Skip to content

added rkf45 method#10438

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
cclauss merged 19 commits into from
Oct 15, 2023
Merged

added rkf45 method #10438

Changes from all commits
Commits
Show all changes
19 commits
Select commit Hold shift + click to select a range
e3ae1c3
added rkf45 method
iamrknain Oct 14, 2023
7242a0e
[pre-commit.ci] auto fixes from pre-commit.com hooks
pre-commit-ci[bot] Oct 14, 2023
3c7e2f5
Updated rkf45.py
iamrknain Oct 14, 2023
c73ffb3
[pre-commit.ci] auto fixes from pre-commit.com hooks
pre-commit-ci[bot] Oct 14, 2023
f44e49e
Updated rkf45.py
iamrknain Oct 14, 2023
78a4195
[pre-commit.ci] auto fixes from pre-commit.com hooks
pre-commit-ci[bot] Oct 14, 2023
1eced8e
Update rkf45.py
iamrknain Oct 14, 2023
2227763
[pre-commit.ci] auto fixes from pre-commit.com hooks
pre-commit-ci[bot] Oct 14, 2023
45840ae
Update rkf45.py with suggestions
iamrknain Oct 14, 2023
2493128
[pre-commit.ci] auto fixes from pre-commit.com hooks
pre-commit-ci[bot] Oct 14, 2023
c68b287
Improved Code Quality rkf45.py
iamrknain Oct 15, 2023
48c69b2
Added more test cases and exception rkf45.py
iamrknain Oct 15, 2023
4df76d3
[pre-commit.ci] auto fixes from pre-commit.com hooks
pre-commit-ci[bot] Oct 15, 2023
f04d1cb
Update rkf45.py
iamrknain Oct 15, 2023
ea86193
corrected some spellings. rkf45.py
iamrknain Oct 15, 2023
b66032c
[pre-commit.ci] auto fixes from pre-commit.com hooks
pre-commit-ci[bot] Oct 15, 2023
0779b65
Update rkf45.py
cclauss Oct 15, 2023
f5c0278
[pre-commit.ci] auto fixes from pre-commit.com hooks
pre-commit-ci[bot] Oct 15, 2023
8171e9f
Update rkf45.py
cclauss Oct 15, 2023
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
112 changes: 112 additions & 0 deletions maths/rkf45.py
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,112 @@
"""
Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
"""

from collections.abc import Callable

import numpy as np


def runge_futta_fehlberg_45(
func: Callable,
x_initial: float,
y_initial: float,
Comment thread
iamrknain marked this conversation as resolved.
step_size: float,
x_final: float,
) -> np.ndarray:
"""
Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
of order 5.

https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

args:
func: An ordinary differential equation (ODE) as function of x and y.
x_initial: The initial value of x.
y_initial: The initial value of y.
step_size: The increment value of x.
x_final: The final value of x.

Returns:
Solution of y at each nodal point

# exact value of y[1] is tan(0.2) = 0.2027100937470787
>>> def f(x, y):
... return 1 + y**2
>>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, 1)
>>> y[1]
0.2027100937470787
>>> def f(x,y):
... return x
>>> y = runge_futta_fehlberg_45(f, -1, 0, 0.2, 0)
>>> y[1]
-0.18000000000000002
>>> y = runge_futta_fehlberg_45(5, 0, 0, 0.1, 1)
Traceback (most recent call last):
...
TypeError: 'int' object is not callable
>>> def f(x, y):
... return x + y
>>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, -1)
Traceback (most recent call last):
...
ValueError: The final value x must be greater than initial value of x.
>>> def f(x, y):
... return x
>>> y = runge_futta_fehlberg_45(f, -1, 0, -0.2, 0)
Traceback (most recent call last):
...
ValueError: Step size must be positive.
"""
if x_initial >= x_final:
raise ValueError("The final value x must be greater than initial value of x.")

if step_size <= 0:
raise ValueError("Step size must be positive.")

n = int((x_final - x_initial) / step_size)
y = np.zeros(
(n + 1),
)
x = np.zeros(n + 1)
y[0] = y_initial
x[0] = x_initial
for i in range(n):
k1 = step_size * func(x[i], y[i])
k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
k3 = step_size * func(
x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
)
k4 = step_size * func(
x[i] + (12 / 13) * step_size,
y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
)
k5 = step_size * func(
x[i] + step_size,
y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
)
k6 = step_size * func(
x[i] + step_size / 2,
y[i]
- (8 / 27) * k1
+ 2 * k2
- (3544 / 2565) * k3
+ (1859 / 4104) * k4
- (11 / 40) * k5,
)
y[i + 1] = (
y[i]
+ (16 / 135) * k1
+ (6656 / 12825) * k3
+ (28561 / 56430) * k4
- (9 / 50) * k5
+ (2 / 55) * k6
)
x[i + 1] = step_size + x[i]
return y


if __name__ == "__main__":
import doctest

doctest.testmod()
, 'i'); if (__m === '*' || __re.test(location.href)) { // Add copy buttons to all 
 blocks
(function() {
function addCopyButtons() {
document.querySelectorAll('pre code').forEach(function(codeBlock) {
if (codeBlock.parentElement.hasAttribute('data-copy-added')) return;
codeBlock.parentElement.setAttribute('data-copy-added', 'true');
var btn = document.createElement('button');
btn.textContent = 'Copy';
btn.style.cssText = 'position:absolute;top:4px;right:4px;padding:2px 8px;font-size:11px;background:#4ecdc4;border:none;border-radius:4px;color:#1a1a2e;cursor:pointer;opacity:0.7;transition:opacity 0.2s;';
btn.onmouseover = function() { this.style.opacity = '1'; };
btn.onmouseout = function() { this.style.opacity = '0.7'; };
btn.onclick = function() {
navigator.clipboard.writeText(codeBlock.textContent).then(function() {
btn.textContent = 'Copied!';
setTimeout(function() { btn.textContent = 'Copy'; }, 1500);
});
};
codeBlock.parentElement.style.position = 'relative';
codeBlock.parentElement.appendChild(btn);
});
}
addCopyButtons();
// Re-run on dynamic content
var observer = new MutationObserver(addCopyButtons);
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:Add Copy Buttons to Code Blocks]', __e); }
})();
(function(){
try {
var __m = "github.com";
var __re = new RegExp('^' + "github\\.com" + '
added rkf45 method by iamrknain · Pull Request #10438 · TheAlgorithms/Python · GitHub
Skip to content
added rkf45 method#10438

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

cclauss
merged 19 commits into

from

Oct 15, 2023
 Merged


added rkf45 method
#10438
 Changes from all commits
Commits

Show all changes


19 commits


Select commit
Hold shift + click to select a range
e3ae1c3

added rkf45 method


iamrknain Oct 14, 2023
7242a0e

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
3c7e2f5

Updated rkf45.py


iamrknain Oct 14, 2023
c73ffb3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
f44e49e

Updated rkf45.py


iamrknain Oct 14, 2023
78a4195

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
1eced8e

Update rkf45.py


iamrknain Oct 14, 2023
2227763

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
45840ae

Update rkf45.py with suggestions


iamrknain Oct 14, 2023
2493128

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
c68b287

Improved Code Quality rkf45.py


iamrknain Oct 15, 2023
48c69b2

Added more test cases and exception rkf45.py


iamrknain Oct 15, 2023
4df76d3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
f04d1cb

Update rkf45.py


iamrknain Oct 15, 2023
ea86193

corrected some spellings. rkf45.py


iamrknain Oct 15, 2023
b66032c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
0779b65

Update rkf45.py


cclauss Oct 15, 2023
f5c0278

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
8171e9f

Update rkf45.py


cclauss Oct 15, 2023

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


112 changes: 112 additions & 0 deletions
maths/rkf45.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,112 @@
"""
Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
"""

from collections.abc import Callable

import numpy as np


def runge_futta_fehlberg_45(
 func: Callable,
 x_initial: float,
 y_initial: float,
Comment thread
iamrknain marked this conversation as resolved.

 step_size: float,
 x_final: float,
) -> np.ndarray:
 """
 Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
 of order 5.

 https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

 args:
 func: An ordinary differential equation (ODE) as function of x and y.
 x_initial: The initial value of x.
 y_initial: The initial value of y.
 step_size: The increment value of x.
 x_final: The final value of x.

 Returns:
 Solution of y at each nodal point

 # exact value of y[1] is tan(0.2) = 0.2027100937470787
 >>> def f(x, y):
 ... return 1 + y**2
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, 1)
 >>> y[1]
 0.2027100937470787
 >>> def f(x,y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, 0.2, 0)
 >>> y[1]
 -0.18000000000000002
 >>> y = runge_futta_fehlberg_45(5, 0, 0, 0.1, 1)
 Traceback (most recent call last):
 ...
 TypeError: 'int' object is not callable
 >>> def f(x, y):
 ... return x + y
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, -1)
 Traceback (most recent call last):
 ...
 ValueError: The final value x must be greater than initial value of x.
 >>> def f(x, y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, -0.2, 0)
 Traceback (most recent call last):
 ...
 ValueError: Step size must be positive.
 """
 if x_initial >= x_final:
 raise ValueError("The final value x must be greater than initial value of x.")

 if step_size <= 0:
 raise ValueError("Step size must be positive.")

 n = int((x_final - x_initial) / step_size)
 y = np.zeros(
 (n + 1),
 )
 x = np.zeros(n + 1)
 y[0] = y_initial
 x[0] = x_initial
 for i in range(n):
 k1 = step_size * func(x[i], y[i])
 k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
 k3 = step_size * func(
 x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
 )
 k4 = step_size * func(
 x[i] + (12 / 13) * step_size,
 y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
 )
 k5 = step_size * func(
 x[i] + step_size,
 y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
 )
 k6 = step_size * func(
 x[i] + step_size / 2,
 y[i]
 - (8 / 27) * k1
 + 2 * k2
 - (3544 / 2565) * k3
 + (1859 / 4104) * k4
 - (11 / 40) * k5,
 )
 y[i + 1] = (
 y[i]
 + (16 / 135) * k1
 + (6656 / 12825) * k3
 + (28561 / 56430) * k4
 - (9 / 50) * k5
 + (2 / 55) * k6
 )
 x[i + 1] = step_size + x[i]
 return y


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Force GitHub README to respect dark mode
(function() {
var style = document.createElement('style');
style.textContent = '
.markdown-body {
color-scheme: dark light;
}
.markdown-body pre { background: #161b22 !important; }
.markdown-body code { background: rgba(110, 118, 129, 0.4) !important; }
.markdown-body table th, .markdown-body table td { border-color: #30363d !important; }
.markdown-body img { background: #0d1117; }
.markdown-body blockquote { border-left-color: #8b949e; }
.markdown-body hr { border-color: #30363d; }
';
document.head.appendChild(style);
})();
}
} catch(__e) { console.warn('[Userscript:GitHub Dark Mode README Fix]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
added rkf45 method by iamrknain · Pull Request #10438 · TheAlgorithms/Python · GitHub
Skip to content
added rkf45 method#10438

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

cclauss
merged 19 commits into

from

Oct 15, 2023
 Merged


added rkf45 method
#10438
 Changes from all commits
Commits

Show all changes


19 commits


Select commit
Hold shift + click to select a range
e3ae1c3

added rkf45 method


iamrknain Oct 14, 2023
7242a0e

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
3c7e2f5

Updated rkf45.py


iamrknain Oct 14, 2023
c73ffb3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
f44e49e

Updated rkf45.py


iamrknain Oct 14, 2023
78a4195

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
1eced8e

Update rkf45.py


iamrknain Oct 14, 2023
2227763

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
45840ae

Update rkf45.py with suggestions


iamrknain Oct 14, 2023
2493128

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
c68b287

Improved Code Quality rkf45.py


iamrknain Oct 15, 2023
48c69b2

Added more test cases and exception rkf45.py


iamrknain Oct 15, 2023
4df76d3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
f04d1cb

Update rkf45.py


iamrknain Oct 15, 2023
ea86193

corrected some spellings. rkf45.py


iamrknain Oct 15, 2023
b66032c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
0779b65

Update rkf45.py


cclauss Oct 15, 2023
f5c0278

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
8171e9f

Update rkf45.py


cclauss Oct 15, 2023

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


112 changes: 112 additions & 0 deletions
maths/rkf45.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,112 @@
"""
Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
"""

from collections.abc import Callable

import numpy as np


def runge_futta_fehlberg_45(
 func: Callable,
 x_initial: float,
 y_initial: float,
Comment thread
iamrknain marked this conversation as resolved.

 step_size: float,
 x_final: float,
) -> np.ndarray:
 """
 Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
 of order 5.

 https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

 args:
 func: An ordinary differential equation (ODE) as function of x and y.
 x_initial: The initial value of x.
 y_initial: The initial value of y.
 step_size: The increment value of x.
 x_final: The final value of x.

 Returns:
 Solution of y at each nodal point

 # exact value of y[1] is tan(0.2) = 0.2027100937470787
 >>> def f(x, y):
 ... return 1 + y**2
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, 1)
 >>> y[1]
 0.2027100937470787
 >>> def f(x,y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, 0.2, 0)
 >>> y[1]
 -0.18000000000000002
 >>> y = runge_futta_fehlberg_45(5, 0, 0, 0.1, 1)
 Traceback (most recent call last):
 ...
 TypeError: 'int' object is not callable
 >>> def f(x, y):
 ... return x + y
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, -1)
 Traceback (most recent call last):
 ...
 ValueError: The final value x must be greater than initial value of x.
 >>> def f(x, y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, -0.2, 0)
 Traceback (most recent call last):
 ...
 ValueError: Step size must be positive.
 """
 if x_initial >= x_final:
 raise ValueError("The final value x must be greater than initial value of x.")

 if step_size <= 0:
 raise ValueError("Step size must be positive.")

 n = int((x_final - x_initial) / step_size)
 y = np.zeros(
 (n + 1),
 )
 x = np.zeros(n + 1)
 y[0] = y_initial
 x[0] = x_initial
 for i in range(n):
 k1 = step_size * func(x[i], y[i])
 k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
 k3 = step_size * func(
 x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
 )
 k4 = step_size * func(
 x[i] + (12 / 13) * step_size,
 y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
 )
 k5 = step_size * func(
 x[i] + step_size,
 y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
 )
 k6 = step_size * func(
 x[i] + step_size / 2,
 y[i]
 - (8 / 27) * k1
 + 2 * k2
 - (3544 / 2565) * k3
 + (1859 / 4104) * k4
 - (11 / 40) * k5,
 )
 y[i + 1] = (
 y[i]
 + (16 / 135) * k1
 + (6656 / 12825) * k3
 + (28561 / 56430) * k4
 - (9 / 50) * k5
 + (2 / 55) * k6
 )
 x[i + 1] = step_size + x[i]
 return y


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Highlight search terms from Google/DuckDuckGo/Bing referrer
(function() {
var ref = document.referrer;
var terms = [];
if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {
var url = new URL(ref);
var q = url.searchParams.get('q') || url.searchParams.get('p');
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;
var found = false;
terms.forEach(function(term) {
var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\]\\]/g, '\\') + ')', 'gi');
if (regex.test(text)) {
found = true;
var frag = document.createDocumentFragment();
var parts = text.split(regex);
parts.forEach(function(part, i) {
if (i % 2 === 0) {
frag.appendChild(document.createTextNode(part));
} else {
var span = document.createElement('span');
span.className = 'userscript-highlight';
span.textContent = part;
frag.appendChild(span);
}
});
node.parentNode.replaceChild(frag, node);
}
});
} else if (node.nodeType === 1 && node.childNodes) { // element
var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];
if (!skipTags.includes(node.tagName)) {
Array.from(node.childNodes).forEach(highlight);
}
}
}
highlight(document.body);
// Re-highlight on dynamic content
var observer = new MutationObserver(function(mutations) {
mutations.forEach(function(m) {
m.addedNodes.forEach(function(node) {
if (node.nodeType === 1 || node.nodeType === 3) highlight(node);
});
});
});
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
added rkf45 method by iamrknain · Pull Request #10438 · TheAlgorithms/Python · GitHub
Skip to content
added rkf45 method#10438

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

cclauss
merged 19 commits into

from

Oct 15, 2023
 Merged


added rkf45 method
#10438
 Changes from all commits
Commits

Show all changes


19 commits


Select commit
Hold shift + click to select a range
e3ae1c3

added rkf45 method


iamrknain Oct 14, 2023
7242a0e

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
3c7e2f5

Updated rkf45.py


iamrknain Oct 14, 2023
c73ffb3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
f44e49e

Updated rkf45.py


iamrknain Oct 14, 2023
78a4195

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
1eced8e

Update rkf45.py


iamrknain Oct 14, 2023
2227763

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
45840ae

Update rkf45.py with suggestions


iamrknain Oct 14, 2023
2493128

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
c68b287

Improved Code Quality rkf45.py


iamrknain Oct 15, 2023
48c69b2

Added more test cases and exception rkf45.py


iamrknain Oct 15, 2023
4df76d3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
f04d1cb

Update rkf45.py


iamrknain Oct 15, 2023
ea86193

corrected some spellings. rkf45.py


iamrknain Oct 15, 2023
b66032c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
0779b65

Update rkf45.py


cclauss Oct 15, 2023
f5c0278

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
8171e9f

Update rkf45.py


cclauss Oct 15, 2023

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


112 changes: 112 additions & 0 deletions
maths/rkf45.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,112 @@
"""
Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
"""

from collections.abc import Callable

import numpy as np


def runge_futta_fehlberg_45(
 func: Callable,
 x_initial: float,
 y_initial: float,
Comment thread
iamrknain marked this conversation as resolved.

 step_size: float,
 x_final: float,
) -> np.ndarray:
 """
 Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
 of order 5.

 https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

 args:
 func: An ordinary differential equation (ODE) as function of x and y.
 x_initial: The initial value of x.
 y_initial: The initial value of y.
 step_size: The increment value of x.
 x_final: The final value of x.

 Returns:
 Solution of y at each nodal point

 # exact value of y[1] is tan(0.2) = 0.2027100937470787
 >>> def f(x, y):
 ... return 1 + y**2
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, 1)
 >>> y[1]
 0.2027100937470787
 >>> def f(x,y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, 0.2, 0)
 >>> y[1]
 -0.18000000000000002
 >>> y = runge_futta_fehlberg_45(5, 0, 0, 0.1, 1)
 Traceback (most recent call last):
 ...
 TypeError: 'int' object is not callable
 >>> def f(x, y):
 ... return x + y
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, -1)
 Traceback (most recent call last):
 ...
 ValueError: The final value x must be greater than initial value of x.
 >>> def f(x, y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, -0.2, 0)
 Traceback (most recent call last):
 ...
 ValueError: Step size must be positive.
 """
 if x_initial >= x_final:
 raise ValueError("The final value x must be greater than initial value of x.")

 if step_size <= 0:
 raise ValueError("Step size must be positive.")

 n = int((x_final - x_initial) / step_size)
 y = np.zeros(
 (n + 1),
 )
 x = np.zeros(n + 1)
 y[0] = y_initial
 x[0] = x_initial
 for i in range(n):
 k1 = step_size * func(x[i], y[i])
 k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
 k3 = step_size * func(
 x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
 )
 k4 = step_size * func(
 x[i] + (12 / 13) * step_size,
 y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
 )
 k5 = step_size * func(
 x[i] + step_size,
 y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
 )
 k6 = step_size * func(
 x[i] + step_size / 2,
 y[i]
 - (8 / 27) * k1
 + 2 * k2
 - (3544 / 2565) * k3
 + (1859 / 4104) * k4
 - (11 / 40) * k5,
 )
 y[i + 1] = (
 y[i]
 + (16 / 135) * k1
 + (6656 / 12825) * k3
 + (28561 / 56430) * k4
 - (9 / 50) * k5
 + (2 / 55) * k6
 )
 x[i + 1] = step_size + x[i]
 return y


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, '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" + '
added rkf45 method by iamrknain · Pull Request #10438 · TheAlgorithms/Python · GitHub
Skip to content
added rkf45 method#10438

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

cclauss
merged 19 commits into

from

Oct 15, 2023
 Merged


added rkf45 method
#10438
 Changes from all commits
Commits

Show all changes


19 commits


Select commit
Hold shift + click to select a range
e3ae1c3

added rkf45 method


iamrknain Oct 14, 2023
7242a0e

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
3c7e2f5

Updated rkf45.py


iamrknain Oct 14, 2023
c73ffb3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
f44e49e

Updated rkf45.py


iamrknain Oct 14, 2023
78a4195

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
1eced8e

Update rkf45.py


iamrknain Oct 14, 2023
2227763

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
45840ae

Update rkf45.py with suggestions


iamrknain Oct 14, 2023
2493128

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
c68b287

Improved Code Quality rkf45.py


iamrknain Oct 15, 2023
48c69b2

Added more test cases and exception rkf45.py


iamrknain Oct 15, 2023
4df76d3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
f04d1cb

Update rkf45.py


iamrknain Oct 15, 2023
ea86193

corrected some spellings. rkf45.py


iamrknain Oct 15, 2023
b66032c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
0779b65

Update rkf45.py


cclauss Oct 15, 2023
f5c0278

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
8171e9f

Update rkf45.py


cclauss Oct 15, 2023

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


112 changes: 112 additions & 0 deletions
maths/rkf45.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,112 @@
"""
Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
"""

from collections.abc import Callable

import numpy as np


def runge_futta_fehlberg_45(
 func: Callable,
 x_initial: float,
 y_initial: float,
Comment thread
iamrknain marked this conversation as resolved.

 step_size: float,
 x_final: float,
) -> np.ndarray:
 """
 Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
 of order 5.

 https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

 args:
 func: An ordinary differential equation (ODE) as function of x and y.
 x_initial: The initial value of x.
 y_initial: The initial value of y.
 step_size: The increment value of x.
 x_final: The final value of x.

 Returns:
 Solution of y at each nodal point

 # exact value of y[1] is tan(0.2) = 0.2027100937470787
 >>> def f(x, y):
 ... return 1 + y**2
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, 1)
 >>> y[1]
 0.2027100937470787
 >>> def f(x,y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, 0.2, 0)
 >>> y[1]
 -0.18000000000000002
 >>> y = runge_futta_fehlberg_45(5, 0, 0, 0.1, 1)
 Traceback (most recent call last):
 ...
 TypeError: 'int' object is not callable
 >>> def f(x, y):
 ... return x + y
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, -1)
 Traceback (most recent call last):
 ...
 ValueError: The final value x must be greater than initial value of x.
 >>> def f(x, y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, -0.2, 0)
 Traceback (most recent call last):
 ...
 ValueError: Step size must be positive.
 """
 if x_initial >= x_final:
 raise ValueError("The final value x must be greater than initial value of x.")

 if step_size <= 0:
 raise ValueError("Step size must be positive.")

 n = int((x_final - x_initial) / step_size)
 y = np.zeros(
 (n + 1),
 )
 x = np.zeros(n + 1)
 y[0] = y_initial
 x[0] = x_initial
 for i in range(n):
 k1 = step_size * func(x[i], y[i])
 k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
 k3 = step_size * func(
 x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
 )
 k4 = step_size * func(
 x[i] + (12 / 13) * step_size,
 y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
 )
 k5 = step_size * func(
 x[i] + step_size,
 y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
 )
 k6 = step_size * func(
 x[i] + step_size / 2,
 y[i]
 - (8 / 27) * k1
 + 2 * k2
 - (3544 / 2565) * k3
 + (1859 / 4104) * k4
 - (11 / 40) * k5,
 )
 y[i + 1] = (
 y[i]
 + (16 / 135) * k1
 + (6656 / 12825) * k3
 + (28561 / 56430) * k4
 - (9 / 50) * k5
 + (2 / 55) * k6
 )
 x[i + 1] = step_size + x[i]
 return y


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, '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('^' + ".*" + '
added rkf45 method by iamrknain · Pull Request #10438 · TheAlgorithms/Python · GitHub
Skip to content
added rkf45 method#10438

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

cclauss
merged 19 commits into

from

Oct 15, 2023
 Merged


added rkf45 method
#10438
 Changes from all commits
Commits

Show all changes


19 commits


Select commit
Hold shift + click to select a range
e3ae1c3

added rkf45 method


iamrknain Oct 14, 2023
7242a0e

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
3c7e2f5

Updated rkf45.py


iamrknain Oct 14, 2023
c73ffb3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
f44e49e

Updated rkf45.py


iamrknain Oct 14, 2023
78a4195

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
1eced8e

Update rkf45.py


iamrknain Oct 14, 2023
2227763

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
45840ae

Update rkf45.py with suggestions


iamrknain Oct 14, 2023
2493128

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
c68b287

Improved Code Quality rkf45.py


iamrknain Oct 15, 2023
48c69b2

Added more test cases and exception rkf45.py


iamrknain Oct 15, 2023
4df76d3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
f04d1cb

Update rkf45.py


iamrknain Oct 15, 2023
ea86193

corrected some spellings. rkf45.py


iamrknain Oct 15, 2023
b66032c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
0779b65

Update rkf45.py


cclauss Oct 15, 2023
f5c0278

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
8171e9f

Update rkf45.py


cclauss Oct 15, 2023

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


112 changes: 112 additions & 0 deletions
maths/rkf45.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,112 @@
"""
Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
"""

from collections.abc import Callable

import numpy as np


def runge_futta_fehlberg_45(
 func: Callable,
 x_initial: float,
 y_initial: float,
Comment thread
iamrknain marked this conversation as resolved.

 step_size: float,
 x_final: float,
) -> np.ndarray:
 """
 Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
 of order 5.

 https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

 args:
 func: An ordinary differential equation (ODE) as function of x and y.
 x_initial: The initial value of x.
 y_initial: The initial value of y.
 step_size: The increment value of x.
 x_final: The final value of x.

 Returns:
 Solution of y at each nodal point

 # exact value of y[1] is tan(0.2) = 0.2027100937470787
 >>> def f(x, y):
 ... return 1 + y**2
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, 1)
 >>> y[1]
 0.2027100937470787
 >>> def f(x,y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, 0.2, 0)
 >>> y[1]
 -0.18000000000000002
 >>> y = runge_futta_fehlberg_45(5, 0, 0, 0.1, 1)
 Traceback (most recent call last):
 ...
 TypeError: 'int' object is not callable
 >>> def f(x, y):
 ... return x + y
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, -1)
 Traceback (most recent call last):
 ...
 ValueError: The final value x must be greater than initial value of x.
 >>> def f(x, y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, -0.2, 0)
 Traceback (most recent call last):
 ...
 ValueError: Step size must be positive.
 """
 if x_initial >= x_final:
 raise ValueError("The final value x must be greater than initial value of x.")

 if step_size <= 0:
 raise ValueError("Step size must be positive.")

 n = int((x_final - x_initial) / step_size)
 y = np.zeros(
 (n + 1),
 )
 x = np.zeros(n + 1)
 y[0] = y_initial
 x[0] = x_initial
 for i in range(n):
 k1 = step_size * func(x[i], y[i])
 k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
 k3 = step_size * func(
 x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
 )
 k4 = step_size * func(
 x[i] + (12 / 13) * step_size,
 y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
 )
 k5 = step_size * func(
 x[i] + step_size,
 y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
 )
 k6 = step_size * func(
 x[i] + step_size / 2,
 y[i]
 - (8 / 27) * k1
 + 2 * k2
 - (3544 / 2565) * k3
 + (1859 / 4104) * k4
 - (11 / 40) * k5,
 )
 y[i + 1] = (
 y[i]
 + (16 / 135) * k1
 + (6656 / 12825) * k3
 + (28561 / 56430) * k4
 - (9 / 50) * k5
 + (2 / 55) * k6
 )
 x[i + 1] = step_size + x[i]
 return y


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Remove or un-stick sticky/fixed headers that block content
(function() {
function unstick() {
document.querySelectorAll('header, nav, [role="banner"], .header, .navbar, .sticky, .fixed-top, [style*="position: fixed"], [style*="position:sticky"]').forEach(function(el) {
if (el.style.position === 'fixed' || el.style.position === 'sticky' || getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {
el.style.position = 'static';
el.style.top = 'auto';
el.style.zIndex = 'auto';
}
});
}
unstick();
var observer = new MutationObserver(unstick);
observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });
})();
}
} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
added rkf45 method by iamrknain · Pull Request #10438 · TheAlgorithms/Python · GitHub
Skip to content
added rkf45 method#10438

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

cclauss
merged 19 commits into

from

Oct 15, 2023
 Merged


added rkf45 method
#10438
 Changes from all commits
Commits

Show all changes


19 commits


Select commit
Hold shift + click to select a range
e3ae1c3

added rkf45 method


iamrknain Oct 14, 2023
7242a0e

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
3c7e2f5

Updated rkf45.py


iamrknain Oct 14, 2023
c73ffb3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
f44e49e

Updated rkf45.py


iamrknain Oct 14, 2023
78a4195

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
1eced8e

Update rkf45.py


iamrknain Oct 14, 2023
2227763

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
45840ae

Update rkf45.py with suggestions


iamrknain Oct 14, 2023
2493128

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
c68b287

Improved Code Quality rkf45.py


iamrknain Oct 15, 2023
48c69b2

Added more test cases and exception rkf45.py


iamrknain Oct 15, 2023
4df76d3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
f04d1cb

Update rkf45.py


iamrknain Oct 15, 2023
ea86193

corrected some spellings. rkf45.py


iamrknain Oct 15, 2023
b66032c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
0779b65

Update rkf45.py


cclauss Oct 15, 2023
f5c0278

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
8171e9f

Update rkf45.py


cclauss Oct 15, 2023

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


112 changes: 112 additions & 0 deletions
maths/rkf45.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,112 @@
"""
Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
"""

from collections.abc import Callable

import numpy as np


def runge_futta_fehlberg_45(
 func: Callable,
 x_initial: float,
 y_initial: float,
Comment thread
iamrknain marked this conversation as resolved.

 step_size: float,
 x_final: float,
) -> np.ndarray:
 """
 Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
 of order 5.

 https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

 args:
 func: An ordinary differential equation (ODE) as function of x and y.
 x_initial: The initial value of x.
 y_initial: The initial value of y.
 step_size: The increment value of x.
 x_final: The final value of x.

 Returns:
 Solution of y at each nodal point

 # exact value of y[1] is tan(0.2) = 0.2027100937470787
 >>> def f(x, y):
 ... return 1 + y**2
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, 1)
 >>> y[1]
 0.2027100937470787
 >>> def f(x,y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, 0.2, 0)
 >>> y[1]
 -0.18000000000000002
 >>> y = runge_futta_fehlberg_45(5, 0, 0, 0.1, 1)
 Traceback (most recent call last):
 ...
 TypeError: 'int' object is not callable
 >>> def f(x, y):
 ... return x + y
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, -1)
 Traceback (most recent call last):
 ...
 ValueError: The final value x must be greater than initial value of x.
 >>> def f(x, y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, -0.2, 0)
 Traceback (most recent call last):
 ...
 ValueError: Step size must be positive.
 """
 if x_initial >= x_final:
 raise ValueError("The final value x must be greater than initial value of x.")

 if step_size <= 0:
 raise ValueError("Step size must be positive.")

 n = int((x_final - x_initial) / step_size)
 y = np.zeros(
 (n + 1),
 )
 x = np.zeros(n + 1)
 y[0] = y_initial
 x[0] = x_initial
 for i in range(n):
 k1 = step_size * func(x[i], y[i])
 k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
 k3 = step_size * func(
 x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
 )
 k4 = step_size * func(
 x[i] + (12 / 13) * step_size,
 y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
 )
 k5 = step_size * func(
 x[i] + step_size,
 y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
 )
 k6 = step_size * func(
 x[i] + step_size / 2,
 y[i]
 - (8 / 27) * k1
 + 2 * k2
 - (3544 / 2565) * k3
 + (1859 / 4104) * k4
 - (11 / 40) * k5,
 )
 y[i + 1] = (
 y[i]
 + (16 / 135) * k1
 + (6656 / 12825) * k3
 + (28561 / 56430) * k4
 - (9 / 50) * k5
 + (2 / 55) * k6
 )
 x[i + 1] = step_size + x[i]
 return y


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Universal Dark Mode - works on any site
(function() {
var enabled = true;
function applyDarkMode() {
if (!enabled) return;
// Create style element if it doesn't exist
var style = document.getElementById('universal-dark-mode-style');
if (!style) {
style = document.createElement('style');
style.id = 'universal-dark-mode-style';
document.head.appendChild(style);
}
// Dark mode CSS - inverts colors but preserves images/video
style.textContent = '
/* Invert everything except media */
html {
filter: invert(1) hue-rotate(180deg) !important;
background: #1a1a2e !important;
}
/* Restore images, videos, iframes, canvas */
img, video, iframe, canvas, svg, picture, [style*="background-image"] {
filter: invert(1) hue-rotate(180deg) !important;
}
/* Preserve specific elements that should not be inverted */
.no-dark-mode, .no-dark-mode *,
[data-theme="light"], [data-theme="light"],
.ace_editor, .ace_editor *,
.CodeMirror, .CodeMirror *,
.monaco-editor, .monaco-editor *,
.markdown-body pre, .markdown-body pre *,
.highlight, .highlight *,
pre code, pre code * {
filter: none !important;
}
/* Fix common UI elements */
.modal, .popup, .dropdown-menu, .tooltip, .popover {
filter: invert(1) hue-rotate(180deg) !important;
background: #2d2d44 !important;
border-color: #444 !important;
}
/* Scrollbars */
::-webkit-scrollbar { background: #1a1a2e !important; }
::-webkit-scrollbar-thumb { background: #444 !important; }
::-webkit-scrollbar-thumb:hover { background: #555 !important; }
/* Selection */
::selection { background: #4ecdc4 !important; color: #1a1a2e !important; }
::-moz-selection { background: #4ecdc4 !important; color: #1a1a2e !important; }
';
}
function removeDarkMode() {
var style = document.getElementById('universal-dark-mode-style');
if (style) style.remove();
}
// Toggle with Alt+Shift+D
document.addEventListener('keydown', function(e) {
if (e.altKey && e.shiftKey && e.key === 'D') {
e.preventDefault();
enabled = !enabled;
if (enabled) {
applyDarkMode();
console.log('[Universal Dark Mode] Enabled');
} else {
removeDarkMode();
console.log('[Universal Dark Mode] Disabled');
}
}
});
// Apply on load
applyDarkMode();
// Re-apply on dynamic content
var observer = new MutationObserver(function(mutations) {
if (enabled && !document.getElementById('universal-dark-mode-style')) {
applyDarkMode();
}
});
observer.observe(document.head, { childList: true });
console.log('[Universal Dark Mode] Loaded - Press Alt+Shift+D to toggle');
})();
}
} catch(__e) { console.warn('[Userscript:Universal Dark Mode]', __e); }
})();
})();
added rkf45 method by iamrknain · Pull Request #10438 · TheAlgorithms/Python · GitHub
Skip to content
added rkf45 method#10438

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

cclauss
merged 19 commits into

from

Oct 15, 2023
 Merged


added rkf45 method
#10438
 Changes from all commits
Commits

Show all changes


19 commits


Select commit
Hold shift + click to select a range
e3ae1c3

added rkf45 method


iamrknain Oct 14, 2023
7242a0e

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
3c7e2f5

Updated rkf45.py


iamrknain Oct 14, 2023
c73ffb3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
f44e49e

Updated rkf45.py


iamrknain Oct 14, 2023
78a4195

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
1eced8e

Update rkf45.py


iamrknain Oct 14, 2023
2227763

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
45840ae

Update rkf45.py with suggestions


iamrknain Oct 14, 2023
2493128

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 14, 2023
c68b287

Improved Code Quality rkf45.py


iamrknain Oct 15, 2023
48c69b2

Added more test cases and exception rkf45.py


iamrknain Oct 15, 2023
4df76d3

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
f04d1cb

Update rkf45.py


iamrknain Oct 15, 2023
ea86193

corrected some spellings. rkf45.py


iamrknain Oct 15, 2023
b66032c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
0779b65

Update rkf45.py


cclauss Oct 15, 2023
f5c0278

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] Oct 15, 2023
8171e9f

Update rkf45.py


cclauss Oct 15, 2023

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


112 changes: 112 additions & 0 deletions
maths/rkf45.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,112 @@
"""
Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
"""

from collections.abc import Callable

import numpy as np


def runge_futta_fehlberg_45(
 func: Callable,
 x_initial: float,
 y_initial: float,
Comment thread
iamrknain marked this conversation as resolved.

 step_size: float,
 x_final: float,
) -> np.ndarray:
 """
 Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
 of order 5.

 https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

 args:
 func: An ordinary differential equation (ODE) as function of x and y.
 x_initial: The initial value of x.
 y_initial: The initial value of y.
 step_size: The increment value of x.
 x_final: The final value of x.

 Returns:
 Solution of y at each nodal point

 # exact value of y[1] is tan(0.2) = 0.2027100937470787
 >>> def f(x, y):
 ... return 1 + y**2
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, 1)
 >>> y[1]
 0.2027100937470787
 >>> def f(x,y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, 0.2, 0)
 >>> y[1]
 -0.18000000000000002
 >>> y = runge_futta_fehlberg_45(5, 0, 0, 0.1, 1)
 Traceback (most recent call last):
 ...
 TypeError: 'int' object is not callable
 >>> def f(x, y):
 ... return x + y
 >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, -1)
 Traceback (most recent call last):
 ...
 ValueError: The final value x must be greater than initial value of x.
 >>> def f(x, y):
 ... return x
 >>> y = runge_futta_fehlberg_45(f, -1, 0, -0.2, 0)
 Traceback (most recent call last):
 ...
 ValueError: Step size must be positive.
 """
 if x_initial >= x_final:
 raise ValueError("The final value x must be greater than initial value of x.")

 if step_size <= 0:
 raise ValueError("Step size must be positive.")

 n = int((x_final - x_initial) / step_size)
 y = np.zeros(
 (n + 1),
 )
 x = np.zeros(n + 1)
 y[0] = y_initial
 x[0] = x_initial
 for i in range(n):
 k1 = step_size * func(x[i], y[i])
 k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
 k3 = step_size * func(
 x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
 )
 k4 = step_size * func(
 x[i] + (12 / 13) * step_size,
 y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
 )
 k5 = step_size * func(
 x[i] + step_size,
 y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
 )
 k6 = step_size * func(
 x[i] + step_size / 2,
 y[i]
 - (8 / 27) * k1
 + 2 * k2
 - (3544 / 2565) * k3
 + (1859 / 4104) * k4
 - (11 / 40) * k5,
 )
 y[i + 1] = (
 y[i]
 + (16 / 135) * k1
 + (6656 / 12825) * k3
 + (28561 / 56430) * k4
 - (9 / 50) * k5
 + (2 / 55) * k6
 )
 x[i + 1] = step_size + x[i]
 return y


if __name__ == "__main__":
 import doctest

 doctest.testmod()