180 5. Approximation and Location of Zeros
Exercises
- Let w be any zero of the polynomial a,t” + a,-lt”-’ +.... Show that
(a) w = o;l[-a,-i - fzn-2wB1 -I. f - ai~-“+~ - a~w~~+l]
@I lwl 5 bnl-l c;=o bkl-
- Let fc,~i,rs ,..., 2, be complex numbers. Show that
(a) IZI +z2++..+znl 6 IzII+I~~I+-~~+I~~I
(b) (zo+rl +-+.+~,a( 2 Izol- Ial- Izzl-...- Iznl.
When does equality occur?
- Cauchy’s estimate. Let p(t) = a,t” + ... + ac and let K be the
maximum of the absolute values of as/a,, al/a,,... ,a,-i/a,. Show
that, if w is a zero of p(t), then
(a) 0 = a,w”[l+(a,~~/a,)w~‘+~~~+(a~/a,)w~”+’+(ao/a,)w~”]
(b) lwl < K + 1.
- (a) Show that, if w is a nonvanishing zero of the polynomial p(t),
then 20-l is a root of the polynomial t”p(t-‘).
(b) Use (a) and th e result of Exercise 3 to determine a lower bound
for the absolute value of a nonzero root of a polynomial equation. - (a) Let n _> 1, and suppose ai (0 5 i 5 n) are complex numbers
with a0 # 0. Show that the polynomial
t” - Jan&-l - lan-21tn-2 -... - lqlt - laoI
has a unique positive zero r.
(b) Prove that every zero of the polynomial
t” + Q,-lP--l +. * * + qt + al-J
satisfies ]w] 5 r, where r is defined as in (a).
- The polynomial at + b has the zero -b/a. In (c), this will be general-
ized to an estimate for the zeros of polynomials of higher degree. In
what follows, n 1 2,
g(t) = b,t” +... + b,t + b.
f(t) = a,P +.. * + a1t + ao.
(a) Let 0 < b, 5 bnwl 5 bn-2 _< e-e < bl 5 bo. Show that every
zero w of the polynomial g(t) must satisfy ]w] > 1.