#### Runge Kutta Gills

p
```"""
Use the Runge-Kutta-Gill's method of order 4 to solve Ordinary Differential Equations.

https://www.geeksforgeeks.org/gills-4th-order-method-to-solve-differential-equations/
Author : Ravi Kumar
"""

from collections.abc import Callable
from math import sqrt

import numpy as np

def runge_kutta_gills(
func: Callable[[float, float], float],
x_initial: float,
y_initial: float,
step_size: float,
x_final: float,
) -> np.ndarray:
"""
Solve an Ordinary Differential Equations using Runge-Kutta-Gills Method of order 4.

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

>>> def f(x, y):
...     return (x-y)/2
>>> y = runge_kutta_gills(f, 0, 3, 0.2, 5)
>>> y[-1]
3.4104259225717537

>>> def f(x,y):
...     return x
>>> y = runge_kutta_gills(f, -1, 0, 0.2, 0)
>>> y
array([ 0.  , -0.18, -0.32, -0.42, -0.48, -0.5 ])

>>> def f(x, y):
...     return x + y
>>> y = runge_kutta_gills(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_gills(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)
y[0] = y_initial
for i in range(n):
k1 = step_size * func(x_initial, y[i])
k2 = step_size * func(x_initial + step_size / 2, y[i] + k1 / 2)
k3 = step_size * func(
x_initial + step_size / 2,
y[i] + (-0.5 + 1 / sqrt(2)) * k1 + (1 - 1 / sqrt(2)) * k2,
)
k4 = step_size * func(
x_initial + step_size, y[i] - (1 / sqrt(2)) * k2 + (1 + 1 / sqrt(2)) * k3
)

y[i + 1] = y[i] + (k1 + (2 - sqrt(2)) * k2 + (2 + sqrt(2)) * k3 + k4) / 6
x_initial += step_size
return y

if __name__ == "__main__":
import doctest

doctest.testmod()
```