CSC-140 Assignment 5

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Joanna Fitzgerald
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1 CSC-140 Assignment 5 Please do not Google a solution to these problems, cause that won t teach you anything about programming - the only way to get good at it, and understand it, is to do it! 1 Introduction In this assignment we get to work more with methods, strings, recursion and finally arrays. We will consider Fibonacci numbers again and also look at matrices. 2 Fibonacci Again We have already seen a couple of different ways to compute the Fibonacci number, but let us try yet another one; this one requires working with array, as it is based on a matrix solution, so let us first look at some matrix math. 2.1 Matrices For now we will limit our study to matrices of size 2 2 (We get to n n later in the assignment). An example is the following ] 1 1 F This matrix is actually the generating matrix for the Fibonacci numbers, as we shall see soon. In order to compute the n th Fibonacci number, we need to be able to multiply matrices. Let us see how we multiply two 2 2 matrices, it is quite simple: ] ] ] b0,0 b 0,1 (a0,0 b 0,0 + a 0,1 b 1,0 ) (a 0,0 b 0,1 + a 0,1 b 1,1 ) b 1,0 b 1,1 (a 1,0 b 0,0 + a 1,1 b 1,0 ) (a 1,0 b 0,1 + a 1,1 b 1,1 ) 1

2 We see that the matrix product of two 2 2 matrices is a 2 2 matrix, where the 4 entries are a sum of 2 products based on entries in the matrices being multiplied. (A word of warning; if A and B are matrices, there is no guarantee that A B B A, so order matters!) Here is an example with some numbers: ] ] ( ) ( ) ( ) ( ) ] We can now easily define exponentiation for matrices. Just a a 0 1 for any value a, we have a similar identity for matrices: ] where 1 a a for any number a, M I M for any matrix M, so I is the equivalent of 1 in matrix algebra when it comes to multiplication. Finally we have : ] I n ] { n ] }} ]{ a0,0 a 0,1 a0,0 a 0,1 ] or we could write it as a recursive formula: ] ] n 0 1 if n 0 ] n 1 ] otherwise 2.2 Representing Matrices with Arrays Let us take a look at a 2 2 matrix again: ] we need a 2-dimensional array of 2 integers: 2

3 int ] ] a new int2]2]; where a0]0] a 0,0 a0]1] a 0,1 a1]0] a 1,0 a1]1] a 1,1 Note how the arrays indices match the matrix subscripts. For example, the I matrix from above can be defined like this: int ] ] I new int ] ] { { 1, 0 }, { 0, 1 } }; Now, what is the relationship between Fibonacci and Matrices? Well, if we denote the n th Fibonacci number F n then the following identity holds: ] n ] 1 1 Fn+1 F n F n F n So, if we compute the n th power of the matrix M entry 1]0] or 0]1] we will have the n th Fibonacci number. ] n and read off Question 1: Implement a method called fibmatrix that has the following interface: public static int fibmatrix(int n) { // your code here } that takes in a value n and returns the n th Fibonacci number. There are two ways to do this: iteratively or recursively. If you do it iteratively you may need a temporary matrix to hold the result in for each loop, and remember an array is a reference, so the actual values are not copied, just the reference (pointer). If you do it recursively you will need to write a method for computing exponentials of 2 2 matrices - here you probably also need either a temporary matrix or 4 integer variables to hold the result of the matrix multiplication before they go (back) into the result matrix. 3

4 3 Prime Numbers A simple way of computing the first n primes is using the Sieve of Erathostenes: 1. Create an array P from 1 to n. (keep in mind that entry P 0] 1, i.e., the value n is at index n We know that 1 is a not a prime, so cross it out it! 3. Next we know that 2 is a prime, but that any multiple of 2 is not, so go through the array and mark 4, 6, 8, 10,... as not primes (you can set the entry equal to 0 to mark that it has been marked as not a prime.) 4. Now we move on one entry at a time and do the following: if the entry is 0, then it was struck out as a non-prime, so just move on. If it is not 0, then it is a prime, and we leave it and strike out all its multiples like we did in the case of when we reach entry (n/2) + 1 we are done. Everything in the array that is not 0 is a prime. Here is a small example: Strike outs Result original: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} 1: {/1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} 2: 4,6,8,10,12,14 {1, 2, 3, /4, 5, //6, 7, /8, 9, /// 10, 11, /// 12, 13, /// 14, 15} 3: 6,9,12,15 {/1, 2, 3, /4, 5, //6, 7, /8, /9, /// 10, 11, /// 12, 13, /// 14, /// 15} 4: {/1, 2, 3, /4, 5, //6, 7, /8, /9, /// 10, 11, /// 12, 13, /// 14, /// 15} 5: 10,15 {/1, 2, 3, /4, 5,//6, 7, /8, /9,/// 10, 11,/// 12, 13, 14, // /// 15} 6: {/1, 2, 3, /4, 5,//6, 7, /8, /9,/// 10, 11,/// 12, 13, 14, // /// 15} 7: 14 {/1, 2, 3, /4, 5,//6, 7, /8, //9,/// 10, 11, 12, // 13, 14,/// 15} final: {/1, 2, 3, /4, 5,//6, 7, /8, //9,/// 10, 11, /// 12, 13, 14, /// /// 15} We can now read the primes from the list and get: {2, 3, 5, 7, 11, 13} Question 2: Implement a method called sieve with the following interface: public static int ] sieve(int n) { // your code her 4

5 } that takes in a value n and returns an array with all the primes less than or equal to n. Also write some code that prints out the list in a nice way like this: There are 25 primes < 100: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97} Note, there is no comma after the last number (97). 4 Matrices - Bigger Ones! If we go back to looking at matrices, we can expand on the idea of matrix multiplication for matrices of any size (almost). If we consider 2 square matrices (same number of columns as rows) A and B of size n n, then we can define the matrix product A B to be another matrix C of the same size (namely n n) like this: where a 0,0 a 0,1 a 0,n 1 a 1,n 1... a n 1,0 a n 1,1 a n 1,n 1 c 0,0 c 0,1 c 0,n 1 c 1,0 c 1,1 c 1,n 1... c n 1,0 c n 1,1 c n 1,n 1 c i,j n 1 k0 b 0,0 b 0,1 b 0,n 1 b 1,0 b 1,1 b 1,n 1... b n 1,0 b n 1,1 b n 1,n 1 a i,k b k,j (Try to convince yourself that this also works for the 2 2 matrices, and that the formula above is exactly the one that I gave you on page 1. C i,j is the 5

6 dot product of the i th row of A and the J th column of B Question 3: Implement a method called checkm atrixdim with the following interface: public static boolean checkmatrixdim(int ] ] M, int n) { // your code here } that takes in a matrix M in the form of a two-dimensional array, and a number (n), and returns true if the array represents an n n matrix (i.e., if the length of M and all its entries Mi] is n.) Question 4: Let us also write a method for printing out a matrix, but first let us write a small method called rjustify(int val, int n) that takes in an integer value (val) and a number n, and returns a string of length n with the value in val right justified. For example executing code like this: System.out.println( Here is 123 right justified 7: + ); + rjustify(123,7) should produce this output: Here is 123 right justified 7: 123 Remember, if a is an integer value, ( + a).length() is an easy way to get the length of a as a string. Question 5: Now you can implement a method called printmatrix that takes in a 2-dimensional integer array and a number n. If the matrix is an n n matrix it should print the matrix. You should justify all the entries by j + 1 where j is the length of the longest number (when viewed as a string; remember, if a is an integer variable, then +a is a string holding the value of a.) Here is an example of calling this procedure and its output: printmatrix(new int ] ] { {1,2,3,4,5}, {6,7,8,9,10}, {11,33,55,77,99}, {100,300,600,877,899}, 6

7 { ,54,12,56,3425}}, 5); Question 6: Finally, write a method called matrixmult that takes in two 2-dimensional arrays A and B representing two matrices, and an integer n representing their dimension. This method should return the matrix product A B as defined above if both matrices are n n matrices, else just return the value null (return null;). Question 7: Use your matrixmult method to compute the 25 th Fibonacci ] number by computing and printing the bottom left entry (1]0]). 5 Topics Covered Methods. Recursion. 1 and 2-dimensional arrays. 6 Work List 1. Make sure you read the Topics Covered section and familiarize yourself with all the topics. 2. Create a file Assignment5.java, and implement the code from the questions above. 3. Test your program with a number of different values. 4. Once you are happy with your solution, go to csgfms.cs.unlv.edu and hand it in. Just the.java file! 7

8 5. Bring to class (along with the printout of your solution) a write up in which you explain how you solved the problem, and show output from a couple of tests; enough to convince me that it works. 7 Due Date The Assignment5.java file must be uploaded to csgfms.cs.unlv.edu no later than Monday March 22nd before midnight (midnight between Monday and Tuesday), and the write up should be brought to class the day after. 8

9 8 Sample Output from my Solution Question 1: fib(0) 0 fib(1) 1 fib(2) 1 fib(3) 2 fib(4) 3 fib(5) 5 fib(6) 8 fib(7) 13 fib(8) 21 fib(9) 34 fib(10) 55 fib(11) 89 fib(12) 144 fib(13) 233 fib(14) 377 Question 2: There are 25 primes less than 100 and they are {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97} Question 4: Here is 123 right justified 7: 123 9

10 Question 5: Question 6: F 0 1 F^ Question 7: Fib(25)

CSC-140 Assignment 4

CSC-140 Assignment 4 Please do not Google a solution to these problem, cause that won t teach you anything about programming - the only way to get good at it, and understand it, is to do it! 1 Introduction