Lecture 5

Warm up exercise

• Write a for loop that calculates the first 20 Fibonacci numbers. Place result in a vector Fibo, so that Fibo(n) is the nth number. Recall that the sequence satisfies:

F(1)=1

F(2)=1

for n>2

F(n)=F(n-1)+F(n-2) 

 

while Statements

Sometime, you need to iterate many times but don't really want to create a vector, or you don't know in advance how many time you need to iterate...for this you can use a while statement.

The syntax for a while statement is

while expression

    a 

    bunch

    of

    statements

end

At the beginning of each iteration, Matlab evaluates the expression at the top of the loop. if it is true, Matlab executes the statements to the end. If it is false, Matlab jumps to the end 

and continues after it.

So, for example:

 

>> x=10;

>> while x>0

x=x-1

end

>>

 

will print out all the numbers from 9 to 0 (including both)  

 

Take note of two things: 

 

• Matlab does not check the expression all the time. Only before starting a new iteration
• If the expression at the top is false when the first iteration should start, it will not. 

Exercises

• Find the value of the 100th Fibonacci number, using a while loop, and no big vectors.
• Write your prime checking program using a while loop instead of a for loop.
• Write a while loop that modifies n. if n is even it changes n to n/2, otherwise, it changes it to 3n+1. if n is 1 the loop stops. Try this out for several different starting values. Count how many steps you need to get to 1.
  

 

Defining your own function

While Matlab has a vast collection of functions, sometimes you want to add some of your own. A function should be thought of as a black box, with a slot in the top (for the input) and a drawer at the bottom for the output. When you give the function input ("call the function"), it calculates the output and puts it in the drawer ("returns it"). Once the function is written, you do not need to care about how it works.

 

Functions must be defined in their own file, and the name of the file must match the name of the function. So, for example, to define a function called example_function_1, create a file named example_function_1.m. That file must start with the keyword function, so

 

function y = example_function_1(x)

 

is the first line of the file. let's dissect this first line:

function	As I said, the first word must be "function".
y =	This declares the local name of a variable whose value will be returned when the execution of the function is over.
example_function_1	The name of the function. This must match the name of the file.
(x)	The local name of the variable that will contain the input value.

So the rest of the file could be 

 

y=2*x;

end

 

admittedly, a very boring function: It takes the input value (locally called x), doubles it, and assigns that result to y. Since y is declared to be the return value and the function has 

ended, the return value is simply twice the input.

 

we call the function by giving it a value:

 

>> example_function_1(17)

 

or

 

>> example_function_1(rand)

 

The return value can then be assigned to another variable:

 

>> z= example_function_1(18);

>> z 

 

Variable Scope

An important difference between a function and a script (the files that don't have "function" as their first word are called scripts) is that of "variable scope".

When executing a script from the command-line, the scripts has access to all the variables that are defined on the command line. For instance, if you have a script called script1.m whose contents are 

x+1

y=y/2

and you run this script from the commandline like so:

>> x=2

>> y = 6

>> script1

>> y 

You will see that the script knows that x=2 and that y=6. Not only this, note that y is changed in the scripts, the new value of y, 3 is also the value of y when check on the command

line. This seems natural, perhaps, but this is not how functions behave. A function with the same content function1.m

function function1

x+1

y=y/2

will fail to run, since within the function there are no such variables x and y. The variables of the function, are different from the variables outside the function, even if they have 

the same name. 

• change your prime checking program so that the input is the number, and the output is 1 if the number is prime and 0 if not.
• change your n-changing program, so that the input is the initial value of n and the output is the number of steps it took to reach 1.
• use your primechecking function in another program (can be a script) that finds (a) the 1000th prime number and (b) the first prime number greater than 1000000.
• use your n-changing function in another program (can be a script) that finds the number of iterations needed for the first 100 numbers, and plots then nicely.