Design for Generative Recursion, and The Cost of Computation

1
Design for Generative Recursion,and The Cost of
Computation

• Or,
• A gentle introduction to numerical methods and
  complexity theory

2
Binary Search

• Example from applied math
• Problem Given f, find r s.t. f(r)0
• Approach "Binary Search". We're given
• rleft lt r ltrright
• f is monotonic, continuous
• "No need for exact answer"
• "Just get us a range Range wide"
• Note Will we need all these assumptions?

3
Binary Search

• (define (find-root f left right)
• (cond
• (lt (- right left) Range) left
• else
• (local ((define mid (/ ( left right) 2)))
  
• (cond (lt (f mid) 0 (f right))
• (find-root mid right)
• else (find-root left mid)))))

4
Termination Argument

• A good argument for termination is to say exactly
  how many steps are needed to terminate. This
  computation terminates in
• log_2 (Difference/Range)
• where Difference is the difference between left
  and right bounds.

5
Time Complexity

• add m n (on naturals) m recursive calls
• append l_m l_n m recursive calls
• Cross l_m l_n mn recursive calls
• map f l_n n recursive calls, and calls to f
• find-root ? recursive calls
• insert l_n upto n recursive calls to insert
• insert-sort l_n upto nn/2 calls to insert
• maxi l_n ?