Appl. Num. Anal. Comp. Math. 2, No. 3, 344 – 356 (2005) / DOI 10.1002/anac.200410044
Dimitrios Christou
∗ and Marilena Mitrouli
∗∗
Department of Mathematics, University of Athens, Panepistimioupolis 15784, Athens, Greece
Received 5 November 2004, revised 10 August 2005, accepted 10 October 2005
Published online 11 November 2005
The computation of the Greatest Common Divisor (GCD) of a set of more than two polynomials is a non-generic problem. There are cases where iterative methods of computing the GCD of many polynomials, based on the
Euclidean algorithm, fail to produce accurate results, when they are implemented in a software programming environment. This phenomenon is very strong especially when floating-point data are being used. The ERES method is an iterative matrix based method, which successfully evaluates an approximate GCD, by performing row transformations and shifting on a matrix, formed directly from the coefficients of the given polynomials.
ERES deals with any kind of real data. However, due to its iterative nature, it is extremely sensitive when performing floating-point operations. It succeeds in producing results with minimal error, if we combine both floating-point and symbolic operations. In the present paper we study the behavior of the ERES method using floating-point and exact symbolic arithmetic. The conclusions derived from our study are useful for any other algorithm involving extended matrix operations.
2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
The computation of the Greatest Common Divisor of a set of polynomials has been a problem to the mathematicians for a long time because there are many applications in mathematics and engineering, where the GCD of a set of polynomials plays an important role. The existence of a common divisor of many polynomials is not always a trivial subject. Therefore, several numerical methods have been proposed for its computation. Some of them are based on Euclid’s algorithm and its generalizations and others are based on procedures involving matrices (matrix-based methods) [3, 5, 11].
Matrix based methods have better performance and quite good numerical stability, especially in the case of large sets of polynomials. In this category we have those, which depend on the properties of generalized resultants
[12]. Such methods tend to be computationally expensive because they create and work with very large matrices.
The ERES method is an iterative matrix based method, which is based on the properties of the GCD as an invariant of the original set of polynomials under extended-row-equivalence and shifting operations and it is very effective, when working with large matrices [4, 5].
Consider a set of polynomials
Π m,d
= { p i
( s ) ∈ < [ s ] , i = 1 . . . m }
, which has m elements and denote by d the maximal degree of the polynomials of the set.
< [ s ] denotes the ring of polynomials with coefficients from the reals, < , and the GCD of Π m,d a basis matrix
P m can be defined by will be denoted by g ( s ) . For any Π m,d set a vector representative p m
( s ) and p m
( s ) = [ p
1
( s ) , p
2
( s ) , . . . , p m
( s )] t = [ p
0
, p
1
, . . . , p d
] e d
( s ) = P m e d
( s ) where, P m e ( s ) d
∈ < m × ( d +1)
= [1 , s, . . . , s d − 1 , s is a matrix formed directly from the coefficients of the polynomials of the set and d = p
1
= 0 and p c c = w (Π m,d
] t
. If c is an integer for which p
0
) is called the order of
Π m,d and s c
= · · · = p c − 1
= 0 , then is an elementary divisor of the GCD. The set is considered to be a c-order set and will be called proper if c = 0
, and non-proper if c ≥ 1
.
Definition 1.1 On any set Π m,d and hence on its basis matrix P m
, the following operations can be defined :
∗
∗∗
Corresponding author: e-mail: dchrist@math.uoa.gr
e-mail: mmitroul@cc.uoa.gr
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345
(i) Elementary row operations with scalars from < .
(ii) Addition or elimination of zero rows on P m
.
(iii) If r t = [0 , . . . , 0 , a k
, . . . , a d +1
] ∈ < d +1 , a k
= 0 is a row of
P m then we define as the Shifting operation shf : shf( r t
) = r
∗ t
= [ a k
, . . . , a d +1
, 0 , . . . , 0] ∈ < d +1
By shf (Π m,d
) = Π ∗ m,d
, we shall denote the set obtained from
Π m,d
P m
.
by applying Shifting on every row of
The combination of the three types of operations described in definition 1.1 are referred to as Extended-Row-
Equivalence and Shifting (ERES). These operations on
P m have an interpretation on the set of polynomials.
Remark 1.2 The shifting operation applied either on vectors or matrices can be implemented in any programming environment very easily. However, in theory, the shifting operation applied on a matrix is a subject, which remains open. Matrix theory does not provide us with enough tools to establish a clear mathematical relation between a matrix and its shifted form. Specifically, we are interested in expressing the matrix shifting as a matrix product. This would have been very useful in the study of the overall error analysis of the ERES method.
The following theorem summarizes a number of properties of the GCD under the above transformations [4].
Theorem 1.3 For any set
Π m,d
, with a basis matrix
P m have the following properties with rank
ρ ( P m
) = r and gcd(Π m,d
) = g ( s ) we
(i) If R is the row space of P m
, then g ( s ) remains invariant after the execution of elementary row operations on P m
, i.e.
g ( s ) is an invariant of R . Furthermore, if r = dim( R ) = d + 1 , then g ( s ) = 1 .
(ii) If w (Π m,d
) = c ≥ 1 and shf (Π m,d
) = Π ∗ m,d and gcd(Π ∗ m,d
) = g ∗ ( s ) then g ( s ) = s c g ∗ ( s )
.
(iii) If Π m,d is proper ( c = 0) , then g ( s ) is invariant under the combined ERES set of operations.
The above results form the basis for the ERES methodology, which is developed in [4]. In [5] can be found a comparison of the ERES method with other existing methodologies. In the present paper we study extensively the behavior of the ERES method using floating-point or exact symbolic arithmetic. More specifically, we are concerned with ERES’s capability of computing the existing non-trivial GCD of a given set of univariate polynomials, with minimal numerical error. From this comparison of behavior we produce useful remarks, which will help us to develop the best performance of the method and build a framework for combining numerical and symbolical computations.
The construction of the algorithm of the ERES method is based on algebraic processes, which are applied iteratively on a matrix formed directly from the coefficients of the polynomials of the original set. We shall denote by
P
( k ) m the matrix, which occurs after the k th
(thus
P
(0) m
≡ P m iteration and it is used during the
). The output is a vector g ∈ < n k th + 1 iteration of the algorithm containing the coefficients of the GCD.
The main target of the ERES algorithm is to reduce the number of the rows of the initial basis matrix
P m and finally to end up to a rank 1 matrix, which contains the coefficients of the GCD. Indeed, if we combine Gaussian elimination with partial pivoting and Shifting and furthermore, by selecting an appropriate numerical accuracy
ε
G such that all the elements of the matrix
P
( k ) m reducing the size of the matrix P
( k ) m with values less than during the iterations.
ε
G
On the other hand, the Singular Value Decomposition of P
( k ) m be set equal to zero, we succeed in provides the ERES algorithm with a termination criterion. This criterion is described in the following theorem [4, 5] :
Theorem 2.1 (Termination Criterion) Let A = [ r
Then for an appropriate accuracy ε t singular values σ m
≤ σ m − 1
A satisfy the conditions
| σ
1
−
> 0 the numerical ε t
≤ · · · ≤ σ m | ≤ ε t
1 of the normalization A and
σ i
≤ ε t
1
, . . . , r m
N
] t ∈ <
= [
, i = 2 , 3 , . . . , m
.
u m × n
1
, m ≤ n, r
1
-rank of A equals to one ( ρ
, . . . , u m
] t
ε t
( A
∈ <
= 0
) = 1 m × n , u
, i i
=
= 1 r i
/
, . . . , m
) if and only if the k r i k
2 of
.
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346 D. Christou and M. Mitrouli: Estimation of the GCD of many polynomials with the ERES method
The specified variables ε t and ε
G will be defined as the termination and the Gaussian accuracy respectively.
It is more appropriate to apply the SVD termination criterion on a normalized matrix
P
( k ) m
, when the polynomials, which correspond to the rows of the matrix, have the same degree, i.e. the last non zero column of P
( k ) m does not contain zero elements.
Therefore, if a rank 1 matrix
P
( k ) m occurs after the k th iteration, the process stops. The termination criterion described in theorem 2.1 is used to check when the numerical rank of a matrix is considered to be equal to 1.
The ERES algorithm.
Form the basis matrix
P m
Check if
P m
∈ < m × n , n = d + 1 is proper or non-proper (specify c).
.
If c ≥ 1 then
P := shf( P m
) else
P := P m
. (Initial matrix)
Let r be the row dimension of
P
. (Initial value r := m
)
Let q be the column dimension of
P
. (Initial value q := n
)
Repeat
STEP 1 : Specify the degrees d i
If d i
= d j of each polynomial row.
Reorder matrix P such that d
1
≤ d
2
, ∀ i, j = 1 , 2 , . . . , r then
≤ · · · ≤ d r
.
Compute the Singular Value Decomposition of P .
P
If
|
:=
σ
1
V
−
Σ W t , Σ = diag { σ
1 r | ≤ ε t then
, . . . , σ r
} , W t = [ w
1
, . . . , w n
] t
Select appropriately the GCD vector g.
quit end if end if
STEP 2 : Scale appropriately matrix P such as | P
1 , 1
| > | P i, 1
| for
Apply Gaussian elimination with partial pivoting to P , i = 2 , . . . , r .
eliminating entries and rows with values ≤ ε
G
Specify the new value of r .
.
STEP 3 : Normalize each row of P using vector 2-norm.
Apply shifting on every row of
P
.
Eliminate the last zero columns of
P
.
Specify the new value of q
.
until r = 1
The ERES algorithm produces either a single row-vector or a rank 1 matrix. In the first case, the yielded row-vector contains the coefficients of the GCD. But if a unity rank matrix occurs, the selection of the most appropriate GCD vector is not trivial. We can pick any row of this matrix as the GCD vector since the matrix is normalized, or take advantage of the SVD according to the following proposition [5] :
1
Proposition 2.2 Let A = V · Σ · W t be the singular value decomposition of a given matrix A ∈ < m × n
. Then a “best” rank 1 approximation to A in the Frobenius norm is given by A
1
= σ
1
· u · w t
, where
, ρ
σ
1
( A ) = is the largest singular value of A and u and w are the first columns of the orthogonal matrices V and W of the singular value decomposition of
A respectively. The vector w is then the “best” representative of the rows of matrix
A in the sense of the rank 1 approximation.
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Example 2.3 Consider the following set of polynomials :
Π
5 , 4
=




 p p p p
1
2
3
4 p
5
( s ) = − s 4
( s ) = s 3
+ s 3
+ 3 s 2
(
( s s
) =
) =
− s 4 s
( s ) = − s
4
4
− 2 s
− 1
− s 3
3
+ 6
− s − 3
+ 2
+ s s 2 s
−
+ 1 s
+ 1
− 5 m = 5 , d = 4 .
with exact GCD, g ( s ) = s 2 − 1
. Let
ε t following form :
= ε
G
= 10 − 16
. The initial basis matrix
P m is proper
( c = 0) and has the
P m
≡ P (0) m
=

− 5 − 1 6 1 − 1
− 3 − 1 3 1 0
1
− 1
1
2 0
0 0
1 0
−
−
2
0
1
−
−
1
1
1

∈ < 5 × 5 , rank( P (0) m
) = 3
After the first iteration of the ERES algorithm, we obtain the matrix :
P (1) m
=

−
−
3
√
10
5
√
26
13
√
2
2
−
3
√
5
√
10
26
26
0
−
3
√
10
5
√
26
√
13
2
2
−
√
5
3
10
√
26
26

∈ < 3 × 4 , rank( P (1) m
) = 3
0
After the second iteration of the ERES algorithm, we obtain the matrix :
P (2) m
=


−
−
√
√
2
2
√
10
10
5
0
0
√
√
2
2
√
10
10
5

 ∈ < 2 × 3 , rank( P (2) m
) = 1
At this point the procedure stops according to the SVD termination criterion and the selected numerical accuracies. By using the result of the proposition 2.2 (or by selecting any row of g = [ −
2
2 0
2
2 ]
, which leads to the polynomial s 2 − 1
, if we divide the elements of g
P
(2) m by
), the GCD vector is
2
2
.
Computational complexity.
For a set of polynomials
Π m,d the amount of floating point operations performed in the k th iteration of the algorithm depends on the size of the matrix P
( k ) m and it is summarized in table 1, [5].
The first iteration is the most computationally expensive iteration since the initial matrix P
(0) m is larger than any
P
( k ) m
. Unless we know exactly the degree of the GCD of the set we cannot specify from the beginning the number of iterations required by the algorithm. The following theoretical bound can be expressed for the number
N it required iterations :
N it
≤ min {
P m i =1 d i
, 2( d − deg g ( s )) } of
Table 1 Required floating-point operations by the matrix
P
( k ) m
∈ < r × q
Gaussian elimination
O ( z
3
3
) , z = min { r − 1 , q }
Normalization
O (2 rq )
SVD
O ( rq 2 + q 3 )
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348 D. Christou and M. Mitrouli: Estimation of the GCD of many polynomials with the ERES method
Numerical stability.
From a theoretical point of view the following theorem establishes the numerical stability of each iteration step of the method [5] :
Theorem 2.4 The matrix P
( k ) m
∈ < m × n
, computed by the method in the k arithmetic with unit round-off u , satisfies the properties th iteration, using floating-point
M ( k ) · P ( k ) m
= P ( k − 1) m
+ E ( k ) , k E ( k ) k
∞
≤ 3 .
003 · ( N it
− 1) · n 3 · u where
M ( k ) matrix.
∈ < m × m the matrix accounting for the performed transformations and
E ( k ) ∈ < m × n
, the error
Main advantages of the ERES algorithm.
•
Quick reduction of the size of the original matrix, which leads to fast data processing and low memory consumption [5].
•
Numerical stability. The normalization of
P
( k ) m keeps the elements bounded.
3.1
Performance of ERES method using floating-point arithmetic
The ERES method shows very interesting numerical properties, when implemented properly in a software programming environment. The careful selection of the accuracies
ε
G and
ε t is very crucial, since they both influence the correctness of the achieved results. Any matrix based method requires appropriate accuracies for its implementation. When row operations are involved (e.g. resultant methods [12]) accuracies such as ε
G
When the nature of the method is iterative (e.g. Blankinship method [5]) accuracies such as ε t are required.
are required. Most of the mathematical software packages, which are available today, use fixed point arithmetic and floating-point arithmetic of a certain number of digits (which in the case of floating-point arithmetic form the mantissa of a real number). In the following we will be concerned with the case of floating-point and exact symbolic arithmetic and throughout the paper we shall use the following notation :
Notation
SFP : Standard Floating-Point arithmetic, where the internal accuracy of the system is fixed and often limited to 16 digits (double precision)
1
.
VFP : Variable Floating-Point arithmetic, where the internal accuracy of the system is determined by the user
(variable precision)
2
.
ES : Exact Symbolic arithmetic, where the system uses rational arithmetic to perform the operations.
HC : The combination of VFP and ES operations will be referred to as Hybrid Computations.
In the following we will show that when using standard floating-point arithmetic, the internal accuracy of the system is not always enough for the ERES method to produce good results. On the other hand, Hybrid
Computations proves to be more appropriate.
Example 3.1 We consider a set
Φ maximum degree 20, and exact GCD,
1 g (
= Π s ) =
11 , 20 s 3 of 11 polynomials with integer coefficients (max. 3 digits),
+ 3 s 2 + 4 s + 2
, [4, 5]. Next, for this polynomial set, we will study three cases of behavior of the ERES method using the above described types of arithmetic. The results are presented in the tables 2, 3 and 4. The relative errors were estimated by using the Frobenius norm.
Generally, the accuracies
ε
G and
ε t have not the same value and it seems that their selection depends strongly on the nature of the given data. Furthermore, there is not a clear relation between the values we assign to them
1
2
It is also referred to as hardware floating-point precision
It is also referred to as software floating-point precision
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349
Table 2 SFP behavior of the ERES algorithm for the set Φ
1
Type of data
Accuracies float double precision, Digits = 15, ε t
= 10 − 15 , ε
G
= 10 − 15
Resulted GCD 1 .
0
Type of data float
Accuracies double precision, Digits = 15,
ε t
= 10 − 15 , ε
G
= 10 − 8
Resulted GCD
1 .
0 s 3 + 2 .
99999999608220 s 2 + 3 .
99999999215816 s + 1 .
99999999215450
Relative error
0 .
215 10 − 8
Type of data float
Accuracies double precision, Digits = 15,
ε t
= 10 − 8 , ε
G
= 10 − 8
Resulted GCD
1 .
0 s 3 + 2 .
99999996640441 s 2 + 3 .
99999993280994 s + 1 .
99999993280184
Relative error
0 .
184 10 − 7 and the properties or the structure of the basis matrix
P m
. Since the selection of the most appropriate values for the Gaussian accuracy
ε
G and the termination accuracy
ε t is non-trivial, we have to pay attention to the type of the initial data (e.g. integers, decimals, fractions, radicals etc.) and the kind of operations, which are performed throughout the whole process of computing the GCD of a given set of polynomials
Π m,d
.
The ERES is an iterative method with no fixed number of iterations and since it is a matrix based method, it often deals with large amount of data. If the initial data has a standard floating-point format, then the performed operations also yield results in standard floating-point format and this causes a lot of round-off errors to occur, especially when we have many iterations. This often leads to results of very low accuracy (see table 2). The number of digits used by the system for the internal representation of the data plays also an important role. Thus, when working with ERES, it would be more preferable to use VFP arithmetic. This kind of arithmetic allows the user to determine by himself the number of digits used for the internal representation of the data (variable precision). We will refer to this kind of accuracy as the software accuracy and it will be denoted by the term
Digits
3
.
Various tests on several polynomial sets with floating-point data, in VFP arithmetic, showed that we can obtain quite satisfactory results from the ERES method, if we allow the system to perform the internal operations with more than 20 digits of accuracy (i.e. Digits ≥ 20 ) and simultaneously assign large values to the accuracies ε
ε
G
(see table 3). Also, it has been observed that it is not worthwhile assigning values to ε t and ε
G less than t
10 and
− 16
.
Furthermore, there is a way to have a value for the termination accuracy ε t as an upper bound, by observing the values that the termination criterion of the algorithm gets during the process of computing the GCD. This works especially when the accuracy ε
G does not affect the data during the matrix transformation. The algorithm can be slightly modified to detect a proper value for the termination accuracy by itself. In most cases, the Gaussian accuracy
ε
G can be deduced from the software accuracy. Generally, the quality of the results obtained by the
ERES algorithm can be controlled by the variable Digits, which determines the software accuracy. But, there is no way to know in advance the proper value of Digits for an accurate computation and there are cases of sets of polynomials where this value must be quite large, which costs in terms of memory and time.
3.2
Performance of ERES method using exact symbolic arithmetic
Another option is to use ES arithmetic for the operations performed with the ERES algorithm. The symbolic operations used by rational arithmetic always yield exact results and it is more preferable to make use of them,
3
The variable Digits is very common amongst various software packages for mathematical applications such as Maple, Mathematica,
Matlab and others.
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350 D. Christou and M. Mitrouli: Estimation of the GCD of many polynomials with the ERES method
Table 3 VFP behavior of the ERES algorithm for the set Φ
1
Type of data float
Accuracies variable precision, Digits = 24, ε t
= 10 − 16 , ε
G
= 10 − 16
Resulted GCD 1 .
0 s 3 + 2 .
99999999999999999998275 s 2 + 3 .
99999999999999999996562 s
+1 .
99999999999999999996565
Relative error 0 .
942 10 − 20
Type of data
Accuracies float variable precision, Digits = 23,
ε t
= 10 − 16 , ε
G
= 10 − 16
Resulted GCD 1 .
0
Type of data float
Accuracies variable precision, Digits = 23,
ε t
= 10 − 16 , ε
G
= 10 − 12
Resulted GCD
1 .
0 s 3 + 2 .
9999999999999997573495 s 2 + 3 .
9999999999999995261066 s
+1 .
9999999999999995238086
Relative error
0 .
130 10 − 15
Type of data float
Accuracies variable precision, Digits = 20,
ε t
= 10 − 16 , ε
G
= 10 − 12
Resulted GCD
1 .
0 s 3 + 3 .
0000000000001436500 s 2 + 4 .
0000000000002876900 s
+2 .
0000000000002882608
Relative error
0 .
788 10 − 13
Type of data float
Accuracies variable precision, Digits = 18, ε t
= 10 − 12 , ε
G
= 10 − 8
Resulted GCD
1 .
0 s + 1 .
00000204413857072
Relative error
0 .
816 when our initial data are type of integer, fraction or radical. For the symbolical implementation of the ERES algorithm it can be used any symbolical package, like Maple
4 or Mathematica.
Table 4 ES behavior of the ERES algorithm for the set Φ
1
Type of data rational
Accuracies symbolic, ε t
≤ 10 − 16 , ε
G
≤ 10 − 16
Resulted GCD s 3 + 3 s 2 + 4 s + 2
Relative error
0
The following examples show clearly that the ERES method yields better results when using symbolic operations.
4
All the results presented in this paper were obtained by using Maple 8 and an Intel Pentium4 PC, 2.0MHz, 512Mb Ram . The source code of the ERES algorithm and the examples of this paper are available from the corresponding author.
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351
Example 3.2 We consider a set Φ
2
= Π
2 , 16 mum degree 16, and exact GCD, g ( s ) = s 4 − of 2 polynomials with integer coefficients (max. 8 digits), maxi-
10 s 3 + 35 s 2 − 50 s + 24 , [3, 5]. The results are presented in table
5.
Table 5 VFP and ES behavior of the ERES algorithm for the set Φ
2
Type of data
Accuracies
Resulted GCD
Relative error rational symbolic, ε t
≤ 10 − 16 , ε
G
≤ 10 − 16 s 4 − 10 s 3 + 35 s 2 − 50 s + 24
0
Iterations / Time 23 / 2.969sec
Type of data
Accuracies
Resulted GCD
Relative error rational variable precision, Digits = 22, ε t
= 10 − 16 , ε
G
= 10 − 16
1 .
0 s 4 − 10 .
00000000006499972615 s 3 + 35 .
00000000058529628077 s 2
− 50 .
00000000169210353285 s + 24 .
00000000156365855379
0 .
358 10 − 10
Iterations / Time 23 / 0.359sec
Example 3.3 We construct from
Φ
2 a new set
Φ
3
= Π
100 , 16 of 100 linearly depended polynomials. For the set s 4
Φ
− 10 s 3
3 of polynomials with integer (max. 10 digits) or real-decimal coefficients and exact GCD, g ( s ) =
+ 35 s 2 − 50 s + 24 , we study two cases of behavior, which are presented in tables 6, 7 and clearly show the difference between VFP and HC operations.
Table 6 VFP behavior of the ERES algorithm for the set Φ
3
Type of data
Accuracies
Resulted GCD rational, float variable precision , Digits = 20, ε t
= 10 − 16 , ε
G
= 10 − 16
1.0
Type of data
Accuracies
Resulted GCD
Relative error rational, float variable precision , Digits = 20, ε t
= 10 − 12 , ε
G
= 10 − 16
1 .
0 s 4 − 10 .
000000235029354837 s 3 + 35 .
000002115340572177 s 2
− 50 .
000006111297186974 s + 24 .
000005641618034996
0 .
129 10 − 6
Iterations / Time 25 / 1.156sec
Type of data
Accuracies
Resulted GCD
Relative error rational, float variable precision , Digits = 22, ε t
= 10 − 16 , ε
G
= 10 − 16
1 .
0 s 4 − 10 .
00000002242983684635 s 3 + 35 .
00000020186794424622 s 2
− 50 .
00000058317165039951 s + 24 .
00000053830905097638
0 .
123 10 − 7
Iterations / Time 25 / 1.234sec
2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

352 D. Christou and M. Mitrouli: Estimation of the GCD of many polynomials with the ERES method
Table 7 HC behavior of the ERES algorithm for the set
Φ
3
Type of data
Accuracies
Resulted GCD
Relative error rational, float hybrid computations, Digits = 20, ε t
≤ 10 − 16 , ε
G
≤ 10 − 16 s 4 − 10 s 3 + 35 s 2 − 50 s + 24
0
Iterations / Time 25 / 5.453sec
We can see that there is greater numerical error when using only VFP arithmetic.
3.3
Estimation of approximate GCD’s with the ERES method
As we have already mentioned, the ERES method produces good results either by using variable precision operations with enough digits or exact symbolic operations. But the question is how “good” are these results, or, in other words, is the GCD obtained by the ERES a good approximation of the true GCD? The numerical computation of the exact GCD is an ill-posed problem in the sense that arbitrary tiny perturbations reduce a non-trivial
GCD to the constant 1. The proper definition of the “approximate” GCD, [7, 9, 10], and the way we can measure the strength of the approximation, have remained open. Regarding ERES, different values to the specified accuracies
ε t and
ε
G
, often leads to the computation of a common divisor of the set of polynomials, which is not the greatest common divisor (see table 3, last case). This can be a real problem, since we do not know in advance the GCD of a given set of polynomials. Unless a mathematical tool to check the strength of approximation of the obtained GCD is available, we ought to double-check the results by assigning more values to the accuracies ε t and ε
G or to the variable Digits.
However, the ERES method proved to be a quite useful mathematical tool in computing an approximate GCD of sets of polynomials with equal root clusters. The polynomials of this kind of sets, theoretically, are considered to be co-prime. By selecting a proper value for the accuracy ε t
, which in most cases must be ≥ 10 − 8
, we often obtain as a GCD a new polynomial, which is approximately equal to the least degree polynomial of the original set (example 3.5).
Example 3.4 We consider a set Φ
4 the polynomials have GCD, g
1
( s ) = s 2
= Π
− 3
4 , 12 s of polynomials with real-decimal or integer coefficients. Two of
+ 2 and the other two have GCD, g
2
( s ) = s 2 − 2 .
999999989 s +
1 .
999999979
. Normally, the polynomials of the set are considered to be co-prime. The results we get from the
ERES method are presented in table 8.
Table 8 VFP and HC behavior of the ERES algorithm for the set Φ
4
Type of data rational, float
Accuracies variable precision, Digits = 20, ε t
= 10 − 8 , ε
G
= 10 − 8
Approximate GCD 1 .
0 s 2 − 2 .
9999999915637740963 s + 1 .
9999999842285569510
Iterations / Time 10 / 0.259sec
Type of data rational, float
Accuracies hybrid computations, Digits = 20,
ε t
= 10 − 16 , ε
G
= 10 − 8
Approximate GCD
1 .
0 s 2 − 2 .
9999999843558725734 s + 1 .
9999999688507700303
Iterations / Time 9 / 0.422sec
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Appl. Num. Anal. Comp. Math. 2, No. 3 (2005) / www.anacm.org
353
Example 3.5 We consider the set Φ
5
= Π
3 , 2
= { s 2 + 3 s + 2 , s 2 − 2 .
998 s + 1 .
997001 , 4 s 2 − 11 .
996 s +
7 .
994001 } where the polynomials have approximately equal root clusters. Theoretically, the exact GCD is, g ( s ) = 1 . The results we get from the ERES method are presented in table 9.
Table 9 VFP and HC behavior of the ERES algorithm for the set Φ
5
Type of data rational, float
Accuracies variable precision, Digits = 20,
ε t
= 10 − 3 , ε
G
= 10 − 16
Approximate GCD
1 .
0 s 2 − 2 .
9989997142432029971 s + 1 .
9984999881743550886
Type of data rational, float
Accuracies hybrid computations, Digits = 20, ε t
= 10 − 3 , ε
G
= 10 − 16
Approximate GCD 1 .
0 s 2 − 2 .
9989997142432029967 s + 1 .
9984999881743550885
Generally, the ERES method always succeeds in computing an approximation of the GCD of a given set of polynomials, and this important feature is not common amongst other iterative methods based on the Euclidean algorithm.
3.4
Computational Results
Notation In the next tables we shall use the following notation : m
: the number of polynomials d
0
: the degree of the GCD
Rel : the relative error according to Frobenius norm
Dig : the number of Digits (software accuracy) d
Iter
Time
: the maximal degree of the polynomials
: the number of iterations of the algorithm
: algorithm’s estimated time of execution (sec)
SRF-ES : square-root-free symbolic operations
Table 10 Behavior of the ERES algorithm for random sets of polynomials
VFP
No.
i ii m
5
8 d
4
7 d
1
2
0
Iter Dig
3
5
5
Rel
17 10 − 17
21
10 − 18
21 10 − 17 iii iv v
5
15
10
15
2
2
6
7
20 10 − 18
22
10 − 15
35 10 − 17 vi 15 15 5 6 vii 20 15 5 5 21 10 − 18
22
10 − 16 viii 30 15 3 ix 2
6
20 4 33 32 10 − 15 x xi xii
10 10 2
15
100
200
20
20
50
4
4
2
7 23 10 − 17
40
10 − 20
7
9 200 10 − 67
HC
Time Dig
0.015
0.032
0.047
Rel
16 10 − 15
17
10 − 16
19 10 − 17
0.078
0.172
0.172
17 10 − 14
20
10 − 15
30 10 − 16
0.203
0.218
18 10 − 16
18
10 − 15
32 10 − 15
0.266
0.234
22 10 − 17
34
10 − 22
1.109
69.032
150 10 − 57
ES SRF-ES
Time
0.016
0.032
0.063
0.109
Time
0.125
0.110
0.219
0.250
Time
0.016
0.047
0.125
0.156
0.187
0.171
0.219
0.656
0.937
1.156
0.328
0.405
0.406
0.265
0.234
1.329
15.031
0.704
4.609
0.265
38.391
14.344
1.922
34.562
14.062
314.390
> 6000 > 6000
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354 D. Christou and M. Mitrouli: Estimation of the GCD of many polynomials with the ERES method
Remark 3.6 Considering the experimental results in table 10, we have to note that all the polynomial sets have been constructed by random GCD’s and random polynomials with integer coefficients ( ≤ 9999) . When using ES error, which is estimated by using the Frobenius vector norm k v k
F
=
³P n i =0
| v i
| 2
´
1
2 for v = ( v
0
, . . . , v n
) t ∈
< n
ε
G
, is 0 for ES operations and less than
10 − 14 for SRF-ES operations. In every case, both accuracies
ε t and are set to
10 − 15
, which is close to the limit of double-precision’s arithmetic. The columns of VFP and HC operations present the lowest number of digits, which is required for an acceptable result, and the corresponding relative error and time of execution.
Table 11 Storage and time requirements for a random set Π
12 , 12 with exact GCD, g ( s ) = s 2 + 11
3 s + 0 .
625
Type of data Type of operations Digits Memory (bytes) Time (secs) Relative error rational rational, float float
ES
HC
VFP
14
28
28
7,735,542
2,853,434
2,868,462
0.547
0.203
0.172
0
0 .
223795 10 − 11
0 .
955775 10 − 11
The ERES method is quite effective, when properly implemented in a programming environment. We can have large sets of real polynomials without restrictions to the type of data. Actually the method proves to be faster, when the polynomials of a given large set Π m,d the Gaussian accuracy ε
G are linearly depended. An appropriate selection of a value for
, helps ERES to reduce dramatically the row dimension of the basis matrix and hence proceed with a smaller set of polynomials. This reduction always takes place in the first iteration of the method.
In fact, for any polynomial set, considering the vectors of the coefficients of each polynomial, there is a way to find the most orthogonal linearly independent representatives of the set, without transforming the original data, and form a base of polynomials, which can give us the GCD of the whole set [5]. Such base is referred to as best
uncorrupted base [4, 6]. However, the process of computing such a base is very demanding in terms of time and memory and it can benefit the ERES method only if the basis matrix m À n
,
( n = d + 1)
.
P m
∈ < m × n is highly rank deficient, or
On the other hand, if the maximal degree of the polynomials of a given set is very large, then the column dimension of the basis matrix P m will be also large. This often leads to many iterations, especially if the degree of the GCD of the set is small. Actually, the ERES algorithm becomes slow, when the maximal degree of the polynomials of a given set is large enough and especially when m ¿ n . Therefore, the dimensions of the basis matrix P m and the degree of the GCD are very important factors to take into account, in order to select either VFP arithmetic or ES arithmetic for the operations, which are performed during the execution of the ERES algorithm in a programming environment.
Variable precision operations, in VFP arithmetic, can always be our choice, since they are faster and more economical in memory bytes than symbolic operations, especially if there is no need to use enough digits to have a reliable software accuracy. But, it has been observed that, when the basis matrix P m has large dimensions and full rank, we must use a lot of digits for the software accuracy in order to avoid results with great numerical error.
This is absolutely necessary if the column dimension of P m is large and the degree of the GCD is small, because this will invoke ERES to perform many iterations and hence, intense the propagation of errors. Additionally, if we increase the number of digits of the software accuracy, the time and memory requirements will also increase.
But still, it is not easy for the user to determine in advance the proper number of digits that are necessary for a successful computation of the GCD.
Exact symbolic operations could have been our first and only choice, since they produce excellent results with minimal or no error at all. But, they are very costly regarding time and memory (see table 11). This problem is more obvious, when the basis matrix P m is large and dense. Indeed, the successive matrix transformations, performed by the method, take enough time and consume a lot of memory bytes. However, if we recall that
2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Appl. Num. Anal. Comp. Math. 2, No. 3 (2005) / www.anacm.org
355 the ERES algorithm decreases the size of the basis matrix during the iterations, the symbolical manipulation of the data can rarely be prohibitively expensive. Symbolic operations are a good choice for the ERES method, especially when the basis matrix P m is sparse.
Since both variable precision operations and symbolic operations have advantages and disadvantages, it would be best if we could combined them appropriately in order to get good performance and stability of the ERES algorithm in every case. Many tests considering large sets of polynomials, which form a basis matrix
P m
< m × n
∈
, m À n, ρ ( P m
) = n , showed that we can have results with quite acceptable errors within a short time, if we use rational data and hence symbolic operations to achieve a reduction of the row dimension of the basis matrix P m in the first iteration and then, after converting the data of the obtained matrix to a floatingpoint format, proceed with VFP operations. The results presented in table 10 (HC columns) derive from this type of hybrid computations. An important fact is that this kind of HC operations requires less Digits (software accuracy) comparing to VFP operations. Additionally, if our initial data are type of radical, it is better to convert them to an approximate rational form (fraction of integers). This conversion can also take place throughout the process of normalization in order to avoid the presence of square roots in our data. This technique reduces the time of execution of the ERES algorithm, when using symbolic operations. The operations performed by using this technique shall be called square-root-free symbolic operations (SRF-ES) (see table 10). Yet again, such techniques cannot be applied generally.
Many software mathematical packages like Maple and Mathematica, allows us to choose freely whether to use variable precision operations or symbolic operations, by converting our initial data to an appropriate type.
The type of the initial data suggests the type of operations. For example, if our initial data are type of rational or radical, then symbolic operations will be performed. However, there are some restrictions, which oblige us to convert our data to floating-point format. Specifically, many built-in routines work only with floatingpoint data. For example, in Maple 8, we have to use floating-point data for the computation of the Singular
Value Decomposition of the matrix P
( k ) m obtained in the k th iteration, because computation of the left and right singular vectors (as columns of matrices) may only be requested for a matrix whose entries can be evaluated to purely floating-point values. This also holds for Mathematica. In other words, the function that we call for the computation of the singular values of a matrix uses by default standard (hardware) floating-point arithmetic
(15 digits) or software floating-point arithmetic, if requested. Therefore, when we are working with the ERES symbolically, it is more preferable to select as a GCD a row from the last matrix P
( k ) m first row of the matrix
W t rather than selecting the from the SVD of
P
( k ) m
, as described in proposition 2.2. The opposite holds, if there are floating-point data. These restrictions bring up to the surface the problem of assigning proper values to the accuracies ε t
, ε
G and Digits. Similar problems with the appropriate selection of accuracies arise in any other matrix based method computing the GCD of polynomials or other required quantities. Thus, we must always have in mind the software accuracy, which determines the numerical accuracy of the system.
Important factors to take into account when working with the ERES method.
•
Dimensions and structure of the original basis matrix
P m
.
• Data type of the elements of P m
.
•
Numerical accuracy of the original data (for floating-point numbers).
•
Appropriate selection of values for the variable Digits and, if it is necessary, for
ε t
, ε
G
.
Therefore, we conclude that the computation of the GCD of a set of polynomials with real coefficients by the ERES method, is a process where floating-point operations and exact symbolic operations must combined together for a better overall performance. We can say that we are looking for a kind of hybrid type of implementation for the ERES method.
The above study helps to answer the following question : Is it worthwhile to use hybrid computations to improve the quality of the results achieved from an algorithm? From our experience we think that hybrid computations can mostly improve the accuracy of the results and thus can be used whenever possible.
2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

356 D. Christou and M. Mitrouli: Estimation of the GCD of many polynomials with the ERES method
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