Hello. I NEED A PYTHON CODE. I have this python code (bellow) and I need to solv
ID: 3856117 • Letter: H
Question
Hello. I NEED A PYTHON CODE. I have this python code (bellow) and I need to solve, still in python, the linear systems Va = y and Vb= y using algorithms created by factorization. 1. LU with and without pivoting, 2. by the polar decomposition of A and 3. QR with and without pivoting.
import random import numpy as np k = 2 # number of equally spaced elements space = 0.3 #value of the spacing between equally spaced elements x = [i"space for i in range(k)] #equally spaced xk elements for i in range(k,6): x.append(-1l+random.random0 ) #randomly spaced rest of the elements y = [np.exp(i) for i imx] #np.sinO, np.cos(), np.expl, np.log0, np.log100, np.logo can be used V = np.vander(x) #constructs vandermonde matrix of the vector X. U = np.vander(y) #constructs vandermonde matrix of the vector 'y.Explanation / Answer
/* method to return prime divisor for n */
long long int PollardRho(long long int n)
{
/* initialize random seed */
srand (time(NULL));
/* no prime divisor for 1 */
if (n==1) return n;
/* even number means one of the divisors is 2 */
if (n % 2 == 0) return 2;
/* we will pick from the range [2, N) */
long long int x = (rand()%(n-2))+2;
long long int y = x;
/* the constant in f(x).
* Algorithm can be re-run with a different c
* if it throws failure for a composite. */
long long int c = (rand()%(n-1))+1;
/* Initialize candidate divisor (or result) */
long long int d = 1;
/* until the prime factor isn't obtained.
If n is prime, return n */
while (d==1)
{
/* Tortoise Move: x(i+1) = f(x(i)) */
x = (modular_pow(x, 2, n) + c + n)%n;
/* Hare Move: y(i+1) = f(f(y(i))) */
y = (modular_pow(y, 2, n) + c + n)%n;
y = (modular_pow(y, 2, n) + c + n)%n;
/* check gcd of |x-y| and n */
d = __gcd(abs(x-y), n);
/* retry if the algorithm fails to find prime factor
* with chosen x and c */
if (d==n) return PollardRho(n);
}
return d;
}
/* driver function */
int main()
{
long long int n = 10967535067;
printf("One of the divisors for %lld is %lld.",
n, PollardRho(n));
return 0;
}
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