Problem statement: Given an array of numbers A=[x0,x1,...x(N-1)] Calculate the sum S(A) using recursion. Here I am deriving the steps and providing Algorithm

1. RECURSION
TECHNIQUES
IN PROGRAMING
By Shck Tchamna: tchamna@gmail.com

2. Problem statement
•Given an array of numbers
•Calculate the sum
 
0 1 1
, ,... N
A x x x 

  0 1 1
... N
S A x x x 
   
     
0 1
0 ; 1 ; , 0, 1
k
A x A x A k x k N
    

3. There are many ways to solve it!
Iterative
Recursive

4. A = [2,8,12,0]
Sum_array = 0
for i in range(len(A)):
Sum_array = Sum_array + A[i]
print(Sum_array)
Iterative

5. Recursive
All we need to find in a recursive formula are:
1. Generalization of the steps: The recurrence
definition
2. Base case: The simplest case of te probe

6.   1 0 1
|step
S A x x
 
     
2 0 1 2 1 2
| |
step step
S A x x x S A x
    
       
0 1 1
| ... |
stepk k k k
step k
S A x x x S A x
 
     
1
k k k
S S x

 
  0 1 1
... N
S A x x x 
   
 |stepk k
Let S A S
 

7.      
1
| |
stepk k
step k
S A S A x

 
1
k k k
S S x

 
So our recursive function depends on A and k
   
, , 1 k
f A k f A k x
  
 
k
x A k

     
, , 1
f A k f A k A k
  

8.      
, , 1
f A k f A k A k
  
Generalization
Base Case
What happen in the above formula when k=0
   
0 1
| ...
stepk k k
S A x x x

   
  0 0
|
S A x

Step k: sum from 0 to k
Step 0: sum from 0 to 0

9.      
, , 1
f A k f A k A k
  
Generalization
Base Case
   
0 0 0
| ,0
S A x f A x
  
def f(A,k):
if k ==0:
return A[0]
else:
return f(A,k-1) + A[k]

10. def f(A,k):
if k == 0:
return A[0]
return f(A,k-1) + A[k]
def f(A,k):
if k ==0:
return A[0]
else:
return f(A,k-1) + A[k]
# Test the code
A = [1, 5, 8, 5 -18]
N = len(A)-1 # we substract 1 because indexes start in python from 0 to len(A)-1
Sum_A = f(A,N)
print(Sum_A)
The above function computes the sum of the k first elements of the array A
If you want to calculate the sum of all the elements of the array, then let k = len(A)