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chore: improve comments and add tests to trapezoidal rule (#11640)
* chore: improve comments and add tests to trapezoidal rule * fix: too much characters in line * Update maths/trapezoidal_rule.py Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> * Update maths/trapezoidal_rule.py Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> * Update maths/trapezoidal_rule.py Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> * Update maths/trapezoidal_rule.py Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> * fix: change function name in calls * modify tests, changes numbers to remove coma * updating DIRECTORY.md * Fix doctest whitespace * Try to fix line length in doctest --------- Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com> Co-authored-by: tianyizheng02 <tianyizheng02@users.noreply.github.com>
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Julien Richard
GitHub
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@ -142,6 +142,7 @@
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* [Haralick Descriptors](computer_vision/haralick_descriptors.py)
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* [Harris Corner](computer_vision/harris_corner.py)
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* [Horn Schunck](computer_vision/horn_schunck.py)
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* [Intensity Based Segmentation](computer_vision/intensity_based_segmentation.py)
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* [Mean Threshold](computer_vision/mean_threshold.py)
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* [Mosaic Augmentation](computer_vision/mosaic_augmentation.py)
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* [Pooling Functions](computer_vision/pooling_functions.py)
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@ -507,6 +508,7 @@
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* [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
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* [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
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* [Karger](graphs/karger.py)
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* [Lanczos Eigenvectors](graphs/lanczos_eigenvectors.py)
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* [Markov Chain](graphs/markov_chain.py)
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* [Matching Min Vertex Cover](graphs/matching_min_vertex_cover.py)
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* [Minimum Path Sum](graphs/minimum_path_sum.py)
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@ -886,6 +888,7 @@
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* [N Body Simulation](physics/n_body_simulation.py)
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* [Newtons Law Of Gravitation](physics/newtons_law_of_gravitation.py)
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* [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
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* [Period Of Pendulum](physics/period_of_pendulum.py)
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* [Photoelectric Effect](physics/photoelectric_effect.py)
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* [Potential Energy](physics/potential_energy.py)
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* [Rainfall Intensity](physics/rainfall_intensity.py)
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@ -1,28 +1,25 @@
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"""
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Numerical integration or quadrature for a smooth function f with known values at x_i
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This method is the classical approach of suming 'Equally Spaced Abscissas'
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method 1:
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"extended trapezoidal rule"
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int(f) = dx/2 * (f1 + 2f2 + ... + fn)
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"""
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def method_1(boundary, steps):
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def trapezoidal_rule(boundary, steps):
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"""
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Apply the extended trapezoidal rule to approximate the integral of function f(x)
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over the interval defined by 'boundary' with the number of 'steps'.
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Implements the extended trapezoidal rule for numerical integration.
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The function f(x) is provided below.
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Args:
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boundary (list of floats): A list containing the start and end values [a, b].
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steps (int): The number of steps or subintervals.
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Returns:
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float: Approximation of the integral of f(x) over [a, b].
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Examples:
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>>> method_1([0, 1], 10)
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0.3349999999999999
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:param boundary: List containing the lower and upper bounds of integration [a, b]
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:param steps: The number of steps (intervals) used in the approximation
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:return: The numerical approximation of the integral
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>>> abs(trapezoidal_rule([0, 1], 10) - 0.33333) < 0.01
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True
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>>> abs(trapezoidal_rule([0, 1], 100) - 0.33333) < 0.01
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True
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>>> abs(trapezoidal_rule([0, 2], 1000) - 2.66667) < 0.01
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True
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>>> abs(trapezoidal_rule([1, 2], 1000) - 2.33333) < 0.01
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True
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"""
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h = (boundary[1] - boundary[0]) / steps
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a = boundary[0]
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@ -31,7 +28,6 @@ def method_1(boundary, steps):
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y = 0.0
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y += (h / 2.0) * f(a)
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for i in x_i:
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# print(i)
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y += h * f(i)
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y += (h / 2.0) * f(b)
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return y
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@ -39,49 +35,66 @@ def method_1(boundary, steps):
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def make_points(a, b, h):
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"""
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Generates points between 'a' and 'b' with step size 'h', excluding the end points.
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Args:
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a (float): Start value
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b (float): End value
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h (float): Step size
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Examples:
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Generates points between a and b with step size h for trapezoidal integration.
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:param a: The lower bound of integration
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:param b: The upper bound of integration
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:param h: The step size
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:yield: The next x-value in the range (a, b)
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>>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
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[0.1, 0.2, 0.30000000000000004, 0.4, 0.5, 0.6, 0.7, 0.7999999999999999, \
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0.8999999999999999]
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>>> list(make_points(0, 10, 2.5))
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[2.5, 5.0, 7.5]
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>>> list(make_points(0, 10, 2))
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[2, 4, 6, 8]
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>>> list(make_points(1, 21, 5))
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[6, 11, 16]
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>>> list(make_points(1, 5, 2))
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[3]
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>>> list(make_points(1, 4, 3))
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[]
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"""
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x = a + h
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while x <= (b - h):
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yield x
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x = x + h
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x += h
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def f(x): # enter your function here
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def f(x):
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"""
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Example:
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>>> f(2)
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4
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This is the function to integrate, f(x) = (x - 0)^2 = x^2.
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:param x: The input value
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:return: The value of f(x)
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>>> f(0)
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0
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>>> f(1)
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1
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>>> f(0.5)
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0.25
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"""
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y = (x - 0) * (x - 0)
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return y
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return x**2
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def main():
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a = 0.0 # Lower bound of integration
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b = 1.0 # Upper bound of integration
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steps = 10.0 # define number of steps or resolution
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boundary = [a, b] # define boundary of integration
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y = method_1(boundary, steps)
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"""
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Main function to test the trapezoidal rule.
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:a: Lower bound of integration
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:b: Upper bound of integration
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:steps: define number of steps or resolution
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:boundary: define boundary of integration
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>>> main()
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y = 0.3349999999999999
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"""
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a = 0.0
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b = 1.0
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steps = 10.0
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boundary = [a, b]
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y = trapezoidal_rule(boundary, steps)
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print(f"y = {y}")
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