One dimensional root finding has many applications, and a common situation is called continuation,

where a sequence of $($related$)$ roots are to be solved for. An elementary example of a continuation

is the numerical construction of a contour $($or level curve$)$ of a $2$D function. A contour plot is a $2$D

representation of a function of two variables, f$($x$,$ y$)$ that plots all curves of the form f$($x$,$ y$)=$ c$,$

for various constants c$.$ Figure $1($a$)$ shows a colour plot of the HI$($x$,$ y$)$ function, including the

zero contour $($HI$($x$,$ y$)=0$ is the white dashed curve$).$ While it may seem that finding solutions

to HI$($x$,$ y$)=0$ is a $2$D problem, there is a way to apply the $1$D methods presented in class.

Specifically, given a point $($xn$,$ yn$)$ on the level curve we seek the next point

xn$+1=$ xn $+$ h cos$(\backslash$theta n$),$ yn$+1=$ yn $+$ h sin$(\backslash$theta n$),$

at a fixed distance h $>0$ and some angle $\backslash$theta satisfying

f$($xn$+1,$ yn$+1)=$ f$($xn $+$ h cos$(\backslash$theta n$),$ yn $+$ h sin$(\backslash$theta n$))=0.$

This is a $1$D root$-$finding problem in the variable $\backslash$theta n$!$ The parameter h is referred to as the step

size, and is a numerical constant. It will determine the accuracy of the computed contour. The

MATLAB script CA$3$ demo.m implements the above procedure using MATLAB$$s fzero command.

For illustration purposes it use a large stepsize value h $=0\mathrm{.}6$ and produces a $24-$point contour that

is$,$ unsurprisingly, a rather poor approximation to the true level curve $($Figure $1,$ red curve$).$

An illustration for our procedure is in Figure $1($b$)$ where the point P$3$ on the contour $($thick blue

curve$)$ has just been located at the angle $\backslash$theta $2.$ Note that there is actually a second point on the

$($green$)$ circle around P$2,$ but it is the previous point P$1!$ Hence, special care needs to be taken

that our root$-$finding algorithm doesn$$t reverse itself. We can now repeat the procedure on the

$($magenta$)$ circle around P$3$ to iterate the search procedure to find P$4.$

But how can we find the first point? We can use the same angle solve procedure provided that we

begin from a point P$0$ that only needs to be within a distance h from our contour. Note in the

illustration, P$0$ is not on the contour, but the general procedure still works. For this assignment,

you can manually set P$0$ as an $$eyeballed$$ point that sits near your contour.

$1$

$($a$)($b$)$

Figure $1$: $($a$)$ The function HI$($x$,$ y$)$ and the trace determined by MATLAB$$s fzero command. $($b$)$

The root$-$finding algorithm illustrated.

The goal of this assignment is to compare the secant method to the MATLAB$$s fzero, in terms

of efficiency, and robustness $($use tic, toc for the former$)$ when computing the zero contour of $2$D

functions. Begin by downloading CA$3$ demo.m$,$ a code that does the root$-$finding using Matlab$$s

fzero command $$ you will need to replace fzero calls inside the contour finding loop. Make sure

you download secant.m from Canvas to help you modify CA$3$ demo.m$.$

The secant method needs an initial interval $$ for this you will need to think about the geometry.

It turns out that the $$right$$ interval for finding the angle $\backslash$theta n is $[\backslash$theta n $\backslash$delta $,\backslash$theta n $+\backslash$delta $],$ where $\backslash$delta gives

HI$($x$(\backslash$theta n$),$ y$(\backslash$theta n$))$ opposite signs at the endpoints. You should start by setting $\backslash$delta $=\backslash$pi $/2,$ but you

will be required to experiment with this parameter, as well as the h parameter. Consider all of the

following when completing your report:

$$ Start by trying to reproduce Figure $1$ using the secant method, with h $=0\mathrm{.}6$ and $\backslash$delta $=\backslash$pi $/2.$

What happens?

$$ Test your root finder for a variety of $($h$,\backslash$delta $)$ values to determine one where you can get all the

way around the curve. How many steps do you need to get around the curve.

$$ What kind of convergence failures do you observe in your experiment? Offer explanations for

the failures.

$$ How do these calculations compare against MATLAB$$s fzero?