\{i:i=1,2,\ldots,n\right\\},$ (15)

associated with the $r$th column, there exists a unique polynomial $p_{r}(x)$
in one variable $x$, of degree at most $m-1$, for all $r=1,2,\ldots,m$,
satisfying the problem

$p_{r}(i)=a_{ir}\textrm{ for all }i=1,2,\ldots,m.$ (16)

Therefore, the matrix $A=(a_{ij})_{m\times n}$ can be represented as
$\begin{pmatrix}p_{1}(x)p_{2}(x)\ldots p_{m}(x)\end{pmatrix}_{1\times n}$.
Again, for the given set of nodes

$\Omega_{n}=\left\\{j:j=1,2,\ldots,n\right\\},$ (17)

there exists a unique polynomial $P(x,y)$ in the variable $y$ (for fixed $x$),
of degree at most $n-1$, satisfying the problem

$P(x,j)=p_{j}(x)\textrm{ for all }j=1,2,\ldots,n.$ (18)

On combing, $P(x,y)$ is a polynomial in $\mathcal{P}_{m}^{n}$, satisfying the
MIP (* ‣ 1.1) with respect to the set
$\mathcal{D}=\Omega_{m}\times\Omega_{n}$. This completes the proof of
existence part.

Uniqueness: The polynomial expressions with respect to the set of nodes (15)
can be written as

$a_{ir}=p_{r}(i)+R_{m,r}(i),\;\;i=1,2,\ldots,m,$ (19)

where $R_{m,r}(i)$ are the remainders for all $r=1,2,\ldots,n$. The
polynomials, satisfy the conditions $a_{kr}=p_{r}(k)$ for all
$k=1,2,\ldots,m$. Therefore, the remainder term $R_{m,r}$; $r=1,2,\ldots,n$
vanishes at $i=k$ for all $k=1,2,\ldots,m$. Thus, the polynomials satisfying
the conditions (16) are unique for all $r=1,2,\ldots,n$. Again, the polynomial
expression with respect to the set of nodes (17) can be written as

$p_{j}(x)=P(x,j)+R_{n}(j),\;\;j=1,2,\ldots,n,$ (20)

where $R_{n}(j)$ is the remainder. The polynomial $P(x,j)$, satisfies the
conditions $p_{r}(k)=P(x,k)$ for all $k=1,2,\ldots,n$. Therefore, the
remainder term $R_{n}$ vanishes at $j=k$ for all $k=1,2,\ldots,n$. Hence,
polynomial satisfying the conditions (18) is unique. This completes the
theorem.

###### Remark 3.1

We have used an approach of uni-variate polynomial interpolation for the
considered MIP(* ‣ 1.1) and the interpolation space coincide with the space
suggested by the author Narumi in the year 1920 in [21]. Narumi had not
mentioned the used approach of interpolation for the existence of the space,
however a method of bivariate divided difference is used to evaluate the
coefficients of the given polynomial form in the suggested space.

###### Remark 3.2

In general, the interpolation methods assume some certain properties on the
sought structure such as smoothness, regularity (nonexistence of some
singularities) in a sufficiently large vicinity and so on. The ignorance of
these types of assumptions would be dangerous from the error point of view,
[24, 25, 26]. We are assuming that the original function is smooth and regular
or ‘well-behaved’ in the given domain. The present work is carried out in the
algebraic direction and approximation or error estimation is not the part of
the article.

In the next theorems, some formulae to construct the unique polynomial
$P\in\mathcal{P}_{m}^{n}$ satisfying the MIP (* ‣ 1.1) with respect to the set
$\mathcal{D}$ for a given matrix in $\mathbb{R}^{m\times n}$ are formulated
using well-known univariate interpolation formulae.

###### Theorem 3.2

Let $A=(a_{ij})\in\mathbb{R}^{m\times n}$ be a given matrix, then there exist
a unique polynomial $P\in\mathcal{P}_{m}^{n}$, satisfying $a_{ij}=P(i,j)$ for
all $i=1,2,\ldots,m$, $j=1,2,\ldots,n$, given by

$P(x,y)=\sum_{r=1}^{n}\left\\{\prod_{\alpha=1,\alpha\neq
r}^{n}{\frac{(y-\alpha)}{(r-\alpha)}}\right\\}{p_{r}(x)},$ (21)

where

$\quad p_{r}(x)=\sum_{k=1}^{m}\left\\{\prod_{\alpha=1,\alpha\neq
k}^{m}{\frac{(x-\alpha)}{(k-\alpha)}}\right\\}{a_{kr}}\textrm{ for all
}r=1,2,\ldots,n.$ (22)

Proof: Let $A=(a_{ij})\in\mathbb{R}^{m\times n}$ be a given matrix, then the
poisedness of the MIP (* ‣ 1.1) with respect to the set $\mathcal{D}$ of nodes
insures that, there exist unique polynomial $P\in\mathcal{P}_{m}^{n}$
satisfying $a_{ij}=P(i,j)$ for all $i=1,2,\ldots,m$, $j=1,2,\ldots,n$. Using
univariate Lagrange interpolation formula, the polynomials with respect to the
set of nodes (15), satisfying the problem (16), can be written in the form

$p_{r}(x)=l_{1,r}(x)a_{1r}+l_{2,r}(x)a_{2r}+\ldots+l_{n,r}(x)a_{nr},$ (23)

where $l_{k,r}(x)$, $k=1,2,\ldots,m$ are the polynomials of one variable $x$,
of degree at most $m-1$ for all $r=1,2,\ldots,n$. The polynomials (23) will
satisfy the interpolating conditions (16), if and only if

$l_{k,r}(x_{i})=\begin{cases}0,&\text{if }k\neq i\\\ 1,&\text{if
}k=i\end{cases}\textrm{ for all }r=1,2,\ldots,n,$ (24)

and the polynomials $l_{k,r}(x)$ for all $r=1,2,\ldots,n$, satisfying the
conditions (24), can be given as

$l_{k,r}(x)=\prod_{\alpha=1,\alpha\neq
k}^{m}{\frac{(x-\alpha)}{(k-\alpha)}}\textrm{ for all }k=1,2,\ldots,m.$ (25)

The equations (23) and (25), insured that

$\quad p_{r}(x)=\sum_{k=1}^{m}\left\\{\prod_{\alpha=1,\alpha\neq
k}^{m}{\frac{(x-\alpha)}{(k-\alpha)}}\right\\}{a_{kr}}\textrm{ for all
}r=1,2,\ldots,n.$

Again, using univariate Lagrange interpolation formula, the polynomial with
respect to the set nodes (17), satisfying the problem (18), can be written in
the form

$P(x,y)=L_{1}(y)p_{1}(x)+L_{2}(y)p_{2}(x)+\ldots+L_{n}(y)p_{n}(x),$ (26)

where $L_{r}(x)$, $r=1,2,\ldots,n$ are the polynomials of one variables $y$,
of degree at most $n-1$, i.e., $L_{r}\in\Pi_{n-1}^{1}$. The polynomial (26)
will satisfy the interpolating conditions (18), if and only if

$L_{r}(y_{j})=\begin{cases}0,&\text{if }r\neq j\\\ 1,&\text{if
}r=j\end{cases},$ (27)

and the polynomials $L_{r}(y)$, satisfying the conditions (27), can be given
as

$L_{r}(y)=\prod_{\alpha=1,\alpha\neq
r}^{n}{\frac{(y-\alpha)}{(r-\alpha)}}\textrm{ for all }r=1,2,\ldots,n.$ (28)

The combination of the equations (26) and (28), completes the result.

In the similar manner, two more formulae using univariate Newton forward and
backward interpolation formulae, for the given set of nodes (15) and (17)
satisfying the interpolation problems (16) and (18) respectively, can be given
as follows:

###### Theorem 3.3

Let $A=(a_{ij})\in\mathbb{R}^{m\times n}$ be a given matrix, then there exist
a unique polynomial $P\in\mathcal{P}_{m}^{n}$, satisfying $a_{ij}=P(i,j)$ for
all $i=1,2,\ldots,m$, $j=1,2,\ldots,n$, given by

$\displaystyle P(x,y)={}$ $\displaystyle p_{1}(x)+\frac{(y-1)}{1!}{\Delta
p_{1}(x)}+\frac{(y-1)(y-2)}{2!}{\Delta^{2}p_{1}(x)}$ (29)
$\displaystyle\quad\quad\quad\quad+\ldots+\frac{(y-1)(y-2)\ldots(y-(n-1))}{(n-1)!}{\Delta^{n-1}p_{1}(x)},$

where

$\Delta^{r}p_{1}(x)=\sum_{\beta=0}^{r}{(-1)^{\beta}}{r\choose\beta}{p_{r+1-\beta}(x)}\textrm{
for all }r=1,2,\ldots,n-1,$ (30)

and

$\displaystyle p_{r}(x)={}$ $\displaystyle a_{1r}+\frac{(x-1)}{1!}{\Delta
a_{1r}}+\frac{(x-1)(x-2)}{2!}{\Delta^{2}a_{1r}}$ (31)
$\displaystyle\quad\quad\quad\quad\quad\quad+\ldots+\frac{(x-1)(x-2)\ldots(x-(m-1))}{(m-1)!}{\Delta^{m-1}a_{1r}},$

for all $r=1,2,\ldots,n$, where

$\Delta^{k}a_{1r}=\sum_{\alpha=0}^{k}{(-1)^{\alpha}}{k\choose\alpha}{a_{(k+1-\alpha)r}}\textrm{
for all }k=1,2,\ldots,m-1.$ (32)

###### Theorem 3.4

Let $A=(a_{ij})\in\mathbb{R}^{m\times n}$ be a given matrix, then there exist
a unique polynomial $P\in\mathcal{P}_{m}^{n}$, satisfying $a_{ij}=P(i,j)$ for
all $i=1,2,\ldots,m$, $j=1,2,\ldots,n$, given by

$\displaystyle P(x,y)={}$ $\displaystyle p_{n}(x)+\frac{(y-n)}{1!}{\nabla
p_{n}(x)}+\frac{(y-n)(y-(n-1))}{2!}{\nabla^{2}p_{n}(x)}$ (33)
$\displaystyle\quad\quad\quad\quad+\ldots+\frac{(y-n)(y-(n-1))\ldots(y-2)}{(n-1)!}{\nabla^{n-1}p_{n}(x)},$

where

$\nabla^{r}p_{n}(x)=\sum_{\beta=0}^{r}{(-1)^{\beta}}{r\choose\beta}{p_{n-\beta}(x)}\textrm{
for all }r=1,2,\ldots,n-1.$ (34)

and

$\displaystyle p_{r}(x)={}$ $\displaystyle a_{mr}+\frac{(x-m)}{1!}{\nabla
a_{mr}}+\frac{(x-m)(x-(m-1))}{2!}{\nabla^{2}a_{nr}}$ (35)
$\displaystyle\quad\quad\quad\quad\quad\quad+\ldots+\frac{(x-m)(x-(m-1))\ldots(x-2)}{(m-1)!}{\nabla^{m-1}a_{mr}},$

for all $r=1,2,\ldots,n$, where

$\nabla^{k}a_{mr}=\sum_{\alpha=0}^{k}{(-1)^{\alpha}}{k\choose\alpha}{a_{(m-\alpha)r}}\textrm{
for all }k=1,2,\ldots,m-1.$ (36)

###### Remark 3.3

Suppose $P\in\mathcal{P}_{m}^{n}$ is the unique polynomial of the form (** ‣
2.1) which satisfy the MIP (* ‣ 1.1) for the given matrix
$(a_{ij})\in\mathbb{R}^{m\times n}$. Then, the $mn$ independent conditions
$a_{ij}=P(i,j)$ for all $i=1,2,\ldots,m$, $j=1,2,\ldots,n$ constitute a non-
homogeneous linear system of $mn$ equations of the form ${\Lambda}X=\mu$,
where $X=(\lambda_{00},\lambda_{01},\ldots,\lambda_{(m-1)(n-1)})^{T}$,
$\mu=(a_{11},a_{12},\ldots,a_{mn})^{T}$, and $\Lambda$ is a coefficient
matrix. The uniqueness of the solution implies that $det(\Lambda)\neq 0$.
Using some suitable method to solve linear system of non-homogeneous
equations, the coefficients $\lambda_{k_{1}k_{2}}$, where
$k_{1}=0,1,\ldots,m-1$ and $k_{2}=0,1,\ldots,n-1$ can be determined uniquely.

##  4 Isomorphism and isomorphic structures between the spaces
$\mathbb{R}^{m\times n}$ and $\mathcal{P}_{m}^{n}$

Suppose the notation $P_{A}(x,y)$ represent the unique polynomial
$P_{A}\in\mathcal{P}_{m}^{n}$ which satisfy the MIP (* ‣ 1.1) for the given
matrix $A\in\mathbb{R}^{m\times n}$. This notation will be used frequently in
the reaming part of the paper. The spaces $\mathbb{R}^{m\times n}$ and
$\mathcal{P}_{m}^{n}$ are isomorphic with respect to the vector spaces
structure, therefore there must exist an isomorphism (invertible linear map)
between them [2].

Using the result of the theorem 3.1, the polynomial map
$D_{p}:\mathbb{R}^{m\times n}\rightarrow\mathcal{P}_{m}^{n}$ can be defined as

$D_{p}(A)=P_{A}(x,y)\textrm{ for all }A\in\mathbb{R}^{m\times n}.$ (37)

Again, let the map $f:\mathbb{R}^{m\times n}\times\mathbb{R}^{n\times
q}\rightarrow\mathbb{R}^{m\times q}$ is defined as $f(A,B)=A.B$ for all
$A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{n\times q}$, then the
composition map $D_{p}of:\mathbb{R}^{m\times n}\times\mathbb{R}^{n\times
q}\rightarrow\mathbb{R}^{m\times q}$ is given as $D_{p}of(A,B)=P_{A.B}(x,y)$
for all $A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{n\times q}$. Also, let
the maps $g:\mathbb{R}^{m\times n}\times\mathbb{R}^{n\times
q}\rightarrow\mathcal{P}_{m}^{n}\times\mathcal{P}_{n}^{q}$ and
$h:\mathcal{P}_{m}^{n}\times\mathcal{P}_{n}^{q}\rightarrow\mathcal{P}_{m}^{q}$
are defined as $g(A,B)=(P_{A}(x,y),P_{B}(x,y))$ for all
$A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{n\times q}$ and
$h(P,Q)=P(x,y)\otimes Q(x,y)$ for all $P\in\mathcal{P}_{m}^{n}$,
$Q\in\mathcal{P}_{n}^{q}$ respectively. Then, the composition map
$hog:\mathbb{R}^{m\times n}\times\mathbb{R}^{n\times
q}\rightarrow\mathcal{P}_{m}^{q}$ is given as $hog(A,B)=P_{A}(x,y)\otimes
P_{B}(x,y)$ for all $A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{n\times
q}$.

In this section, by proving the invertibility and linearity, it will be proved
that the polynomial map (37) is an isomorphism. Further, the property
$P_{A.B}(x,y)=P_{A}(x,y)\otimes P_{B}(x,y)$ for all $A\in\mathbb{R}^{m\times
n}$, $B\in\mathbb{R}^{n\times q}$ will be proved to establish the discussed
product structure. At last, the existence of the identity polynomial under the
operation ‘$\otimes$’ will be ensured and it will be proved that the algebraic
structures $\langle\mathbb{R}^{n\times n},.\rangle$ and
$\langle\mathcal{P}_{n}^{n},\otimes\rangle$ are isomorphic.

###### Theorem 4.1

For all $P\in\mathcal{P}_{m}^{n}$, there exists a unique matrix
$A=(a_{ij})\in\mathbb{R}^{m\times n}$ such that $a_{ij}=P(i,j)$ for all
$i=1,2,\ldots,m$, $j=1,2,\ldots,n$.

###### Theorem 4.2

Let $A,B\in\mathbb{R}^{m\times n}$ and $\alpha$ be a real scalar, then the
following properties holds:

  1. (i)

$P_{A+B}(x,y)=P_{A}(x,y)+P_{B}(x,y)$ in $\mathcal{P}_{m}^{n}$.

  2. (ii)

$P_{\alpha A}(x,y)=\alpha P_{A}(x,y)$ in $\mathcal{P}_{m}^{n}$.

Proof: Let $A,B\in\mathbb{R}^{m\times n}$ be two matrices given as
$A=(a_{ij})_{m\times n}$ and $B=(b_{ij})_{m\times n}$. Then,
$A+B=(a_{ij}+b_{ij})_{m\times n}$ and $\alpha A=(\alpha a_{ij})_{m\times n}$,
for some scalar $\alpha$. Therefore, there exists unique
$P_{A}\in\mathcal{P}_{m}^{n}$, $P_{B}\in\mathcal{P}_{m}^{n}$,
$P_{A+B}\in\mathcal{P}_{m}^{n}$, and $P_{\alpha A}\in\mathcal{P}_{m}^{n}$,
which satisfy the MIP’s

$P_{A}(i,j)=a_{ij}\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,n,$ (38)
$P_{B}(i,j)=b_{ij}\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,n,$ (39)
$P_{A+B}(i,j)=a_{ij}+b_{ij}\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,n,$
(40) $\textrm{and }P_{\alpha A}(i,j)=\alpha a_{ij}\textrm{ for all
}i=1,2,\ldots,m,\;j=1,2,\ldots,n,$ (41)

respectively. Again, the definition of usual addition and scalar
multiplication of the polynomials in $\mathcal{P}_{m}^{n}$ implies that

$\displaystyle(i)\;\;P_{A}(i,j)+P_{B}(i,j)$
$\displaystyle=a_{ij}+b_{ij}\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,n$
$\displaystyle=P_{a+B}(i,j)\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,n,$
$\displaystyle and\;\;(ii)\;\;\alpha P_{A}(i,j)$ $\displaystyle=\alpha
a_{ij}\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,n$ $\displaystyle=\alpha
P_{\alpha A}(i,j)\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,n.$

The MIP (40) and (41) have unique solution in the space $\mathcal{P}_{m}^{n}$
with respect to the set $\mathcal{D}$ by the theorem 3.1. This completes the
proof.

###### Remark 4.1

If $A_{1},A_{2},...,A_{k}\in M_{n,n}$, and
$\alpha_{1},\alpha_{2},...,\alpha_{k}$ be some real scalars, then

$P_{\alpha_{1}{A_{1}}+\alpha_{2}{A_{2}}+...+\alpha_{k}{A_{k}}}(x,y)={\sum_{i=1}^{k}}{\alpha_{i}{P_{A_{i}}(x,y)}}.$

On combining the theorems 3.1, 4.1, and 4.2, the properties of the polynomial
map (37) can be combine as follows.

###### Theorem 4.3

The polynomial map $D_{p}:\mathbb{R}^{m\times
n}\rightarrow\mathcal{P}_{m}^{n}$ defined by

$D_{p}(A)=P_{A}(x,y)\textrm{ for all }A\in\mathbb{R}^{m\times n}.$ (42)

is an ispmprphism.

###### Remark 4.2

The inverse linear map
${D_{p}}^{-1}:\mathcal{P}_{m}^{n}\rightarrow\mathbb{R}^{m\times n}$ is given
by

${D_{p}}^{-1}(P(x,y))={\left[P(i,j)\right]}_{m\times n}\textrm{ for all
}P\in\mathcal{P}_{m}^{n}.$ (43)

###### Theorem 4.4

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times q}$, then
$P_{AB}(x,y)=P_{A}(x,y)\otimes P_{B}(x,y)\textrm{ in }\mathcal{P}_{m}^{q}$.

Proof: Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times q}$ be
two matrices given as $A=(a_{ij})_{m\times n}$ and $B=(b_{ij})_{n\times p}$.
Then,

$AB=\left({\sum_{k=1}^{n}}{\left(a_{ik}\times b_{kj}\right)}\right)_{m\times
q}$

Therefore, there exists unique $P_{A}\in\mathcal{P}_{m}^{n}$,
$P_{B}\in\mathcal{P}_{m}^{n}$ and $P_{AB}\in\mathcal{P}_{m}^{p}$ satisfying
the MIP’s

$P_{A}(i,j)=a_{ij}\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,n,$ (44)
$P_{B}(i,j)=b_{ij}\textrm{ for all }i=1,2,\ldots,n,\;j=1,2,\ldots,q,$ (45)
$P_{AB}(i,j)={\sum_{k=1}^{n}}{\left(a_{ik}\times b_{kj}\right)}\textrm{ for
all }i=1,2,\ldots,m,\;j=1,2,\ldots,q,$ (46)

with respect to the sets $\\{(i,j):i=1,2,\ldots,m$, $j=1,2,\ldots,n\\}$,
$\\{(i,j):i=1,2,\ldots,n$, $j=1,2,\ldots,q\\}$, and $\\{(i,j):i=1,2,\ldots,m$,
$j=1,2,\ldots,q\\}$ respectively. Again, the definition of DP product (2.2)
implies that

$\displaystyle P_{A}(i,j)\otimes P_{B}(i,j)$
$\displaystyle={\sum_{k=1}^{n}}{\left(P_{A}(i,k)\times
P_{B}(k,j)\right)}\textrm{ for all }i=1,2,\ldots,m,\;j=1,2,\ldots,p$
$\displaystyle={\sum_{k=1}^{n}}{\left(a_{ik}\times b_{kj}\right)}\textrm{ for
all }i=1,2,\ldots,m,\;j=1,2,\ldots,p$ $\displaystyle=P_{AB}(i,j)\textrm{ for
all }i=1,2,\ldots,m,\;j=1,2,\ldots,q.$

Using Theorem 3.1, the MIP (46) with respect to the set
$\\{(i,j):i=1,2,\ldots,m$, $j=1,2,\ldots,q\\}$ will have unique solution in
$\mathcal{P}_{m}^{q}$. This completes the proof.

###### Remark 4.3

If $A_{1},A_{2},...,A_{k}\in\mathbb{R}^{m\times n}$, $B_{1},B_{2},...,B_{k}\in
M_{n,p}$, and $\alpha_{1},\alpha_{2},...,\alpha_{k}$ be some real scalars,
then

$P_{\alpha_{1}{A_{1}}{B_{1}}+\alpha_{2}{A_{2}}{B_{2}}+...+\alpha_{k}{A_{k}}{B_{k}}}(x,y)={\sum_{i=1}^{k}}{\alpha_{i}{\left(P_{A_{i}}(x,y)\otimes
P_{B_{i}}(x,y)\right)}}\in\mathcal{P}_{m}^{p}.$

######  Corollary 4.1 (Existence of identity polynomial under DP product)

Suppose $P_{A}\in\mathcal{P}_{m}^{n}$, then

  1. (i)

there exist a $P_{I_{n}}\in\mathcal{P}_{n}^{n}$, such that $P_{A}(x,y)\otimes
P_{I_{n}}(x,y)=P_{A}(x,y)$,

  2. (ii)

there exist a $P_{I_{m}}\in\mathcal{P}_{m}^{m}$, such that
$P_{I_{m}}(x,y)\otimes P_{A}(x,y)=P_{A}(x,y)$.

The combination of the theorems 4.3 and 4.4 establish a very important result,
that can be summarize as follows.

###### Theorem 4.5

The binary structures $\langle\mathbb{R}^{n\times n},.\rangle$ and
$\langle\mathcal{P}_{n}^{n},\otimes\rangle$ are isomorphic.

For $m=n$, the combination of the theorems 2.1, 2.2, 2.5, 2.6 and the
corollary 4.1 can be resulted as follows.

###### Theorem 4.6

The binary structure $\langle\mathcal{P}_{n}^{n},+,\otimes\rangle$ form a ring
with unity.

##  5 Numerical verification

In this section, five examples will be included to illustrate and verify the
results.

###### Example 5.1

Let $\delta\in\mathbb{R}^{1\times 3}$, $\eta\in\mathbb{R}^{3\times 1}$,
$\vartheta,\psi\in\mathbb{R}^{2\times 2}$, $\zeta\in\mathbb{R}^{2\times 3}$
and $\theta\in\mathbb{R}^{3\times 2}$ be six matrices
$\delta=\begin{pmatrix}1&-1&-2\end{pmatrix}$, $\eta=\begin{pmatrix}-1\\\ 1\\\
3\end{pmatrix}$, $\vartheta=\begin{pmatrix}-15&36\\\ -1&96\end{pmatrix}$,
$\psi=\begin{pmatrix}-216&2\\\ 540&-5\end{pmatrix}$,
$\zeta=\begin{pmatrix}1&0&2\\\ -1&2&-3\end{pmatrix}$ and
$\theta=\begin{pmatrix}1&0\\\ 0&-1\\\ 1&0\end{pmatrix}$ respectively.
Therefore, there exists unique $P_{\delta}(1,y)\in\mathcal{P}_{2}^{1}$,
$P_{\eta}(x,1)\in\mathcal{P}_{2}^{1}$,
$P_{\vartheta}(x,y)\in\mathcal{P}_{2}^{2}$,
$P_{\psi}(x,y)\in\mathcal{P}_{2}^{2}$, $P_{\zeta}(x,y)\in\mathcal{P}_{2}^{3}$
and $P_{\theta}(x,y)\in\mathcal{P}_{3}^{2}$ given by
$P_{\delta}(1,y)=\frac{1}{2}y^{2}-\frac{7}{2}y+4$, $P_{\eta}(x,1)=2x-3$,
$P_{\vartheta}(x,y)=46xy-32x+5y-34$, $P_{\psi}(x,y)=-763xy+1519x+981y-1953$,
$P_{\zeta}(x,y)=-\frac{11}{2}xy^{2}+7y^{2}+\frac{41}{2}xy-17x-26y+22$, and
$P_{\theta}(x,y)=x^{2}-4x-y+5$ respectively.

(a) $P_{\delta}\in{\mathcal{P}_{2}^{1}}$

(b) $P_{\eta}\in{\mathcal{P}_{2}^{1}}$

(c) $P_{\vartheta}\in{\mathcal{P}_{2}^{2}}$

(d) $P_{\psi}\in{\mathcal{P}_{2}^{2}}$

(e) $P_{\zeta}\in{\mathcal{P}_{2}^{3}}$

(f) $P_{\theta}\in{\mathcal{P}_{3}^{2}}$

Figure 1: Three-dimensional interpolating surfaces of the polynomials
associated to the MIP’s for the given matrices $\delta$, $\eta$, $\vartheta$,
$\psi$, $\zeta$ and $\theta$ indicating the data points.

###### Example 5.2

Let $A_{1}\in\mathbb{R}^{2\times 3}$, $A_{2}\in\mathbb{R}^{3\times 2}$,
$A_{3}\in\mathbb{R}^{3\times 3}$, and $A_{4}\in\mathbb{R}^{3\times 3}$ be four
matrices given by $A_{1}=\begin{pmatrix}1&-1&2\\\ -1&0&-2\end{pmatrix}$,
$A_{2}=\begin{pmatrix}1&-1\\\ -1&0\\\ 2&-2\end{pmatrix}$,
$A_{3}=\begin{pmatrix}1&2&3\\\ 2&0&4\\\ 3&4&-1\end{pmatrix}$, and
$A_{4}=\begin{pmatrix}0&1&2\\\ -1&0&3\\\ -2&-3&0\end{pmatrix}$ respectively.
Then, there exists unique $P_{A_{1}}\in\mathcal{P}_{2}^{3}$,
$P_{A_{2}}\in\mathcal{P}_{3}^{2}$, $P_{A_{3}}\in\mathcal{P}_{3}^{3}$, and
$P_{A_{4}}\in\mathcal{P}_{3}^{3}$ given as

$\displaystyle
P_{A_{1}}(x,y)=-4xy^{2}+\frac{13}{2}y^{2}+15xy-13x-\frac{49}{2}y+21,$ (47)
$\displaystyle
P_{A_{2}}(x,y)=-4x^{2}y+\frac{13}{2}x^{2}+15xy-\frac{49}{2}x-13y+21,$ (48)
$\displaystyle
P_{A_{3}}(x,y)=-\frac{9}{2}x^{2}y^{2}+\frac{33}{2}x^{2}y+\frac{33}{2}xy^{2}-\frac{123}{2}xy-12x^{2}-12y^{2}$
(49) $\displaystyle+46x+46y-34,$ $\displaystyle\textrm{and
}P_{A_{4}}(x,y)=-x^{2}y+xy^{2}+x^{2}-y^{2}-2x+2y,$ (50)

respectively. Clearly, $P_{A_{1}}(x,y)=P_{A_{2}}^{T}(x,y)$,
$P_{A_{3}}(x,y)=P_{A_{3}}^{T}(x,y)$, and $P_{A_{4}}(x,y)=-P_{A_{4}}^{T}(x,y)$.
Since $A_{1}=A_{2}^{T}$, $A_{3}=A_{3}^{T}$, and $A_{4}=-A_{4}^{T}$, where
$A^{T}$ represents the transpose of the matrix $A$, i.e., the properties of
transpose, symmetric and skew-symmetric matrices holds in the associated
polynomial subspaces of two variables.

(a) $P_{A_{1}}\in{\mathcal{P}_{2}^{3}}$

(b) $P_{A_{2}}\in{\mathcal{P}_{3}^{2}}$

(c) $P_{A_{3}}\in{\mathcal{P}_{3}^{3}}$

(d) $P_{A_{4}}\in{\mathcal{P}_{3}^{3}}$

(e) $P_{A_{1}}\in\mathcal{P}_{2}^{3}$ and $P_{A_{2}}\in\mathcal{P}_{3}^{2}$

Figure 2: Three-dimensional interpolating surfaces validating geometric
properties of the associated matrices $A_{1}$ $A_{2}$, $A_{3}$ and $A_{4}$
respectively.

###### Example 5.3

Let $\tau=\begin{pmatrix}-1&2\\\ 3&-4\end{pmatrix}$ and
$I_{2}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ be the identity matrix in
$\mathbb{R}^{2\times 2}$. Then, $\tau^{2}=\begin{pmatrix}7&-10\\\
-15&22\end{pmatrix}$ and there exists unique
$P_{\tau},P_{\tau^{2}},\textrm{and }P_{I_{2}}\in\mathcal{P}_{2}^{2}$ given as
$P_{\tau}(x,y)=-10xy+14x+13y-18$, $P_{\tau^{2}}(x,y)=54xy-76x-71y+100$, and
$P_{I_{2}}(x,y)=2xy-3x-3y+5$ respectively. Here,
$P_{\tau^{2}}(x,y)+5P_{\tau}(x,y)-2P_{I}(x,y)=0$, i.e.,

$P_{\tau^{2}}(x,y)-tr(\tau).P_{\tau}(x,y)+det(\tau).P_{I_{2}}(x,y)=0,$ (51)

i.e., the Cayley-Hamilton theorem holds in the associated polynomial space
$\mathcal{P}_{2}^{2}$.

Additionally, since the polynomial $P_{\tau}(x,y)=-10xy+14x+13y-18$, is
invertible (see example 2.7). Let $P_{\tau^{-1}}\in\mathcal{P}_{2}^{2}$ be the
inverse polynomial. Then, using the equation (51) and definition (2.2), we get

$P_{\tau^{-1}}(x,y)=\frac{1}{det(\tau)}[-P_{\tau}(x,y)+tr(\tau).P_{I}(x,y)]=-\frac{1}{2}x-y+\frac{7}{2}\in\mathcal{P}_{2}^{2}.$
(52)

Therefore, $\tau^{-1}=\begin{pmatrix}2&1\\\
\frac{3}{2}&\frac{1}{2}\end{pmatrix}$ and it is the required inverse of the
matrix $\tau\in\mathbb{R}^{2\times 2}$.

###### Example 5.4

Let us consider the polynomial map $D_{p}:\mathbb{R}^{2\times
2}\rightarrow\mathcal{P}_{2}^{2}$ defined as

$\displaystyle D_{p}{\left(\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\right)}=(4a-2b$ $\displaystyle-2c+d)+(-2a+2b+c-d)x$ (53)
$\displaystyle-(2a-b-2c+d)y+(a-b-c+d)xy.$

Since, ${D_{p}}{(kA+B)}=k{D_{p}}(A)+{D_{p}}(B)$ for all
$A,B\in\mathbb{R}^{2\times 2}$ and non-zero scalar $k$. The map (53) is
linear. Again, $B_{1}=\left\\{\begin{bmatrix}1&0\\\
0&0\end{bmatrix},\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\\
1&0\end{bmatrix},\begin{bmatrix}0&0\\\ 0&1\end{bmatrix}\right\\}$ and
$B_{2}={\\{1,x,y,xy\\}}$ are the standard bases of the spaces
$\mathbb{R}^{2\times 2}$ and $\mathcal{P}_{2}^{2}$ respectively, thus

$\displaystyle D_{p}{\left(\begin{bmatrix}1&0\\\
0&0\end{bmatrix}\right)}=4-2x-2y+xy,D_{p}{\left(\begin{bmatrix}0&1\\\
0&0\end{bmatrix}\right)}=-2+x+2y-xy,$ $\displaystyle
D_{p}{\left(\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}\right)}=-2+2x+y-xy\textrm{
and }D_{p}{\left(\begin{bmatrix}0&0\\\ 0&1\end{bmatrix}\right)}=1-x-y+xy.$

Therefore, the corresponding co-ordinate matrix with respect to the bases
$B_{1}$ and $B_{2}$ is

$[D_{p}]_{B_{1}}^{B_{2}}=\begin{bmatrix}4&-2&-2&1\\\ -2&1&2&-1\\\ -2&2&1&-1\\\
1&-1&-1&1\end{bmatrix}_{4\times 4}.$ (54)

Here, $det\left([D_{p}]_{B_{1}}^{B_{2}}\right)=-1\neq 0$, i.e.,
$ker(D_{p})=\left\\{0\right\\}$.

###### Note 5.1

The inverse linear transformation
$D_{p}^{-1}:\mathcal{P}_{2}^{2}\rightarrow\mathbb{R}^{2\times 2}$ is given by

$D_{p}^{-1}{\left(a+bx+cy+dxy\right)}=\begin{bmatrix}a+b+c+d&a+b+2c+2d\\\
a+2b+c+2d&a+2b+2c+4d\end{bmatrix}.$

The corresponding co-ordinate matrix with respect to the bases $B_{2}$ and
$B_{1}$ is

$[D_{p}^{-1}]_{B_{2}}^{B_{1}}=\begin{bmatrix}1&1&1&1\\\ 1&1&2&2\\\ 1&2&1&2\\\
1&2&2&4\end{bmatrix}_{4\times 4}.$ (55)

Clearly,

$[D_{p}]_{B_{1}}^{B_{2}}.[D_{p}^{-1}]_{B_{2}}^{B_{1}}=\begin{bmatrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{bmatrix}_{4\times
4}=[D_{p}^{-1}]_{B_{2}}^{B_{1}}.[D_{p}]_{B_{1}}^{B_{2}}.$

###### Example 5.5

Let $A_{1}\in\mathbb{R}^{2\times 3}$, $A_{2}\in\mathbb{R}^{3\times 2}$,
$A_{3}\in\mathbb{R}^{3\times 3}$, and $A_{4}\in\mathbb{R}^{3\times 3}$ be four
matrices given in the example 5.2. Then,
${A_{1}}{A_{2}}=\begin{pmatrix}6&-5\\\ -5&5\end{pmatrix}$,
${A_{1}}{A_{3}}=\begin{pmatrix}5&10&-3\\\ -7&-10&-1\end{pmatrix}$,
${A_{1}}{A_{4}}=\begin{pmatrix}-3&-5&-1\\\ 4&5&-2\end{pmatrix}$,
${A_{2}}{A_{1}}=\begin{pmatrix}2&-1&4\\\ -1&1&-2\\\ 4&-2&8\end{pmatrix}$,
${A_{3}}{A_{4}}=\begin{pmatrix}-8&-8&8\\\ -8&-10&4\\\ -2&6&18\end{pmatrix}$,
and ${A_{4}}{A_{1}}=\begin{pmatrix}8&8&2\\\ 8&10&-6\\\
-8&-4&-18\end{pmatrix}$. Therefore, there exists unique
$P_{{A_{1}}{A_{2}}}\in\mathcal{P}_{2}^{2}$,
$P_{{A_{1}}{A_{3}}}\in\mathcal{P}_{2}^{3}$,
$P_{{A_{1}}{A_{4}}}\in\mathcal{P}_{2}^{3}$,
$P_{{A_{2}}{A_{1}}}\in\mathcal{P}_{3}^{3}$,
$P_{{A_{3}}{A_{4}}}\in\mathcal{P}_{3}^{3}$, and
$P_{{A_{4}}{A_{3}}}\in\mathcal{P}_{3}^{3}$ given as

$\displaystyle P_{{A_{1}}{A_{2}}}(x,y)=$ $\displaystyle{}21xy-32x-32y+49,$
$\displaystyle P_{{A_{1}}{A_{3}}}(x,y)=$ $\displaystyle
15xy^{2}-24y^{2}-53xy+26x+85y-44,$ $\displaystyle P_{{A_{1}}{A_{4}}}(x,y)=$
$\displaystyle-7xy^{2}+10y^{2}+24xy-10x-35y+15,$ $\displaystyle
P_{{A_{2}}{A_{1}}}(x,y)=$
$\displaystyle\frac{17}{2}x^{2}y^{2}-32x^{2}y-32xy^{2}+\frac{241}{2}xy+\frac{55}{2}x^{2}+\frac{55}{2}y^{2}-\frac{207}{2}x-\frac{207}{2}y+89,$
$\displaystyle P_{{A_{3}}{A_{4}}}(x,y)=$
$\displaystyle-3x^{2}y^{2}+15x^{2}y+9xy^{2}-47xy-9x^{2}+2y^{2}+29x+8y-12,$
$\displaystyle\textrm{and }P_{{A_{4}}{A_{3}}}(x,y)=$ $\displaystyle
3x^{2}y^{2}-9x^{2}y-15xy^{2}+47xy-2x^{2}+9y^{2}-8x-29y+12,$

respectively. The equations (47), (48), (49), and (50) and the definition
(2.2) verify that
$P_{{A_{1}}{A_{2}}}(x,y)={P_{A_{1}}(x,y)}\otimes{P_{A_{2}}(x,y)}$,
$P_{{A_{1}}{A_{3}}}(x,y)={P_{A_{1}}(x,y)}\otimes{P_{A_{3}}(x,y)}$,
$P_{{A_{1}}{A_{4}}}(x,y)={P_{A_{1}}(x,y)}\otimes P_{A_{4}}(x,y)$,
$P_{{A_{2}}{A_{1}}}(x,y)={P_{A_{2}}(x,y)}\otimes P_{A_{1}}(x,y)$,
$P_{{A_{3}}{A_{4}}}(x,y)={P_{A_{3}}(x,y)}\otimes P_{A_{4}}(x,y)$, and
$P_{{A_{4}}{A_{3}}}(x,y)={P_{A_{4}}(x,y)}\otimes P_{A_{3}}(x,y)$.

##  6 Concluding remarks

In this paper, the well known $mn$-dimensional two-variate space of tensor-
product polynomials, $\mathcal{P}_{m}^{n}$, of degree at most $(m-1,n-1)$,
over the field $\mathbb{R}$ is considered. A theory of two-variable
polynomials is developed in $\mathcal{P}_{m}^{n}$, parallel to the basic
algebraic operations and geometric properties of the space
$\mathbb{R}^{m\times n}$, by establishing the basic algebra and some algebraic
properties with respect to the usual addition, scalar multiplication, and a
newly defined algebraic operation ‘$\otimes$’. The space $\mathcal{P}_{m}^{n}$
form a subspace of two variable polynomials, of degree at most $m+n-2$ and is
isomorphic to $\mathbb{R}^{m\times n}$ with respect to vector algebra
structure under usual addition and scalar multiplication operation. Using
counterexamples, it is verified that space $\mathcal{P}_{n}^{n}$ is non-
commutative and possess zero-divisor property under the operation $\otimes$.
It is proved that $\mathcal{P}_{m}^{n}$ is closed under the operation
$\otimes$, if $m=n$. Some more algebraic properties of the polynomials in
$\mathcal{P}_{m}^{n}$ are discussed under the operation $\otimes$ and it is
proved that the associative and distributive law hold. Some definitions are
included which define identity, inverse, eigenvalue, and power of the
polynomials in $\mathcal{P}_{m}^{n}$ under the operation $\otimes$. In
addition, the transpose, symmetry, skew-symmetry, and orthogonality of the
polynomials in $\mathcal{P}_{n}^{n}$ is also defined.

In the next section, the existence of the space $\mathcal{P}_{m}^{n}$ is
established using tensor-product and univariate polynomial interpolation
approach with respect to the MIP(* ‣ 1.1) for a given matrix
$(a_{ij})\in\mathbb{R}^{m\times n}$. The poisedness of the MIP(* ‣ 1.1) in
$\mathcal{P}_{m}^{n}$ is also ensured. Further, three formulae are derived to
construct the polynomial in $\mathcal{P}_{m}^{n}$ which satisfy the MIP(* ‣
1.1) for a given matrix $(a_{ij})\in\mathbb{R}^{m\times n}$ using well-known
univariate interpolation formulae. The method of the non-homogeneous system of
linear equations is also remarked to obtain the coefficients of the required
polynomial.

The next section focuses on some properties of the established polynomial map
from $\mathbb{R}^{m\times n}$ to $\mathcal{P}_{m}^{n}$ on behalf of the
construction formulae. It is achieved that the obtained polynomial map is
invertible and linear, i.e., is an isomorphism. The respective invertible
linear map is also defined. Finally, the multiplicative structure of the
polynomials with respect to given matrices is discussed and it is proved that
$\langle\mathbb{R}^{n\times n},.\rangle$ and
$\langle\mathcal{P}_{n}^{n},\otimes\rangle$ are isomorphic, i.e., the spaces
$\mathbb{R}^{n\times n}$ and $\mathcal{P}_{n}^{n}$ are isomorphic with respect
to the algebra structure. The existence of the identity polynomial in
$\mathcal{P}_{m}^{n}$ under the operation $\otimes$ is also ensured which
endorses that $\langle\mathcal{P}_{n}^{n},+,\otimes\rangle$ form a ring with
unity over $\mathbb{R}$.

In the last section, five examples are included to validate the results and
for geometrical observations. The first example is validating the theorems
3.1, 3.2, 3.3, and 3.4 geometrically by surface diagrams in the figures 1. The
second example shows that the polynomials in the associated subspaces preserve
the geometric properties of the respective matrices such as transpose,
symmetry, and skew-symmetry. In the third example, it is verified that the
polynomials in the associated subspace $\mathcal{P}_{2}^{2}$ satisfy the
result of Cayley-Hamilton theorem with respect to the given matrix in
$\mathbb{R}^{2\times 2}$ and the inverse of the given matrix is also obtained
using polynomial algebra. In fourth example, the polynomial map from
$\mathbb{R}^{2\times 2}$ to $\mathcal{P}_{2}^{2}$ is defined using the result
of the theorems 3.2, 3.3 or 3.4. It is verified that the defined map is linear
as well as invertible. The inverse linear map is also included for the cross
verification. Finally, last example verify that
$P_{{A_{i}}{A_{j}}}(x,y)=P_{A_{i}}(x,y)\otimes P_{A_{j}}(x,y)$ (wherever
defined) for $i,j=1,2,3,4$ with respect to the given matrices
$A_{1}\in\mathbb{R}^{2\times 3}$, $A_{2}\in\mathbb{R}^{3\times 2}$,
$A_{3}\in\mathbb{R}^{3\times 3}$, and $A_{4}\in\mathbb{R}^{3\times 3}$.
Theoretical findings and numerical observations assert that space
$\mathcal{P}_{m}^{n}$ preserves almost every algebraic and geometrical
structure of the space $\mathbb{R}^{m\times n}$.

The transformation of the finite order real matrices
$a_{ij}\in\mathbb{R}^{m\times n}$ into $\mathcal{P}_{m}^{n}$, algebraic
operations, respective algebra, algebraic and geometric structures, and
isomorphism are building some additional tools for algebra, matrix theory,
cryptography, and numerical linear algebra community. Considering the rapid
growth in the development of the efficient algorithms for the polynomial
computation (over matrices), the outcomes can also be beneficial for
scientific and numerical computations. In addition, a matrix is a most common
data structure to store the complete information of the objects, independent
of the content of the data, in the various models of computer graphics, signal
processing, image processing, computer-aided designs in control systems,
thermal engineering to name a few. Therefore, the generation of the respective
three-dimensional smooth interpolating surface (preserving the geometric
nature) in the established subspace of two variable polynomials could play
some important roles in the stages of analysis, interpretation, and
applications.

## References

  * [1] F. S. Alfio Quarteroni, Riccardo Sacco, Numerical Mathematics.  Texts in Applied Mathematics, Springer, Berlin, Heidelberg, 2007.
  * [2] S. J. Axler, Linear Algebra Done Right.  Undergraduate Texts in Mathematics, Springer International Publishing, third ed., 2015.
  * [3] C. De Boor and A. Ron, “Computational aspects of polynomial interpolation in several variables,” Mathematics of Computation, vol. 58, no. 198, pp. 705–727, 1992.
  * [4] T. Sauer and Y. Xu, “On multivariate lagrange interpolation,” Mathematics of Computation, vol. 64, no. 211, pp. 1147–1170, 1995.
  * [5] T. Sauer, “Polynomial Interpolation in Several Variables: Lattices, Differences, and Ideals,” Studies in Computational Mathematics, vol. 12, pp. 191–230, 2006.
  * [6] K. Saniee, “A Simple Expression for Multivariate Lagrange Interpolation,” SIAM Undergraduate Research Online, 2008.
  * [7] P. J. Olver, “On Multivariate Interpolation,” Studies in Applied Mathematics, vol. 116, no. 4, pp. 201–240, 2006.
  * [8] M. Gasca and T. Sauer, “Polynomial Interpolation in Several Variables,” Advances in Computational Mathematics, vol. 12, pp. 377–410, 2000.
  * [9] A. L. M´ehaut´e, “On some aspects of multivariate polynomial interpolation,” Advances in Computational Mathematics, vol. 12, pp. 311–333, 2000.
  * [10] S. Levy, L. Velho, A. Frery, and J. Gomes, Image Processing for Computer Graphics and Vision.  Texts in Computer Science, Springer London, 2009.
  * [11] M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis, and Machine Vision.  Cengage Learning, 2014.
  * [12] P. Larsen and N. Hansen, Computer Aided Design in Control and Engineering Systems: Advanced Tools for Modern Technology.  IFAC proceedings series, Elsevier Science, 2014.
  * [13] Z. Chen, Computer Aided Design in Control Systems 1988: Selected Papers from the 4th IFAC Symposium, Beijing, PRC, 23-25 August 1988.  IFAC Symposia Series, Elsevier Science, 2017.
  * [14] E. Sciubba, G. Manfrida, and U. Desideri, ECOS 2012 The 25th International Conference on Efficiency, Cost, Optimization and Simulation of Energy Conversion Systems and Processes (Perugia, June 26th-June 29th, 2012).  Proceedings e report, Firenze University Press, 2012.
  * [15] J. C. Carr, W. R. Fright, and R. K. Beatson, “Surface interpolation with radial basis functions for medical imaging,” IEEE transactions on medical imaging, vol. 16, no. 1, pp. 96–107, 1997.
  * [16] D. Bini and V. Y. Pan, Polynomial and Matrix Computations (Vol. 1): Fundamental Algorithms.  Basel, Switzerland, Switzerland: Birkhauser Verlag, 1994.
  * [17] M. G. Anne Bourlioux, Modern Methods in Scientific Computing and Applications.  75, Springer Netherlands, 2002.
  * [18] U. Langer and P. Paule, Numerical and Symbolic Scientific Computing: Progress and Prospects.  Texts & Monographs in Symbolic Computation, Springer Vienna, 2011.
  * [19] S. Dharm Prakash and U. Amit, “Matrix interpolation problem,” https://arxiv.org/abs/1712.08165, 2017.
  * [20] P. Kergin, “A natural interpolation of ck functions,” Journal of Approximation Theory, vol. 29, no. 4, pp. 278–293, 1980.
  * [21] S. NARUMI, “Some formulas in the theory of interpolation of many independent variables,” Tohoku Mathematical Journal, First Series, vol. 18, pp. 309–321, 1920.
  * [22] K. Atkinson, Elementary Numerical Analysis.  Wiley, New York, 1985.
  * [23] J. R. Jain MK, Iyengar SRK, Numerical methods for scientific and engineering computation.  2nd ed, New Delhi: Wiley Eastern Limited, 1991.
  * [24] B. Einarsson, Accuracy and Reliability in Scientific Computing.  Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 2005\.
  * [25] M. Heath, Scientific computing: an introductory survey.  McGraw-Hill Higher Education, McGraw-Hill, 2002.
  * [26] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing.  New York, NY, USA: Cambridge University Press, 3 ed., 2007.


# Is your forecaster smarter than an energy engineer: a deep dive into
electricity price forecasting

Maria Margarida Mascarenhas  ELECTA, KU Leuven
Belgium
<EMAIL_ADDRESS>Hussain Kazmi  ELECTA,
KU Leuven
Belgium
<EMAIL_ADDRESS>
###### Abstract

The field of electricity price forecasting has seen significant advances in
the last years, including the development of new, more accurate forecast
models. These models leverage statistical relationships in previously observed
data to predict the future; however, there is a lack of analysis explaining
these models, which limits their real world applicability in critical
infrastructure. In this paper, using data from the Belgian electricity
markets, we explore a state-of-the-art forecasting model to understand if its
predictions can be trusted in more general settings than the limited context
it is trained in. If the model produces poor predictions in extreme conditions
or if its predictions are inconsistent with reality, it cannot be relied upon
in real-world where these forecasts are used in downstream decision-making
activities. Our results show that, despite being largely accurate enough in
general, even state of the art forecasts struggle with remaining consistent
with reality.

###### Index Terms:

electricity price forecasting, LEAR, explainability models

##  I Introduction

With the liberalization of the European electricity system, market prices
serve several important purposes. On the one hand, they allow grid operators
to promote competition and ensure grid stability. On the other, they allow
market participants, such as generators and demand aggregators etc., to
maximize their profits through new services and algorithmic innovations. In
other words, they allow scheduling algorithms to determine the optimal time to
consume or produce electricity [1].

Forecasting electricity prices is therefore becoming increasingly important
due to the competition in the market. An accurate forecasting algorithm
therefore increases the utility of relevant stakeholders. In this paper, we
investigate the behavior of a state-of-the-art electricity price forecasting
algorithm [2]. More concretely, we address the question of whether this
forecaster is physically consistent, i.e. whether it attributes the right
importance to the relevant input features when making forecasts for the
future, and if its predictions can be trusted in extreme market conditions?
For dependable and mission critical infrastructure, it may actually be
preferable to have a slightly less accurate model that makes its predictions
giving a physically consistent importance to input features, than a more
accurate model that makes more accurate predictions for dubious reasons. The
latter model will often fail to generalize in unexpected conditions, such as
the ones that arose in the aftermath of pandemic-related disruptions.

##  II Data and method

We investigate the performance of a state of the art price forecaster (LEAR
model) for the EPEX-BE electricity market. The EPEX-BE market prices
demonstrate high volatility with several spikes and negative prices. Section
II-A shows a detailed list with the features used for this market to make the
price forecasts, and where they were obtained. Section II-B presents a
detailed description of the forecast model.

###  II-A Data

The data used by the forecaster to predict EPEX-BE spot prices was obtained
from the ENTSO-E transparency platform website [3] for the Belgium features,
and the Rte website [4] for the French features. The data presented below are
the input features used by the forecast model ranging from 01/01/2015 until
31/12/2021.

  * •

Electricity day-ahead prices

  * •

Day-ahead generation forecast (FR)

  * •

Day-ahead system load forecast (FR)

  * •

Day-Ahead generation forecast for wind (BE)

  * •

Day-Ahead generation forecast for solar (BE)

  * •

Day-ahead system load forecast (BE)

There were some missing days in these fields, which were substituted with the
average of the day before and after the missing data.

###  II-B The LEAR Model

The LEAR model uses past values as predictors and has exogenous variables as
explanatory variables. LEAR uses the least absolute shrinkage and selection
operator (LASSO), a feature selection algorithm, to choose among the variables
and estimate their parameters [5]. The model has previously been shown to
offer excellent accuracy to performance ratio [2]. The input features that the
model needs to predict the 24 day-ahead prices for day $d$
($p_{d}=[p_{d,1},...,p_{d,24}]$) include:

  * •

The historical day-ahead prices of the previous three days and seven days ago:
$p_{d-1},p_{d-2},p_{d-3},p_{d-7}$

  * •

The day-ahead forecasts of the exogenous variables considered from day $d$,
$d-1$ and $d-7$: $x^{i}_{d}=[x^{i}_{d,1},...,x^{i}_{d,24}]$,
$x^{i}_{d-1}=[x^{i}_{d-1,1},...,x^{i}_{d-1,24}]$,
$x^{i}_{d-7}=[x^{i}_{d-7,1},...,x^{i}_{d-7,24}]$ where $i$ is the exogenous
variable in question.

  * •

A binary dummy variable that represents the day of the week:
$z=[z_{d,1},...,z_{d,7}]$ where each parameter is 0 except the one
corresponding to the predicted day of the week. For example, in a Tuesday the
variable would be $z=[0,1,0,0,0,0,0]$

The LEAR equation’s coefficients will be analysed since they indicate the
weight that the LEAR model assigns to each feature. Of course, the actual
contribution will be the product between these coefficients and the feature
value itself. This model was trained using a calibration window of one year
and then used to predict hourly prices from the beginning of 2016 until the
end of 2021. The input features are normalized before being used by the
forecaster. The actual implementation borrows heavily from the epftoolbox,
outsourced in [2], but supplements the implemented price forecaster with more
input features, as explained previously.

##  III Results and Discussion

Before analyzing the physical consistency of the forecast model, it is
important to understand that the LEAR model fails to predict accurate prices
when there are: (1) negative or unusually low electricity prices and (2)
unusually high prices. Outside of these two scenarios, the model is
significantly accurate. This is illustrated in Fig. 2. In Fig. 1 it is
possible to observe the product between the model’s coefficients and the
feature value of four hours with spikes and four hours with negative prices.
The real price (underlined) and the predicted price are also presented in each
bar. These eight hours were chosen at random to illustrate the importance that
the LEAR model attributes to each feature.

Figure 1: Product between LEAR model’s coefficients and feature value, which
the sum plus the b-value of the regression for each hour equals the predicted
price.

###  III-A Negative prices

The model, in general, seems incapable of modelling negative electricity
prices accurately. Negative electricity prices in Belgium are often due to a
high energy generation surplus (e.g. through renewable sources) combined with
low demand. This implies that the model should learn a negative causal link
between (predicted) renewable energy generation and electricity prices.
However, for most of the experimental period, the LEAR model places rather low
importance to both solar and wind. In fact, the features with the most
significant influence include the French load and generation forecasts.
Likewise, the coefficients assigned to French electricity generation are
supposed to be negative since larger imports would decrease the price of
electricity in Belgium. However, this is not evidenced during 2021, when
prices crashed in the post-Covid lockdown periods. Therefore, for the negative
prices to be more accurately predicted, renewable energy generation should
have a more considerable influence and the load forecast in France a smaller
one.

###  III-B High prices

For high prices, the model makes accurate predictions when the prices of the
previous days are high as well, owing to the autoregressive component.
However, when there is a spike in electricity prices, i.e., an isolated high
price, the LEAR model fails to predict this spike, and instead predicts much
lower price values. Regarding the spikes, the features that most influence the
prediction of the LEAR model are the French generation and load forecasts.
Since most of the analysed spikes take place during winter evenings, it makes
sense for the Belgian wind generation forecast to be a key determinant.
However, this would not necessarily fix under-predictions, since the
predictions are often lower than the real price. What would make the
predictions more accurate is assigning greater importance in this case to the
load forecast in Belgium and the load forecast in France for the last two
days.

##  IV Conclusion

This paper shows the importance of analyzing a model’s accuracy in greater
detail, and studying the forecasting models’ predictions for underlying
causality. Thus, it is evident from this brief analysis that the LEAR model is
quite accurate, but it fails during extreme conditions, as seen in Fig. 2.
This should be foreseeable by a domain expert, but is not necessarily
something the model generalizes well on. Moreover, with the pandemic, the
model’s accuracy appears to have decreased.

Creating new models is not always the solution to a more reliable forecast
algorithm, but understanding where the model fails and consequently ensuring
that the importance given to input features is physically consistent (and
causally accurate). Therefore, more work should be put into understanding the
models and improving their interpretability. The paper includes preliminary
ideas for how the models could have performed more accurately, but these
suggestions are limited by the issue of multicollinearity of features, which
makes it important to interpret the model’s behaviour. A more thorough
analysis can address this, in addition to also investigating how other state
of the art models, such as deep neural networks, perform on this metric.

## References

  * [1] Catalão, J.P.S., Mariano, S.J.P.S., Mendes, V.M.F., and Ferreira, L.A.F.M. “Short-term electricity prices forecasting in a competitive market: A neural network approach”. In: Electric Power Systems Research 77.10 (2007), pp. 1297–1304. ISSN: 0378-7796. DOI: https://doi.org/10.1016/j.epsr. 2006.09.022. URL: https://www.sciencedirect.com/science/article/pii/ S0378779606002422.
  * [2] Lago, J., Marcjasz, G., De Schutter, B., & Weron, R. (2021). EPFTOOLBOX: The first open-access PYTHON library for driving research in electricity price forecasting (EPF). WORMS Software (WORking papers in Management Science Software).
  * [3] ENTSO-E. ENTSO-E Transparency Platform. 2022. URL: https://transparency.entsoe.eu/dashboard/show.
  * [4] Rte. View data published by RTE. 2022. URL: https://www.services- rte.com/en/view-data-published-by-rte.html.
  * [5] Uniejewski, Bartosz, Nowotarski, Jakub, and Weron, Rafał. “Automated Variable Selection and Shrinkage for Day-Ahead Electricity Price Forecasting”. In: Energies 9.8 (2016). ISSN: 1996-1073. DOI: 10 . 3390 / en9080621. URL: https://www.mdpi.com/1996-1073/9/8/621

## Appendix

Figure 2: Scatter plot of the error (difference between the real and predicted
price) vs real price.