Unknown
5.1. Approximation of Roots 163 unknown zero T of p(t). This method will enable us to (often) find a better approximation to r. ...
164 5. Approximation and Location of Zeros 0) (ii) (iii) Y = P(X) *X Let c # 1. Show that Newton’s Method applied to the polyno ...
5.1. Approximation of Roots 165 approximation ~1 to the solution, and let un = f(u,,-1) for n 2 2. With suitable choice off and ...
166 5. Approximation and Location of Zeros Explorations E.42. Convergence of Newton Approximations. It is not always the case th ...
5.1. Approximation of Roots 167 and there arises 6, 3r2+ll, 16196r+O, 000541708 = 0 almost, or (rejecting 6, 3r2) r = -0,0000485 ...
168 (ii) d(a, b) = d(b, a); (iii) d(a, c) 2 d(a, b) + d(b, c). Approximation and Location of Zeros The third property is the t ...
5.1. Approximation of Roots 169 1 r=a+ 1 b+ 1 I c+ d+-+... where a, b, c, d, e,... are integers. Such an expression is called a ...
170 5. Approximation and Location of Zeros 5.2 Tests for Real Zeros How many real zeros does the polynomial t5 - 3t4 - t2 -4t + ...
5.2. Tests for Real Zeros 171 Descartes’ rule of signs. Let p(t) be a polynomial over R. Prove that p(t) cannot have more posit ...
172 5. Approximation and Location of Zeros (d) Deduce that, if r is any positive zero of f(t), then r<l+ I- M c -. Qn Suppos ...
5.2. Tests for Real Zeros 173 (a) If a # band q(a) = q(b), s h ow f rom Rolle’s Theorem that p(t) = q’(t) has a zero in the inte ...
174 5. Approximation and Location of Zeros (a) t2-t u=-1 v=2 (b) t2-t+l u=-1 v=2. Let p(t) = 2t6 - 7t5 + t4 + t3 - 12t2 - 5t + ...
5.2. Tests for Real Zeros 175 by induction on the degree of the polynomials. Suppose it holds for all de- grees less than n = de ...
176 5. Approximation and Location of Zeros n(t) = pl(t)ql(t) - dt), degpz < degpl n(t) = n(Mt) - n(t), degp3 < dem m(t) = ...
5.2. Tests for Real Zeros 177 Secondly, observe that if we divide the terms of Sturm’s sequence by the same polynomial, the numb ...
178 5. Approximation and Location of Zeros Sketch the graph of the equation y = f(x) when a = l/2, 1, 2, 3, 4, 8. On the same ax ...
5.3. Location of Complex Roots 179 There are values of a for which the fixed point is an unstable population density. For exampl ...
180 5. Approximation and Location of Zeros Exercises Let w be any zero of the polynomial a,t” + a,-lt”-’ +.... Show that (a) w ...
5.3. Location of Complex Roots 181 (b) Let b, 1 b,-l >_... >_ bo > 0. Show that every zero w of the polynomial g(t) mus ...
182 5. Approximation and Location of Zeros (a) Show that, if all the zeros of f(t) have negative real part, then a, b, c, d are ...
«
5
6
7
8
9
10
11
12
13
14
»
Free download pdf