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gibson:teaching:spring-2018:math445:lab7

# Lab 7: Nonlinear equations and Newton's method

In this lab you will

• write your first numerical algorithm
• learn the most widely-used algorithm for solving nonlinear equations
• 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.

1. Use a for loop to perform the Newton-search iteration . Take up to ten Newton-search iterations.

2. Use an if statement inside the for loop to test if either or for some specified tolerance .

3. If either of those conditions 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 error function, computed by the built-in Matlab function erf.

If is in meters and is in seconds, the thermal conductivity of soil is . 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. 