(1) (a) For a nonlinear equation, $f(x) = 0$ and an approximate root, $x^{(k)}$, does a small
residual, $|f(x^{(k)})|$, guarantee that $x^{(k)}$ is an accurate approximation to the actual root,
$x^*$? Why or why not?
(b) Suppose you are using an iterative method to solve a nonlinear equation, $f(x) = 0$, for
a root that is ill-conditioned, and you need to choose a test to decide when an approximate
root is is sufficiently accurate. Would it be better to terminate the iteration when you find
an iterate, $x^{(k)}$, for which $|f(x^{(k)})|$ is small, or when $|x^{(k)} - x^{(k-1)}|$ is small? Why?
(c) What is meant by a bracket for a nonlinear function in one dimension? Why is this
concept related to finding roots of a nonlinear function?
(d) Suggest an approach for safeguarding the secant method for solving a one-dimensional
nonlinear equation so that the method will converge even if we start with an initial guess
that is far from the root.
