2.7 Bracketing

• The bisection method is a bracketed root-finding method: this means that at every iteration, we know that the root we are looking for is within the current interval.

• We’ll apply this to the secant and Newton methods, such that we keep track of an interval at each iteration that is guaranteed to contain a root.

Idea: Suppose that $f$ is continuous on $[a,b]$, and that $f(a)$ and $f(b)$ have different signs. Then we know that $f$ has a root in this interval.

Applying the secant method to iterates $a$ and $b$ will produce the next iterate $x_k$ to be in between $a$ and $b$. (why?) IVT on secant line.

Then we take either the interval $(a,x_k)$ or $(x_k,b)$, depending on which one contains the sign-change.

Newton’s method applied to $x_k$ may give an $x_{k+1}$ that is outside $[a,b]$.

• If $x_{k+1}$ is inside $[a,b]$, then take it as the next iterate.

• If $x_{k+1}$ is outside $[a,b]$, then apply a bisection step instead.