Give big-O estimates for the factorial function and the logarithm of the factorial function, where the factorial function f(n) = n! is defined by n! = 1 · 2·3· ... ·n whenever n is a positive integer, and O! = 1 . For example, 1 ! = 1, 2! = 1 · 2=2, 3 ! =1· 2·3 = 6, 4! = 1 · 2·3· 4=24. Note that the function n! grows rapidly. For instance, 20! = 2,432,902,008,176,640,000.

Give big-O estimates for the factorial function and the logarithm of the factorial function, where the factorial function f(n) = n! is defined by n! = 1 · 2·3· ... ·n whenever n is a positive integer, and O! = 1 . For example, 1 ! = 1, 2! = 1 · 2=2, 3 ! =1· 2·3 = 6, 4! = 1 · 2·3· 4=24. Note that the function n! grows rapidly. For instance, 20! = 2,432,902,008,176,640,000.

Programming Logic & Design Comprehensive

9th Edition

ISBN:9781337669405

Author:FARRELL

Publisher:FARRELL

Chapter4: Making Decisions

Section: Chapter Questions

Problem 9RQ

See similar textbooks

Similar questions

2. Prove from the definition that 2n2 + 100n log n + 1000 = O(n2) 3. For what values of n is 50 n lg n greater than 0.5 n2? Why do we say that 0.5 n2 is asymptotically larger, if 50 n lg n is larger for many values? (Hint: you may need to graph the functions or play around with your calculator.)
Rank the following functions by asymptotic growth rate in non-decreasing order:f1(n) = 2^2^1000000; f2(n) = 2^1000000n; f3(n) = (n 2) f4(n) = n√n: Justify your answer!Example: The function f(n) = n grows asymptotically slower then the functiong(n) = n2; that is, f(n) 2 O(g(n)), but g(n) 2 O = (f(n)).(ii) Using big O-notation show that n1+0;001 2 O = (n).
Graph the functions 8n, 4nlogn, 2n2, n3, and 2n using a logarithmic scale for the x- and y-axes. That is, if the function is f(n) is y, plot this as a point with x-coordinate at log n and y-coordinate at log y.
In RSA, given that the primes p and q are approximately the samesize, approximately how big is φ(n) compared to n?
Factor 262063, 9420457, and 181937053 using the POLLARD RHO ALGORITHM, if the function f is defined to be f (x) = x2 +1. How many iterations are needed to factor each of these three integers? I need iterations for each that are needed to factor
Order the following functions by asymptotic growth rate: 4n log n +2n 3n + 100 n log n n2 + 10n 210 4n n3 2log n 2n n log n
Let A = {x ∈ Z : x ≤ 3} and let B = {x ∈ Q : x2 = 9}. Is B ⊆ A? Give a brief reason for your answer.
Transform the following sentences to CNF. 1. ꓯ x (P(x) ꓦ Q(x)) → R(x) 2. ꓯ x ꓯ y ꓯ z (A(x,y) ꓥ A(y,z)) → A(x,z) 3. ꓯ x ꓯ y ꓱ z P(x,z) ꓥ Q(y,z) 4. ꓯ x ⌐ [(P(x) ꓥ Q(x)) ꓦ (R(x) ꓥ S(x))]
Q-2 Suppose that you pick a bit string from the set of all bit strings of length ten. What is the probability that the bit string has more 0s than 1s?
A student is interested in computing the area under the function f(x)=x over the interval [0,1] and decides to get an accurate answer using a Monte Carlo experiment (by this we are assuming the student doesn't know the answer to such an elementary mathematical question). Which of the following R codes will give such an approximate answer? M = 10^6x = runif(M)y = runif(M)mean( y <= x ) M = 10^6x = runif(M)mean( x ) M = 10^6x = rnorm(M)mean( x ) M = 10^6x = rnorm(M)y = runif(M)mean( y <= x )
Prove that the following shows LHS = RHS : (+ 2 1 ) = 3 Given: 1=λf.λy.f y 2=λf.λy.f(f y) 3=λf.λy.f(f(f y)) M+N=λx.λy.(M x)((N x)y)
% Define the function f(x) f = @(x) -1*(x < 0) + 1*(x >= 0 & x <= 2); % Set the maximum value of N Nmax = 512; % Initialize the x-axis values for plotting x = linspace(-1,3,1000); % Compute the Fejér sums for various values of N F = zeros(length(x), Nmax); for N = 1:Nmax % Compute the Nth Fejér sum Sn = @(x) 0; for n = 1:N bn = (-1)^(n+1) / (n*pi); Sn = @(x) Sn(x) + bn*sin(n*pi*x); end FN = @(x) (1/N) * Sn(x); % Evaluate the Fejér sum at each point on the x-axis I tried to submit this to MATLAB, but can't seem to get it right.