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\centerline{{\it Class 11}: {\bf Monday September 28}}

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\item[SUMMARY] Introduction to Root Finding
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\item[READING] Burden \& Faires, 47--54
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{\bf Example}\\
Consider a ball constructed of wood which has a density of $\rho = 0.638$ grams per cubic cm and the radius is $r=10$ cm. How much of the ball will be submerged when it is in water (with unit density)?

$M_w$ = Mass of water displaced = $\displaystyle \int_0^d \pi (r^2 - (x-r)^2) dx$\\

$M_b$ = Mass of ball = $4 \pi r^3 \rho/3$

What's the equation which must be solved to find $d$, the distance below the surface the ball will float? (Produce an equation for $d$ of the form $f(d)=0$ with $d$ being the only letter present.)

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\underline{\bf Question}\\
How would you solve this equation for $d$?
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{\large\bf Root-Finding}\\
We will be looking at algorithms for the solution of equations of one variable, i.e. equations of the form $f(x) = 0$. This is often referred to as finding the {\bf roots} of the equation $f(x)=0$ or finding the {\bf zeroes} of the function $f(x)$.\\

{\large\bf Bracketing The Root}\\
How do we know where the roots of a function $f(x)$ are? How can we ``bracket'' a zero of $f(x)$?
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\underline{\sc Groupwork}\\
The \Matlab function {\bf brackplo} will do this for us. Go to the computers and run {\tt brackplo} on the function you need to find zeroes of to find $d$. I have made a function called {\bf sphere.m} which you can use to help you. What do you see? How many roots are there? What range did you ask {\tt brackplo} to search on?
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{\large\bf The Bisection Method of Bolzano}\\
The bisection algorithm produces a sequence of approximations $\lbrace p_n \rbrace$ to the zero of the function $f(x)$\\
where ${\displaystyle p_n = a_n + \frac{b_n-a_n}{2} = \frac{a_n + b_n}{2}}$
and the $n$-th bracket is described by $[a_n,b_n]$

Write down the Bisection Algorithm in pseudocode here:
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{\large\bf bisect.m}\\
In the NMM Toolbox, we have an implementation of the bisection algorithm in {\bf bisect.m}. Use \matlab to find the value of $d$ which we have been looking for which tells us how much of the pine sphere is submerged.\\

$d = $

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