CoCalc -- 5.ipynb

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Project: Akin - Spring 2022

Path: Computer Problems/Section 5.1/5.ipynb

Views: ⁵¹
Image: ubuntu2004

Kernel: Python 3 (system-wide)

In [1]:

import numpy as np
import matplotlib.pyplot as plt

In [2]:

def three_point(x,h,f):
    return (f(x-h) - f(x+h))/(2*h)

In [3]:

def f(x):
    return np.sin(x)-np.cos(x)

In [4]:

x = 0
x

Out[4]:

0

In [5]:

df = (np.sin(0)+np.cos(0))
df

Out[5]:

1.0

In [6]:

p1 = np.abs(three_point(x,(10**(-1)),f)) 
p1

Out[6]:

0.998334166468281

In [7]:

p2 = np.abs(three_point(x,(10**(-2)),f)) 
p2

Out[7]:

0.9999833334166652

In [8]:

p3 = np.abs(three_point(x,(10**(-3)),f)) 
p3

Out[8]:

0.9999998333333759

In [9]:

p4 = np.abs(three_point(x,(10**(-4)),f)) 
p4

Out[9]:

0.99999999833289

In [10]:

p5 = np.abs(three_point(x,(10**(-5)),f)) 
p5

Out[10]:

0.9999999999843466

In [11]:

p6 = np.abs(three_point(x,(10**(-6)),f)) 
p6

Out[11]:

0.9999999999732445

In [12]:

p7 = np.abs(three_point(x,(10**(-7)),f)) 
p7

Out[12]:

0.9999999994736442

In [13]:

p8 = np.abs(three_point(x,(10**(-8)),f)) 
p8

Out[13]:

0.9999999994736442

In [14]:

p9 = np.abs(three_point(x,(10**(-9)),f)) 
p9

Out[14]:

1.0000000272292198

In [15]:

p10 = np.abs(three_point(x,(10**(-10)),f)) 
p10

Out[15]:

1.000000082740371

In [16]:

p11 = np.abs(three_point(x,(10**(-11)),f)) 
p11

Out[16]:

1.000000082740371

In [17]:

p12 = np.abs(three_point(x,(10**(-12)),f)) 
p12

Out[17]:

1.0000333894311098

In [18]:

p = np.array([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12]) 
p

Out[18]:

array([0.99833417, 0.99998333, 0.99999983, 1.        , 1.        ,
       1.        , 1.        , 1.        , 1.00000003, 1.00000008,
       1.00000008, 1.00003339])

In [19]:

E1 = df - p1
E1

Out[19]:

0.0016658335317190076

In [20]:

E2 = df - p2
E2

Out[20]:

1.6666583334767893e-05

In [21]:

E3 = df - p3
E3

Out[21]:

1.6666662405739885e-07

In [22]:

E4 = df - p4
E4

Out[22]:

1.6671100055987154e-09

In [23]:

E5 = df - p5
E5

Out[23]:

1.565336749109747e-11

In [24]:

E6 = df - p6
E6

Out[24]:

2.6755486715046572e-11

In [25]:

E7 = df - p7
E7
E7

Out[25]:

5.26355847796367e-10

In [26]:

E8 = df - p6
E8

Out[26]:

2.6755486715046572e-11

In [27]:

E9 = df - p9
E9

Out[27]:

-2.7229219767832546e-08

In [29]:

E10 = df - p10
E10

Out[29]:

-8.274037099909037e-08

In [30]:

E11 = df - p11
E11

Out[30]:

-8.274037099909037e-08

In [31]:

E12 = df - p12
E12

Out[31]:

-3.3389431109753787e-05

In [32]:

E = np.array([E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,E11,E12])
E

Out[32]:

array([ 1.66583353e-03,  1.66665833e-05,  1.66666624e-07,  1.66711001e-09,
        1.56533675e-11,  2.67554867e-11,  5.26355848e-10,  2.67554867e-11,
       -2.72292198e-08, -8.27403710e-08, -8.27403710e-08, -3.33894311e-05])

In [33]:

F1 = np.stack((p,E),axis=1)
F1

Out[33]:

array([[ 9.98334166e-01,  1.66583353e-03],
       [ 9.99983333e-01,  1.66665833e-05],
       [ 9.99999833e-01,  1.66666624e-07],
       [ 9.99999998e-01,  1.66711001e-09],
       [ 1.00000000e+00,  1.56533675e-11],
       [ 1.00000000e+00,  2.67554867e-11],
       [ 9.99999999e-01,  5.26355848e-10],
       [ 9.99999999e-01,  2.67554867e-11],
       [ 1.00000003e+00, -2.72292198e-08],
       [ 1.00000008e+00, -8.27403710e-08],
       [ 1.00000008e+00, -8.27403710e-08],
       [ 1.00003339e+00, -3.33894311e-05]])

In [34]:

h=np.array([10**(-1),10**(-2),10**(-3),10**(-4),10**(-5),10**(-6),10**(-7),10**(-8),10**(-9),10**(-10),10**(-11),10**(-12)])
h

Out[34]:

array([1.e-01, 1.e-02, 1.e-03, 1.e-04, 1.e-05, 1.e-06, 1.e-07, 1.e-08,
       1.e-09, 1.e-10, 1.e-11, 1.e-12])

In [35]:

plt.plot(h,E,color='red')

Out[35]:

[<matplotlib.lines.Line2D at 0x7f1cd6eaf5e0>]

In [0]:

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