Editing Numerical differentiation (section)

=== Code ===
The following is an example of a [[Python (programming language)|Python]] implementation for finding derivatives numerically for <math>f(x) = \frac{2x}{1+\sqrt{x}}</math> using the various three-point difference formulas at <math>x_{0} = 4</math>. The function <code>func</code> has derivative <code>func_prime</code>.

{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
|'''Example implementation in [[Python (programming language)|Python]]'''
|-
|<syntaxhighlight lang="python3" line="1">
import math

def func(x):
    return (2*x) / (1 + math.sqrt(x))

def func_prime(x):
    return (2 + math.sqrt(x)) / ((1 + math.sqrt(x))**2)

def three_point_forward(value, h):
    return ((-3/2) * func(value) + 2*func(value + h) - (1/2)*func(value + 2*h)) / h

def three_point_backward(value, h):
    return ((-1/2)*func(value - h) + (1/2)*func(value + h)) / h

def three_point_central(value, h):
    return ((1/2)*func(value - 2*h) - 2*func(value - h) + (3/2)*func(value)) / h

value = 4
actual = func_prime(value)

print("Actual value " + str(actual))
print("============================================")

for step_size in [0.1, 0.01, 0.001, 0.0001]:
    
    print("Step size " + str(step_size))

    forward = three_point_forward(value, step_size)
    backward = three_point_backward(value, step_size)
    central = three_point_central(value, step_size)
    
    print("Forward  {:>12}, Error = {:>12}".format(str(forward), str(abs(forward - actual))))
    print("Backward {:>12}, Error = {:>12}".format(str(forward), str(abs(backward - actual))))
    print("Central  {:>12}, Error = {:>12}".format(str(forward), str(abs(central - actual))))
    print("============================================")
</syntaxhighlight>
|}

{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
|'''Output'''
|-
|<syntaxhighlight lang="python3" line="1">
Actual value 0.4444444444444444
============================================
Step size 0.1
Forward  0.4443963018050967, Error = 4.814263934771468e-05
Backward 0.4443963018050967, Error = 2.5082646145202503e-05
Central  0.4443963018050967, Error = 5.231976394060034e-05
============================================
Step size 0.01
Forward  0.4444439449793336, Error = 4.994651108258807e-07
Backward 0.4444439449793336, Error = 2.507721614808389e-07
Central  0.4444439449793336, Error = 5.036366184096863e-07
============================================
Step size 0.001
Forward  0.4444444394311464, Error = 5.013297998957e-09
Backward 0.4444444394311464, Error = 2.507574814458735e-09
Central  0.4444444394311464, Error = 5.017960935660426e-09
============================================
Step size 0.0001
Forward  0.4444444443896245, Error = 5.4819926376126205e-11
Backward 0.4444444443896245, Error = 2.5116131396885066e-11
Central  0.4444444443896245, Error = 5.037903427762558e-11
============================================
</syntaxhighlight>
|}