Runge Kutta Fehlberg 45

p
```"""
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()
```