maths.numerical_analysis.adams_bashforth
========================================

.. py:module:: maths.numerical_analysis.adams_bashforth

.. autoapi-nested-parse::

   Use the Adams-Bashforth methods to solve Ordinary Differential Equations.

   https://en.wikipedia.org/wiki/Linear_multistep_method
   Author : Ravi Kumar



Classes
-------

.. autoapisummary::

   maths.numerical_analysis.adams_bashforth.AdamsBashforth


Module Contents
---------------

.. py: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.


   .. py:method:: __post_init__() -> None


   .. py:method:: step_2() -> numpy.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.



   .. py:method:: step_3() -> numpy.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()
      >>> float(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.



   .. py:method:: step_4() -> numpy.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()
      >>> float(y[4])
      0.30699999999999994
      >>> float(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.



   .. py:method:: step_5() -> numpy.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()
      >>> float(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.



   .. py:attribute:: func
      :type:  collections.abc.Callable[[float, float], float]


   .. py:attribute:: step_size
      :type:  float


   .. py:attribute:: x_final
      :type:  float


   .. py:attribute:: x_initials
      :type:  list[float]


   .. py:attribute:: y_initials
      :type:  list[float]