Using the pseudocodes provided in lecture, write functions that implement the false position $($FP$)$ and secant methods to find the roots of single equations.

Then use them! Find the first positive roots of the following two equations:

$f(x)=-185\mathrm{.}377-566\mathrm{.}734x+184\mathrm{.}162x^{{?}^{{?}^{?}}}2+40\mathrm{.}32x^{{?}^{{?}^{?}}}3-15\mathrm{.}6x^{{?}^{{?}^{?}}}4+x^{{?}^{{?}^{?}}}5$

$g(x)=x-tan(\frac{x}{3})+1$

For both equations, use the interval $3,4\mathrm{.}5$ for FP and the starting point $[3]$ for Secant. Does the single starting point provide a positive root for secant in each case? If not, explain why and try the starting point $[4].$

Provide the number of iterations performed to reach a root in all four cases $($each equation, each method$).$ Comment on the counts required for each method.

Automated Bounds Finding

Using the pseudocode provided, write a wrapper for FP that, when given the single starting point $[3],$ will find bounds for FP$,$ then find the root of $f(x)$ using that method. Provide the number of iterative steps to find the bounds $($i$.$e$.$ how many times you "adjusted" a and$/$or b before you bounded the root$),$ and what the bounds provided to FP were.

NOTE! Some of the guesses WILL give you negative roots or diverge! This is intentional. As I write above, "describe what happens for each starting guess".

Multiple Root Problems

Use the MATLAB function fsolve $G$ to solve the $2-$dimensional root problem

$y^2=-x^2+34$

$y=x^2-6$

The steps for this are:

Develop a "zeros" function for the two equations as shown in the example problem for fsolve. Note that you will have to use a single vector to pass $x$ and $y,$ that is$,$ rename these as $x=[x_1,x_2]$

Use an initial guess of $x0=[2,2]$

Call fsolve with a handle to the function you created and the initial guess.

Note that you can $($fairly easily$)$ get an idea where the true roots are by graphing the functions.