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In [220]:



    


import numpy as np
import numpy.linalg as LA
import matplotlib.pyplot as plt
from pandas import *
%matplotlib inline



    











In [296]:



    


def Arnodi_algo(A, b, k):
    N = A.shape[0]
    Q = np.zeros((N, k), dtype='complex')
    H = np.zeros((k, k), dtype='complex')
    Q[:, [0]] = b / LA.norm(b)
    
    for j in range(k - 1):
        r = A.dot(Q[:,[j]])
        for i in range(j + 1):
            H[i][j] = Q[:, [i]].T.conj().dot(r)[0][0]
            r -= Q[:, [i]] * H[i, j]
        H[j + 1, j] = LA.norm(r)
        Q[:, [j + 1]] = r / H[j + 1, j]
    for i in range(k):
        H[i, k - 1] = Q[:, [i]].T.conj().dot(r)[0][0]
        r -= Q[:, [i]] * H[i, k - 1]
    return Q, H

def Lanczos_algo(A, b, k):
    N = A.shape[0]
    Q = np.zeros((N, k), dtype='complex')
    alpha = np.zeros(k, dtype='complex')
    beta = np.zeros(k - 1, dtype='complex')
    Q[:, [0]] = b / LA.norm(b)
    r = np.dot(A, Q[:,[0]])
    for i in range(k - 1):
        alpha[i] = np.dot(Q[:, [i]].T.conj(), r)[0][0]
        r -= Q[:, [i]] * alpha[i]
        beta[i] = LA.norm(r)
        Q[:, [i + 1]] = r / beta[i]
        r = np.dot(A, Q[:,[i + 1]]) - beta[i] * Q[:,[i]]
    alpha[k - 1] = np.dot(Q[:, [k - 1]].T.conj(), r)[0][0]
    return Q, alpha, beta



    











In [302]:



    


np.random.seed(13)
N = 200
k = 200
A = np.random.randn(N, N) + 1j * np.random.randn(N, N)
A = (A + A.T.conj()) / np.sqrt(2)
b = (np.random.randn(N, 1) + 1j * np.random.randn(N, 1)) / np.sqrt(2)



    











In [303]:



    


Q_ar, H_ar = Arnodi_algo(A, b, k)
my_id_ar = np.dot(Q_ar.T.conj(), Q_ar)
my_id_ar[np.absolute(my_id_ar) < 0.0001] = 0
DataFrame(np.absolute(my_id_ar));



    











In [304]:



    


Q_lan, alpha, beta = Lanczos_algo(A, b, k)
H_lan = np.diag(beta, -1) + np.diag(alpha) + np.diag(beta, 1)
my_id_lan = np.dot(Q_lan.T.conj(), Q_lan)
my_id_lan[np.absolute(my_id_lan) < 0.0001] = 0
DataFrame(np.absolute(my_id_lan));



    











In [305]:



    


lam_true = LA.eigvalsh(A)
lam_ar = LA.eigvals(H_ar)
lam_lan = LA.eigvals(H_lan)



    











In [306]:



    


plt.plot(lam_true, 'x')
plt.plot(sorted(lam_ar.real))
plt.plot(sorted(lam_lan.real))



    


















    
Out[306]:








[<matplotlib.lines.Line2D at 0x7f38f03a9b00>]










    
























In [307]:



    


A[np.random.random(A.shape) + np.eye(N) < .7] = 0



    











In [308]:



    


Q_ar, H_ar = Arnodi_algo(A, b, k)
my_id_ar = np.dot(Q_ar.T.conj(), Q_ar)
my_id_ar[np.absolute(my_id_ar) < 0.0001] = 0
DataFrame(np.absolute(my_id_ar));



    











In [309]:



    


Q_lan, alpha, beta = Lanczos_algo(A, b, k)
H_lan = np.diag(beta, -1) + np.diag(alpha) + np.diag(beta, 1)
my_id_lan = np.dot(Q_lan.T.conj(), Q_lan)
my_id_lan[np.absolute(my_id_lan) < 0.0001] = 0
DataFrame(np.absolute(my_id_lan))



    


















    
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      182
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      ...
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      183
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      184
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      ...
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      185
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      186
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      ...
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      187
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      188
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      189
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      190
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      191
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      192
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      193
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      194
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      195
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      196
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      197
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      198
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      199
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200 rows × 200 columns



















In [310]:



    


lam_true = LA.eigvalsh(A)
lam_ar = LA.eigvals(H_ar)
lam_lan = LA.eigvals(H_lan)



    











In [312]:



    


plt.plot(lam_true, 'x')
plt.plot(sorted(lam_ar.real))
plt.plot(sorted(lam_lan.real))



    


















    
Out[312]:








[<matplotlib.lines.Line2D at 0x7f38f026ce10>]










    
























In [275]:



    


#DataFrame(A.real)



    


















    
Out[275]:











  
    

      
      0
      1
      2
      3
      4
      5
      6
      7
      8
      9
      ...
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      391
      392
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      395
      396
      397
      398
      399
    

  
  
    

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      11
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      12
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      16
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      17
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      18
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      27
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      28
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      29
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      ...
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400 rows × 400 columns



















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Content source: LorenzoBi/courses

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