Numpy库练习

Exercise 9.1: Matrix operations 

Calculate A + A, AA>,A>A and AB. Write a function that computes A(B−λI) for any λ. 

import numpy

def calc(A,B,x):
    I=numpy.eye(m)
    return numpy.dot(A,B-I*x)
    
n=200
m=500
A=numpy.random.normal(size=(n,m))
B=numpy.random.normal(size=(m,m))
for i in range(m):
    for j in range(m):
        if ((i

Exercise 9.2: Solving a linear system 

Generate a vector b with m entries and solve Bx = b.

import numpy

n=200
m=500
A=numpy.random.normal(size=(n,m))
B=numpy.random.normal(size=(m,m))
for i in range(m):
    for j in range(m):
        if ((i

Exercise 9.3: Norms 

Compute the Frobenius norm of A: kAkF and the infinity norm of B: kBk∞. Also find the largest and smallest singular values of B.

import numpy
import scipy

n=200
m=500
A=numpy.random.normal(size=(n,m))
B=numpy.random.normal(size=(m,m))
for i in range(m):
    for j in range(m):
        if ((i

Exercise 9.4: Power iteration 

Generate a matrix Z, n × n, with Gaussian entries, and use the power iteration to find the largest eigenvalue and corresponding eigenvector of Z. How many iterations are needed till convergence?

Optional: use the time.clock() method to compare computation time when varying n.

import numpy
import time

def infi_norm(x):
    res=0
    for i in range(n):
        res=max(res,abs(x[i]))
    return res
    
n=6
Z=numpy.random.normal(size=(n,n))
old=numpy.ones(n,dtype=float)
t0=time.clock()
r=1
new=numpy.dot(Z,old.T)
new=new/infi_norm(old)
while (infi_norm(old-new)>1e-6 and r<1e+6):
    r=r+1
    old=new 
    new=numpy.dot(Z,old.T)
    new=new/infi_norm(old)
print(time.clock()-t0)

Exercise 9.5: Singular values 

Generate an n×n matrix, denoted by C, where each entry is 1 with probability p and 0 otherwise. Use the linear algebra library of Scipy to compute the singular values of C. What can you say about the relationship between n, p and the largest singular value?

import numpy
import scipy

n=5
p=0.3
A=numpy.random.rand(n,n)
C=numpy.where(A>p,0,1)
D=scipy.linalg.svdvals(C)
e=D.max()

Exercise 9.6: Nearest neighbor 

Write a function that takes a value z and an array A and finds the element in A that is closest to z. The function should return the closest value, not index.

Hint: Use the built-in functionality of Numpy rather than writing code to find this value manually. In particular, use brackets and argmin.

import numpy
import scipy

def find_nearest(A,z):
    A=A-z
    B=numpy.abs(A)
    i=B.argmin()
    return A.flatten()[i]+z