Fibonacci

1. Fibonacci Problem Solving and Thinking in Engineering Programming H. James de St. Germain

2. 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)

3. What is the Requirements • High Level English Description (or Pseudocode Version 1) • Calculate and Display the first ‘X’ Fibonacci Numbers

4. 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

5. Do it by Hand! 0  add the first number 1  to the second number 1  to get the next number Now What?

6. Do it by Hand! 0 1  now add this number 1  to this number 2  to get the next number Now What?

7. Do it by Hand! 0 1 1  now add this number 2  to this number 3  to get the next number Now What?

8. Do it by Hand! 0 1 1 2  now add this number 3  to this number 5  to get the next number Now What?

9. Do it by Hand! 0 1 1 2 3  now add this number 5  to this number 8  to get the next number Now What?

10. What does the Programneed to know at Each step? 0 1 1 2 3  now add this number 5  to this number 8  to get the next number • the previous number • the number before that • the current number

11. What happens at each step? • Pseudocode Version 2: • set the first number to 0 • set the second number to 1 • Add previous two numbers together to get current number • repeat step 3 until done • Question: • Are the “last two numbers” always the same?

12. Transform Repeat to While • Add previous two numbers together to get current number • repeat step 1 until done • while not done • Add previous two numbers together to get current number end

13. 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

14. Create Variables for our Program • second_previous = 0; • previous = 1; • current_number = ???? • current_number = second_previous + … previous;

15. What happens at each step? • Add previous and 2nd previous numbers to get the current Fibonacci number • 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?

16. 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;

17. 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

18. 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

19. Pseudocode ( 3rd Version) • print “0,1”: • set the first two values to 0 and 1 • While we haven’t reached our goal • add these values to get the next (or current) value • print the current value: • update the previous two values

20. Onward to Code • fprintf(“0, 1, “); • second_previous = 0 • previous = 1 • current = previous + second_previous; • fprintf(“%d, ”, current);

21. 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);

22. 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);

23. 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);

24. 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!

25. While Loop • while ( current < some large number) • Use a while loop because we want all Fibonacci numbers less than some number

26. FOR loop • for ith_fib_number = 3:1000 • Use a for loop because we want the first 1000 Fibonacci numbers

27. 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

28. 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

29. Thoughts • Is the variable I used in the loop? • Nope! Its just a place holder. for I = 3:total_fib_numbers current = second_previous + previous; fprintf(‘%d ‘, current); second_previous = previous; previous = current; end % the for loop

30. 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?

31. 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

32. New Code with Array % Pre-allocate (save buckets for) % enough space for all the numbers fib_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;

33. 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?

34. 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

35. Let me Repeat! • NEVER use: array = 5 + 6; • ALWAYS use: array( position ) = 5 + 6; You must always “index” into an array!

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

37. Draw a Black Box • You have 1 minutes to draw a black box for this function

38. Function as Black Box Function Input Output

39. Fibonacci as Black Box Fibonacci Fibonacci Numbers Count Array of Numbers Integer Compute the first “count” fibonacci numbers

40. 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

41. 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

42. Function Design Pattern • You have one minute to write the function design pattern for this function

43. Function Design Pattern function result_array = compute_fib( how_many ) result_array(1) = 0; end %function

44. 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

45. Function Code 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 loop end % function

46. How many… • 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 loop end % function

47. How many… • 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 loop end % function

48. 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

49. How many… • 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 loop end % function

50. End Fibonacci • Questions?