In this lab you will
fsolve
functionfor
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.
for
loop to perform the Newton-search iteration . Take up to ten Newton steps.if
statement inside the for
loop to test if either
or for some specified tolerance . If either condition is true,
use a break
statement to terminate the iteration and return from the function. For our
purposes 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 for which .
(b) Find the cube root of 72 by devising and solving an equation of the form whose solution is . Is there a simpler way to calculate in Matlab? Do that, and compare your answers.
(c) Find an for which .
Hint: find good initial guesses for the Newton search by plotting each function and roughly estimating an position at which 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 at distance below the surface and time after the beginning of a cold spell is given approximately by
where
* is the constant surface temperature during the cold spell,
* is the initial soil temperature before the cold spell started,
* is the thermal conductivity of the soil, and
* is the {\it error function}, computed by the built-in Matlab function erf
.
If is in meters and is in seconds, the . Let and , and recall that water freezes at . 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.