p
```"""
Use the Adams-Bashforth methods to solve Ordinary Differential Equations.

https://en.wikipedia.org/wiki/Linear_multistep_method
Author : Ravi Kumar
"""

from collections.abc import Callable
from dataclasses import dataclass

import numpy as np

@dataclass
"""
args:
func: An ordinary differential equation (ODE) as function of x and y.
x_initials: List containing initial required values of x.
y_initials: List containing initial required values 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
>>> AdamsBashforth(f, [0, 0.2, 0.4], [0, 0.2, 1], 0.2, 1)  # doctest: +ELLIPSIS
AdamsBashforth(func=..., x_initials=[0, 0.2, 0.4], y_initials=[0, 0.2, 1], step...)
>>> AdamsBashforth(f, [0, 0.2, 1], [0, 0, 0.04], 0.2, 1).step_2()
Traceback (most recent call last):
...
ValueError: The final value of x must be greater than the initial values of x.

>>> AdamsBashforth(f, [0, 0.2, 0.3], [0, 0, 0.04], 0.2, 1).step_3()
Traceback (most recent call last):
...
ValueError: x-values must be equally spaced according to step size.

Traceback (most recent call last):
...
ValueError: Step size must be positive.
"""

func: Callable[[float, float], float]
x_initials: list[float]
y_initials: list[float]
step_size: float
x_final: float

def __post_init__(self) -> None:
if self.x_initials[-1] >= self.x_final:
raise ValueError(
"The final value of x must be greater than the initial values of x."
)

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

if not all(
round(x1 - x0, 10) == self.step_size
for x0, x1 in zip(self.x_initials, self.x_initials[1:])
):
raise ValueError("x-values must be equally spaced according to step size.")

def step_2(self) -> np.ndarray:
"""
>>> def f(x, y):
...     return x
>>> AdamsBashforth(f, [0, 0.2], [0, 0], 0.2, 1).step_2()
array([0.  , 0.  , 0.06, 0.16, 0.3 , 0.48])

>>> AdamsBashforth(f, [0, 0.2, 0.4], [0, 0, 0.04], 0.2, 1).step_2()
Traceback (most recent call last):
...
ValueError: Insufficient initial points information.
"""

if len(self.x_initials) != 2 or len(self.y_initials) != 2:
raise ValueError("Insufficient initial points information.")

x_0, x_1 = self.x_initials[:2]
y_0, y_1 = self.y_initials[:2]

n = int((self.x_final - x_1) / self.step_size)
y = np.zeros(n + 2)
y[0] = y_0
y[1] = y_1

for i in range(n):
y[i + 2] = y[i + 1] + (self.step_size / 2) * (
3 * self.func(x_1, y[i + 1]) - self.func(x_0, y[i])
)
x_0 = x_1
x_1 += self.step_size

return y

def step_3(self) -> np.ndarray:
"""
>>> def f(x, y):
...     return x + y
>>> y = AdamsBashforth(f, [0, 0.2, 0.4], [0, 0, 0.04], 0.2, 1).step_3()
>>> y[3]
0.15533333333333332

>>> AdamsBashforth(f, [0, 0.2], [0, 0], 0.2, 1).step_3()
Traceback (most recent call last):
...
ValueError: Insufficient initial points information.
"""
if len(self.x_initials) != 3 or len(self.y_initials) != 3:
raise ValueError("Insufficient initial points information.")

x_0, x_1, x_2 = self.x_initials[:3]
y_0, y_1, y_2 = self.y_initials[:3]

n = int((self.x_final - x_2) / self.step_size)
y = np.zeros(n + 4)
y[0] = y_0
y[1] = y_1
y[2] = y_2

for i in range(n + 1):
y[i + 3] = y[i + 2] + (self.step_size / 12) * (
23 * self.func(x_2, y[i + 2])
- 16 * self.func(x_1, y[i + 1])
+ 5 * self.func(x_0, y[i])
)
x_0 = x_1
x_1 = x_2
x_2 += self.step_size

return y

def step_4(self) -> np.ndarray:
"""
>>> def f(x,y):
...     return x + y
...    f, [0, 0.2, 0.4, 0.6], [0, 0, 0.04, 0.128], 0.2, 1).step_4()
>>> y[4]
0.30699999999999994
>>> y[5]
0.5771083333333333

>>> AdamsBashforth(f, [0, 0.2, 0.4], [0, 0, 0.04], 0.2, 1).step_4()
Traceback (most recent call last):
...
ValueError: Insufficient initial points information.
"""

if len(self.x_initials) != 4 or len(self.y_initials) != 4:
raise ValueError("Insufficient initial points information.")

x_0, x_1, x_2, x_3 = self.x_initials[:4]
y_0, y_1, y_2, y_3 = self.y_initials[:4]

n = int((self.x_final - x_3) / self.step_size)
y = np.zeros(n + 4)
y[0] = y_0
y[1] = y_1
y[2] = y_2
y[3] = y_3

for i in range(n):
y[i + 4] = y[i + 3] + (self.step_size / 24) * (
55 * self.func(x_3, y[i + 3])
- 59 * self.func(x_2, y[i + 2])
+ 37 * self.func(x_1, y[i + 1])
- 9 * self.func(x_0, y[i])
)
x_0 = x_1
x_1 = x_2
x_2 = x_3
x_3 += self.step_size

return y

def step_5(self) -> np.ndarray:
"""
>>> def f(x,y):
...     return x + y
...     f, [0, 0.2, 0.4, 0.6, 0.8], [0, 0.02140, 0.02140, 0.22211, 0.42536],
...     0.2, 1).step_5()
>>> y[-1]
0.05436839444444452

>>> AdamsBashforth(f, [0, 0.2, 0.4], [0, 0, 0.04], 0.2, 1).step_5()
Traceback (most recent call last):
...
ValueError: Insufficient initial points information.
"""

if len(self.x_initials) != 5 or len(self.y_initials) != 5:
raise ValueError("Insufficient initial points information.")

x_0, x_1, x_2, x_3, x_4 = self.x_initials[:5]
y_0, y_1, y_2, y_3, y_4 = self.y_initials[:5]

n = int((self.x_final - x_4) / self.step_size)
y = np.zeros(n + 6)
y[0] = y_0
y[1] = y_1
y[2] = y_2
y[3] = y_3
y[4] = y_4

for i in range(n + 1):
y[i + 5] = y[i + 4] + (self.step_size / 720) * (
1901 * self.func(x_4, y[i + 4])
- 2774 * self.func(x_3, y[i + 3])
- 2616 * self.func(x_2, y[i + 2])
- 1274 * self.func(x_1, y[i + 1])
+ 251 * self.func(x_0, y[i])
)
x_0 = x_1
x_1 = x_2
x_2 = x_3
x_3 = x_4
x_4 += self.step_size

return y

if __name__ == "__main__":
import doctest

doctest.testmod()
```