[2022 Fall] Assignment6-2¶

Course: AP3021

Q6-2¶

1. Determine the condition number for the following system using the row-sum norm. Do not normalize the system.¶

In [1]:

import numpy as np

import numpy as np

In [2]:

a = np.array([[ 1,  4,  9, 16, 25],
              [ 4,  9, 16, 25, 36],
              [ 9, 16, 25, 36, 49],
              [16, 25, 36, 49, 64],
              [25, 36, 49, 64, 81]])
a_determinant = np.linalg.det(a)

print("Condition number of A:", np.linalg.cond(a, p=None))

# print('determinant of A =', a_determine)

a = np.array([[ 1, 4, 9, 16, 25], [ 4, 9, 16, 25, 36], [ 9, 16, 25, 36, 49], [16, 25, 36, 49, 64], [25, 36, 49, 64, 81]]) a_determinant = np.linalg.det(a) print("Condition number of A:", np.linalg.cond(a, p=None)) # print('determinant of A =', a_determine)

Condition number of A: 9.915718256577622e+16

2. Repeat (a) but scale the matrix by making the max element in each row to 1.¶

In [3]:

a = np.float32([[ 1,  4,  9, 16, 25],
              [ 4,  9, 16, 25, 36],
              [ 9, 16, 25, 36, 49],
              [16, 25, 36, 49, 64],
              [25, 36, 49, 64, 81]])

for i in range(0, 5) :
    for j in range(0, 5) :
        a[i][j] = float(a[i][j] / a[i][4])
        # print(a[i][j])
        
print("Making the max element in each row to 1:\n")
print(a)

a = np.float32([[ 1, 4, 9, 16, 25], [ 4, 9, 16, 25, 36], [ 9, 16, 25, 36, 49], [16, 25, 36, 49, 64], [25, 36, 49, 64, 81]]) for i in range(0, 5) : for j in range(0, 5) : a[i][j] = float(a[i][j] / a[i][4]) # print(a[i][j]) print("Making the max element in each row to 1:\n") print(a)

Making the max element in each row to 1:

[[0.04       0.16       0.36       0.64       1.        ]
 [0.11111111 0.25       0.44444445 0.6944444  1.        ]
 [0.18367347 0.3265306  0.5102041  0.7346939  1.        ]
 [0.25       0.390625   0.5625     0.765625   1.        ]
 [0.30864197 0.44444445 0.60493827 0.79012346 1.        ]]

3. 想想看，由condition number可知，這是一個什麼樣的矩陣呢?¶

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Last update: 2024-04-27