mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-10-16 18:51:14 +00:00
777eca813a
* Corrected typo in function name and doctests. rkf45.py There was a mistake in name of function (runge_futta_fehlberg instead of runge_kutta_fehlberg) . I have corrected this in function name and also doctest. * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Rename rkf45.py to runge_kutta_fehlberg_45.py * Update runge_kutta_fehlberg_45.py --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com> Co-authored-by: Christian Clauss <cclauss@me.com>
115 lines
3.1 KiB
Python
115 lines
3.1 KiB
Python
"""
|
|
Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
|
|
"""
|
|
|
|
from collections.abc import Callable
|
|
|
|
import numpy as np
|
|
|
|
|
|
def runge_kutta_fehlberg_45(
|
|
func: Callable,
|
|
x_initial: float,
|
|
y_initial: float,
|
|
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_kutta_fehlberg_45(f, 0, 0, 0.2, 1)
|
|
>>> y[1]
|
|
0.2027100937470787
|
|
>>> def f(x,y):
|
|
... return x
|
|
>>> y = runge_kutta_fehlberg_45(f, -1, 0, 0.2, 0)
|
|
>>> y[1]
|
|
-0.18000000000000002
|
|
>>> y = runge_kutta_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_kutta_fehlberg_45(f, 0, 0, 0.2, -1)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: The final value of x must be greater than initial value of x.
|
|
>>> def f(x, y):
|
|
... return x
|
|
>>> y = runge_kutta_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 of 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()
|