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

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.

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.

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.

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.