Definition of the ERES operations

Let us consider the set of univariate polynomials Ph+1,n =n

a(s), bi(s)∈ R[s], i = 1, 2, . . . , h with n = deg{a(s)}, p = max_i deg{bⁱ(s)}

≤ n and h, n ≥ 1o (3.1) We represent the polynomials a(s), bi(s) with respect to the highest degrees (n, p) as

a(s) = ansⁿ+ an−1sⁿ⁻¹+ . . . + a1s + a0 , an6= 0

bi(s) = bi,ps^p+ . . . + bi,1s + bi,0, i = 1, 2, . . . , h (3.2) The set Ph+1,n will be called an (n, p)-ordered polynomial set.

Definition 3.1. For anyPh+1,n set, we define a vector representative (vr), p

h+1(s) and a basis matrix Ph+1 represented as

ph+1(s) = [ p₁(s), . . . , p_h+1(s) ]^t= [ p

1, . . . , p

m−1, p

h+1]· en(s) = P_h+1· en(s) where Ph+1 ∈ R(h+1)×(n+1), e_n(s) = [1, s, . . . , sⁿ⁻¹, sⁿ]^t and p_i ∈ Rⁿ⁺¹ for all i = 1, . . . , h + 1.

The matrix P_h+1 is formed directly from the coefficients of the polynomials of the setP^h+1,n and it has the least possible dimensions.

Definition 3.2. If c is the integer for which p

1 = . . . = p

c−1 = 0 and p_c 6= 0, then c = w(P^h+1,n) is called the order of P^h+1,n and s^c is an elementary divisor of the GCD. The set P^h+1,n is considered to be a c-order set and will be called proper if c = 0, and nonproper if c≥ 1.

If we have a nonproper set Ph+1,n with w(Ph+1,n) = c, then we can always consider the corresponding proper onePh+1, n−c by dismissing the c leading zero columns. Then

gcd{Ph+1,n} = s^c· gcd{Ph+1, n−c} (3.3) Therefore, in the following and without loss of generality, we assume that Ph+1,n

is proper.

Definition 3.3 (ERES operations). Given a setPh+1,n of many polynomials with a basis matrix Ph+1 the following operations are defined [40] :

a) Elementary row operations with scalars from R on Ph+1. b) Addition or elimination of zero rows on Ph+1.

c) If a^t= [0, . . . , 0, al, . . . , an+1]∈ Rⁿ⁺¹, al 6= 0 is a row of P^h+1 then we define as the Shifting operation

shf : shf(a^t) = [al, . . . , an+1, 0, . . . , 0]∈ Rⁿ⁺¹

By shf(P^h+1,n), we shall denote the set obtained from P^h+1,n by applying shifting to every row of Ph+1 (Matrix Shifting).

Type (a), (b) and (c) operations are referred to as Extended-Row-Equivalence and Shifting (ERES) operations.

REMARK 3.1. The ERES operations without applying the Shifting operation will be referred to as ERE operations.

The following theorem describes the properties characterising the GCD of any given P^h+1,n.

Theorem 3.1 ([40]). For any set Ph+1,n, with a basis matrix Ph+1, ρ(Ph+1) = r and gcd{Ph+1,n} = φ(s) we have the following properties :

i) If R^P is the row space of Ph+1, then φ(s) is an invariant of R^P (e.g. φ(s) remains invariant after the execution of elementary row operations on Ph+1).

Furthermore if r = dim(R^P) = n + 1, then φ(s) = 1.

ii) If w(P^h+1,n) = c≥ 1 and Ph+1,n^∗ = shf(P^h+1,n), then φ(s) = gcd{Ph+1,n} = s^c · gcd

Ph+1,n^∗

iii) If Ph+1,n is proper, then φ(s) is invariant under the combined ERES set of operations.

The GCD of any set of polynomials is a property of the row space of the basis matrix of the set. This property indicates that not all polynomials are required for the computation of the GCD [40]. Thus, the computation of the GCD requires selection of a base that is best suited for such computations. The already known methods for finding bases for given sets of vectors are based on the fact that they virtually transform the original data by using mostly Gaussian or orthogonal techniques (Gram-Schmidt, Householder etc) [18, 27]. Evidently, they obtain new sets and amongst the new vectors they choose the required ones that span the

original set. Thus, the base will be consisted of vectors completely different from the given ones. Furthermore, the numerical transformation of the original data always introduce round-off errors which in many cases can affect the quality of the final results very badly, especially in nongeneric computations.

◮ The selection of the “best uncorrupted base”

The issue of selecting the best possible base from all those vectors provided by the rows of the basis matrix without transforming the original data is critical for nongeneric computations and this problem is referred to as the selection of the best uncorrupted base.

Definition 3.4. Let A = {a1, a₂, . . . , a_m} be a set of m vectors in Rⁿ. Then, a subset B = {b1, b₂, . . . , b_r} of A with r < m vectors is an uncorrupted base of A, if B consists of the original vectors of A, (i.e. bj ∈ {a1, a₂, . . . , a_m} for every j = 1, 2, . . . , r), all b_j for j = 1, 2, . . . , r are linearly independent, andB spans A.

The setA can be expressed in terms of a matrix A = [a1, a₂, . . . , a_m]^t∈ R^m×n. Then, the problem of finding an uncorrupted base for the set A is transferred into finding an uncorrupted base for the row space of the matrix A. Therefore, for a given set Ph+1,n with a basis matrix Ph+1 and R the row space of Ph+1, an uncorrupted base ofR is defined by the rows of Ph+1without being transformed. If vector orthogonality is used to characterise the “best” selection of an uncorrupted base, then the best uncorrupted base ofR is defined from the rows of Ph+1 which are orthogonal. However, such a base is not uniquely defined [60] and a procedure for the selection of the “best orthogonal” (or “most orthogonal”) subset of R requires an appropriate quantitative numeric indicator that defines the degree of orthogonality of the selected set of vectors.

Such a procedure for the selection of the best uncorrupted base of the row space of a matrix has been presented in [57] and aims at the construction of a base that contains vectors that are mostly orthogonal, i.e. they form a set with the highest degree of orthogonality. This method relies on the properties of the Gram matrix and uses tools from the theory of compound matrices (Section 2.2.1).

Definition 3.5. Let A = {a1, a₂, . . . , a_m} be a set of m given vectors ai ∈ Rⁿ, i = 1, 2, . . . , m. The matrix defined by

GA=









(a₁· a1) (a₁ · a2) . . . (a₁· am) (a₂· a1) (a₂ · a2) . . . (a₂· am)

... ... ... ...

(a_m· a1) (a_m· a2) . . . (a_m· am)







∈ R^m×m

where (a_i· aj) denotes the inner product of the vectors a_i, a_j, is called the Gram matrix of A and the determinant det{GA} is called the Grammian of A.

The Grammian provides us with an important criterion about the linear independence of vectors.

Theorem 3.2 ([60]). The vectors a₁, a₂, . . . , a_m are linearly independent if and only if their Grammian is positive and not equal to zero.

Theorem 3.3 ([60]). For any set A = {a1, a₂, . . . , a_m} with kaik2 = 1 for all i = 1, 2, . . . , m we have

0≤ det{GA} ≤ 1

where the left equality holds when the set is linearly dependent and the right holds when the set is orthogonal.

Definition 3.6. If A = [a₁, a₂, . . . , a_m]^t ∈ R^m×n, then the normalization of A is a matrix AN = [v₁, v₂, . . . , v_m]^t ∈ R^m×n with the property v_i = _ka^ai

ik2, i = 1, 2, . . . , m.

The next proposition gives the outline of the procedure that defines an indicator of the degree of orthogonality for a given set of vectors and computes the most orthogonal uncorrupted base of the set (or the matrix).

Proposition 3.1 ([46, 57]). Let A = [a₁, a₂, . . . , a_m]^t ∈ R^m×n, ρ(A) = r ≤ min{m, n}, A^N = [v₁, v₂, . . . , v_m]^t∈ R^m×n the normalization of A. Suppose G∈ R^m×n the Gram matrix of the vectors v₁, v₂, . . . , v_m and Cr(G) = [ci,j]∈ R(^mr)^×(^mr) the r^th compound matrix of G. If cii= det(G[a/a]), a = (i1, i2, . . . , ir)∈ Q^r,m1 is the maximal diagonal element of Cr(G), then a most orthogonal uncorrupted base for the row space of A, consists of the vectors {ai1, a_i2, . . . , a_ir}.

Obviously, the maximal diagonal element of the r^th compound matrix of the Grammian G defines the best degree of orthogonality. The main advantage of this procedure is that it does not alter the elements of the original basis matrix Ph+1. It just indicates the best (most orthogonal) combination of linearly independent rows of Ph+1 according to the largest diagonal element of an r-order compound matrix (r = rank(Ph+1)) associated with the Gram matrix, which is created by the rows of Ph+1. The selected best combination of the most orthogonal, linearly independent row vectors forms a base for Ph+1,n which is represented by a new matrix Pr ∈ R^r×(n+1).

◮ The formulation of the ERES method

From Theorem 3.1 it is evident that ERES operations preserve the GCD of any Ph+1,n and thus can be easily applied in order to obtain a modified basis matrix with much simpler structure. The successive application of these operations on a

1Q_r,m denotes the set of strictly increasing sequences of r integers (1≤ r ≤ m) chosen from 1, 2, . . . , m.

basis matrix of a set of polynomials leads to the formulation of the ERES method for computing the GCD of a set of polynomials [57]. After successive applications of ERES operations on an initial basis matrix, the maximal degree of the resulting set of polynomials is reduced and after a finite number of steps the resulting matrix has rank 1. At this stage, the process is terminated and considering that all the arithmetic operations are performed accurately (symbolic-rational operations), any row of the last obtained matrix specifies the coefficients of the required GCD of the set.

Therefore, from a theoretical point of view, the ERES method in its simplest form consists of three basic procedures:

1. Computation of the best uncorrupted base for the set P^h+1,n.

2. Application of elementary row operations to the processed matrix, which practically involves row reordering, triangularisation, and elimination of zero rows (ERE operations).

3. Application of the Shifting operation to the nonzero rows of the processed matrix.

The iterative application of the process of triangularisation and Shifting is actually the core of the ERES method and we shall refer to it as the main procedure of the method. Conversely, the computation of the best uncorrupted base ofPh+1,n is necessary only when the row dimension of Ph+1is larger than its column dimension and it is performed only once before the main procedure in order to get a more concrete set of polynomials. The problem that will be considered next is the formulation of an algebraic expression which will represent the relation between the initial basis matrix Ph+1 and the final matrix, which occurs after the iterative application of the ERES operations.

The matrix Br ∈ R^r×(h+1), which corresponds to the selection of the best uncorrupted base of Ph+1, is actually a simple permutation matrix, which allows us to work with r < h + 1 independent rows of Ph+1 and thus starting ERES with a matrix Pr ∈ R^r×(n+1) of shorter dimensions.

Pr = Br· Ph+1 (3.4)

The ERE row operations, i.e. triangularisation, deletion of zero rows and reordering of rows, can be represented by a matrix R∈ R^r1×r [18, 27], which converts the initial rectangular matrix Pr into an upper trapezoidal form. However, the matrix representation of the Shifting operation is not straightforward. This problem has to do with the connectivity of the matrices, which are obtained after the process of triangularisation in each iteration of the main procedure of the ERES method.

In [57] and related work the problem of the matrix representation of the Shifting operation for real matrices remained open. Solving this problem is crucial for establishing an overall matrix representation of the ERES method which will allow us to study in more detail the numerical stability of the method not only for a single iteration of the main procedure of the method as in [57], but for all the performed iterations. Therefore, the problem that we will study in the following section is to find the simplest possible algebraic relation between a matrix and its shifted form.