Additional Problems

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

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?

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?

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 1 1. if ( even(m)) then d = 2 else 1.1 d = 3 1.2 do while ((m mod d /= 0 ) and (d < sqrt(m))) 1.2.1 d = d + 2 2. if (m mod d /= 0) then d = 1 end algorithm

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?

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)

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

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

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

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.

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?

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

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

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

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