91Ó°ÊÓ

Find the forward and backward error for the following functions, where the root is \(3 / 4\) and the approximate root is \(x_{a}=0.74\) : (a) \(f(x)=4 x-3\) (b) \(f(x)=(4 x-3)^{2}\) (c) \(f(x)=(4 x-3)^{3}\) (d) \(f(x)=(4 x-3)^{1 / 3}\)

Apply two steps of the Secant Method to the following equations with initial guesses \(x_{0}=1\) and \(x_{1}=2\). (a) \(x^{3}=2 x+2\) (b) \(e^{x}+x=7\) (c) \(e^{x}+\sin x=4\)

Find all fixed points of the following \(g(x)\). (a) \(\frac{x+6}{3 x-2}\) (b) \(\frac{8+2 x}{2+x^{2}}\) (c) \(x^{5}\)

Use the Intermediate Value Theorem to find an interval of length one that contains a root of the equation. (a) \(x^{5}+x=1\) (b) \(\sin x=6 x+5\) (c) \(\ln x+x^{2}=3\)

Use Theorem \(1.11\) or \(1.12\) to estimate the error \(e_{i+1}\) in terms of the previous error \(e_{i}\) as Newton's Method converges to the given roots. Is the convergence linear or quadratic? (a) \(x^{5}-2 x^{4}+2 x^{2}-x=0 ; r=-1, r=0, r=1\) (b) \(2 x^{4}-5 x^{3}+3 x^{2}+x-1=0\); \(r=-1 / 2, r=1\)

(a) Find the multiplicity of the root \(r=0\) of \(f(x)=1-\cos x\). (b) Find the forward and backward errors of the approximate root \(x_{a}=0.0001\).

Find the relation between forward and backward error for finding the root of the linear function \(f(x)=a x-b\).

Consider the equation \(8 x^{4}-12 x^{3}+6 x^{2}-x=0\). For each of the two solutions \(x=0\) and \(x=1 / 2\), decide which will converge faster (say, to eight-place accuracy), the Bisection Method or Newton's Method, without running the calculation.

If the Secant Method converges to \(r, f^{\prime}(r) \neq 0\), and \(f^{\prime \prime}(r) \neq 0\), then the approximate error relationship \(e_{i+1} \approx\left|f^{\prime \prime}(r) /\left(2 f^{\prime}(r)\right)\right| e_{i} e_{i-1}\) can be shown to hold. Prove that if in addition \(\lim _{i \rightarrow \infty} e_{i+1} / e_{i}^{\alpha}\) exists and is nonzero for some \(\alpha>0\), then \(\alpha=(1+\sqrt{5}) / 2\) and \(e_{i+1} \approx\left|\left(f^{\prime \prime}(r) / 2 f^{\prime}(r)\right)\right|^{\alpha-1} e_{i}^{\alpha} .\)

Sketch a function \(f\) and initial guess for which Newton's Method diverges.