The document provides an explanation of recursion using the example of calculating factorials. Recursion involves a function calling itself, with each call depending on the result of the next call with a smaller input. This continues recursively until a stopping condition is reached. Then results are passed back up to calculate the final output. The key aspects are that a function calls itself, results depend on subsequent recursive calls, and a stopping condition is needed to avoid infinite looping.

1. Recursion101 Simple explanations for AS F452

2. Recursive routine It calls itself. Repetitive problems simplified. Must have an exit case to avoid endless loop.

3. Non-Geek explanation A recursive problem could be represented by the dolls. The same problem is used by each but it depends on working it the answer from the next smaller. When you have the answer to the calculation of the smallest you can work out the rest by going back up through to the largest doll. Russian Dolls

4. An example Factorial in mathematics The result of successively multiplying an integer in reductions of 1 Indicated by an exclamation mark (!) Example 5! = 5 * 4 * 3 * 2 * 1 = 120 The function is very useful in statistics See an example here

5. Rewriting the Factorial 5! = 5 * 4 * 3 * 2 * 1 = 120 We can rewrite the section 4 * 3 * 2 * 1 as 4! It’s Factorial 4

6. Rewriting the Factorial These are all equivalent 5! = 5 * 4 * 3 * 2 * 1 (120)  5 * 4! (24)  5 * 4 * 3! (6)  5 * 4 * 3 * 2! (2)  5 * 4 * 3 * 2 * 1! (1)  5 * 4 * 3 * 2 * 1 * 0! (1)

7. Expressing in code Using pseudo code (modelled on Visual Basic) Create a function to return the outcome: Function Factorial (n as integer) as Integer Function identifier = Factorial. That’s the name that will be called Parametern is the input value (suppliedas Integer) The ‘as Integer’ defines the type to be returned

8. Executing the code The code runs normally until: It meets the call to itself The code pauses and a new call is started The parameter passed acts as Stopping Condition when it reaches a set value

9. Executing continued When stopping condition reached The value is calculated Control is passed back to the previous call This is done successively back to the original calls, all stored on The Stack

10. Pseudo Code (1) Function Factorial (n as Integer) as Integer Return n * Factorial (n-1) End Function n is the variable holding the values If 5 was the value passed it would be Return 5 * Factorial (4)

11. Pseudo code (2) 01 Function Factorial (n as Integer) as Integer 02 If n = 1 then 03 Return 1 04 Else 05 Return n * Factorial (n-1) 06End if 07 End function RECURSIVE CALL IS LINE 05

12. Step-by-step Example We trace the calculation of 3! -> Factorial 3 We’ll follow the execution of the code Concentrating on the recursive call (yellow highlight)

13. Execution (1) Imagine your program analyses sets of data At some stage in your code: you need to calculate 3! (Factorial 3) This is 3 * 2 * 1 and evaluates to 6 You want to assign it to variable intAnswer The function is called by: intAnswer = Factorial(3) intAnswer is assigned the outcome of Factorial(3)

14. Execution (2) of Factorial(3) n = 3 Line 02 is FALSE Jumps to Line 05 Return means ‘calculate’ Has to calculate: 3 * Factorial(3-1) = 3 * Factorial(2) That’s the recursive call!

15. Execution (3) of Factorial(3) We got to 3 * Factorial(2) To work out Factorial(2) Returns to Line 02 n = 2 -> Line 02 = False Jump to Line 05 Return 2 * Factorial(2-1) = 2 * Factorial(1) Another recursive call

16. Execution (4) of Factorial(3) We got to 2 * Factorial(1) To work out Factorial(1) Returns to Line 02 n = 2 -> Line 02 = True Function returns value = 1 STOPPING CONDITION REACHED It can now go back through the stack

17. Execution (5) of Factorial(3) Working back through the stack Factorial(1) = 1 Factorial(2) -> 2 * Factorial(1) -> 2*1 = 2 Factorial(3) -> 3 * Factorial(2) -> 3 * 2 = 6 Factorial(3) execution is complete Variable assignment in the code can complete intAnswer = Factorial(3) = 6

18. Essential nature This is another form of iteration loop It is sometimes thought as ’Elegant’ The next slide compares iteration and recursion This is where the higher marks lie Moving past straight facts and evaluating

19. Factorial as iterative WHILE loop Function Factorial (n as Integer) as integer Dim intTemp as integer intTemp = 1 WHILE n > 1 intTemp = intTemp * n n = n -1 LOOP RETURN intTemp End function

20. Compare Iteration/Recursion

21. The essential bits It calls itself Calls depend on the one below Needs a stopping condition When next call met pause and start new call Progressive until stopping condition Control passed back ‘up-the-chain’ Every recursion can be written as an iteration loop Remember the comparisons for exam You should be able to identify recursive use