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tianyizheng02 merged 12 commits into from
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chore: improve comments and add tests to trapezoidal rule #11640

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2e67da3
chore: improve comments and add tests to trapezoidal rule
jurichar Oct 1, 2024
aa5a885
fix: too much characters in line
jurichar Oct 1, 2024
79ed4e7
Update maths/trapezoidal_rule.py
jurichar Oct 6, 2024
bda4d82
Update maths/trapezoidal_rule.py
jurichar Oct 6, 2024
9190304
Update maths/trapezoidal_rule.py
jurichar Oct 6, 2024
2409e25
Update maths/trapezoidal_rule.py
jurichar Oct 6, 2024
242b208
fix: change function name in calls
jurichar Oct 6, 2024
d8c8ace
modify tests, changes numbers to remove coma
jurichar Oct 6, 2024
911943b
Merge branch 'master' into issue_fix
tianyizheng02 Dec 31, 2024
8d7d6f1
updating DIRECTORY.md
tianyizheng02 Dec 31, 2024
1c0debe
Fix doctest whitespace
tianyizheng02 Dec 31, 2024
0c5b78d
Try to fix line length in doctest
tianyizheng02 Dec 31, 2024
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3 changes: 3 additions & 0 deletions DIRECTORY.md
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Original file line numberDiff line numberDiff line change
Expand Up@@ -142,6 +142,7 @@
* [Haralick Descriptors](computer_vision/haralick_descriptors.py)
* [Harris Corner](computer_vision/harris_corner.py)
* [Horn Schunck](computer_vision/horn_schunck.py)
* [Intensity Based Segmentation](computer_vision/intensity_based_segmentation.py)
* [Mean Threshold](computer_vision/mean_threshold.py)
* [Mosaic Augmentation](computer_vision/mosaic_augmentation.py)
* [Pooling Functions](computer_vision/pooling_functions.py)
Expand DownExpand Up@@ -507,6 +508,7 @@
* [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
* [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
* [Karger](graphs/karger.py)
* [Lanczos Eigenvectors](graphs/lanczos_eigenvectors.py)
* [Markov Chain](graphs/markov_chain.py)
* [Matching Min Vertex Cover](graphs/matching_min_vertex_cover.py)
* [Minimum Path Sum](graphs/minimum_path_sum.py)
Expand DownExpand Up@@ -886,6 +888,7 @@
* [N Body Simulation](physics/n_body_simulation.py)
* [Newtons Law Of Gravitation](physics/newtons_law_of_gravitation.py)
* [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
* [Period Of Pendulum](physics/period_of_pendulum.py)
* [Photoelectric Effect](physics/photoelectric_effect.py)
* [Potential Energy](physics/potential_energy.py)
* [Rainfall Intensity](physics/rainfall_intensity.py)
Expand Down
97 changes: 55 additions & 42 deletions maths/trapezoidal_rule.py
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Original file line numberDiff line numberDiff line change
@@ -1,28 +1,25 @@
"""
Numerical integration or quadrature for a smooth function f with known values at x_i

This method is the classical approach of suming 'Equally Spaced Abscissas'

method 1:
"extended trapezoidal rule"
int(f) = dx/2 * (f1 + 2f2 + ... + fn)

"""


def method_1(boundary, steps):
def trapezoidal_rule(boundary, steps):
"""
Apply the extended trapezoidal rule to approximate the integral of function f(x)
over the interval defined by 'boundary' with the number of 'steps'.

Args:
boundary (list of floats): A list containing the start and end values [a, b].
steps (int): The number of steps or subintervals.
Returns:
float: Approximation of the integral of f(x) over [a, b].
Examples:
>>> method_1([0, 1], 10)
0.3349999999999999
Implements the extended trapezoidal rule for numerical integration.
The function f(x) is provided below.

:param boundary: List containing the lower and upper bounds of integration [a, b]
:param steps: The number of steps (intervals) used in the approximation
:return: 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
"""
h = (boundary[1] - boundary[0]) / steps
a = boundary[0]
Expand All@@ -31,57 +28,73 @@ def method_1(boundary, steps):
y = 0.0
y += (h / 2.0) * f(a)
for i in x_i:
# print(i)
y += h * f(i)
y += (h / 2.0) * f(b)
return y


def make_points(a, b, h):
"""
Generates points between 'a' and 'b' with step size 'h', excluding the end points.
Args:
a (float): Start value
b (float): End value
h (float): Step size
Examples:
Generates points between a and b with step size h for trapezoidal integration.

:param a: The lower bound of integration
:param b: The upper bound of integration
:param h: The step size
:yield: The next x-value in the range (a, b)

>>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
[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))
[]
"""
x = a + h
while x <= (b - h):
yield x
x = x + h
x += h


def f(x): # enter your function here
def f(x):
"""
Example:
>>> f(2)
4
This is the function to integrate, f(x) = (x - 0)^2 = x^2.

:param x: The input value
:return: The value of f(x)

>>> f(0)
0
>>> f(1)
1
>>> f(0.5)
0.25
"""
y = (x - 0) * (x - 0)
return y
return x**2


def main():
a = 0.0 # Lower bound of integration
b = 1.0 # Upper bound of integration
steps = 10.0 # define number of steps or resolution
boundary = [a, b] # define boundary of integration
y = method_1(boundary, steps)
"""
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
"""
a = 0.0
b = 1.0
steps = 10.0
boundary = [a, b]
y = trapezoidal_rule(boundary, steps)
print(f"y = {y}")


Expand Down
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chore: improve comments and add tests to trapezoidal rule by jurichar · Pull Request #11640 · TheAlgorithms/Python · GitHub
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chore: improve comments and add tests to trapezoidal rule#11640

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tianyizheng02
merged 12 commits into

from

Dec 31, 2024
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chore: improve comments and add tests to trapezoidal rule
#11640
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2e67da3

chore: improve comments and add tests to trapezoidal rule


jurichar Oct 1, 2024
aa5a885

fix: too much characters in line


jurichar Oct 1, 2024
79ed4e7

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
bda4d82

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
9190304

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
2409e25

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
242b208

fix: change function name in calls


jurichar Oct 6, 2024
d8c8ace

modify tests, changes numbers to remove coma


jurichar Oct 6, 2024
911943b

Merge branch 'master' into issue_fix


tianyizheng02 Dec 31, 2024
8d7d6f1

updating DIRECTORY.md


tianyizheng02 Dec 31, 2024
1c0debe

Fix doctest whitespace


tianyizheng02 Dec 31, 2024
0c5b78d

Try to fix line length in doctest


tianyizheng02 Dec 31, 2024

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3 changes: 3 additions & 0 deletions
DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -142,6 +142,7 @@
 * [Haralick Descriptors](computer_vision/haralick_descriptors.py)
 * [Harris Corner](computer_vision/harris_corner.py)
 * [Horn Schunck](computer_vision/horn_schunck.py)
 * [Intensity Based Segmentation](computer_vision/intensity_based_segmentation.py)
 * [Mean Threshold](computer_vision/mean_threshold.py)
 * [Mosaic Augmentation](computer_vision/mosaic_augmentation.py)
 * [Pooling Functions](computer_vision/pooling_functions.py)
Expand DownExpand Up@@ -507,6 +508,7 @@
 * [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
 * [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
 * [Karger](graphs/karger.py)
 * [Lanczos Eigenvectors](graphs/lanczos_eigenvectors.py)
 * [Markov Chain](graphs/markov_chain.py)
 * [Matching Min Vertex Cover](graphs/matching_min_vertex_cover.py)
 * [Minimum Path Sum](graphs/minimum_path_sum.py)
Expand DownExpand Up@@ -886,6 +888,7 @@
 * [N Body Simulation](physics/n_body_simulation.py)
 * [Newtons Law Of Gravitation](physics/newtons_law_of_gravitation.py)
 * [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
 * [Period Of Pendulum](physics/period_of_pendulum.py)
 * [Photoelectric Effect](physics/photoelectric_effect.py)
 * [Potential Energy](physics/potential_energy.py)
 * [Rainfall Intensity](physics/rainfall_intensity.py)
Expand Down

97 changes: 55 additions & 42 deletions
maths/trapezoidal_rule.py

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -1,28 +1,25 @@
"""
Numerical integration or quadrature for a smooth function f with known values at x_i

This method is the classical approach of suming 'Equally Spaced Abscissas'

method 1:
"extended trapezoidal rule"
int(f) = dx/2 * (f1 + 2f2 + ... + fn)

"""


def method_1(boundary, steps):
def trapezoidal_rule(boundary, steps):
 """
 Apply the extended trapezoidal rule to approximate the integral of function f(x)
 over the interval defined by 'boundary' with the number of 'steps'.

 Args:
 boundary (list of floats): A list containing the start and end values [a, b].
 steps (int): The number of steps or subintervals.
 Returns:
 float: Approximation of the integral of f(x) over [a, b].
 Examples:
 >>> method_1([0, 1], 10)
 0.3349999999999999
 Implements the extended trapezoidal rule for numerical integration.
 The function f(x) is provided below.

 :param boundary: List containing the lower and upper bounds of integration [a, b]
 :param steps: The number of steps (intervals) used in the approximation
 :return: 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
 """
 h = (boundary[1] - boundary[0]) / steps
 a = boundary[0]
Expand All@@ -31,57 +28,73 @@ def method_1(boundary, steps):
 y = 0.0
 y += (h / 2.0) * f(a)
 for i in x_i:
 # print(i)
 y += h * f(i)
 y += (h / 2.0) * f(b)
 return y


def make_points(a, b, h):
 """
 Generates points between 'a' and 'b' with step size 'h', excluding the end points.
 Args:
 a (float): Start value
 b (float): End value
 h (float): Step size
 Examples:
 Generates points between a and b with step size h for trapezoidal integration.

 :param a: The lower bound of integration
 :param b: The upper bound of integration
 :param h: The step size
 :yield: The next x-value in the range (a, b)

 >>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
 [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))
 []
 """
 x = a + h
 while x <= (b - h):
 yield x
 x = x + h
 x += h


def f(x): # enter your function here
def f(x):
 """
 Example:
 >>> f(2)
 4
 This is the function to integrate, f(x) = (x - 0)^2 = x^2.

 :param x: The input value
 :return: The value of f(x)

 >>> f(0)
 0
 >>> f(1)
 1
 >>> f(0.5)
 0.25
 """
 y = (x - 0) * (x - 0)
 return y
 return x**2


def main():
 a = 0.0 # Lower bound of integration
 b = 1.0 # Upper bound of integration
 steps = 10.0 # define number of steps or resolution
 boundary = [a, b] # define boundary of integration
 y = method_1(boundary, steps)
 """
 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
 """
 a = 0.0
 b = 1.0
 steps = 10.0
 boundary = [a, b]
 y = trapezoidal_rule(boundary, steps)
 print(f"y = {y}")


Expand Down

, 'i');
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chore: improve comments and add tests to trapezoidal rule by jurichar · Pull Request #11640 · TheAlgorithms/Python · GitHub
Skip to content
chore: improve comments and add tests to trapezoidal rule#11640

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 Merged

tianyizheng02
merged 12 commits into

from

Dec 31, 2024
 Merged


chore: improve comments and add tests to trapezoidal rule
#11640
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2e67da3

chore: improve comments and add tests to trapezoidal rule


jurichar Oct 1, 2024
aa5a885

fix: too much characters in line


jurichar Oct 1, 2024
79ed4e7

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
bda4d82

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
9190304

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
2409e25

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
242b208

fix: change function name in calls


jurichar Oct 6, 2024
d8c8ace

modify tests, changes numbers to remove coma


jurichar Oct 6, 2024
911943b

Merge branch 'master' into issue_fix


tianyizheng02 Dec 31, 2024
8d7d6f1

updating DIRECTORY.md


tianyizheng02 Dec 31, 2024
1c0debe

Fix doctest whitespace


tianyizheng02 Dec 31, 2024
0c5b78d

Try to fix line length in doctest


tianyizheng02 Dec 31, 2024

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3 changes: 3 additions & 0 deletions
DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -142,6 +142,7 @@
 * [Haralick Descriptors](computer_vision/haralick_descriptors.py)
 * [Harris Corner](computer_vision/harris_corner.py)
 * [Horn Schunck](computer_vision/horn_schunck.py)
 * [Intensity Based Segmentation](computer_vision/intensity_based_segmentation.py)
 * [Mean Threshold](computer_vision/mean_threshold.py)
 * [Mosaic Augmentation](computer_vision/mosaic_augmentation.py)
 * [Pooling Functions](computer_vision/pooling_functions.py)
Expand DownExpand Up@@ -507,6 +508,7 @@
 * [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
 * [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
 * [Karger](graphs/karger.py)
 * [Lanczos Eigenvectors](graphs/lanczos_eigenvectors.py)
 * [Markov Chain](graphs/markov_chain.py)
 * [Matching Min Vertex Cover](graphs/matching_min_vertex_cover.py)
 * [Minimum Path Sum](graphs/minimum_path_sum.py)
Expand DownExpand Up@@ -886,6 +888,7 @@
 * [N Body Simulation](physics/n_body_simulation.py)
 * [Newtons Law Of Gravitation](physics/newtons_law_of_gravitation.py)
 * [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
 * [Period Of Pendulum](physics/period_of_pendulum.py)
 * [Photoelectric Effect](physics/photoelectric_effect.py)
 * [Potential Energy](physics/potential_energy.py)
 * [Rainfall Intensity](physics/rainfall_intensity.py)
Expand Down

97 changes: 55 additions & 42 deletions
maths/trapezoidal_rule.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -1,28 +1,25 @@
"""
Numerical integration or quadrature for a smooth function f with known values at x_i

This method is the classical approach of suming 'Equally Spaced Abscissas'

method 1:
"extended trapezoidal rule"
int(f) = dx/2 * (f1 + 2f2 + ... + fn)

"""


def method_1(boundary, steps):
def trapezoidal_rule(boundary, steps):
 """
 Apply the extended trapezoidal rule to approximate the integral of function f(x)
 over the interval defined by 'boundary' with the number of 'steps'.

 Args:
 boundary (list of floats): A list containing the start and end values [a, b].
 steps (int): The number of steps or subintervals.
 Returns:
 float: Approximation of the integral of f(x) over [a, b].
 Examples:
 >>> method_1([0, 1], 10)
 0.3349999999999999
 Implements the extended trapezoidal rule for numerical integration.
 The function f(x) is provided below.

 :param boundary: List containing the lower and upper bounds of integration [a, b]
 :param steps: The number of steps (intervals) used in the approximation
 :return: 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
 """
 h = (boundary[1] - boundary[0]) / steps
 a = boundary[0]
Expand All@@ -31,57 +28,73 @@ def method_1(boundary, steps):
 y = 0.0
 y += (h / 2.0) * f(a)
 for i in x_i:
 # print(i)
 y += h * f(i)
 y += (h / 2.0) * f(b)
 return y


def make_points(a, b, h):
 """
 Generates points between 'a' and 'b' with step size 'h', excluding the end points.
 Args:
 a (float): Start value
 b (float): End value
 h (float): Step size
 Examples:
 Generates points between a and b with step size h for trapezoidal integration.

 :param a: The lower bound of integration
 :param b: The upper bound of integration
 :param h: The step size
 :yield: The next x-value in the range (a, b)

 >>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
 [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))
 []
 """
 x = a + h
 while x <= (b - h):
 yield x
 x = x + h
 x += h


def f(x): # enter your function here
def f(x):
 """
 Example:
 >>> f(2)
 4
 This is the function to integrate, f(x) = (x - 0)^2 = x^2.

 :param x: The input value
 :return: The value of f(x)

 >>> f(0)
 0
 >>> f(1)
 1
 >>> f(0.5)
 0.25
 """
 y = (x - 0) * (x - 0)
 return y
 return x**2


def main():
 a = 0.0 # Lower bound of integration
 b = 1.0 # Upper bound of integration
 steps = 10.0 # define number of steps or resolution
 boundary = [a, b] # define boundary of integration
 y = method_1(boundary, steps)
 """
 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
 """
 a = 0.0
 b = 1.0
 steps = 10.0
 boundary = [a, b]
 y = trapezoidal_rule(boundary, steps)
 print(f"y = {y}")


Expand Down

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Highlight search terms from Google/DuckDuckGo/Bing referrer
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var ref = document.referrer;
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if (q) {
terms = q.split(/\s+/).filter(function(t) { return t.length > 2; });
}
}
if (terms.length === 0) return;
var style = document.createElement('style');
style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';
document.head.appendChild(style);
function highlight(node) {
if (node.nodeType === 3) { // text node
var text = node.textContent;
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terms.forEach(function(term) {
var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\]\\]/g, '\\') + ')', 'gi');
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found = true;
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chore: improve comments and add tests to trapezoidal rule by jurichar · Pull Request #11640 · TheAlgorithms/Python · GitHub
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chore: improve comments and add tests to trapezoidal rule


jurichar Oct 1, 2024
aa5a885

fix: too much characters in line


jurichar Oct 1, 2024
79ed4e7

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
bda4d82

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
9190304

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
2409e25

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
242b208

fix: change function name in calls


jurichar Oct 6, 2024
d8c8ace

modify tests, changes numbers to remove coma


jurichar Oct 6, 2024
911943b

Merge branch 'master' into issue_fix


tianyizheng02 Dec 31, 2024
8d7d6f1

updating DIRECTORY.md


tianyizheng02 Dec 31, 2024
1c0debe

Fix doctest whitespace


tianyizheng02 Dec 31, 2024
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Try to fix line length in doctest


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DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -142,6 +142,7 @@
 * [Haralick Descriptors](computer_vision/haralick_descriptors.py)
 * [Harris Corner](computer_vision/harris_corner.py)
 * [Horn Schunck](computer_vision/horn_schunck.py)
 * [Intensity Based Segmentation](computer_vision/intensity_based_segmentation.py)
 * [Mean Threshold](computer_vision/mean_threshold.py)
 * [Mosaic Augmentation](computer_vision/mosaic_augmentation.py)
 * [Pooling Functions](computer_vision/pooling_functions.py)
Expand DownExpand Up@@ -507,6 +508,7 @@
 * [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
 * [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
 * [Karger](graphs/karger.py)
 * [Lanczos Eigenvectors](graphs/lanczos_eigenvectors.py)
 * [Markov Chain](graphs/markov_chain.py)
 * [Matching Min Vertex Cover](graphs/matching_min_vertex_cover.py)
 * [Minimum Path Sum](graphs/minimum_path_sum.py)
Expand DownExpand Up@@ -886,6 +888,7 @@
 * [N Body Simulation](physics/n_body_simulation.py)
 * [Newtons Law Of Gravitation](physics/newtons_law_of_gravitation.py)
 * [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
 * [Period Of Pendulum](physics/period_of_pendulum.py)
 * [Photoelectric Effect](physics/photoelectric_effect.py)
 * [Potential Energy](physics/potential_energy.py)
 * [Rainfall Intensity](physics/rainfall_intensity.py)
Expand Down

97 changes: 55 additions & 42 deletions
maths/trapezoidal_rule.py

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -1,28 +1,25 @@
"""
Numerical integration or quadrature for a smooth function f with known values at x_i

This method is the classical approach of suming 'Equally Spaced Abscissas'

method 1:
"extended trapezoidal rule"
int(f) = dx/2 * (f1 + 2f2 + ... + fn)

"""


def method_1(boundary, steps):
def trapezoidal_rule(boundary, steps):
 """
 Apply the extended trapezoidal rule to approximate the integral of function f(x)
 over the interval defined by 'boundary' with the number of 'steps'.

 Args:
 boundary (list of floats): A list containing the start and end values [a, b].
 steps (int): The number of steps or subintervals.
 Returns:
 float: Approximation of the integral of f(x) over [a, b].
 Examples:
 >>> method_1([0, 1], 10)
 0.3349999999999999
 Implements the extended trapezoidal rule for numerical integration.
 The function f(x) is provided below.

 :param boundary: List containing the lower and upper bounds of integration [a, b]
 :param steps: The number of steps (intervals) used in the approximation
 :return: 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
 """
 h = (boundary[1] - boundary[0]) / steps
 a = boundary[0]
Expand All@@ -31,57 +28,73 @@ def method_1(boundary, steps):
 y = 0.0
 y += (h / 2.0) * f(a)
 for i in x_i:
 # print(i)
 y += h * f(i)
 y += (h / 2.0) * f(b)
 return y


def make_points(a, b, h):
 """
 Generates points between 'a' and 'b' with step size 'h', excluding the end points.
 Args:
 a (float): Start value
 b (float): End value
 h (float): Step size
 Examples:
 Generates points between a and b with step size h for trapezoidal integration.

 :param a: The lower bound of integration
 :param b: The upper bound of integration
 :param h: The step size
 :yield: The next x-value in the range (a, b)

 >>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
 [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))
 []
 """
 x = a + h
 while x <= (b - h):
 yield x
 x = x + h
 x += h


def f(x): # enter your function here
def f(x):
 """
 Example:
 >>> f(2)
 4
 This is the function to integrate, f(x) = (x - 0)^2 = x^2.

 :param x: The input value
 :return: The value of f(x)

 >>> f(0)
 0
 >>> f(1)
 1
 >>> f(0.5)
 0.25
 """
 y = (x - 0) * (x - 0)
 return y
 return x**2


def main():
 a = 0.0 # Lower bound of integration
 b = 1.0 # Upper bound of integration
 steps = 10.0 # define number of steps or resolution
 boundary = [a, b] # define boundary of integration
 y = method_1(boundary, steps)
 """
 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
 """
 a = 0.0
 b = 1.0
 steps = 10.0
 boundary = [a, b]
 y = trapezoidal_rule(boundary, steps)
 print(f"y = {y}")


Expand Down

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Strip utm_, fbclid, gclid, etc. from all links on page
(function() {
var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',
'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',
'ref', 'ref_src', 'source', 'medium', 'campaign'];
function cleanUrl(url) {
try {
var u = new URL(url, window.location.origin);
var changed = false;
trackingParams.forEach(function(p) {
if (u.searchParams.has(p)) {
u.searchParams.delete(p);
changed = true;
}
});
return changed ? u.toString() : url;
} catch (e) {
return url;
}
}
function cleanLinks() {
document.querySelectorAll('a[href]').forEach(function(a) {
var clean = cleanUrl(a.href);
if (clean !== a.href) a.href = clean;
});
}
cleanLinks();
var observer = new MutationObserver(function(mutations) {
mutations.forEach(function(m) {
m.addedNodes.forEach(function(node) {
if (node.nodeType === 1) {
if (node.tagName === 'A') cleanLinks();
node.querySelectorAll('a[href]').forEach(function(a) {
var clean = cleanUrl(a.href);
if (clean !== a.href) a.href = clean;
});
}
});
});
});
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
})();
(function(){
try {
var __m = "youtube.com";
var __re = new RegExp('^' + "youtube\\.com" + '
chore: improve comments and add tests to trapezoidal rule by jurichar · Pull Request #11640 · TheAlgorithms/Python · GitHub
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chore: improve comments and add tests to trapezoidal rule#11640

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#11640
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2e67da3

chore: improve comments and add tests to trapezoidal rule


jurichar Oct 1, 2024
aa5a885

fix: too much characters in line


jurichar Oct 1, 2024
79ed4e7

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
bda4d82

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
9190304

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
2409e25

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
242b208

fix: change function name in calls


jurichar Oct 6, 2024
d8c8ace

modify tests, changes numbers to remove coma


jurichar Oct 6, 2024
911943b

Merge branch 'master' into issue_fix


tianyizheng02 Dec 31, 2024
8d7d6f1

updating DIRECTORY.md


tianyizheng02 Dec 31, 2024
1c0debe

Fix doctest whitespace


tianyizheng02 Dec 31, 2024
0c5b78d

Try to fix line length in doctest


tianyizheng02 Dec 31, 2024

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DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -142,6 +142,7 @@
 * [Haralick Descriptors](computer_vision/haralick_descriptors.py)
 * [Harris Corner](computer_vision/harris_corner.py)
 * [Horn Schunck](computer_vision/horn_schunck.py)
 * [Intensity Based Segmentation](computer_vision/intensity_based_segmentation.py)
 * [Mean Threshold](computer_vision/mean_threshold.py)
 * [Mosaic Augmentation](computer_vision/mosaic_augmentation.py)
 * [Pooling Functions](computer_vision/pooling_functions.py)
Expand DownExpand Up@@ -507,6 +508,7 @@
 * [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
 * [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
 * [Karger](graphs/karger.py)
 * [Lanczos Eigenvectors](graphs/lanczos_eigenvectors.py)
 * [Markov Chain](graphs/markov_chain.py)
 * [Matching Min Vertex Cover](graphs/matching_min_vertex_cover.py)
 * [Minimum Path Sum](graphs/minimum_path_sum.py)
Expand DownExpand Up@@ -886,6 +888,7 @@
 * [N Body Simulation](physics/n_body_simulation.py)
 * [Newtons Law Of Gravitation](physics/newtons_law_of_gravitation.py)
 * [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
 * [Period Of Pendulum](physics/period_of_pendulum.py)
 * [Photoelectric Effect](physics/photoelectric_effect.py)
 * [Potential Energy](physics/potential_energy.py)
 * [Rainfall Intensity](physics/rainfall_intensity.py)
Expand Down

97 changes: 55 additions & 42 deletions
maths/trapezoidal_rule.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -1,28 +1,25 @@
"""
Numerical integration or quadrature for a smooth function f with known values at x_i

This method is the classical approach of suming 'Equally Spaced Abscissas'

method 1:
"extended trapezoidal rule"
int(f) = dx/2 * (f1 + 2f2 + ... + fn)

"""


def method_1(boundary, steps):
def trapezoidal_rule(boundary, steps):
 """
 Apply the extended trapezoidal rule to approximate the integral of function f(x)
 over the interval defined by 'boundary' with the number of 'steps'.

 Args:
 boundary (list of floats): A list containing the start and end values [a, b].
 steps (int): The number of steps or subintervals.
 Returns:
 float: Approximation of the integral of f(x) over [a, b].
 Examples:
 >>> method_1([0, 1], 10)
 0.3349999999999999
 Implements the extended trapezoidal rule for numerical integration.
 The function f(x) is provided below.

 :param boundary: List containing the lower and upper bounds of integration [a, b]
 :param steps: The number of steps (intervals) used in the approximation
 :return: 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
 """
 h = (boundary[1] - boundary[0]) / steps
 a = boundary[0]
Expand All@@ -31,57 +28,73 @@ def method_1(boundary, steps):
 y = 0.0
 y += (h / 2.0) * f(a)
 for i in x_i:
 # print(i)
 y += h * f(i)
 y += (h / 2.0) * f(b)
 return y


def make_points(a, b, h):
 """
 Generates points between 'a' and 'b' with step size 'h', excluding the end points.
 Args:
 a (float): Start value
 b (float): End value
 h (float): Step size
 Examples:
 Generates points between a and b with step size h for trapezoidal integration.

 :param a: The lower bound of integration
 :param b: The upper bound of integration
 :param h: The step size
 :yield: The next x-value in the range (a, b)

 >>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
 [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))
 []
 """
 x = a + h
 while x <= (b - h):
 yield x
 x = x + h
 x += h


def f(x): # enter your function here
def f(x):
 """
 Example:
 >>> f(2)
 4
 This is the function to integrate, f(x) = (x - 0)^2 = x^2.

 :param x: The input value
 :return: The value of f(x)

 >>> f(0)
 0
 >>> f(1)
 1
 >>> f(0.5)
 0.25
 """
 y = (x - 0) * (x - 0)
 return y
 return x**2


def main():
 a = 0.0 # Lower bound of integration
 b = 1.0 # Upper bound of integration
 steps = 10.0 # define number of steps or resolution
 boundary = [a, b] # define boundary of integration
 y = method_1(boundary, steps)
 """
 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
 """
 a = 0.0
 b = 1.0
 steps = 10.0
 boundary = [a, b]
 y = trapezoidal_rule(boundary, steps)
 print(f"y = {y}")


Expand Down

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Auto-enable theater mode on YouTube
(function() {
function tryTheater() {
var btn = document.querySelector('button[aria-label="Theater mode"], ytd-player #player button[title="Theater mode"]');
if (btn && !btn.classList.contains('activated')) {
btn.click();
}
}
// Try immediately
tryTheater();
// Try after navigation (SPA)
var lastUrl = location.href;
setInterval(function() {
if (location.href !== lastUrl) {
lastUrl = location.href;
setTimeout(tryTheater, 500);
}
}, 1000);
// Also try on player load
var observer = new MutationObserver(tryTheater);
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
chore: improve comments and add tests to trapezoidal rule by jurichar · Pull Request #11640 · TheAlgorithms/Python · GitHub
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chore: improve comments and add tests to trapezoidal rule#11640

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tianyizheng02
merged 12 commits into

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Dec 31, 2024
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chore: improve comments and add tests to trapezoidal rule
#11640
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2e67da3

chore: improve comments and add tests to trapezoidal rule


jurichar Oct 1, 2024
aa5a885

fix: too much characters in line


jurichar Oct 1, 2024
79ed4e7

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
bda4d82

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
9190304

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
2409e25

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
242b208

fix: change function name in calls


jurichar Oct 6, 2024
d8c8ace

modify tests, changes numbers to remove coma


jurichar Oct 6, 2024
911943b

Merge branch 'master' into issue_fix


tianyizheng02 Dec 31, 2024
8d7d6f1

updating DIRECTORY.md


tianyizheng02 Dec 31, 2024
1c0debe

Fix doctest whitespace


tianyizheng02 Dec 31, 2024
0c5b78d

Try to fix line length in doctest


tianyizheng02 Dec 31, 2024

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3 changes: 3 additions & 0 deletions
DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -142,6 +142,7 @@
 * [Haralick Descriptors](computer_vision/haralick_descriptors.py)
 * [Harris Corner](computer_vision/harris_corner.py)
 * [Horn Schunck](computer_vision/horn_schunck.py)
 * [Intensity Based Segmentation](computer_vision/intensity_based_segmentation.py)
 * [Mean Threshold](computer_vision/mean_threshold.py)
 * [Mosaic Augmentation](computer_vision/mosaic_augmentation.py)
 * [Pooling Functions](computer_vision/pooling_functions.py)
Expand DownExpand Up@@ -507,6 +508,7 @@
 * [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
 * [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
 * [Karger](graphs/karger.py)
 * [Lanczos Eigenvectors](graphs/lanczos_eigenvectors.py)
 * [Markov Chain](graphs/markov_chain.py)
 * [Matching Min Vertex Cover](graphs/matching_min_vertex_cover.py)
 * [Minimum Path Sum](graphs/minimum_path_sum.py)
Expand DownExpand Up@@ -886,6 +888,7 @@
 * [N Body Simulation](physics/n_body_simulation.py)
 * [Newtons Law Of Gravitation](physics/newtons_law_of_gravitation.py)
 * [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
 * [Period Of Pendulum](physics/period_of_pendulum.py)
 * [Photoelectric Effect](physics/photoelectric_effect.py)
 * [Potential Energy](physics/potential_energy.py)
 * [Rainfall Intensity](physics/rainfall_intensity.py)
Expand Down

97 changes: 55 additions & 42 deletions
maths/trapezoidal_rule.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -1,28 +1,25 @@
"""
Numerical integration or quadrature for a smooth function f with known values at x_i

This method is the classical approach of suming 'Equally Spaced Abscissas'

method 1:
"extended trapezoidal rule"
int(f) = dx/2 * (f1 + 2f2 + ... + fn)

"""


def method_1(boundary, steps):
def trapezoidal_rule(boundary, steps):
 """
 Apply the extended trapezoidal rule to approximate the integral of function f(x)
 over the interval defined by 'boundary' with the number of 'steps'.

 Args:
 boundary (list of floats): A list containing the start and end values [a, b].
 steps (int): The number of steps or subintervals.
 Returns:
 float: Approximation of the integral of f(x) over [a, b].
 Examples:
 >>> method_1([0, 1], 10)
 0.3349999999999999
 Implements the extended trapezoidal rule for numerical integration.
 The function f(x) is provided below.

 :param boundary: List containing the lower and upper bounds of integration [a, b]
 :param steps: The number of steps (intervals) used in the approximation
 :return: 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
 """
 h = (boundary[1] - boundary[0]) / steps
 a = boundary[0]
Expand All@@ -31,57 +28,73 @@ def method_1(boundary, steps):
 y = 0.0
 y += (h / 2.0) * f(a)
 for i in x_i:
 # print(i)
 y += h * f(i)
 y += (h / 2.0) * f(b)
 return y


def make_points(a, b, h):
 """
 Generates points between 'a' and 'b' with step size 'h', excluding the end points.
 Args:
 a (float): Start value
 b (float): End value
 h (float): Step size
 Examples:
 Generates points between a and b with step size h for trapezoidal integration.

 :param a: The lower bound of integration
 :param b: The upper bound of integration
 :param h: The step size
 :yield: The next x-value in the range (a, b)

 >>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
 [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))
 []
 """
 x = a + h
 while x <= (b - h):
 yield x
 x = x + h
 x += h


def f(x): # enter your function here
def f(x):
 """
 Example:
 >>> f(2)
 4
 This is the function to integrate, f(x) = (x - 0)^2 = x^2.

 :param x: The input value
 :return: The value of f(x)

 >>> f(0)
 0
 >>> f(1)
 1
 >>> f(0.5)
 0.25
 """
 y = (x - 0) * (x - 0)
 return y
 return x**2


def main():
 a = 0.0 # Lower bound of integration
 b = 1.0 # Upper bound of integration
 steps = 10.0 # define number of steps or resolution
 boundary = [a, b] # define boundary of integration
 y = method_1(boundary, steps)
 """
 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
 """
 a = 0.0
 b = 1.0
 steps = 10.0
 boundary = [a, b]
 y = trapezoidal_rule(boundary, steps)
 print(f"y = {y}")


Expand Down

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chore: improve comments and add tests to trapezoidal rule by jurichar · Pull Request #11640 · TheAlgorithms/Python · GitHub
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#11640
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2e67da3

chore: improve comments and add tests to trapezoidal rule


jurichar Oct 1, 2024
aa5a885

fix: too much characters in line


jurichar Oct 1, 2024
79ed4e7

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
bda4d82

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
9190304

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
2409e25

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
242b208

fix: change function name in calls


jurichar Oct 6, 2024
d8c8ace

modify tests, changes numbers to remove coma


jurichar Oct 6, 2024
911943b

Merge branch 'master' into issue_fix


tianyizheng02 Dec 31, 2024
8d7d6f1

updating DIRECTORY.md


tianyizheng02 Dec 31, 2024
1c0debe

Fix doctest whitespace


tianyizheng02 Dec 31, 2024
0c5b78d

Try to fix line length in doctest


tianyizheng02 Dec 31, 2024

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DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -142,6 +142,7 @@
 * [Haralick Descriptors](computer_vision/haralick_descriptors.py)
 * [Harris Corner](computer_vision/harris_corner.py)
 * [Horn Schunck](computer_vision/horn_schunck.py)
 * [Intensity Based Segmentation](computer_vision/intensity_based_segmentation.py)
 * [Mean Threshold](computer_vision/mean_threshold.py)
 * [Mosaic Augmentation](computer_vision/mosaic_augmentation.py)
 * [Pooling Functions](computer_vision/pooling_functions.py)
Expand DownExpand Up@@ -507,6 +508,7 @@
 * [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
 * [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
 * [Karger](graphs/karger.py)
 * [Lanczos Eigenvectors](graphs/lanczos_eigenvectors.py)
 * [Markov Chain](graphs/markov_chain.py)
 * [Matching Min Vertex Cover](graphs/matching_min_vertex_cover.py)
 * [Minimum Path Sum](graphs/minimum_path_sum.py)
Expand DownExpand Up@@ -886,6 +888,7 @@
 * [N Body Simulation](physics/n_body_simulation.py)
 * [Newtons Law Of Gravitation](physics/newtons_law_of_gravitation.py)
 * [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
 * [Period Of Pendulum](physics/period_of_pendulum.py)
 * [Photoelectric Effect](physics/photoelectric_effect.py)
 * [Potential Energy](physics/potential_energy.py)
 * [Rainfall Intensity](physics/rainfall_intensity.py)
Expand Down

97 changes: 55 additions & 42 deletions
maths/trapezoidal_rule.py

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Original file line numberDiff line numberDiff line change
@@ -1,28 +1,25 @@
"""
Numerical integration or quadrature for a smooth function f with known values at x_i

This method is the classical approach of suming 'Equally Spaced Abscissas'

method 1:
"extended trapezoidal rule"
int(f) = dx/2 * (f1 + 2f2 + ... + fn)

"""


def method_1(boundary, steps):
def trapezoidal_rule(boundary, steps):
 """
 Apply the extended trapezoidal rule to approximate the integral of function f(x)
 over the interval defined by 'boundary' with the number of 'steps'.

 Args:
 boundary (list of floats): A list containing the start and end values [a, b].
 steps (int): The number of steps or subintervals.
 Returns:
 float: Approximation of the integral of f(x) over [a, b].
 Examples:
 >>> method_1([0, 1], 10)
 0.3349999999999999
 Implements the extended trapezoidal rule for numerical integration.
 The function f(x) is provided below.

 :param boundary: List containing the lower and upper bounds of integration [a, b]
 :param steps: The number of steps (intervals) used in the approximation
 :return: 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
 """
 h = (boundary[1] - boundary[0]) / steps
 a = boundary[0]
Expand All@@ -31,57 +28,73 @@ def method_1(boundary, steps):
 y = 0.0
 y += (h / 2.0) * f(a)
 for i in x_i:
 # print(i)
 y += h * f(i)
 y += (h / 2.0) * f(b)
 return y


def make_points(a, b, h):
 """
 Generates points between 'a' and 'b' with step size 'h', excluding the end points.
 Args:
 a (float): Start value
 b (float): End value
 h (float): Step size
 Examples:
 Generates points between a and b with step size h for trapezoidal integration.

 :param a: The lower bound of integration
 :param b: The upper bound of integration
 :param h: The step size
 :yield: The next x-value in the range (a, b)

 >>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
 [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))
 []
 """
 x = a + h
 while x <= (b - h):
 yield x
 x = x + h
 x += h


def f(x): # enter your function here
def f(x):
 """
 Example:
 >>> f(2)
 4
 This is the function to integrate, f(x) = (x - 0)^2 = x^2.

 :param x: The input value
 :return: The value of f(x)

 >>> f(0)
 0
 >>> f(1)
 1
 >>> f(0.5)
 0.25
 """
 y = (x - 0) * (x - 0)
 return y
 return x**2


def main():
 a = 0.0 # Lower bound of integration
 b = 1.0 # Upper bound of integration
 steps = 10.0 # define number of steps or resolution
 boundary = [a, b] # define boundary of integration
 y = method_1(boundary, steps)
 """
 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
 """
 a = 0.0
 b = 1.0
 steps = 10.0
 boundary = [a, b]
 y = trapezoidal_rule(boundary, steps)
 print(f"y = {y}")


Expand Down

, 'i');
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function applyDarkMode() {
if (!enabled) return;
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filter: invert(1) hue-rotate(180deg) !important;
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border-color: #444 !important;
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chore: improve comments and add tests to trapezoidal rule by jurichar · Pull Request #11640 · TheAlgorithms/Python · GitHub
Skip to content
chore: improve comments and add tests to trapezoidal rule#11640

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 Merged

tianyizheng02
merged 12 commits into

from

Dec 31, 2024
 Merged


chore: improve comments and add tests to trapezoidal rule
#11640
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2e67da3

chore: improve comments and add tests to trapezoidal rule


jurichar Oct 1, 2024
aa5a885

fix: too much characters in line


jurichar Oct 1, 2024
79ed4e7

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
bda4d82

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
9190304

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
2409e25

Update maths/trapezoidal_rule.py


jurichar Oct 6, 2024
242b208

fix: change function name in calls


jurichar Oct 6, 2024
d8c8ace

modify tests, changes numbers to remove coma


jurichar Oct 6, 2024
911943b

Merge branch 'master' into issue_fix


tianyizheng02 Dec 31, 2024
8d7d6f1

updating DIRECTORY.md


tianyizheng02 Dec 31, 2024
1c0debe

Fix doctest whitespace


tianyizheng02 Dec 31, 2024
0c5b78d

Try to fix line length in doctest


tianyizheng02 Dec 31, 2024

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DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -142,6 +142,7 @@
 * [Haralick Descriptors](computer_vision/haralick_descriptors.py)
 * [Harris Corner](computer_vision/harris_corner.py)
 * [Horn Schunck](computer_vision/horn_schunck.py)
 * [Intensity Based Segmentation](computer_vision/intensity_based_segmentation.py)
 * [Mean Threshold](computer_vision/mean_threshold.py)
 * [Mosaic Augmentation](computer_vision/mosaic_augmentation.py)
 * [Pooling Functions](computer_vision/pooling_functions.py)
Expand DownExpand Up@@ -507,6 +508,7 @@
 * [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
 * [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
 * [Karger](graphs/karger.py)
 * [Lanczos Eigenvectors](graphs/lanczos_eigenvectors.py)
 * [Markov Chain](graphs/markov_chain.py)
 * [Matching Min Vertex Cover](graphs/matching_min_vertex_cover.py)
 * [Minimum Path Sum](graphs/minimum_path_sum.py)
Expand DownExpand Up@@ -886,6 +888,7 @@
 * [N Body Simulation](physics/n_body_simulation.py)
 * [Newtons Law Of Gravitation](physics/newtons_law_of_gravitation.py)
 * [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
 * [Period Of Pendulum](physics/period_of_pendulum.py)
 * [Photoelectric Effect](physics/photoelectric_effect.py)
 * [Potential Energy](physics/potential_energy.py)
 * [Rainfall Intensity](physics/rainfall_intensity.py)
Expand Down

97 changes: 55 additions & 42 deletions
maths/trapezoidal_rule.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -1,28 +1,25 @@
"""
Numerical integration or quadrature for a smooth function f with known values at x_i

This method is the classical approach of suming 'Equally Spaced Abscissas'

method 1:
"extended trapezoidal rule"
int(f) = dx/2 * (f1 + 2f2 + ... + fn)

"""


def method_1(boundary, steps):
def trapezoidal_rule(boundary, steps):
 """
 Apply the extended trapezoidal rule to approximate the integral of function f(x)
 over the interval defined by 'boundary' with the number of 'steps'.

 Args:
 boundary (list of floats): A list containing the start and end values [a, b].
 steps (int): The number of steps or subintervals.
 Returns:
 float: Approximation of the integral of f(x) over [a, b].
 Examples:
 >>> method_1([0, 1], 10)
 0.3349999999999999
 Implements the extended trapezoidal rule for numerical integration.
 The function f(x) is provided below.

 :param boundary: List containing the lower and upper bounds of integration [a, b]
 :param steps: The number of steps (intervals) used in the approximation
 :return: 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
 """
 h = (boundary[1] - boundary[0]) / steps
 a = boundary[0]
Expand All@@ -31,57 +28,73 @@ def method_1(boundary, steps):
 y = 0.0
 y += (h / 2.0) * f(a)
 for i in x_i:
 # print(i)
 y += h * f(i)
 y += (h / 2.0) * f(b)
 return y


def make_points(a, b, h):
 """
 Generates points between 'a' and 'b' with step size 'h', excluding the end points.
 Args:
 a (float): Start value
 b (float): End value
 h (float): Step size
 Examples:
 Generates points between a and b with step size h for trapezoidal integration.

 :param a: The lower bound of integration
 :param b: The upper bound of integration
 :param h: The step size
 :yield: The next x-value in the range (a, b)

 >>> list(make_points(0, 1, 0.1)) # doctest: +NORMALIZE_WHITESPACE
 [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))
 []
 """
 x = a + h
 while x <= (b - h):
 yield x
 x = x + h
 x += h


def f(x): # enter your function here
def f(x):
 """
 Example:
 >>> f(2)
 4
 This is the function to integrate, f(x) = (x - 0)^2 = x^2.

 :param x: The input value
 :return: The value of f(x)

 >>> f(0)
 0
 >>> f(1)
 1
 >>> f(0.5)
 0.25
 """
 y = (x - 0) * (x - 0)
 return y
 return x**2


def main():
 a = 0.0 # Lower bound of integration
 b = 1.0 # Upper bound of integration
 steps = 10.0 # define number of steps or resolution
 boundary = [a, b] # define boundary of integration
 y = method_1(boundary, steps)
 """
 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
 """
 a = 0.0
 b = 1.0
 steps = 10.0
 boundary = [a, b]
 y = trapezoidal_rule(boundary, steps)
 print(f"y = {y}")


Expand Down