Normas

Normas#

import numpy as np
import numpy.linalg as la

x = np.random.randint(10, size=(5, 5))

x

array([[4, 4, 6, 3, 9],
       [0, 9, 8, 1, 8],
       [9, 6, 6, 7, 9],
       [0, 0, 3, 7, 4],
       [6, 8, 9, 0, 4]])

la.norm(x)

30.364452901377952

la.norm(x=x, ord=np.inf)

37.0

np.sum(a=x, axis=1)  # por filas

array([26, 26, 37, 14, 27])

np.sum(x[-1, :])  # Primera fila

27

la.norm(x=x, ord=-np.inf)

14.0

A = np.random.randint(10, size=(5, 5))
dA = np.random.random(size=(5, 5)) * 1e-2

A

array([[2, 7, 6, 3, 8],
       [8, 1, 3, 6, 1],
       [2, 6, 1, 5, 5],
       [3, 9, 4, 5, 4],
       [3, 8, 5, 0, 9]])

dA

array([[1.94124179e-03, 9.23888470e-03, 2.62745731e-03, 1.50224370e-03,
        8.85435469e-06],
       [8.25181745e-03, 3.38225574e-03, 1.90457534e-03, 6.35744944e-04,
        9.66298323e-03],
       [3.14087161e-03, 3.06781757e-03, 9.56458574e-03, 2.73470562e-03,
        7.93938146e-03],
       [3.95234892e-03, 6.09382071e-03, 8.92778119e-03, 4.17657127e-03,
        9.18758691e-03],
       [7.64912670e-03, 7.60160904e-03, 7.74530024e-03, 6.88571667e-03,
        4.28865041e-03]])

A + dA

array([[2.00194124e+00, 7.00923888e+00, 6.00262746e+00, 3.00150224e+00,
        8.00000885e+00],
       [8.00825182e+00, 1.00338226e+00, 3.00190458e+00, 6.00063574e+00,
        1.00966298e+00],
       [2.00314087e+00, 6.00306782e+00, 1.00956459e+00, 5.00273471e+00,
        5.00793938e+00],
       [3.00395235e+00, 9.00609382e+00, 4.00892778e+00, 5.00417657e+00,
        4.00918759e+00],
       [3.00764913e+00, 8.00760161e+00, 5.00774530e+00, 6.88571667e-03,
        9.00428865e+00]])

b = np.random.randint(10, size=5)
db = np.random.random(size=5) * 1e-2

b = b.reshape(5, 1)

db

array([0.00881303, 0.00460518, 0.00232491, 0.0094839 , 0.00863815])

b + db

array([[4.00881303, 4.00460518, 4.00232491, 4.0094839 , 4.00863815],
       [6.00881303, 6.00460518, 6.00232491, 6.0094839 , 6.00863815],
       [5.00881303, 5.00460518, 5.00232491, 5.0094839 , 5.00863815],
       [8.00881303, 8.00460518, 8.00232491, 8.0094839 , 8.00863815],
       [3.00881303, 3.00460518, 3.00232491, 3.0094839 , 3.00863815]])

np.concatenate((A, b), axis=1 # matriz aumentada

  Cell In[17], line 1
    np.concatenate((A, b), axis=1 # matriz aumentada
                                                    ^
SyntaxError: incomplete input

A.shape

(5, 5)

b.shape

(5, 1)

b

array([[3],
       [3],
       [6],
       [8],
       [3]])

la.cond(A)

60.930772057178615

la.cond(A + dA)

60.21344278652582
\[ \frac{\|\Delta \mathbf{x}\|}{\|\mathbf{x}+\Delta \mathbf{x}\|} \leq\|\mathbf{A}\|\left\|\mathbf{A}^{-1}\right\| \frac{\|\Delta \mathbf{A}\|}{\|\mathbf{A}\|} \]
la.norm(db) / la.norm(b + db)

0.00044789747999778414

la.norm(A) * la.norm(la.inv(A)) * la.norm(dA) / la.norm(A)

0.07290285213747842

M = np.zeros((8, 8))

M + 0.1

array([[0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1],
       [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1],
       [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1],
       [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1],
       [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1],
       [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1],
       [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1],
       [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1]])