1. Additional Problems

2. Smallest Divisor of an integer
Problem: Given an integer m, write a program that will find its smallest exact divisor other than 1.
Is the problem clear and well-defined?
what is a divisor?
For any number m, there are divisors
There are at least two divisors
All divisors of m can be totally ordered
Can m be 0?
Can it be negative?
What if m is prime?
Assume: m > 0
Assert: if m is prime, return 1 else the smallest divisor other than 1

3. Towards a strategy All divisors of m are between 1 and m Strategy 1:
Enumerate all divisors of m
choose the smallest one
Better Strategy:
Starting from 2, check one by one whether it is a divisor of m
If one is found before m is reached then return.
Otherwise, return 1.
How to check a divisor?
Can we do better?

4. Towards a Better Strategy
The above inspects (m-1) numbers, in the worst case
for even numbers, the answer is 2
for odd numbers, no need to check an even number
How to check whether a number is even or not?
Do we have go up to m?

5. The Algorithm
Algorithm st_divisor Input m Assume: m > 0 Output d Assert: if m is prime, d is 1; otherwise,
d is the smallest divisor other than if ( even(m)) then d = 2 else d = do while ((m mod d /= 0 ) and (d < sqrt(m)))
d = d if (m mod d /= 0) then d = 1 end algorithm

6. Observation How to compute Square roots?
Better to remove the square root computation before the loop
Look for removing computations inside loops
Note m mod d can not be removed
Is the algorithm correct?
Does it terminate?

7. Loop invariant The invariant condition is: Bound Function: (m – d)
m is odd, d is odd, 1 < d <= sqroot(m) + 1
for no x: 1 < x < d: x is a divisor of m
Bound Function:
(m – d)

8. Improvements How many iterations? Is there a better algorithm?
In the worst case, the largest integer less than or equal to sqroot(m)/2
0, if m is even
Is there a better algorithm?
A lot of redundant checks are still made
Look for prime divisors

9. The square root problem
Problem: Write a program that computes the square root of a given number.
Is the problem definition clear?
If 25 is the input, then 5 is the output
If 81 is the input, then 9 is the output
If 42 is the input, then ?
For non perfect squares, the square root is a real number
So the output should be close to the real square root
How close? to a given accuracy

10. A more precise specification
Problem: Write a program that given a number m outputs a real value r s.t. r*r differs from m by a given accuracy value e
More precisely, the program outputs r s.t.
|r*r - m| < e

11. Solution Strategy ... Guess and Correct Strategy:
Choose an initial guess r less than m
If r*r > m then keep decreasing r by 1 until r*r is less than or equal to m.
If r*r < m then keep increasing r by 0.1, … until r*r exceeds or equals m
If r*r > m then decrease r by 0.01 until r*r exceeds or equals m.
...
Terminate the computation when r*r equals m or differs from m by a given small number.

12. Idea Number of iteration depends upon the initial guess
If m is 10,00,000 and the initial guess is 300 then over 700 steps are needed
Can we have a better strategy?

13. Towards a better strategy
The basic idea of the strategy is to obtain a series of guesses that
falls on either side of the actual value
narrows down closer and closer
To make the guess fall on either side
increase/decrease the guess systematically
To narrow the guess
the amount of increase/decrease is reduced
Improving the strategy
faster ways of obtaining new guess from the old one

14. One Strategy Given a guess a for square root of m
m/a falls on the opposite side
(a + m/a)/2, can be the next guess
This gives rise to the following solution
start with an arbitrary guess, r_0
generate new guesses r_1, r_2, etc by using the averaging formula.
When to terminate?
when the successive guesses differ by a given small number

15. The algorithm Algorithm Sq_root Input m, e: real
assume: m>0, 1 > e > 0
Output r1, r2: real
assert:: |(r2 * r2 - m) | <= |(r1 * r1 - m)|, |r1 - r2| < e   
1. r1 = m/2
2. r2 = r1
3. Do while (|r1 - r2| > e) steps 3.1 and 3.2
3.1 r1 = r2
3.2 r2 = (r1+m/r1)/2
end Algorithm

16. Analysis of the algorithm
Is it correct? Find the loop invariant and bound function
Can the algorithm be improved?
More general techniques available
Numerical analysis