Python Numpy 矩阵乘法与向量收敛循环

Question

I need to write a code in python using numpy to iterate and loop through to multiply it a matrix by a new vector 72 times. Any help will be appreciated.

The matrix is shown as P and the vector is pop. As you can see, I have multiplied the matrix by the vector using B=P.dot(pop) and then I have printed B below and we get B=[135,165]. So what I want to do is setup a loop such that the matrix P is multiplied by B, giving a vector C. And then the matrix P is multiplied by the new vector C etc etc.... and keeps going 72 times. I know that it will converge to [100,200] and stay there, but how do I write a loop code to do this.

So this is what I have so far:

import numpy as np
P=np.array([[0.8,0.1],[0.2,0.9]])
pop=([150,150])

B=P.dot(pop)

C=P.dot(B)

print B
print C

Furthermore how can I show that after each iteration, the sum is always 300 of the 2 components in the vector?

Thanks

Answer 1

This is called a power iteration for finding eigenvectors:

The for loop definitely works but it is somewhat trivial and boring:). If you know that the series converges, as in this case, you can use a while loop with a sense of convergence to avoid unnecessary iterations.

enter code here

import numpy as np
P=np.array([[0.8,0.1],[0.2,0.9]])
pop=([150,150])
B=P.dot(pop)
'''
for i in range(60):
    C=P.dot(B)
    err=np.linalg.norm(B-C,2)
    B=C
print(B)
'''
err=np.inf
iter=0
while err>10E-10:
    iter+=1
    C=P.dot(B)
    err=np.linalg.norm(B-C,2)
    B=C

print(B)
print(err)
print(iter)
> [100. 200.]
> 8.874916220860865e-10
> 67