Fibonacci

Fibonacci

Problem Solving and Thinking in Engineering Programming

H. James de St. Germain

Understand the Problem

• The Fibonacci Series is of interest and excitement to Mathematicians and Scientists.

• The Series is:– 0,1,1,2,3,5,8,13,21,34,55,89,…

• To calculate a Fibonacci Number simply add the two previous numbers together.

• We always start with zero and one (0 and 1)

What is the Requirements

• High Level English Description (or Pseudocode Version 1)

– Calculate and Display the first ‘X’ Fibonacci Numbers

Really Understand the Problem• Start with 0 and 1 (by definition)• Start of sequence is: 0,1• Add these two together: 1• Expanded sequence is: 0,1,1• Add last two numbers together

– 1+1 = 2• Expanded sequence is: 0,1,1,2• Add last two numbers together

– 1+2 = 3• Expanded sequence is: 0,1,1,2,3

Do it by Hand!

0 add the first number1 to the second number1 to get the next number

Now What?

Do it by Hand!

01 now add this number1 to this number2 to get the next number

Now What?

Do it by Hand!

011 now add this number2 to this number3 to get the next number

Now What?

Do it by Hand!0112 now add this number3 to this number5 to get the next number

Now What?

Do it by Hand!01123 now add this number5 to this number8 to get the next number

Now What?

What does the Programneed to know at Each step?

01123 now add this number5 to this number8 to get the next number

1. the previous number2. the number before that3. the current number

What happens at each step?• Pseudocode Version 2:

1. set the first number to 02. set the second number to 13. Add previous two numbers together to get current

number4. repeat step 3 until done

• Question:

– Are the “last two numbers” always the same?

Transform Repeat to While

1. Add previous two numbers together to get current number

2. repeat step 1 until done

1. while not done– Add previous two numbers together to get

current number

end

What informatino do we need to “know” or “compute” at Each Step?• 2nd Previous Number• Previous Number• Current Number

• We need VARIABLES to store each of these

Create Variables for our Program

• second_previous = 0;• previous = 1;• current_number = ????• current_number = second_previous + …

previous;

What happens at each step?

1. Add previous and 2nd previous numbers to get the current Fibonacci number

2. Then update our “previous” variables to contain the “new” previous numbers

– Question: Is the ordering of these two steps important?

– Is the ordering of the two operations in step 2 important?

Which of these produces the correct values in our variables?

• Now is it:– current = second_previous + previous;– previous = current;– second_previous = previous;

• Or is it:– current = second_previous + previous;– second_previous = previous;– previous = current;

Lets Confirm our Understanding:previous = 1, second_previous=1;

• Case 1:– current = second_previous + previous;– % current is assigned the value 2– previous = current;– % previous is assigned the value 2– second_previous = previous;– % second_previous is assigned the value 2

Lets Confirm our Understanding:previous = 1, second_previous=1;

Case 2:– current = second_previous + previous;– % current is assigned the value 2– second_previous = previous;– % 2nd previous is assigned the value 1– previous = current;– % previuos is assigned the value 2

Pseudocode ( 3rd Version)

1. print “0,1”:2. set the first two values to 0 and 13. While we haven’t reached our goal

1. add these values to get the next (or current) value

2. print the current value:3. update the previous two values

Onward to Code

• fprintf(“0, 1, “);• second_previous = 0• previous = 1• current = previous + second_previous;• fprintf(“%d, ”, current);

Sample Code

• second_previous = 0• previous = 1• current = previous + second_previous;• fprintf(“%d, ”, current);• second_previous = previous; • previous = current; • current = previous + second_previous;• fprintf(“%d, ”, current);

Sample Code• second_previous = 0• previous = 1• current = previous + second_previous;• fprintf(“%d, ”, current);• second_previous = previous; • previous = current; • current = previous + second_previous;• fprintf(“%d, ”, current);• second_previous = previous; • previous = current; • current = previous + second_previous;• fprintf(“%d, ”, current);

Sample Code• second_previous = 0• previous = 1• current = previous + second_previous;• fprintf(“%d, ”, current);• second_previous = previous; • previous = current; • current = previous + second_previous;• fprintf(“%d, ”, current);• second_previous = previous; • previous = current; • current = previous + second_previous;• fprintf(“%d, ”, current);• second_previous = previous; • previous = current; • current = previous + second_previous;• fprintf(“%d, ”, current);• second_previous = previous; • previous = current; • current = previous + second_previous;• fprintf(“%d, ”, current);• second_previous = previous; • previous = current; • current = previous + second_previous;• fprintf(“%d, ”, current);• second_previous = previous; • previous = current; • current = previous + second_previous;• fprintf(“%d, ”, current);

Seems like the same old same old, over and over and over

• This implies that we want a loop!

• Remember: A Loop lets the computer do things over and over again so we don’t have to!

• What loop to use?– For loop or While loop?– Give a valid reason to use either!

While Loop

• while ( current < some large number)

– Use a while loop because we want all Fibonacci numbers less than some number

FOR loop

• for ith_fib_number = 3:1000

– Use a for loop because we want the first 1000 Fibonacci numbers

Pseudocode (4th version)Very Close to Code

• Set second_previous to 0• Set previous to 1• Starting with 3, go until ‘X’ (by ones)

– Current value is set to second_previous + previous

– Print current value– Set second_previous to previous– Set previous to current

Code

second_previous = 0;previous = 1;fprintf(‘%d %d ‘, second_previous, previous);for I = 3:total_fib_numbers

current = second_previous + previous;fprintf(‘%d ‘, current);second_previous = previous;previous = current;

end % the for loop

Thoughts

• Is the variable I used in the loop?– Nope! Its just a place holder.

for I = 3:total_fib_numberscurrent = second_previous + previous;fprintf(‘%d ‘, current);second_previous = previous;previous = current;

end % the for loop

Thoughts

• Are we calculating anything?– Sort of, but when the program is over, does

the computer have anything it can use?• Nope

• How would we write code to save these values?– What data type?

Saving the values

• What would we do if we needed to save the values instead of simply printing them to the screen?

• Answer:– Use an Array– Note: now the variable I is important

New Code with Array

% Pre-allocate (save buckets for)% enough space for all the numbersfib_numbers = zeros(1,total_fib_numbers);

% Set up the first two fib numbers from memory% (your memory)fib_numbers(1) = 0;

fib_numbers(2) = 1;

New Code with Array

for i = 3:total_fib_numbers

fib_numbers(i) = fib_numbers(i-1) + …fib_numbers(i-2);

end % for

% where did the previous and % 2nd previous variables go?

What is wrong with this code?

fib_numbers = fib_numbers(i-1) + …fib_numbers(i-2);

Corrected:

fib_numbers(i) = fib_numbers(i-1) + …fib_numbers(i-2);

Notice the Update of the Array uses the “(i)” next to the array variable

Let me Repeat!

• NEVER use:array = 5 + 6;

• ALWAYS use:array( position ) = 5 + 6;

You must always “index” into an array!

Function

• How would we turn this code into a function?– What are the inputs?– What are the outputs?

Draw a Black Box

• You have 1 minutes to draw a black box for this function

Function as Black Box

FunctionInput Output

Fibonacci as Black Box

FibonacciCount

Fibonacci Numbers

Compute the first “count” fibonacci numbers

Array of Numbers

Integer

Comment Your Function

• You have 1 minute to write a brief comment that would go at the top of your .m file for the Fibonacci function

Function Comment

% array_of_fib_numbers = compute_fib(…% how_many)%% Author: H. James de St. Germain% Date: Fall 2007%% This function produces an array of the % first “how many” Fibonacci Numbers

Function Design Pattern

• You have one minute to write the function design pattern for this function

Function Design Pattern

function result_array = compute_fib( how_many )

result_array(1) = 0;

end %function

Function Code

• From your memory and your notes write out the code for this function.

• … you have 1 minute….• Pseudocode:

– set up first two values in array– loop updating the “current” value based on the

previous two values

Function Codefunction result_array = compute_fib( count ) result_array(1) = 0; result_array(2) = 1; for counter = 3 : count result_array(counter) = …

result_array(counter-1) + …result_array(counter-2);

end %for loopend % function

How many…1. semicolons (;s) in the function? Where?

function result_array = compute_fib( count ) result_array(1) = 0; result_array(2) = 1; for counter = 3 : count result_array(counter) = …

result_array(counter-1) + …result_array(counter-2);

end %for loopend % function

How many…1. fprintfs and input statements?

function result_array = compute_fib( count ) result_array(1) = 0; result_array(2) = 1; for counter = 3 : count result_array(counter) = …

result_array(counter-1) + …result_array(counter-2);

end % for loopend % function

NEVER…

• use fprintf in a function– unless told that the function “communicates”

with the user of the program• use input in a function

– unless told that the function “recieves” input from the user of the program

How many…1. variables? (parameters, return variables, local

variables)

function result_array = compute_fib( count ) result_array(1) = 0; result_array(2) = 1; for counter = 3 : count result_array(counter) = …

result_array(counter-1) + …result_array(counter-2);

end % for loopend % function

End Fibonacci

– Questions?