maths.numerical_analysis.runge_kutta_fehlberg_45
================================================

.. py:module:: maths.numerical_analysis.runge_kutta_fehlberg_45

.. autoapi-nested-parse::

   Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.



Functions
---------

.. autoapisummary::

   maths.numerical_analysis.runge_kutta_fehlberg_45.runge_kutta_fehlberg_45


Module Contents
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.. py:function:: runge_kutta_fehlberg_45(func: collections.abc.Callable, x_initial: float, y_initial: float, step_size: float, x_final: float) -> numpy.ndarray

   Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
   of order 5.

   https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

   args:
   func: An ordinary differential equation (ODE) as function of x and y.
   x_initial: The initial value of x.
   y_initial: The initial value of y.
   step_size: The increment value of x.
   x_final: The final value of x.

   Returns:
       Solution of y at each nodal point

   # exact value of y[1] is tan(0.2) = 0.2027100937470787
   >>> def f(x, y):
   ...     return 1 + y**2
   >>> y = runge_kutta_fehlberg_45(f, 0, 0, 0.2, 1)
   >>> float(y[1])
   0.2027100937470787
   >>> def f(x,y):
   ...     return x
   >>> y = runge_kutta_fehlberg_45(f, -1, 0, 0.2, 0)
   >>> float(y[1])
   -0.18000000000000002
   >>> y = runge_kutta_fehlberg_45(5, 0, 0, 0.1, 1)
   Traceback (most recent call last):
       ...
   TypeError: 'int' object is not callable
   >>> def f(x, y):
   ...     return x + y
   >>> y = runge_kutta_fehlberg_45(f, 0, 0, 0.2, -1)
   Traceback (most recent call last):
       ...
   ValueError: The final value of x must be greater than initial value of x.
   >>> def f(x, y):
   ...     return x
   >>> y = runge_kutta_fehlberg_45(f, -1, 0, -0.2, 0)
   Traceback (most recent call last):
       ...
   ValueError: Step size must be positive.