Function: polrootsbound
Section: polynomials
C-Name: polrootsbound
Prototype: GDG
Help: polrootsbound(T,{tau=0.01}): return a sharp upper bound for the
 modulus of the largest complex root of the polynomial T with relative error
 tau.
Doc: return a sharp upper bound $B$ for the modulus of
 the largest complex root of the polynomial $T$ with complex coefficients
 with relative error $\tau$. More precisely, we have $|z| \leq B$ for all roots
 and there exist one root such that $|z_{0}| \geq B \exp(-2\tau)$. Much faster
 than either polroots or polrootsreal.
 \bprog
 ? T=poltchebi(500);
 ? vecmax(abs(polroots(T)))
 time = 5,706 ms.
 %2 = 0.99999506520185816611184481744870013191
 ? vecmax(abs(polrootsreal(T)))
 time = 1,972 ms.
 %3 = 0.99999506520185816611184481744870013191
 ? polrootsbound(T)
 time = 217 ms.
 %4 = 1.0098792554165905155
 ? polrootsbound(T, log(2)/2) \\ allow a factor 2, much faster
 time = 51 ms.
 %5 = 1.4065759938190154354
 ? polrootsbound(T, 1e-4)
 time = 504 ms.
 %6 = 1.0000920717983847741
 ? polrootsbound(T, 1e-6)
 time = 810 ms.
 %7 = 0.9999960628901692905
 ? polrootsbound(T, 1e-10)
 time = 1,351 ms.
 %8 = 0.9999950652993869760
 @eprog
