Suppose we define the sequence of polynomials Po(t) = 1 P(t) = d - t P(t)...
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Suppose we define the sequence of polynomials Po(t) = 1 P₁(t) = d₁ - t P₂(t) = (d₂ t)p₁(t) - bipo(t) : pi(t) : Pn(t) = (dnt)pn-1(t) − b ² -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 b₁,b₂, . 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) (d₁ – t)p₁-1(t) —– b²-1P₁-2(t) 1. Let c be some real number. 2. Calculate the numbers po(c), p₁(c), . . . , Pn(c) 3. Let N(c) be the number of agreements in sign of adjacent terms in the sequence po(c), P₁(c), . . . , Pn(c). 4. N(c) is equal to the number of roots of pn(t) = 0 that are in the interval [c, ∞). Suppose we define the sequence of polynomials Po(t) = 1 P₁(t) = d₁ - t P₂(t) = (d₂ t)p₁(t) - bipo(t) : pi(t) : Pn(t) = (dnt)pn-1(t) − b ² -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 b₁,b₂, . 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) (d₁ – t)p₁-1(t) —– b²-1P₁-2(t) 1. Let c be some real number. 2. Calculate the numbers po(c), p₁(c), . . . , Pn(c) 3. Let N(c) be the number of agreements in sign of adjacent terms in the sequence po(c), P₁(c), . . . , Pn(c). 4. N(c) is equal to the number of roots of pn(t) = 0 that are in the interval [c, ∞).
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The algorithm of Givens is a method used to isolate the roots of the polynomial equation pt 0 where ... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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