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

Author: ravi-ivar-7

maths/numerical_analysis/adams_bashforth.py

@@ -0,0 +1,172 @@
+"""
+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
+
+import numpy as np
+
+class AdamsBashforth:
+    def __init__(self, func: Callable[[float, float], float], x_initials: list[float], y_initials: list[float], step_size: float, x_final: float):
+
+        if x_initials[-1] >= x_final:
+            raise ValueError("The final value of x must be greater than the initial values of x.")
+
+        if step_size <= 0:
+            raise ValueError("Step size must be positive.")
+
+        if not all(x1 - x0 == step_size for x0, x1 in zip(x_initials, x_initials[1:])):
+            raise ValueError("x-values must be equally spaced according to step size.")
+
+        self.func = func
+        self.x_initials = x_initials
+        self.y_initials = y_initials
+        self.step_size = step_size
+        self.x_final = x_final
+
+    """
+    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, 0], 0.2, 1).step_2()
+    >>> y
+    array([0.      0.      0.06    0.178   0.3654  0.63722])
+
+    >>> 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[-1]
+    0.57710833
+
+    >>> def f(x, y):
+    ...     return x + y
+    >>> y = AdamsBashforth(f, [0, 0.2, 1], [0, 0, 0.04], 0.2, 1).step_3()
+    Traceback (most recent call last):
+        ...
+    ValueError: The final value of x must be greater than the all initial values of x.
+
+    >>> def f(x, y):
+    ...     return x + y
+    >>> y = 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.
+
+    >>> def f(x, y):
+    ...     return x
+    >>> y = 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.
+    
+    >>> def f(x, y):
+    ...     return (x -y)/2
+    >>> y = AdamsBashforth(f, [0, 0.2, 0.4], [0, 0, 0.04], 0.2, 1).step_2()
+    Traceback (most recent call last):
+        ...
+    ValueError: Insufficient nodal points values information.
+    """
+
+    def step_2(self) -> np.ndarray:
+        if len(self.x_initials) != 2 or len(self.y_initials) != 2:
+            raise ValueError("Insufficient nodal points values 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 = x_1 + self.step_size
+
+        return y
+
+    def step_3(self) -> np.ndarray:
+        if len(self.x_initials) != 3 or len(self.y_initials) != 3:
+            raise ValueError("Insufficient nodal 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 = x_2 + self.step_size
+
+        return y
+
+    def step_4(self) -> np.ndarray:
+        if len(self.x_initials) != 4 or len(self.y_initials) != 4:
+            raise ValueError("Insufficient nodal 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 = x_3 + self.step_size
+
+        return y
+
+    def step_5(self) -> np.ndarray:
+        if len(self.x_initials) != 5 or len(self.y_initials) != 5:
+            raise ValueError("Insufficient nodal 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 = x_4 + self.step_size
+
+        return y
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()