Suppose we define the sequence of polynomials Po(t) = 1 P(t) = d - t P(t) = (d t)p(t) - bipo(t) : pi(t) : Pn(t) = (dnt)pn-1(t) b -1Pn-2(t). It is an easy exercise to show (Problem 6) that pn(t) is the characteristic polynomial for H. If the subdiagonal entries b,b, . bn-1 are all nonzero, then the al- gorithm of Givens can be used to isolate the roots of pn(t) = 0. The algorithm proceeds as follows. = (3.36) (d t)p-1(t) b-1P-2(t) 1. Let c be some real number. 2. Calculate the numbers po(c), p(c), . . . , Pn(c) 3. Let N(c) be the number of agreements in sign of adjacent terms in the sequence po(c), P(c), . . . , Pn(c). 4. N(c) is equal to the number of roots of pn(t) = 0 that are in the interval [c, ).