Added adams-bashforth method of order 2, 3, 4, 5 (#10969)

* added runge kutta gills method

* added adams-bashforth method of order 2, 3, 4, 5

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* Update adams_bashforth.py

* Deleted extraneous file, maths/numerical_analysis/runge_kutta_gills.py

* Added doctests to each function adams_bashforth.py

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* Update adams_bashforth.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>

maths/numerical_analysis/adams_bashforth.py

@@ -0,0 +1,230 @@
+"""
+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
+class AdamsBashforth:
+    """
+    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.
+
+    >>> AdamsBashforth(f,[0,0.2,0.4,0.6,0.8],[0,0,0.04,0.128,0.307],-0.2,1).step_5()
+    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
+        >>> y = AdamsBashforth(
+        ...    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
+        >>> y = AdamsBashforth(
+        ...     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()