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3.1. EQUATIONS IN ONE VARIABLE 63
Similarly, V f (−∞) is the number of sign changes in the sequence of leading coefficients
of the Sturm sequence with the signs of the odd-degree terms reversed.
Hence V f (−∞)−V f (∞), the total number of real roots, is easily computed from
the Sturm sequence.
While the obvious way of computing V f (a) is by the definition, i.e. evaluating
f 0 (a), . . ., this turns out not to be the most efficient. Rather, while computing
the Sturm sequence f 0 . . . , we should also store the quotients q i , so that f i (x) =
− (f i−2 (x) − q i (x)f i−1 (x)). We then compute as follows.
Algorithm 7 (Sturm Sequence evaluation)
a: A number
Input: f n (x): Last non-zero element of Sturm sequence of f
q i (x): Quotient sequence from Sturm sequence of f
Output: Sequence L of f n (a), f n−1 (a), . . . , f 0 (a).
L[n] := f n (a);
L[n − 1] := q n+1 (a)L[n];
for i = n . . . 2
L[i − 2] := q i (a)L[i − 1] − L[i];
return L
If f has degree n, coefficients of bit-length at most τ, a has numerator
and denominator of bit-length σ, this algorithm has asymptotic complexity
Õ(d 2 max(σ, τ)) [LR01].
Since it is possible to say how big the roots of a polynomial can be (propositions
71, 72 and 73), we can determine, as precisely as we wish, the location of
the real roots of a univariate polynomial: every time the Sturm sequence says
that there are more than one root in an interval, we divide the interval in two,
and re-compute V (a) − V (b) for each half.
This is far from the only way of counting and locating real roots, i.e. solving
problem 1: other methods are based on Descartes’ 5 rule of signs (Theorem 37:
the number of roots of f in (0, ∞) is less than or equal to, by an even number,
the number of sign changes in the coefficients of f) [CA76], its generalisation
the Budan–Fourier theorem [Hur12] (Corollaries 29 and 30: the number of roots
of f in 6 [a, b] is less than or equal to, by an even number, the number of sign
changes in the derivatives of f evaluated at a (i.e. f(a), f ′ (a), f ′′ (a) . . . ) less the
same evaluated at b), on continued fractions [TE07], or on numerical methods
[Pan02].
All such algorithms take time polynomial in d, the degree of the polynomial
whose real roots are being counted/determined. However, since the number of
sign changes in a sequence of t terms is at most t−1, there are at most 2t−1 real
roots (t − 1 positive, t − 1 negative and zero: consider x 3 − x) for a polynomial
with t non-zero terms, irrespective of d.
5 This rule is always called after Descartes, though the proof actually seems to be due to
Gauss [BF93].
6 We assume neither a nor b are roots.