Section 3.4

• 4.

  procedure max(a1,a2,...,an : integers)
  if n = 1 then max(a1,a2,...,an) := a1
  else
  begin
    m := max(a1,a2,...,an-1)
    if m > an then max(a1,a2,...,an) := m
    else max(a1,a2,...,an) := an
  end
  

• 6.

  procedure power(x,n,m: positive integers)
  if n = 1 then power(x,n,m) := x mod m
  else power(x,n,m) := (x power(x,n-1,m)) mod m
  

• 8. Input is a list a1,...,an of integers, call it L. If n = 1, then the output is that the mode is a1 and it appears 1 time. For n > 1, form a new list L' by deleting from L the term an and all terms in L equal to an. Let k be the number of terms deleted. If k = n, then the output is that the mode is an and it appears n times. Otherwise, apply the algorithm recursively to L', obtaining a mode m, which appears t times. Now if t >= k, then the output is that the mode is m and it appears t times; otherwise the output is that the mode is an and it appears k times.
• 12.

  procedure fastpower(n: positive integer, a: real number)
  if n = 1 then fastpower(n,a) := a
  else if n is even then fastpower(n,a) := fastpower(n/2,a)^2
  else fastpower(n,a) := a fastpower((n-1)/2,a)^2
  

• 14. To compute f7, algorithm 7 requires f8 - 1 = 20 additions, and algorithm 8 requires 7 - 1 = 6 additions.
• 20. The iterative algorithm is much more efficient. If we compute with the recursive algorithm, we end up computing the small vlues over and over and over again. Try it for n = 5.