91Ó°ÊÓ

Use the bisection method with a hand calculator or computer to find the indicated roots of the following equations. Use an error tolerance of \(\epsilon=0.0001\). (a) The real root of \(x^{3}-x^{2}-x-1=0\). (b) The root of \(x=1+0.3 \cos (x)\) (c) The smallest positive root of \(\cos (x)=\frac{1}{2}+\sin (x)\). (d) The root of \(x=e^{-x}\). (e) The smallest positive root of \(e^{-x}=\sin (x)\). (f) The real root of \(x^{3}-2 x-2=0\). (g) All real roots of \(x^{4}-x-1=0\).

Convert the equation \(x^{2}-5=0\) to the fixed-point problem $$ x=x+c\left(x^{2}-5\right) \equiv g(x) $$ with \(c\) a nonzero constant. Determine the possible values of \(c\) to ensure convergence of $$ x_{n+1}=x_{n}+c\left(x_{n}^{2}-5\right) $$ to \(\alpha=\sqrt{5}\).

Let the initial interval used in the bisection method have length \(b-a=3\). Find the number of midpoints \(c_{n}\) that must be calculated with the bisection method to obtain an approximate root within an error tolerance of \(10^{-9}\).

Let \(f(x)=1-z x\) for some \(z>0 .\) Solving \(f(x)=0\) is equivalent to calculating \(1 / z\), thus doing a division. (a) Give an interval \([a, b]\), or a way to calculate it, guaranteed to contain \(1 / z\). Do not use division in calculating or defining \([a, b]\). (b) Assume \(10\), consider calculating \(1 / z\) in the single precision arithmetic of the IEEE standard floating-point arithmetic. Using the bisection method, give a way to calculate \(1 / z\) to the full accuracy of the arithmetic.