maths.trapezoidal_rule

Numerical integration or quadrature for a smooth function f with known values at x_i

Functions

f(x)	This is the function to integrate, f(x) = (x - 0)^2 = x^2.
main()	Main function to test the trapezoidal rule.
make_points(a, b, h)	Generates points between a and b with step size h for trapezoidal integration.
trapezoidal_rule(boundary, steps)	Implements the extended trapezoidal rule for numerical integration.

Module Contents

maths.trapezoidal_rule.f(x)

This is the function to integrate, f(x) = (x - 0)^2 = x^2.

Parameters:

x – The input value

Returns:

The value of f(x)

>>> f(0)
0
>>> f(1)
1
>>> f(0.5)
0.25

maths.trapezoidal_rule.main()

Main function to test the trapezoidal rule. :a: Lower bound of integration :b: Upper bound of integration :steps: define number of steps or resolution :boundary: define boundary of integration

>>> main()
y = 0.3349999999999999

maths.trapezoidal_rule.make_points(a, b, h)

Generates points between a and b with step size h for trapezoidal integration.

Parameters:

• a – The lower bound of integration

• b – The upper bound of integration

• h – The step size

Yield:

The next x-value in the range (a, b)

>>> list(make_points(0, 1, 0.1))    
[0.1, 0.2, 0.30000000000000004, 0.4, 0.5, 0.6, 0.7, 0.7999999999999999,     0.8999999999999999]
>>> list(make_points(0, 10, 2.5))
[2.5, 5.0, 7.5]
>>> list(make_points(0, 10, 2))
[2, 4, 6, 8]
>>> list(make_points(1, 21, 5))
[6, 11, 16]
>>> list(make_points(1, 5, 2))
[3]
>>> list(make_points(1, 4, 3))
[]

maths.trapezoidal_rule.trapezoidal_rule(boundary, steps)

Implements the extended trapezoidal rule for numerical integration. The function f(x) is provided below.

Parameters:

• boundary – List containing the lower and upper bounds of integration [a, b]

• steps – The number of steps (intervals) used in the approximation

Returns:

The numerical approximation of the integral

>>> abs(trapezoidal_rule([0, 1], 10) - 0.33333) < 0.01
True
>>> abs(trapezoidal_rule([0, 1], 100) - 0.33333) < 0.01
True
>>> abs(trapezoidal_rule([0, 2], 1000) - 2.66667) < 0.01
True
>>> abs(trapezoidal_rule([1, 2], 1000) - 2.33333) < 0.01
True