Mathematical Analysis of Recursive Algorithm.

Mathematical Analysis of Recursive Algorithm.

• 1.
  Mathematical analysis of recursivealgorithm Bharati. C M.tech
• 2.
  General plans foranalysis of recursive algorithm 1) decide the input size. 2) identify the basic operation. 3) check how many time the basic operation performed and determine the best worst and average case. 4) set up a recursive relation with some initial condition expressing the basic operation. 5)solve recursion with using forward and backward substitution and correct formula using mathematical induction.
• 3.
  Algorithm: factorial(n) if(n==0) return 1 else returnfact(n-1)*n
• 4.
  Mathematical analysis: Step 1:input size is n step 2 :basic operation is multiplication step 3 :recursive function is F(n)=F(n-1)*n where n>0 Step 4: multiplication is given as M(n)=M(n-1)+1
• 5.
  Step 4: Forwordssubstitution: M(n)=M(n-1)+1 M(0)=0 n=0 M(1)=M(1-1)+1=1.............. n=1 M(2)=M(2-1)+1=2.................n=2 M(3)=M(3-1)+1=3
• 6.
  Backward substitution: M(n)=M(n-1)+1 M(n)=[M(n-2)+1]+1 =M(n-2)+2 M(n)=[M(n-3)+1]+2 =M(n-3)+3 M(n)=M(n-i)+i
• 7.
  Now we correctthe formula Prove: M(n)=n by using mathematical induction Let n=0 M(n)=0. M(0)=0=n Induction: M(n-1)=n-1 M(n)=M(n-1)+1 =n-1+1 =n M(n)=n Time complexity. (n)
• 8.
  Thank you