Computation of the LCM of several polynomials using the
6.2 Computation of the LCM using the GCD
6.2.1 Factorisation of polynomials using ERES Division
pω(s) : pω(s) =
h−1Y
k=1
pik(s),∀ ω = (i1, i2, . . . , ih−1)∈ Qh−1,h
) ,
p(s) =Qh
i=1pi(s) and g(s) = gcd{T }, then l(s) satisfies the following properties:
i) For any ω ∈ Qh−1,h, if we can write
pω(s) = g(s)· rω(s) then
l(s) = pωb(s)· gbω(s) where gbω(s) = rω(s).
ii) The polynomials l(s), g(s) and p(s) satisfy the identity :
l(s)· g(s) = p(s) (6.4)
At this point, we need a procedure that can handle the factorisation of a polynomial into two factors, when the one is given. This factorisation can be implemented either as a polynomial division or a system of linear equations. These two approaches will be presented in the following.
6.2.1 Factorisation of polynomials using ERES Division
The identity (6.4) implies that the LCM l(s), is actually the quotient of the division
p(s)
g(s) = l(s) (6.5)
which suggests an algorithmic procedure for the computation of the LCM based on polynomial division. The development of such an LCM algorithm requires:
a) the GCD of the given set of polynomials, and
b) a computational procedure for the division of two polynomials.
Both of the above requirements can be handled effectively in the context of the ERES methodology. As described in Chapters 3 and 4, the ERES method with the developed Hybrid ERES algorithm is an efficient numerical tool for the computation of the GCD of polynomials. Furthermore, the developed ERES Division algorithm in Chapter 3 can also be used to calculate the result of
the division (6.5). Specifically, the vector of polynomials p(s) and g(s) can be transformed through ERES operations into a new vector of polynomials such that
"
p(s) g(s)
#
− − − − −− >ERES
"
r(s) g(s)
#
The coefficients of the LCM l(s), are obtained implicitly during the ERES Division process and the polynomial r(s) represents the remainder of the division p(s)g(s).
Moreover, in the case of sets of polynomials with numerical inaccuracies it has been proved in chapters 4 and 5 that the Hybrid ERES algorithm is capable of producing approximate GCDs of good quality within a specified range of numerical accuracy. The use of an approximate GCD in the identity (6.5) naturally leads to an approximate solution for the LCM. However, the final result may also be affected by additional numerical errors introduced by the ERES operations. But this complication can be avoided, if symbolic-rational operations are used during the application of the ERES Division algorithm. Under this assumption, the result obtained from the polynomial division (6.5), when an approximate GCD is present, will be considered as an approximate LCM. However, we will not proceed further our analysis for the approximate LCM problem using this methodology. A different approach will be presented in the following.
Consequently, the ERES method appears to be a significant part in the process of computing the LCM of a set of several polynomials. However, other methods, such as the Matrix Pencil method [45], can also be used for the computation of the GCD of polynomials. The following algorithm is developed in the context of symbolic-rational computations for the computation of the LCM of a set P of several polynomials and it is based on the results derived from the previous Theorem 6.2. The next Symbolic-Rational (S-R) LCM algorithm is actually a variation of the numerical LCM algorithm, which is described in [47].
ALGORITHM 6.1. The S-R LCM Algorithm Input : P =
pi(s)∈ R[s], i = 1, 2, . . . , h Step 1 : Compute p(s) = p1(s) p2(s)· · · ph(s).
Step 2 : Find the set T =n
pωi(s) : pωi(s) =Qh−1
k=1pik(s), i = 1, . . . , ho for all ωi = (i1, i2, . . . , ih−1)∈ Qh−1,h.
Step 3 : Compute g(s) = gcd{T }.
Step 4 : Compute l(s) = p(s)
g(s) by applying the
ERES Division Algorithm 3.1 to the pair (p(s), g(s)).
Output : l(s) = lcm{P}
◮ Computational complexity and numerical behaviour of the S-R LCM algorithm
The previous Algorithm 6.1 behaves very well when the polynomials have integer coefficients and they are processed by using exact rational operations. However, the amount of operations (addition or multiplication) required for the computation of the initial polynomial p(s) and the polynomials of the setT can be prohibitively high. Specifically, for h polynomials with average degree ¯d ≥ 2, the algorithm must perform:
f l( ¯d, h) = ( ¯d + 1)h+ h ( ¯d + 1)h−1
= ( ¯d + 1)h
1 + h d + 1¯
(6.6)
operations. If we use the Hybrid ERES algorithm to compute the GCD of the setT , then the dimensions of the initial matrix will be equal to h×( ¯d h+ 1) and the total number of the performed operations is about O 13h3+ 2 ¯d2h3− 23d¯3h3
. Finally, the ERES Division algorithm requires about O ¯dh(k + 2)− k2
operations, where k denotes here the degree of the GCD. Therefore, we conclude that the S-R LCM Algorithm 6.1 can be computationally efficient only for moderate sets of polynomials.
Regarding its numerical efficiency, if the original data are given inexactly in floating-point format, it is obvious that it is not wise to perform numerical floating-point operations to compute the polynomial p(s) and the polynomials of the set T , because it is very likely to have many unnecessary numerical errors during the process. Thus, the construction of the initial polynomials of the set T and also the ERES Division algorithm is better to be implemented by using symbolic-rational operations in order to minimize the risk of getting erroneous results. Therefore, considering the S-R LCM algorithm, the computation of an approximate LCM of the set P, relies on the computation of an approximate GCD given by the Hybrid ERES algorithm. In the following, we shall introduce an alternative method for the computation of an approximate LCM without computing the GCD and we will compare the results of the two methods.
Example 6.1.Demonstrate now the steps of the S-R LCM Algorithm 6.1, consider the polynomial setP4 =
pi(s)∈ R[s], i = 1, 2, 3, 4 with p1(s) = (s− 1)(s + 2)2 = s3+ 3 s2− 4 p2(s) = (s + 2)(s− 3)2 = s3− 4 s2− 3 s + 18 p3(s) = (s− 1)(s − 3) = s2− 4 s + 3
p4(s) = (s + 2)(s− 4) = s2− 2 s − 8
(6.7)
Step 1: We compute the polynomial
p(s) = p1(s)· p2(s)· p3(s)· p4(s) = (6.8)
= s10− 7 s9− 6 s8+ 118 s7− 55 s6− 759 s5+ 556 s4+ 2168 s3− 1584 s2− 2160 s + 1728 In the generic case, this polynomial would be the actual LCM of the original set P, unless the polynomials have common factors.
Step 2: Obviously, h = 4 and hence, we get 4 sequences ωi ∈ Q3,4, which will be used in order to formulate the new set
T = (
pωi(s) : pω(s) = Y3 k=1
pik(s), i = 1, . . . , 4 )
Therefore,
ω1 = (1, 2, 3) : pω1(s) = p1(s)· p2(s)· p3(s) ω2 = (1, 2, 4) : pω2(s) = p1(s)· p2(s)· p4(s) ω3 = (1, 3, 4) : pω3(s) = p1(s)· p3(s)· p4(s) ω4 = (2, 3, 4) : pω4(s) = p2(s)· p3(s)· p4(s) and
pω1(s) = s8− 5 s7− 8 s6+ 62 s5+ 5 s4 − 253 s3+ 90 s2+ 324 s− 216 pω2(s) = s8− 3 s7− 21 s6+ 43 s5 + 180 s4− 168 s3− 656 s2+ 48 s + 576 pω3(s) = s7− 3 s6− 15 s5+ 31 s4 + 78 s3− 84 s2− 104 s + 96
pω4(s) = s7− 10 s6+ 24 s5+ 50 s4− 245 s3+ 72 s2+ 540 s− 432 Step 3: The basis matrix of the polynomial set T is
T =
−216 324 90 −253 5 62 −8 −5 1
576 48 −656 −168 180 43 −21 −3 1
96 −104 −84 78 31 −15 −3 1 0
−432 540 72 −245 50 24 −10 1 0
∈ R4×9 (6.9)
and, using the Hybrid ERES algorithm, we obtain the GCD ofT :
g(s) = s4− 9 s2− 4 s + 12 (6.10) Step 4: The ERES Division Algorithm 3.1 applied to the pair (p(s), g(s)) gives the final solution:
l(s) = s6− 7 s5+ 3 s4+ 59 s3− 68 s2− 132 s + 144 (6.11) which is a 6th degree polynomial representing the exact LCM of P4 in (6.7).