====== Lab 7: Solving nonlinear equations with Newton search ======

In this lab you will
  * write your first numerical algorithm
  * learn the most widely-used algorithm for solving nonlinear equations, and which underlies Matlab's ''fsolve'' function
  * solve a practical, real-world problem involving a nonlinear equation
  * gain more experience in programming with ''for'' loops and ''if'' statements


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** Problem 1:**  Write a function ''x = newtonsearch(f, xguess)'' that finds the solution 
''x'' of the equation ''f(x) = 0'' for an input function ''f'' and an initial guess 
''xguess'' using the Newton search algorithm. 

  -  Use a ''for'' loop to perform the Newton-search iteration $x_{n+1} = x_n + \Delta x$. Take up to ten Newton steps.
  -  Use an ''if'' statement inside the ''for'' loop to test if either $|f(x)| < \epsilon$ 
or $|\Delta x| < \epsilon$ for some specified tolerance $\epsilon$. If either condition is true, 
use a ''break'' statement to terminate the iteration and return from the function. For our 
purposes $\epsilon = 10^{-7}$ is a decent choice. 

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**Problem 2:** Test your Newton-search algorithm by solving the following problems. Check your 
answers by plugging the answer ''x'' back into ''f'' and verifying that ''f(x)'' is approximately 
zero.

(a) Find an $x$ for which $x^3 - 7x - 13 = 0$.

(b) Find the cube root of 72 by devising and solving an equation of the form $f(x) = 0$ whose 
solution is $x = \sqrt[3]{72}$. Is there a simpler way to calculate $\sqrt[3]{72}$ in Matlab?
Do that, and compare your answers.

%%(c)%% Find an $x$ for which $\sqrt{4-x^2} = x \tan x$. 

Hint: find good initial guesses for the Newton search by plotting each function and roughly 
estimating an $x$ position at which $f(x)$ is zero. 

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**Problem 3:** Use your Newton-search algorithm to solve the following problem. Utility 
companies must avoid freezing water mains in cold weather. If we assume uniform soil 
conditions, the temperature $T(x,t)$ at distance $x$ below the surface and time $t$ 
after the beginning of a cold spell is given approximately by

\begin{eqnarray*}
\frac{T(x,t) - T_s}{T_i-T_s} = \text{erf} \left(\frac{x}{\sqrt{2 \alpha t}}\right)
\end{eqnarray*}

where
 * $T_s$ is the constant surface temperature during the cold spell,
 * $T_i$ is the initial soil temperature before the cold spell started,
 * $\alpha$ is the thermal conductivity of the soil, and 
 * $\erf$ is the {\it error function}, computed by the built-in Matlab function ''erf''.

If $x$ is in meters and $t$ is in seconds, the $\alpha = 0.138 \times 10^{-6} \; m^2/s$.
Let $T_i= 20\,C$ and $T_S = -15\,C$, and recall that water freezes at $T = 0\,C$. 
Use your Newton-search algorithm to determine how deep a water main must be 
buried so that it will not freeze until at least 60 days' exposure to these 
conditions.