Contents - Student subdomain for University of Bath

2.3. GREATEST COMMON DIVISORS 49
a 4 = −3722432068x − 8393738634,
a 5 = 1954124052188.
Here the numbers are much smaller, and indeed it can be proved that the
coefficient growth is only linear in the step number. a 2 has a content of 3,
which the primitive p.r.s. would eliminate, but this content in fact disappears
later. Similarly a 3 has a content of 5, which again disappears later. Hence we
have the following result.
Theorem 6 Let R be a g.c.d. domain. Then there is an algorithm to calculate
the g.c.d. of polynomials in R[x]. If the original coefficients have length bounded
by B, the length at the i-th step is bounded by iB.
This algorithm is the best method known for calculating the g.c.d., of all those
based on Euclid’s algorithm applied to polynomials with integer coefficients. In
chapter 4 we shall see that if we go beyond these limits, it is possible to find
better algorithms for this calculation.
2.3.3 The Extended Euclidean Algorithm
We can in fact do more with the Euclidean algorithm. Consider the following
variant of Algorithm 1, where we have added a few extra lines, marked (*),
manipulating (a, b), (c, d) and (e, e ′ ).
Algorithm 4 (Extended Euclidean)
Input: f, g ∈ K[x].
Output: h ∈ K[x] a greatest common divisor of f and g, and c, d ∈ K[x] such
that cf + dg = h.
i := 1;
if deg(f) < deg(g)
then a 0 := g; a 1 := f;
(*) a := 1; d := 1; b := c := 0
else a 0 := f; a 1 := g;
(*) c := 1; b := 1; a := d := 0
while a i ≠ 0 do
(*) #Loop invariant: a i = af + bg; a i−1 = cf + dg;
a i+1 = rem(a i−1 , a i );
q i :=the corresponding quotient: #a i+1 = a i−1 − q i a i
(*) e := c − q i a; e ′ := d − q i b; #a i+1 = ef + e ′ g
i := i + 1;
(*) (c, d) = (a, b);
(*) (a, b) = (e, e ′ )
return (a i−1 , c, d);
The comments essentially form the proof of correctness. In particular, if f and
g are relatively prime, there exist c and d such that
cf + dg = 1 : (2.10)