mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-12-05 02:40:16 +00:00
55 lines
1.4 KiB
Python
55 lines
1.4 KiB
Python
from collections.abc import Callable
|
|
|
|
import numpy as np
|
|
|
|
|
|
def euler_modified(
|
|
ode_func: Callable, y0: float, x0: float, step_size: float, x_end: float
|
|
) -> np.ndarray:
|
|
"""
|
|
Calculate solution at each step to an ODE using Euler's Modified Method
|
|
The Euler Method is straightforward to implement, but can't give accurate solutions.
|
|
So, some changes were proposed to improve accuracy.
|
|
|
|
https://en.wikipedia.org/wiki/Euler_method
|
|
|
|
Arguments:
|
|
ode_func -- The ode as a function of x and y
|
|
y0 -- the initial value for y
|
|
x0 -- the initial value for x
|
|
stepsize -- the increment value for x
|
|
x_end -- the end value for x
|
|
|
|
>>> # the exact solution is math.exp(x)
|
|
>>> def f1(x, y):
|
|
... return -2*x*(y**2)
|
|
>>> y = euler_modified(f1, 1.0, 0.0, 0.2, 1.0)
|
|
>>> float(y[-1])
|
|
0.503338255442106
|
|
>>> import math
|
|
>>> def f2(x, y):
|
|
... return -2*y + (x**3)*math.exp(-2*x)
|
|
>>> y = euler_modified(f2, 1.0, 0.0, 0.1, 0.3)
|
|
>>> float(y[-1])
|
|
0.5525976431951775
|
|
"""
|
|
n = int(np.ceil((x_end - x0) / step_size))
|
|
y = np.zeros((n + 1,))
|
|
y[0] = y0
|
|
x = x0
|
|
|
|
for k in range(n):
|
|
y_get = y[k] + step_size * ode_func(x, y[k])
|
|
y[k + 1] = y[k] + (
|
|
(step_size / 2) * (ode_func(x, y[k]) + ode_func(x + step_size, y_get))
|
|
)
|
|
x += step_size
|
|
|
|
return y
|
|
|
|
|
|
if __name__ == "__main__":
|
|
import doctest
|
|
|
|
doctest.testmod()
|