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* Remove eval from arithmetic_analysis/newton_raphson.py
* Relocate contents of arithmetic_analysis/
Delete the arithmetic_analysis/ directory and relocate its files because
the purpose of the directory was always ill-defined. "Arithmetic
analysis" isn't a field of math, and the directory's files contained
algorithms for linear algebra, numerical analysis, and physics.
Relocated the directory's linear algebra algorithms to linear_algebra/,
its numerical analysis algorithms to a new subdirectory called
maths/numerical_analysis/, and its single physics algorithm to physics/.
* updating DIRECTORY.md
---------
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
115 lines
3.1 KiB
Python
115 lines
3.1 KiB
Python
"""
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Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
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"""
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from collections.abc import Callable
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import numpy as np
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def runge_kutta_fehlberg_45(
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func: Callable,
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x_initial: float,
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y_initial: float,
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step_size: float,
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x_final: float,
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) -> np.ndarray:
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"""
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Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
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of order 5.
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https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method
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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_initial: The initial value of x.
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y_initial: The initial value 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:
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Solution of y at each nodal point
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# exact value of y[1] is tan(0.2) = 0.2027100937470787
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>>> def f(x, y):
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... return 1 + y**2
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>>> y = runge_kutta_fehlberg_45(f, 0, 0, 0.2, 1)
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>>> y[1]
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0.2027100937470787
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>>> def f(x,y):
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... return x
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>>> y = runge_kutta_fehlberg_45(f, -1, 0, 0.2, 0)
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>>> y[1]
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-0.18000000000000002
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>>> y = runge_kutta_fehlberg_45(5, 0, 0, 0.1, 1)
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Traceback (most recent call last):
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...
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TypeError: 'int' object is not callable
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>>> def f(x, y):
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... return x + y
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>>> y = runge_kutta_fehlberg_45(f, 0, 0, 0.2, -1)
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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 initial value of x.
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>>> def f(x, y):
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... return x
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>>> y = runge_kutta_fehlberg_45(f, -1, 0, -0.2, 0)
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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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if x_initial >= x_final:
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raise ValueError(
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"The final value of x must be greater than initial value of x."
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)
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if step_size <= 0:
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raise ValueError("Step size must be positive.")
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n = int((x_final - x_initial) / step_size)
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y = np.zeros(
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(n + 1),
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)
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x = np.zeros(n + 1)
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y[0] = y_initial
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x[0] = x_initial
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for i in range(n):
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k1 = step_size * func(x[i], y[i])
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k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
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k3 = step_size * func(
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x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
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)
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k4 = step_size * func(
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x[i] + (12 / 13) * step_size,
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y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
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)
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k5 = step_size * func(
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x[i] + step_size,
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y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
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)
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k6 = step_size * func(
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x[i] + step_size / 2,
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y[i]
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- (8 / 27) * k1
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+ 2 * k2
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- (3544 / 2565) * k3
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+ (1859 / 4104) * k4
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- (11 / 40) * k5,
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)
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y[i + 1] = (
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y[i]
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+ (16 / 135) * k1
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+ (6656 / 12825) * k3
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+ (28561 / 56430) * k4
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- (9 / 50) * k5
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+ (2 / 55) * k6
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)
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x[i + 1] = step_size + x[i]
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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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