While loops

11.2. While loops#

For loops are useful for when we know how many times we want to repeat a set of commands. Sometimes we might not know this and would like to loop until a certain condition is met. This is where while loops come in. The MATLAB syntax for a while loop is

while condition
    % commands
end

The lines that are betwee the while loop declaration and end will be executed while the condition returns a true result. For example, to print the numbers 0 to 4 as we did with a for loop add the following code into your program.

% While loops
i = 1;

fprintf("\n")
while i < 6
    fprintf("%d\n", i)
    i = i + 1;
end

With while loops we don’t have a loop variable so we have defined the variable i and set it equal to 1. Then we have declared a while loop to loop whilst i is less than 6. Inside the while loop we use a fprintf() command to print the current value of i before incrementing it by 1. Run your program and you should see the following added to the console output.

Warning

A common programming error when dealing with while loops is forgetting to update the while loop condition. This can result in the condition always being true and the commands in the while loop being executed forever. This is known as an infinite loop.

To demonstrate this, delete or comment out the i = i + 1 command in the while loop you have just written and rerun the program. You should set lots of 1s being printed to the console since the i variable is always 1 which is of course less than 5. To stop the program click on the stop icon $\blacksquare$.

11.2.1. Approximating the Golden ratio using Fibonacci numbers#

The golden ratio is the ratio between two numbers, $a$ and $b$, where $a+b$ is to $b$ as $b$ is to $a$ and is denoted by $\varphi$. The actual value of $\varphi$ is

\[ \varphi = \frac{1 + \sqrt{5}}{2} = 1.618034\ldots\]

A property of the Fibonacci sequence is that the ratio between two successive numbers in the sequence, $F_{n} / F_{n-1}$, converges to $\varphi$ as $n$ gets large (note that this does not work for the first two numbers since $F_0 = 0$ and we cannot divide by zero), for example

\[ \begin{align*} \frac{F_2}{F_1} &= \frac{1}{1} = 1, & \frac{F_3}{F_2} &= \frac{2}{1} = 2, & \frac{F_4}{F_3} &= \frac{3}{2} = 1.5, & \frac{F_5}{F_4} &= \frac{5}{3} = 1.\dot{6}, & &\ldots \end{align*} \]

Lets use a while loop to calculate the approximation of $\varphi$ until two successive approximations are within $10^{-8}$ of each other. Enter the following code into your program.

% Approximating the golden ratio
a = 1;
b = 1;
new_phi = 1;
old_phi = 0;
fprintf("\n%0.6f", new_phi)

while abs(new_phi - old_phi) > 1e-6
    c = a + b;
    a = b;
    b = c;
    old_phi = new_phi;
    new_phi = b / a;
    fprintf("%0.6f\n", new_phi)
end

Here we start by declaring a and b to be the second and third Fibonacci numbers (we can’t use the first Fibonacci number as that would mean dividing by zero) and declare two variables new_phi and old_phi which contain the successive approximations of $\varphi$. We then use a while loop to repeat calculations until the absolute difference between new_phi and old_phi is less than $10^{-6}$. Inside the while loop we calculate the next Fibonacci number, update the approximations of $\varphi$ and print out the current approximation.

Run your program and you should see the following added to the console.

So our program finishes with an approximation of $\varphi \approx 1.618034$ which agrees with the actual value correct to 6 decimal places. Using while loops to iterate a solution until convergence is a common technique used in numerical methods.

11.2.2. Exercises#

Exercise 11.4

Use a while loop to print “hello again” ten times.

xkcd Collatz cartoon — Fig. 11.1 https://xkcd.com/710#

Exercise 11.5

The Collatz conjecture states that the series generated by the following rules will eventually reach 1.

\[\begin{split} x_{n+1} = \begin{cases} x_n/2 & \textsf{if $x_n$ is even}, \\ 3x_n + 1 & \textsf{if $x_n$ is odd}. \end{cases} \end{split}\]

For example, for $x_0 = 10$ the sequence is

\[ \begin{align*} 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1 \end{align*} \]

and it took 6 steps to reach 1. Write a program that uses a while loop to print the numbers in the Collatz sequence for a given starting number. How many steps does it take for a starting number of 100 to reach 1?

While loops

Contents

11.2. While loops#

11.2.1. Approximating the Golden ratio using Fibonacci numbers#

11.2.2. Exercises#