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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 * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Update adams_bashforth.py * Deleted extraneous file, maths/numerical_analysis/runge_kutta_gills.py * Added doctests to each function adams_bashforth.py * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * 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>
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Ravi Kumar
GitHub
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maths/numerical_analysis/adams_bashforth.py
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230
maths/numerical_analysis/adams_bashforth.py
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"""
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Use the Adams-Bashforth methods to solve Ordinary Differential Equations.
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https://en.wikipedia.org/wiki/Linear_multistep_method
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Author : Ravi Kumar
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"""
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from collections.abc import Callable
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from dataclasses import dataclass
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import numpy as np
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@dataclass
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class AdamsBashforth:
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"""
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args:
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func: An ordinary differential equation (ODE) as function of x and y.
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x_initials: List containing initial required values of x.
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y_initials: List containing initial required values of y.
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step_size: The increment value of x.
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x_final: The final value of x.
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Returns: Solution of y at each nodal point
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>>> def f(x, y):
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... return x + y
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>>> AdamsBashforth(f, [0, 0.2, 0.4], [0, 0.2, 1], 0.2, 1) # doctest: +ELLIPSIS
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AdamsBashforth(func=..., x_initials=[0, 0.2, 0.4], y_initials=[0, 0.2, 1], step...)
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>>> AdamsBashforth(f, [0, 0.2, 1], [0, 0, 0.04], 0.2, 1).step_2()
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Traceback (most recent call last):
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...
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ValueError: The final value of x must be greater than the initial values of x.
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>>> AdamsBashforth(f, [0, 0.2, 0.3], [0, 0, 0.04], 0.2, 1).step_3()
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Traceback (most recent call last):
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...
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ValueError: x-values must be equally spaced according to step size.
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>>> AdamsBashforth(f,[0,0.2,0.4,0.6,0.8],[0,0,0.04,0.128,0.307],-0.2,1).step_5()
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Traceback (most recent call last):
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...
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ValueError: Step size must be positive.
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"""
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func: Callable[[float, float], float]
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x_initials: list[float]
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y_initials: list[float]
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step_size: float
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x_final: float
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def __post_init__(self) -> None:
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if self.x_initials[-1] >= self.x_final:
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raise ValueError(
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"The final value of x must be greater than the initial values of x."
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)
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if self.step_size <= 0:
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raise ValueError("Step size must be positive.")
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if not all(
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round(x1 - x0, 10) == self.step_size
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for x0, x1 in zip(self.x_initials, self.x_initials[1:])
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):
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raise ValueError("x-values must be equally spaced according to step size.")
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def step_2(self) -> np.ndarray:
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"""
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>>> def f(x, y):
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... return x
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>>> AdamsBashforth(f, [0, 0.2], [0, 0], 0.2, 1).step_2()
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array([0. , 0. , 0.06, 0.16, 0.3 , 0.48])
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>>> AdamsBashforth(f, [0, 0.2, 0.4], [0, 0, 0.04], 0.2, 1).step_2()
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Traceback (most recent call last):
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...
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ValueError: Insufficient initial points information.
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"""
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if len(self.x_initials) != 2 or len(self.y_initials) != 2:
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raise ValueError("Insufficient initial points information.")
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x_0, x_1 = self.x_initials[:2]
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y_0, y_1 = self.y_initials[:2]
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n = int((self.x_final - x_1) / self.step_size)
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y = np.zeros(n + 2)
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y[0] = y_0
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y[1] = y_1
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for i in range(n):
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y[i + 2] = y[i + 1] + (self.step_size / 2) * (
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3 * self.func(x_1, y[i + 1]) - self.func(x_0, y[i])
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)
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x_0 = x_1
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x_1 += self.step_size
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return y
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def step_3(self) -> np.ndarray:
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"""
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>>> def f(x, y):
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... return x + y
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>>> y = AdamsBashforth(f, [0, 0.2, 0.4], [0, 0, 0.04], 0.2, 1).step_3()
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>>> y[3]
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0.15533333333333332
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>>> AdamsBashforth(f, [0, 0.2], [0, 0], 0.2, 1).step_3()
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Traceback (most recent call last):
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...
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ValueError: Insufficient initial points information.
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"""
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if len(self.x_initials) != 3 or len(self.y_initials) != 3:
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raise ValueError("Insufficient initial points information.")
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x_0, x_1, x_2 = self.x_initials[:3]
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y_0, y_1, y_2 = self.y_initials[:3]
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n = int((self.x_final - x_2) / self.step_size)
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y = np.zeros(n + 4)
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y[0] = y_0
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y[1] = y_1
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y[2] = y_2
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for i in range(n + 1):
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y[i + 3] = y[i + 2] + (self.step_size / 12) * (
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23 * self.func(x_2, y[i + 2])
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- 16 * self.func(x_1, y[i + 1])
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+ 5 * self.func(x_0, y[i])
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)
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x_0 = x_1
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x_1 = x_2
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x_2 += self.step_size
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return y
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def step_4(self) -> np.ndarray:
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"""
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>>> def f(x,y):
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... return x + y
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>>> y = AdamsBashforth(
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... f, [0, 0.2, 0.4, 0.6], [0, 0, 0.04, 0.128], 0.2, 1).step_4()
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>>> y[4]
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0.30699999999999994
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>>> y[5]
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0.5771083333333333
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>>> AdamsBashforth(f, [0, 0.2, 0.4], [0, 0, 0.04], 0.2, 1).step_4()
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Traceback (most recent call last):
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...
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ValueError: Insufficient initial points information.
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"""
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if len(self.x_initials) != 4 or len(self.y_initials) != 4:
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raise ValueError("Insufficient initial points information.")
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x_0, x_1, x_2, x_3 = self.x_initials[:4]
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y_0, y_1, y_2, y_3 = self.y_initials[:4]
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n = int((self.x_final - x_3) / self.step_size)
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y = np.zeros(n + 4)
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y[0] = y_0
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y[1] = y_1
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y[2] = y_2
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y[3] = y_3
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for i in range(n):
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y[i + 4] = y[i + 3] + (self.step_size / 24) * (
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55 * self.func(x_3, y[i + 3])
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- 59 * self.func(x_2, y[i + 2])
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+ 37 * self.func(x_1, y[i + 1])
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- 9 * self.func(x_0, y[i])
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)
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x_0 = x_1
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x_1 = x_2
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x_2 = x_3
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x_3 += self.step_size
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return y
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def step_5(self) -> np.ndarray:
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"""
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>>> def f(x,y):
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... return x + y
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>>> y = AdamsBashforth(
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... f, [0, 0.2, 0.4, 0.6, 0.8], [0, 0.02140, 0.02140, 0.22211, 0.42536],
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... 0.2, 1).step_5()
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>>> y[-1]
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0.05436839444444452
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>>> AdamsBashforth(f, [0, 0.2, 0.4], [0, 0, 0.04], 0.2, 1).step_5()
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Traceback (most recent call last):
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...
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ValueError: Insufficient initial points information.
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"""
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if len(self.x_initials) != 5 or len(self.y_initials) != 5:
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raise ValueError("Insufficient initial points information.")
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x_0, x_1, x_2, x_3, x_4 = self.x_initials[:5]
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y_0, y_1, y_2, y_3, y_4 = self.y_initials[:5]
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n = int((self.x_final - x_4) / self.step_size)
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y = np.zeros(n + 6)
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y[0] = y_0
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y[1] = y_1
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y[2] = y_2
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y[3] = y_3
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y[4] = y_4
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for i in range(n + 1):
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y[i + 5] = y[i + 4] + (self.step_size / 720) * (
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1901 * self.func(x_4, y[i + 4])
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- 2774 * self.func(x_3, y[i + 3])
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- 2616 * self.func(x_2, y[i + 2])
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- 1274 * self.func(x_1, y[i + 1])
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+ 251 * self.func(x_0, y[i])
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)
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x_0 = x_1
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x_1 = x_2
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x_2 = x_3
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x_3 = x_4
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x_4 += self.step_size
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return y
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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