Journal of Computational and Applied Mathematics 261 (2014) 30–38

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Journal of Computational and AppliedMathematics

journal homepage: www.elsevier.com/locate/cam

A tabular methodology to identify minimal row degrees forMatrix Padé Approximants✩

Celina Pestano-Gabino ∗, Concepción González-Concepción,María Candelaria Gil-FariñaDepartment of Applied Economics, University of La Laguna, 38071 La Laguna, Spain

a r t i c l e i n f o

Article history:Received 15 April 2013Received in revised form 7 October 2013

MSC:41

Keywords:Matrix Padé ApproximationMinimal row degreesCanonical representation

a b s t r a c t

In this paper we propose a tabular procedure for identifying sets of minimal row degreescorresponding to Matrix Padé Approximants. The objective is to make it easier to interpretand properly apply them in various fields. When considering formal matrix power serieswith k × m coefficients, it is useful to examine the linearly dependent rows that are inthe last k rows in certain Hankel matrices corresponding to the coefficients of the series. Alist of properties and suitable examples are added not only by way of illustration but alsobecause they are an original part of themethod, having been chosen to teach the procedureand to underscore certain precautions to be taken.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

A canonical Matrix Padé Approximants (MPA) with minimal row degrees is defined and studied in [1], inspired by theoriginal way in which Tiao and Tsay in [2] resolved the identification of time series VARMA models, through what theydefine as Scalar Component Models (SCM). In fact, the objective of [1] was: to find rational approximations with matrixpolynomials of a matrix power series, under certain conditions of minimality on the degrees of the rows of these polynomi-als, with the constant term in the denominator being invertible and the proposed representation having no free parameters(i.e. is canonical). However, given a set of minimal row degrees (m.r.d.) for an approximant, we propose in [1, Section 5] aunique canonical representation, but an approximant could have several sets of m.r.d. Therefore, we propose in [1] alter-native canonical representations for an approximant. In addition, given a set of m.r.d. for an approximant we could defineother canonical representations different from thosewe have defined in [1]. Depending on the field to be used, some of themmay be better than others.

This work addresses an open question in [1]: automatically finding all of the possible sets of minimal row degrees ofa MPA, for those users who are not familiar with the more mathematical aspects of the results and proofs in [1]. A semi-automatic procedure is proposed in this paper that makes it easier to identify the sets of minimal row degrees to be appliedin several fields.

We highlight the difficulties involved in finding the proper uniqueness and minimality of a MPA and the guidanceprovided by studying these concepts using a tabular methodology. By way of examples related to the process used in thispaper, in the scalar case, Table C provides a fundamental tool by characterizing a rational function and the degrees of the

✩ This work was partially funded by ‘‘Ministerio de Educación y Ciencia’’ (MTM2009-14039-C06-01 and MTM2011-28781).∗ Corresponding author. Tel.: +34 922 31 70 26; fax: +34 922317204.

E-mail addresses: cpestano@ull.es (C. Pestano), cogonzal@ull.es (C. González), mgil@ull.es (M.C. Gil).

0377-0427/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cam.2013.10.012

C. Pestano-Gabino et al. / Journal of Computational and Applied Mathematics 261 (2014) 30–38 31

polynomials that represent it in its unique irreducible or minimal form. In the matrix case, the concepts of rationality,minimality and uniqueness take on a greater complexity and independence. For example, the Hankel determinants shownin Table C lose importance because, among other reasons, they only make sense when the matrices are square. But notonly because of this; even if the matrices are square, Table C does not present a structure that allows for a characterizationof rationality (see [3]). Then, [4] provides an answer to this question by using the rank of Hankel matrices. Furthermore,given a matrix rational function, the ‘‘smallest’’ degrees of the matrix polynomials which represent it are not necessarilyunique. Various types have been defined in the literature involving the degrees of the polynomials that appear in a rationalfunction. For instance, [4] studies aminimal overall degrees, [1] a type of row degrees, etc. In addition, given that the rationalrepresentation of the function for the same pair of minimal overall degrees need not be unique, [4] gives conditions to studythe uniqueness of said representation. All the results obtained are presented graphically in tables.

This paper is divided into sections as follows. Section 2 outlines the preliminary definitions and notation. Section 3comprising the bulk of the paper, the tabular methodology to address the question of what sets of row degrees are minimalfor the approximants of interest. We illustrate the last step of the tabular procedure with a list of properties and suitableexamples chosen to make the methodology easier to understand and apply. Section 4 includes computational aspects of theMathematica procedure developed to study the examples in the previous section. Section 5 introduces a possible applicationof this paper. Finally, we present the conclusions, some open questions and the references.

2. Preliminary definitions and notation

Next we summarize the definitions and notation necessary for our contribution, namely, the practical interpretation ofthe main results in [1], with the help of a suitable tabular methodology.

We will rely on the MPA notation that is most commonly used in the literature.

Definition 1. Starting with F(z) =

∞

i=0 cizi, ci ∈ Ck×m, z ∈ C , assuming that there exist P(z) and Q (z), of degrees p and q

respectively, and of suitable dimensions such that P(z)F(z) − Q (z) = O(zp+q+1), then (P(z),Q (z)) is said to be a left MPAof degrees (p, q), which we shall denote L

[q/p]F . If P(z) is invertible, then equivalently if F(z) − P−1(z)Q (z) = O(zp+q+1), itis said that P−1(z)Q (z) is a left Padé MPA of degrees (p, q).

Analogously, a right MPA of degrees (p, q) is defined, which we shall denote R[q/p]F . We will only deal with the left MPA,

though similar results could be obtained for the right MPA.The polynomial P(z) = P0 + P1z +· · ·+ Ppzp is called the denominator and Q (z) = Q0 +Q1z +· · ·+Qqzq the numerator

of the corresponding approximant.As a consequence of the definition, L[q/p]F exists if and only if the homogeneous system

(PpPp−1 . . . P0)M(p, q, p + q) = 0 (1)

has a solution, where the Hankel matrix M(i, j, h) =

cj−i+1 cj−i+2 · · ch−icj−i+2 cj−i+3 · · ch−i+1

· · · · ·

· · · · ·

cj cj+1 · · ch−1cj+1 cj+2 · · ch

for i, j ∈ N, i ≥ 0, j ≥ 0 and h ≥ i + j.

By convention, cn = 0 ∈ Ck×m if n < 0 and M(0, j, j) = 0 ∈ Ck×m. In particular, L[q/p]F with P0 invertible exists if and onlyif (1) has a solution with P0 invertible, which is equivalent to rankM(p − 1, q − 1, p + q − 1) = rankM(p, q, p + q).

It is a trivial observation that any approximant of degrees (p, q), P−1(z)Q (z), with P0 = I (I the identitymatrix) but invert-ible, has another representation, (P−1

0 P(z))−1(P−10 Q (z)), with the same degrees (p, q) but with constant term P−1

0 P(0) = Iin the denominator.

The main concepts in this paper are in the following definitions.

Definition 2 ([4]). Let A(z) = P−1(z)Q (z) a matrix rational function and P0 = I . The degrees p and q of the matrix polyno-mials P(z) and Q (z), are considered to be minimal overall (to the left) if and only if for any two other polynomials D(z) andN(z) of degrees d and n respectively, which satisfy D(0) = I and A(z) = D−1(z)N(z), it holds that n < q implies d > p andd < p implies n > q.

Definition 3 ([1]). A(z), a matrix rational function of dimension k × m, has at least one row representation (a, b), if thereexist P(z) = P0 + P1z + · · · + Ppzp and Q (z) = Q0 + Q1z + · · · + Qqzq, matrix polynomials of dimensions k × k and k × m,respectively, P0 being invertible, A(z) = P−1(z)Q (z) and there exists i ∈ {1, 2, . . . , k} such that the degree of the i-th rowof P(z) is a and the degree of the i-th row of Q (z) is b.

Moreover, we say that: (a, b) is a pair of row degrees, the set C with the k pairs of row degrees is a set of row degrees forA(z) and we use µC (a, b) to denote the times that (a, b) is repeated in C .

In general, a row representation is not associated with a specific row (see, for instance Example 3.1 in [1]).

32 C. Pestano-Gabino et al. / Journal of Computational and Applied Mathematics 261 (2014) 30–38

Consequently, we say that the vector (v0, v1, . . . , va; u0, u1, . . . , ub), of dimension k(a+ 1) +m(b+ 1), is one (possible)row representation (a, b) for A(z) if said vector is specifically one row, assume the i-th, of (P0, P1, . . . , Pa; Q0,Q1, . . . ,Qb)and the i-th row of Pa+1, . . . , Pp,Qb+1, . . . , and Qq is zero.

Definition 4 ([1]). The k-vector v0 in (v0, v1, . . . , va; u0, u1, . . . , ub) is called the base vector of (v0, v1, . . . , va; u0, u1,. . . , ub), such a row representation (a, b). Moreover, two row representations are said to be linearly independent (l.i.) iftheir two associated base vectors are l.i. (see, for instance, Example 3.2 in [1]).

Therefore, k l.i. row representations (vi0, v

i1, . . . , v

ipi; u

i0, u

i1, . . . , u

iqi) for i = 1, 2, . . . , k, result in polynomials P(z) and

Q (z) with P0 invertible and A(z) = P−1(z)Q (z). We can assume, without loss of generality, P(z) = P0 + P1z + · · · + Ppzp

such that the i-th row of Pj (for j = 0, 1, 2, . . . , pi) is vij and zero if j > pi; and Q (z) = Q0 + Q1z + · · · + Qqzq such that the

i-th row of Qj (for j = 0, 1, 2, . . . , qi) is uij and zero if j > qi.

In keeping with the goal of this paper, we are interested in finding sets of minimal row degrees, as defined below:

Definition 5 ([1]). The set C = {(p1, q1), (p2, q2), . . . , (pk, qk)} is a set of minimal row degrees (set of m.r.d.) for A(z) if andonly if:

(a) C is a set of row degrees,(b) there exist k l.i. row representations for A(z) with degrees (p1, q1), (p2, q2), . . . , and (pk, qk), and(c) there is no other set {(a1, b1), (a2, b2), . . . , (ak, bk)} of k pairs of degrees corresponding to k l.i. row representations for

A(z) such thatk

i=0(pi + qi) >k

i=0(ai + bi).

Moreover, we say thatk

i=0(pi + qi) is the degree of set C (denoted deg C) and that a pair of row degrees is minimal if itbelongs to a set of m.r.d. If the set of row degrees {(p1, q1), (p2, q2), . . . , (pk, qk)} is minimal, any other set resulting fromchanging the order of pairs involved remains the same. When studying possible sets of m.r.d., two sets of row degrees areassumed to be different if at least one of their pairs differ.

We note that in the scalar case, there exist neither diverse possibilities for pairs of minimal degrees nor the various waysto define other types on minimality, in particular in terms of overall degrees. In the matrix case, the Definition 2 given forminimal overall degrees is unaffected if P0 = I or P0 is invertible in general. In contrast, the definition of the set ofm.r.d. doesdepend on the choice of P0 (see, for instance, Example 2.1 in [1]). Therefore, we consider the case in which P0 is invertible,where P0 = I would appear as a possible particular case, because this leaves the possibility open of obtaining smaller valuesfor the row degrees. In order to achieve our objective, the invertible constant term in the denominator must be constructedat the end of the process, without a predefined form (see [1, Section 5]).

We believe that the following definition is the Padé problem that corresponds with the SCM problem, constituting a newapproach to the Matrix Padé Approximation.

Definition 6 ([1]). Given the matrix power series F(z), (P(z)Q (z)) is said to be a canonical MPA (p, q) with minimal rowdegrees {(p1, q1), . . . , (pk, qk)} if and only if:

(a) P(z)F(z) − Q (z) = O(zp+q+1), where max {pi} ≤ p and max{qi} ≤ q,(b) P0 is invertible,(c) each pair of degrees (pi, qi), i = 1, 2, . . . , k, is associated with a different row of (P(z),Q (z)),(d) {(p1, q1), . . . , (pk, qk)} is a set of m.r.d. for A(z) = P−1(z)Q (z) and(e) the rational representation (P(z),Q (z)) of A(z) is canonical, that is, no free coefficients remain in P(z) nor in Q (z).

In terms of item (e), as we know, the approximant L[q/p]F exists if and only if rankM(p − 1, q − 1, p + q − 1) =

rankM(p, q, p + q), and is unique if R[q/p]F exists (see [5]), or, equivalently, if rankM(p − 1, q − 1, p + q − 1) =

rankM(p − 1, q − 1, p + q). However, even if it is unique, the system to find the coefficients of the denominator (that is,(Pp, . . . , P0)M(p, q, p+q) = 0, inwhich the j-th row of Pi is zero if i > pj, for j = 1, 2, . . . , k) may not have a unique solutionwith P0 invertible and m.r.d. {(p1, q1), . . . , (pk, qk)}. Therefore, a canonical representation is proposed in [1, Section 5].

We use the following notation. Let h = p + q ≥ 1, 0 ≤ a ≤ p and 0 ≤ b ≤ q:N(a, b, h): maximum number of l.i. row representations (a, b) of the approximant L

[q/p]F .I(a, b, h) = {f ∈ N/the (ka + f ) − th row of M(a, b, h) is linearly dependent (l.d.) on the preceding rows in the matrix

and f ∈ I(r, s, h) such that r ≤ a, s ≤ b, (r, s) = (a, b)}.n(a, b, h): cardinality of I(a, b, h).So as to set up the main results in [1], we need to distinguish between two types of sets of row degrees, which are the

only possible candidates for the sets of m.r.d. and we refer to them as types 1 and 2.

Definition 7 ([1]). If L[q/p]F exists, then a set C , with exactly k pairs of row degrees for this approximant, is of type 1 if the

pairs can be ordered such that C = {(p1, q1), (p2, q2), . . . , (pk, qk)} and for each i ∈ {1, 2, . . . , k}, i ∈ I(pi, qi, p + q). Or,equivalently, C is of type 1 if

ki=1 I(pi, qi, p + q) = {1, 2, . . . , k}.

C. Pestano-Gabino et al. / Journal of Computational and Applied Mathematics 261 (2014) 30–38 33

Definition 8 ([1]). If L[q/p]F exists, then a set C = {(p1, q1), (p2, q2), . . . , (pk, qk)}, with exactly k pairs of row degrees for

this approximant, is of type 2 if: µC (pi, qi) ≤ n(pi, qi, p + q) andk

i=1 I(pi, qi, p + q) = {1, 2, . . . , k}.

Examples 4.2 and 4.3 in [1] illustrate the last definitions, respectively.The next section is the core of this paper and it is based on the following two paragraphs:From Proposition 4.8 in [1] we can choose candidate pairs of m.r.d., these being the pairs (a, b)/n(a, b, h) = 0, and we

can be sure that if n(a, b, h) = 0 then (a, b) is not a pair of m.r.d.We cannot state, in general, that if n(a, b, h) = 0 then (a, b)is a pair of m.r.d. for an approximant of order h + 1. (See Example 4.4 in [1].) As a result of Proposition 4.8 and Lemma 4.9in [1], we are able to delimit sets (types 1 and 2) and be sure that they will be the only candidate sets of m.r.d. The keydifference between types 1 and 2 sets is that:

From Proposition 5.2 in [1] if the set is of type 1, we can have k row representations associatedwith an invertible constantterm in the denominator by following the guidelines in the canonical form proposed in [1].

• If the set is of type 2, we have to determine whether or not we can find k l.i. row representations associated withsaid set.

3. Tabular methodology. Sets of m.r.d. for MPA

In this section we propose two types of tables. In the first case, we study the sets of m.r.d. associated with a givenapproximant L[q/p]F , using a square or rectangular table. In the second case,we study, at the same time, all the approximantsof a certain order using a triangular table. In the first case we start with values p and q for the degrees of the polynomials,and in the second we start with the value h associated with the order of approximation.

3.1. First case: if L[q/p]F exists, what sets of m.r.d. are associated with L[q/p]F?

Suppose that there exists L[q/p]F (z), i.e., that the condition rank M(p − 1, q − 1, p + q − 1) = rankM(p, q, p + q) is

satisfied.Step 1. Build a table with p + 1 rows (from 0 to p) and q + 1 columns (from 0 to q). For each (i, j)/i ≤ p and j ≤ q:

• place the value n(i, j, p + q) at the intersection of row i and column j and• add the set I (i, j, p + q) in cell (i, j) if n (i, j, p + q) = 0.

Step 2. From the table, deduce the candidates to be sets of m.r.d. (the sets with k pairs {(p1, q1), (p2, q2), . . . , (pk, qk)},such that each pair (pi, qi) does not appear more than n(pi, qi, p + q) times in each set).

• Of them, the sets of type 1 are which verifyk

i=1 I(pi, qi, p + q) = {1, 2, . . . , k} and the other ones are sets of type 2.• Write down the type and degree of each set.

Step 3. Decide which sets are minimal, taking into account the properties below.Note: Step 2 of this methodology has been deduced from Theorem 4.3(a), Proposition 4.8 and Lemma 4.9 of [1]. Also, we

will consider the following properties which can be deduced directly from the same results or they are trivial.

Property 1. If the approximant exists, at least one set of m.r.d. is associated with it.

Property 2. If we obtain only a set in Step 2, regardless of the type, then it is the set of m.r.d.

Example 1. Suppose that c0 =

0 00 0

, c1 = c2 = c3 = c4 =

1 1

−1 −1

are the first coefficients of the series F(z) and we

study L[2/2]F . Note that the approximant exists because rankM(1, 1, 3) = rankM(2, 2, 4). The associated table is:

From Step 2, C = {(0, 0), (1, 1)} is the unique candidate. From Property 2 it is the set of m.r.d. In this case, note that C isa type 1 set.

Property 3. For an approximant L[q/p]F : If C is a set of type 1, and there is not a set B (of type 1 or 2) such that, deg B < deg C,

then C is a set of m.r.d. for L[q/p]F .

34 C. Pestano-Gabino et al. / Journal of Computational and Applied Mathematics 261 (2014) 30–38

Example 2. Suppose k = 3, and the associated table of L[2/2]F is:

The candidates to be set of m.r.d are: P1 = {(1, 1), (2, 0), (0, 2)}, P2 = {(1, 1), (1, 2), (0, 2)}, P3 = {(1, 1), (2, 0), (2, 1)}and P4 = {(1, 1), (2, 1), (1, 2)}. The set with minimal degree, 6, is P1. It is of type 1, then from Property 3 it is the set ofm.r.d. The other candidates have degree greater than 6.

Property 4. If a set of type 2 has minimal degree among all the candidates, it is a set of m.r.d. iff it has associated k l.i. rowrepresentations.

Example 3. Suppose the first coefficients of F(z) are:

c0 =

2 31 2

, c1 =

1 11 2

, c2 =

2 41 2

.

The table for the approximant L[1/1]F is:

The candidates of type 1 are P1 = {(0, 1), (1, 1)} and P2 = {(1, 0), (1, 1)} with degree 3 and, of type 2, only P3 =

{(0, 1), (1, 0)} with degree 2. Since P3 has a lower degree than the other candidates, P1 and P2 would be the sets of m.r.d. ifP3 were not. From Property 4, to know if P3 is the minimal, the question is: has P3 associated two l.i. row representations?

To answer this question we could, for instance, obtain the canonical rows representations proposed in [1]. The canonicalrow representation (0, 1) has v0 = (−0.5 1) and the canonical row representation (1, 0) has v0 = (0 1). They are l.i. andtherefore the set P3 is the set of m.r.d.

3.2. Second case: what sets of m.r.d. are associated with the Padé approximant of order h + 1 and which are their associatedapproximants?

It is the following properties that allow us in this section to propose the second type of table for studying all equal-orderapproximants at the same time. The basic idea is that some row representations of one approximant could be utilized foranother of the sameorder, i.e., given the order of approximation, for each approximantwe combine k l.i. rows representationswith suitable degrees for numerator and denominator.

Property 5. The values N(i, j, h) and n(i, j, h) and the set I(i, j, h) are the same for L[q/p]F and L

[b/a]F if : h = a + b =

p + q, a ≥ i, b ≥ j, p ≥ i and q ≥ j.

Property 6. The approximant L[q/p]F exists if and only if

i≤p,j≤q I(i, j, p + q) = {1, 2, . . . , k}.

Step 1. Given a non negative integer h, build a triangular table with h + 1 rows (from 0 to h) and h + 1 columns (from 0 to h).For each (i, j)/i + j ≤ h:• place the value n(i, j, h) at the intersection of row i and column j and• add the set I(i, j, h) in cell (i, j) if n(i, j, h) = 0.

Step 2. Write down which approximants of order h + 1 exist taking into account Property 6.• Assign to each approximant their candidates to be sets of m.r.d., as explained in Step 2 of the first case of table.• Write down the type and degree of each set.

Step 3. Decide which sets are sets of m.r.d., taking into account the properties in this section.

C. Pestano-Gabino et al. / Journal of Computational and Applied Mathematics 261 (2014) 30–38 35

Example 4. Suppose that the first coefficients of the series F(z) are:

c0 =

1 00 10 0

, c1 =

1 01 01 0

, c2 =

1 00 01 0

.

The table for the approximants of order 3 (h = 2) is:

From Property 6, all the approximants of order 3 exist: L[2/0]F , L

[1/1]F and L[0/2]F .

From Step 2, the candidate sets of m.r.d. for each approximant are:– {(0, 0), (0, 1), (0, 2)} for L

[2/0]F . From Property 2 it is the set of m.r.d. because it is the unique candidate.– {(0, 0), (1, 0), (1, 0)} and {(0, 0), (0, 1), (1, 0)} for L

[1/1]F . All of them are of type 1 and degree 2. From Property 3, theyare sets of m.r.d., because the other candidate, {(1 0), (1 0), (0 1)} (of type 2) has degree 3.

– {(0, 0), (1, 0), (1, 0)} for L[0/2]F , From Property 2 it is the set of m.r.d. because it is the unique candidate.

Example 5. Suppose the F(z) in Example 3.The table for the approximants of order 3 (h = 2) is:

From Property 6 all the approximants with order 3 exist.From Step 2, the candidates to be sets of m.r.d. are:

• {(0, 1), (0, 2)} for L[2/0]F . From Property 2 it is the set of m.r.d. because it is the unique candidate.

• {(1, 0), (2, 0)} for L[0/2]F . From Property 2 it is the set of m.r.d. because it is the unique candidate.

• In Example 3 we already studied L[1/1]F .

In the following example, we illustrate the additional difficulty that could result from studying type 2 sets.

Example 6. Suppose that the first coefficients of the series F(z) are:

c0 =

3 00 3

, c1 =

−1 −11 1

, c2 =

1 12 2

.

The table for L[1/1]F is:

For L[1/1]F there is a candidate of type 2, C1 = {(1, 0), (0, 1)}, and two candidates of type 1, C2 = {(0, 1), (1, 1)} and

C3 = {(1, 0), (1, 1)}. The degree of C1 is the smallest and we doubt if C1 is minimal or not. Then wemust study if there existl.i. row representations (0, 1) and (1, 0).

36 C. Pestano-Gabino et al. / Journal of Computational and Applied Mathematics 261 (2014) 30–38

The canonical row representation (0, 1) obtained – following [1, Section 5] – has v0 = (−2 1) and the canonical rowrepresentation (1, 0) obtained has (v1 v0) = (−1 − 1, −2 1); they are l.d. Due to the fact that any row representation forthese pairs of degrees has v0 = (−2y y), with y = 0, then the set C1 is not minimal. Therefore, C2 and C3 are the sets of m.r.d.for this approximant.

The examples that follow aim to underscore certain precautions to be taken so as to use the methodology properly.

Example 7. Suppose that the first coefficients of the series F(z) are:

c0 =

1 10 0

, c1 =

0 01 1

, c2 =

1 20 0

and considering the following associated table

Taking into account I(1, 0, 2) = I(0, 1, 2) = {2}, it is not true that the approximant L[1/1]F has a unique, hence minimal

candidate of type 2 namely {(1, 0), (0, 1)} because by Property 6 the approximant L[1/1]F does not exist.

Example 8. Let F(z) be the function in Example 4. It is wrong to say in Example 4 that {(0, 0), (1, 0), (1, 0)} is a set of m.r.d.for the approximant L

[0/1]F because hwas 2. If h = 1 the triangular table is:

and therefore the set of m.r.d. for L[0/1]F is {(0, 0), (0, 0), (1, 0)}.

4. Computational aspects

The computational part of the tables requires finding matrix ranks. The basic properties of this section aim to addressthis limitation.

Property 7 (Lemma 4.6 in [1]).• N(0, j, h) = k − rankM(0, j, h), for j > 0.

• N(i, 0, h) = k − rankM(i, 0, h) + rank

c−i+1 c−i+2 · · · ch−ic−i+2 c−i+3 ch−i+1

.

.

....

c0 c1 · · · ch−1

for i > 0.

• N(i, j, h) = k − rankM(i, j, h) + rankM(i − 1, j − 1, h − 1), for i, j > 0.• n(a, b, h) = N(a, b, h) − cardinal

i≤a and j≤b(i,j)=(a,b)

I(i, j, h) if a ≥ 0 and b ≥ 0.

Property 8. N(0, 0, h) = n(0, 0, h) and for i, j > 0:• if N(0, j, h) − N(0, j − 1, h) = 0 then n(0, j, h) = 0,• if N(i, 0, h) − N(i − 1, 0, h) = 0 then n(i, 0, h) = 0, and• if N(i, j, h) − N(i − 1, j, h) = 0 or N(i, j, h) − N(i, j − 1, h) = 0 then n(i, j, h) = 0.Proof. From definitions of N(i, j, h) and n(i, j, h). �

The procedure was programmed using Mathematica, meaning that the results for the ranks shown in the tables stemfrom the MatrixRank command. Other typical commands, such as Table, ArrayFlatten, Append and Complement, were usedfor the intermediate calculations. Grid was used to format the final output as a table. A preliminary version is available athttp://cogonzal.webs.ull.es/Examples.rar.

C. Pestano-Gabino et al. / Journal of Computational and Applied Mathematics 261 (2014) 30–38 37

The proposal of other properties, such as those shown below, is important when designing a program that can moreefficiently handle problems with large dimensions, since it allows for a reduced computation of the number of operationsto perform.

Property 9. If n(i, j, h) = k then I(i, j, k) = {1, 2, . . . , k}.

Proof. From the definition of n(i, j, h). �

Property 10. If

i≤a,j≤b I(i, j, h) = {1, 2, . . . , k} then n(a + u, b + v, h) = 0 for u, v ≥ 0, (u, v) = (0, 0).

Property 11. If n(a, b, h) = k then n(i, j, h) = 0 for any i ≤ a, j ≤ b, (i, j) = (a, b) and for any i ≥ a, j ≥ b, (i, j) = (a, b).

Proof. From the definition of n(i, j, h). �

5. A possible application in multivariate time series models

Presumably this methodology is applicable in all fields in which the Matrix Padé Approximation, with minimal rowdegrees, is relevant. In each particular field, advantages and disadvantages should be established. In the Introduction of [1]we have commented on some references about the theoretical and applied contexts to place our approach.

For instance, in dynamic models as in multivariate time series models a key problem is to understand their structure andparameterization. The parameterization of multivariate time series models encounter several difficulties, for instance themodel may not be unique for a given process or the model contains an excessive number of parameters. There remains theproblem of identifiability (or exchangeable models). It is necessary to find identifiable and parsimonious representation(s)because the statistical procedures for estimatingmodels lose properties when the number of parameters to be estimated in-creases and it is not possible to estimate amodel with free parameters. Moreover, achieving this objective leads to improvedmodel predictions. By a canonical form we mean a well-specified model: the model contains no redundant parameters andthe orders of the polynomial involved are as small as possible so that the total number of parameters that need estimationcould be notably reduced.

An example of this time series models is the VARMAX systems (see, for instance, [6,7]). An ARMAX(p, s, r) system(autoregressive moving-average model with exogenous variables) can be expressed as

A(L)yt + B(L)xt = M(L)εtwhere L is de backshift operator, yt denotes a v component observable output process, xt is a u component vector ofobservable exogenous input variables, εt is a v vector of zero mean, serially uncorrelated random variables, with positivedefinite covariance;A(z) = A0+A1z+· · ·+Apzp, B(z) = B0+B1z+· · ·+Bszs andM(z) = M0+M1z+· · ·+Mrzr are v×v, v×uand v × v matrix operators, respectively and q = max{s, r}. Isolating the input variables xt and εt can be expressed as

A(L)yt = N(L)wt

where w′t = (−x′

tε′t)

′ and N(z) = (B(z) M(z)).Assuming the usual conditions required for convergence and stability and denoting by

F(z) = A−1(z)N(z),

with the tabular methodology in this paper, jointly with the canonical representations in [1, Section 5], we can identify thecanonical MPA(p, q) with minimal row degrees for F(z), which is equivalent to identifying a new canonical form for thisVARMAX model.

To show these ideas in practical examples requires previously developing specific tools to estimate statistically the tablesin this paper.

6. Conclusions

We summarize and conclude:• This work addresses an open question in [1]: automatically finding all of the possible sets of m.r.d. of a MPA, for those

users who are not familiar with the more mathematical aspects of the results and proofs of [1].• The originality of this paper is its methodology. It is an additional contribution of [1] in the form of original tables, new

properties and the way to apply them in carefully chosen examples to complete the methodology.• We propose two types of tables. With the first type we study the sets of m.r.d. for a particular approximant. With

the second type we study the sets of m.r.d. for all of approximants with the same order and the basic idea is that a rowrepresentation could be utilized for several approximants with the same order.

• Considering all the row representations of all the approximants with the same order, for each particular approximantwe combine k l.i. rows representations that are compatible with the degrees of the polynomials of the approximant inquestion.

• The basis for our reasoning was to study the number of linearly dependent rows that are in the last k rows of thematrixM(i, j, h), and the positions of these rows.

38 C. Pestano-Gabino et al. / Journal of Computational and Applied Mathematics 261 (2014) 30–38

• Once the possible candidate sets are localized, what determined their minimality is the degree of the set and,additionally, only for a type 2, if an associated invertible P0 can be found.

• We have managed our objective ‘‘easily’’ in all the situations that resolve the properties. The procedure needs theintervention of the user when we have a candidate of type 2 and with Properties 2 and 6 we cannot resolve automaticallythe following doubts:

• If a set C is of type 1 or 2 and there exists a set B of type 2 that satisfies deg B < deg C , a doubt arises as to whether Bcould be a set of minimal degrees. We have to ensure somehow that we can find or not k l.i. row representations associatedwith said set. We can ensure it in each example, but not systematically in all situations.

Then it is desirable, as noted in [1]:• To resolve automatically this doubt.• To consider the results in the context of the theory of eigenvalues and eigenvectors in order to yield new results and

applications.• To translate the results to the field of time series, which was the inspiration for the work proposed herein.• Since the work we are doing in the context of Matrix Padé Approximation relies on the tools of matrix algebra, its

results are applicable not only to the field of origin, Multivariate Time Series, but also to other fields related to rationalmatrix functions.

• To find an efficient way to compute the tables.

Acknowledgments

We wish to thank Professor F. Marcellán, for letting the first author a research stay in his Department and to ProfessorA. Bultheel for constructive comments on preliminary results of this paper.

References

[1] C. Pestano-Gabino, C. González-Concepción, M.C. Gil-Fariña, A type of matrix Padé approximants inspired by scalar component models, Journal ofComputational and Applied Mathematics 236 (2012) 3360–3372.

[2] G.C. Tiao, R.S. Tsay, Model specification in multivariate time series, Journal of the Royal Statistical Society. Series B 51 (2) (1989) 157–213.[3] A. Draux, On the non-normal Padé table in a non-commutative algebra, Journal of Computational and Applied Mathematics 21 (1988) 271–288.[4] C. Pestano, C. González, Matrix Padé approximation of rational functions, Numerical Algorithms 15 (1997) 1–26.[5] G.L. Xu, A. Bultheel, Matrix Padé approximation: definitions and properties, Linear Algebra and its Applications 137–138 (1990) 67–136.[6] D.S. Poskitt, A note on the specification and estimation of ARMAX systems, Journal of Time Series Analysis 26 (2) (2005) 157–183.[7] D.S. Poskitt, On the identification and estimation of nonstationary and cointegrated ARMAX systems, Econometric Theory 22 (2006) 1138–1175.