Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 4.2, Problem 4E
Program Plan Intro
To find the largest value of ksuch that multiply
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Generate random matrices of size n × n where n = 100, 200, . . . , 1000.Also generate a random b ∈ Rnfor each case. Each number must beof the form m.dddd (Example : 4.5444) which means it has 5 Significant digits in total. Perform Gaussian elimination with and withoutpartial pivoting for each n value (10 cases) above. Report the numberof additions, divisions and multiplications for each case in the form ofa table. No need of the code and the matrices / vectors. Deliverable(s): Two tabular columns indicating the number of additions, multiplications and divisions for each value of n, for with andwithout pivoting in Python
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Chapter 4 Solutions
Introduction to Algorithms
Ch. 4.1 - Prob. 1ECh. 4.1 - Prob. 2ECh. 4.1 - Prob. 3ECh. 4.1 - Prob. 4ECh. 4.1 - Prob. 5ECh. 4.2 - Prob. 1ECh. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - Prob. 4ECh. 4.2 - Prob. 5E
Ch. 4.2 - Prob. 6ECh. 4.2 - Prob. 7ECh. 4.3 - Prob. 1ECh. 4.3 - Prob. 2ECh. 4.3 - Prob. 3ECh. 4.3 - Prob. 4ECh. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - Prob. 7ECh. 4.3 - Prob. 8ECh. 4.3 - Prob. 9ECh. 4.4 - Prob. 1ECh. 4.4 - Prob. 2ECh. 4.4 - Prob. 3ECh. 4.4 - Prob. 4ECh. 4.4 - Prob. 5ECh. 4.4 - Prob. 6ECh. 4.4 - Prob. 7ECh. 4.4 - Prob. 8ECh. 4.4 - Prob. 9ECh. 4.5 - Prob. 1ECh. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Prob. 4ECh. 4.5 - Prob. 5ECh. 4.6 - Prob. 1ECh. 4.6 - Prob. 2ECh. 4.6 - Prob. 3ECh. 4 - Prob. 1PCh. 4 - Prob. 2PCh. 4 - Prob. 3PCh. 4 - Prob. 4PCh. 4 - Prob. 5PCh. 4 - Prob. 6P
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- Following code is a part of an algorithm where A = [a; ;] is an N x N matrix with a;i = a(i, j), x, y, b are vectors of length N. Count the total number of operations (addition/subtraction and multi- plication/division) for the code below in terms of N (Give a simplified answer in terms of N): Begin Algorithm fct = 2 sum for i = 1 to N x(i) y(i) = Fa end i loop for j = 1 to N suml = 0 for k = j+1 to N suml = suml+a(j,k) * y(k) end k loop r(j) end j loop sum = 10 + sum1 * fct b(j)-sum a(j, j) End Algorithmarrow_forwardGiven an n-element sequence of integers, an algorithm executes an O(n)-time computation for each even number in the sequence, and an O(logn)-time computation for each odd number in the sequence. What are the best-case and worst-case running times of this algorithm? Why? Show with proper notations.arrow_forwardYou are given a two-dimensional array A with n rows and n columns such that every element is either 1 or 0, and for every row, the 1s are placed ahead of 0s. Find the quickest algorithm to find a row with the most significant number of 1s. Analyze the time complexity of your algorithm.arrow_forward
- Generate random matrices of size n ×n where n = 100, 200, . . . , 1000.Also generate a random b ∈ Rnfor each case. Each number must beof the form m.dddd (Example : 4.5444) which means it has 5 Signif-icant digits in total. Perform Gaussian elimination with and withoutpartial pivoting for each n value (10 cases) above. Report the numberof additions, divisions and multiplications for each case in the form ofa table. No need of the code and the matrices / vectors.arrow_forwardGive a Θ(lg n) algorithm that computes the remainder when xn is divided byp. For simplicity, you may assume that n is a power of 2. That is, n = 2k forsome positive integer k.arrow_forwardGiven an n-element array X of integers, Algorithm A executes an O(n) time computation for each even number in X and an O(log-n) time computation for each odd number in X. What are the best case and worst case for running time of algorithm C?arrow_forward
- Give a recursive (decrease-by-one) algorithm for finding the position of the smallest element in an array of n real numbers. and Determine the running time complexity of this algorithm.arrow_forwardDetermine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n smaller m whether gcd(n,m) = 1. (You do not have to apply Euclid’s algorithm.)arrow_forwardSparse matrix-vector multiplication in Python is similar to numpy.dot operation, however the provided @ operator is used. Let us consider multiplying a 5 by 5 sparse matrix with a 5 by 3 sparse matrix shown below. 01000 00010 10001 10000 01000 100 001 x 000 010 100 The Python code to implement multiplication of these two sparse matricesarrow_forward
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