Topic Contents. Factoring Methods. Unit 3: Factoring Methods. Finding the square root of a number
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1 Topic Contents Factoring Methods Unit 3 The smallest divisor of an integer The GCD of two numbers Generating prime numbers Computing prime factors of an integer Generating pseudo random numbers Raising a number to a large power Problem Given a number m devise an algorithm to compute its square root. Algorithm Description Let us understand what is meant by square root of a number. Taking specific examples we know that the square root of 4 is 2, square root of 9 is 3 and the square root of 16 is 4 and so on. That is From these examples we can conclude that in general case the square root on n, of a another number m must satisfy the equation n x n = m (1) Suppose, for example, we do not know the square root of 36. We might guess that 9 could be its square root. Using equation (1) to check our guess we find that 9X 9 = 81 which is greater than 36. Our guess of 9 is too high so we might next try 8. For example, 8x 8 = 64 which is still greater than 36 but closer than our original guess. 1
2 The investigation we have made suggests that we could adopt the following systematic approach to solve the problem. Choose a number n less than the number m we want the square of. Square n and if it is greater than m decrease n by 1 and repeat step 2, else go to step 3. When the square of our guess at the square root is less than m we can start increasing n by 0.1 until we again compute a guess greater than m. At this point, we start decreasing our guess by 0.01 and so on until we have computed the square root we require to the desired accuracy. In above algorithm the number of iterations required depends critically on how good our initial guess is (e.g. if m is 10,000 and our initial guess n is 500 we will need over 400 iterations). To try to make a better algorithm, let us again to the problem of finding the square root of 36, we found that If 9 divides into 36 to give 4. If we choose 4 as our square root candidate, we would have found. In other words, the 9 and the 4 tend to cancel out each other by deviating from the square m in opposite directions. Thus, We find that = which is greater than 36. Dividing this into 36: The square root of 36 must lie somewhere between 9, which is too big and 4, which is too small. Taking the average of 9 and 4: we see that it again has a complementary value (i.e. 5.53) that is less than 36. Thus If we are still unsure as to what to do next, we can take a guess at the square root and then use the equation (1) to check whether or not we have guessed correctly. For example: Suppose we don t know the square root of 36. We might guess that 9 could be the square root. Using equation 1 we find that 9 x 9 = 81 which is greater than 36. Another guess like 8 will result to 8 x 8 = 64 which is again greater than 36 but closer than the previous result. We now have two estimates of the square root, one on either side, that are closer than our first two estimates. We can proceed to get an even better estimate of the square root by averaging these two most recent guesses: Where = which is only slightly greater than the square we are seeking. However, we will assume that it is a good strategy and the development of the algorithm. 2
3 To perform the average of g1 and g2 is Algorithm Description To achieve this repetitive interchanging of roles by setting up the following loop We can therefore terminate the algorithm when the difference between g1 and g2 becomes less than some fixed error ( eg: ). This absolute difference is our termination condition. Algorithm ch03pg01.txt Problem Given an integer n devise and algorithm that we will find its smallest exact divisor other than one. Algorithm Development Taken a face value this problems seems to be trivial. We can take the set of numbers 2,4,6,.,n and divide each one in turn into n. As soon as we encounter a number in the set that exactly divides into n our algorithm can terminate as we have found the smallest exact divisor of n. All this looks very cool and straight. Now say suppose n is a prime number. May be n is a very large prime number (1013). So are we going to count all the way upto 1012 starting from 2 and check whether 1013 gets divided by any one of these numbers on the way? That sounds little stupid and requires a lot of work. 3
4 As a starting point for this investigation let us work out and examine the complete set of divisors for a particular number. Choosing the number 36 as our example, we find that its complete set of divisors are {2,3,4,6,9,12,18} We know that an exact divisor of a number divides into that number leaving no remainder. For example we require 9 fours to get 36. It also follows that the bigger number 9 also divides exactly into 36. That is Similarly if we choose the divisor 3, we find that it tells us that there is a bigger number 12 that is also an exact divisor of a number. For our complete set of divisors of 36, we see that: Or in other words, we want an s and b that satisfy. From this set, we see that the smallest divisor (2) is linked with the largest divisor (18), the second smallest divisor (3) is linked with the second largest divisor (12) and so on. From the above logic we can see that The smaller factor s The bigger factor b and or which s is less than b. The crossover point and limiting value must occur when s=b, i.e. when It follows that it is not necessary to look for smaller divisors of n that are greater than the square root of n. So lets review our design again. We know that all even numbers are divisible by 2. It follows that is the number we are testing is not divisible by 2 then it certainly will not be divisible by 4,6,8,10,. If the number we are testing is odd, we need to only consider odd number as potential smallest divisor. The overall strategy for out algorithm can be summarized : If the number n is even then the smallest divisor is 2. Else Compute the square root r of n. While no exact divisor less than square root of n do Test next divisor in sequence 3,5,7, All that is left to do now is work out the implementation details The divisors to test can be generated by starting d at 3 and using 4
5 To determine whether or not a number is an exact divisor of another number, we can check if there is any remainder after division. For this we use mod function. If n mod d = 0 then d is an exact divisor of n. The two conditions for termination can be n mod d = 0 and d>=r Algorithm Description Algorithm Smallestdivisor.txt Problem Given two positive non-zero n and m design an algorithm for finding their greatest common divisor (GCD). Algorithm Development Our first step might therefore be to break the problem down and find all the divisors of the two n and m independently. Once we have these two lists of divisors we soon realize that we must do is select the largest element common to both lists. This element must be the greatest common divisor for the two n and m. We may expect that this algorithm will be relatively time consuming for large values of n and m because of the tedious step of having to generate all factors of two. It was the Greek philosopher, Euclid, who more than 2000 years ago first published a better way to solve this problem. To embark our search for a better algorithm, a good place to start is with a careful look at just what is the greatest common divisor of two numbers. The gcd of two numbers is the largest integer that will divide exactly into the two with no remainder 5
6 So our basic strategy for computing the gcd of two numbers is: The mod function allows us to compute the remainder resulting from an integer division. We use: To try to formulate this construct, let us return to gcd(18,30) problem let us see the calculations Algorithm Description Algorithm gcd.txt Applications Reducing a fraction to its lowest terms. Problem Use the linear congruentaial method to generate a uniform set of pseudo random numbers. Algorithm Development Random number are frequently used in computing science for generators among other things, testing and analysing the behavior of algorithms. A sequence of random numbers should exhibit the following behavior. The sequence should appear as though each number had occurred by chance. Each number should have a specified probability of falling within a given range. 6
7 The approach generally taken in computing science for random number generation is to simulate random processes by producing a sequence of numbers that appear to exhibit random behavior. These sequences are predictable in advance and for this reason they are usually referred to as pseudorandom sequences. There are many methods for generating pseudorandom numbers. Perhaps the most widely used of these algorithms is the linear congruential method. When this method is appropriately parameterized it will generate pseudo-random sequences that, for practical purposes. The implementation of the linear congruential method is very straight-forward. Successive members of the linear congruential sequence {x} are generated using the expression: where the parameters a, b, m and x 0 must be carefully chosen in advance according to certain criteria. The parameters a, b, and m are referred to as the multiplier, increment, and modulus respectively. All parameters should be greater than or equal to zero and m should be greater than x 0,a and b. Parameter x 0 : The parameter x0 can be chosen arbitrarily within the range 0--x0<m. Parameter m: The value of m should e greater than or equal to the length of the random sequence required. In addition it must be possible to do the computation (a*x)+b mod m without roundoff. Parameter a: The choice of a depends on the choice of m. If m is a power of 2 then a should satisfy the condition: a mod 8 = 5 If m is a power of 10, then a should be chosen such that: a mod 200 = 21 Further requirements on a are that it should be larger than m and less than m- m, (a-1) should be a multiple of every prime dividing into m, and if m is a multiple of 4 then (a-1) should also be a multiple of 4. These conditions together with the requirements that b should relatively prime to m are needed to guarantee that the sequence has a period of m. Paramter b: The constant b should be odd and not a multiple of 5. When a, b, and m are chosen according to the conditions outlined above a sequence of m pseudo-random numbers in the range 0 to (m-1) can be generated before the sequence begins to repeat. Algorithm Algorithm for the linear congruential method is given below for an m value of Applications Analysis of algorithm Simulation problems and games 7
8 Problem Given some integer x, compute the value of xn where n is a positive integer considerably greater than 1. Algorithm Development Evaluating the expression p = x n here x and n are given is a straightforward task. A simple method of evaluation is to repeatedly multiply an accumulating product p by x for n iterations, for example, p := 1 for i := 1 to n do p := p*x In most cases, evaluating x n in this way is satisfactory. Let us therefore concern ourselves with trying to discover a more efficient algorithm for power evaluation. Consider the evaluation of x n. In our stepwise approach we have: Studying these steps carefully, we see that x is increasing by one power at each step. In trying to evaluate x 10 more efficiently, what are we really trying to do? To evaluate x 10 more efficiently implies that we will need to take fewer steps to complete the task. Looking back at our example, we see that we could have generated x 4 by simply multiplying x 2 by itself, for example, x 4 = x 2 x x 2 This requires just one multiplication rather than the two multiplications used in our example above. In terms of power addition we have 2+2 = 4 compared with 2+1+1= 4. In one sense, what this most recent example tells us is that we will have solved our problem in the most efficient way when we discover the set of numbers that add to 10 in the least number of steps. Some sets of numbers that add up to 10 are: What we observe is that one of these possibilities we choose must solve the problem more efficiently. Suppose we choose 8 and 2 as our pair of powers to generate x 10 (i.e. x 8 * x 2 = x 10 ). We need to ask what is the "smallest" pair of numbers that we can choose that we can choose that add up to 10? The answer to this question has to be two 5 s. To generate x 5 we square x 2 to give x 4 and then multiplying that result with x. 8
9 The steps to compute x 10 are therefore: With this scheme we have used only four rather than nine multiplications to evaluate x 10 We can try the same idea with second example: Here we can see that only 7 multiplications have been required to raise a number to the power 23. At each stage in the power generation process, one of two conditions apply: When we have an odd power it must have been generated from the power that is one less (eg: x 23 = x 22 * x). Where we have an even power, it can be computed from a power that is half its size (e.g. x 22 = x 11 * x 11 ). These last two statements capture the essence of the algorithm. This means that our algorithm will need to be in two parts: a part that determines the multiplication strategy, and that does the a second part actually power evaluation. To map out the multiplication procedure, we can start with the power required and determine whether it is even or odd. The next step is to integer-divide the current power by 2 and repeat the even/odd determination procedure. For example for x 23 we would have : To carry out the second of the algorithm (i.e. the power evaluation part), we need to work forwards rather than backwards. Which means we must start with the highest element in the d array (in our example d[4]) and work back down to d[1]. Starting out with our product p equal to 1 we can proceed with the power evaluation using the following rule: if the current d array element is zero then (a) we simply square the current power product p, else (a') we square the current product p and multiply by x to generate an odd power. For the evaluation of x 23 the steps are: Referring to the binary representation for x23 we see that it can be written as We can generate binary representation of a number by dividing the power n by 2. For example the binary representation of 23 is : Which suggests that power evaluation can be coupled directly with binary representation of the power of n. 9
10 Each of these powers can be generated by multiplying its predecessors by itself For example: Studying the example once again we see that power evaluation involves the following steps. Successive generation of numbers of the power sequence Inclusion of the current power member into the accumulated product when the corresponding binary digit is one. Table below summarizes the power evaluation for x 23 by our new proposal. The test n mod 2 = 1 can be used to check if the next most significant binary digit is one. If the total product is named product and the power sequences are named psequences then to store the current power we use the following standard form: The variable product will initially will be set to 1. The power sequence variable psequence will need to be initialized to x. With successive iterations the power value of psequence will need to be doubled. This can be accomplished by using The other step that must take place with each iteration is that n must be divided by 2. For example: The algorithm must terminate when n is reduced to zero. Algorithm Description 1. Establish n, the integer power, and x the integer to be raised to the power n. 2. Initialize the power sequence and product variable for the zero power case. 3. While the power n is greater than zero do 1. if next most significant binary digit of power n is one then multiply by current sequence accumulated product power value; 2. reduce power n by a factor of two using integer division; 3. get next power sequence member by multiplying current value by itself. 4. Return x raised to the power n. Algorithm Applications Encryption (secret coding of information), And testing for non-primary of numbers 10
11 Problem Given a number n generate the n th member of the Fibonacci sequence. Algorithm development Remembering back to algorithm (2.6) we are given that the n th member of the Fibonacci sequence f n is defined recursively as follows: We have previously seen how this definition can be used in conjunction with an iterative construct to generate Fibonacci numbers. What we are concerned with here is whether or not there is a more efficient way of generating the n th Fibonacci number rather than the complete sequence of n Fibonacci numbers. It sometimes happens when one member of a set is required the job can be done more efficiently than generating the (n 1) predecessors. In the power generation problem, our task was straightforward because the relationship between powers of i and the powers of 2i was simple and easily defined (e.g. x 6 = x 3 * x 3 ). In our present problem we are not sure whether such a "doubling relationship exists. To explore this possibility let us write down the first few members of the Fibonacci sequence. Here we can try is some combination of two Fibonacci numbers to generate the "doubled' Fibonacci number. This might have more chance of success since in the original problem for generating Fibonacci numbers, two numbers are used to generate the next. Let us therefore try to write f8 in terms of only f4 and f5. To do this we will need to use the original definition for the sequence. That is Checking with f 10, f 5 and f 6 to see if the formula generalizes, we get Without going into a detailed proof, we might be satisfied that in general: Collecting terms we get Since f 5 = 3 and f 4 = 2 the last expression suggests: To generate the f 2n+1 Fibonacci number we need f 2n and f 2n-1 11
12 The problem is we do not have f 2n-1 and so we are not much better off. Therefore we to do this in previous example we need to generate f 9 from f 5 and f 4. Applying the same substitution method as we used to establish: after some substitution and checking, we finally get: We now have the basis of a method for generating the n th Fibonacci number in a similar manner to raising x to the power n as in algorithm (3.7). In this case our algorithm will need to be in two parts: a part that determines the doubling strategy by generating the binary representation of n and a second part that actually computes the nth Fibonacci number according to the doubling strategy. Algorithm description Algorithm nthfibonacci.txt 12
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