:mod:`lamberthub.universal_solvers.vallado`
===========================================

.. py:module:: lamberthub.universal_solvers.vallado
   :noindex:

.. autoapi-nested-parse::

   This module holds all methods devised by David A. Vallado.



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


Functions
~~~~~~~~~

.. autoapisummary::


   lamberthub.universal_solvers.vallado.vallado2013


   lamberthub.universal_solvers.vallado._tof_vallado


   lamberthub.universal_solvers.vallado._X_at_psi


   lamberthub.universal_solvers.vallado._get_A


   lamberthub.universal_solvers.vallado._y_at_psi



.. py:function:: vallado2013(mu, r1, r2, tof, M=0, prograde=True, low_path=True, maxiter=100, atol=1e-05, rtol=1e-07, full_output=False)

   Vallado's algorithm makes use of the universal formulation to solve for the
   Lambert's problem. By making use of a bisection method, it guarantees the
   convergence to the solution but the amount of iterations require
   dramatically increases.

   :param mu: Gravitational parameter, equivalent to :math:`GM` of attractor body.
   :type mu: float
   :param r1: Initial position vector.
   :type r1: numpy.array
   :param r2: Final position vector.
   :type r2: numpy.array
   :param M: Number of revolutions. Must be equal or greater than 0 value.
   :type M: int
   :param prograde: If `True`, specifies prograde motion. Otherwise, retrograde motion is imposed.
   :type prograde: bool
   :param low_path: If two solutions are available, it selects between high or low path.
   :type low_path: bool
   :param maxiter: Maximum number of iterations.
   :type maxiter: int
   :param atol: Absolute tolerance.
   :type atol: float
   :param rtol: Relative tolerance.
   :type rtol: float
   :param full_output: If True, the number of iterations and time per iteration are also returned.
   :type full_output: bool

   :returns: * **v1** (*numpy.array*) -- Initial velocity vector.
             * **v2** (*numpy.array*) -- Final velocity vector.
             * **numiter** (*int*) -- Number of iterations.
             * **tpi** (*float*) -- Time per iteration in seconds.

   .. rubric:: Notes

   This algorithm is presented as an alternative to the one developed by Bate
   in 1971. Bate did not impose a particular numerical solver for his algorithm
   but cited both bisection and Newton's one. However, for some values of the
   boundary problem, the initial guess might diverge if Newton's solver is
   used. That's why Vallado decided to employ a bisection method instead.
   Although detrimental from the point of view of performance, this algorithm
   properly reaches solution in the majority of the cases.

   All credits of the implementation go to Juan Luis Cano Rodríguez and the
   poliastro development team, from which this routine inherits. Some changes
   were made to adapt it to `lamberthub` API.

   Copyright (c) 2012-2021 Juan Luis Cano Rodríguez and the poliastro
   development team.

   .. rubric:: References

   [1] Vallado, D. A. (2001). Fundamentals of astrodynamics and applications
   (Vol. 12). Springer Science & Business Media.


.. py:function:: _tof_vallado(mu, psi, X, A, y)

   Evaluates universal Kepler's equation.

   :param mu: The gravitational parameter.
   :type mu: float
   :param psi: The free-parameter or independent variable.
   :type psi: float
   :param X: Auxiliary variable.
   :type X: float
   :param A: The transfer angle parameter.
   :type A: float
   :param y: Auxiliary variable.
   :type y: float

   :returns: **tof** -- The computed time of flight.
   :rtype: float


.. py:function:: _X_at_psi(psi, y)

   Computes the value of X at given psi.

   :param psi: The free-parameter or independent variable.
   :type psi: float
   :param y: Auxiliary variable.
   :type y: float

   :returns: **X** -- Auxiliary variable.
   :rtype: float


.. py:function:: _get_A(r1_norm, r2_norm, dtheta)

   Computes the value of the A constant.

   :param r1_norm: Initial position vector norm.
   :type r1_norm: float
   :param r2_norm: Final position vector norm.
   :type r2_norm: float
   :param dtheta: The transfer angle in radians.
   :type dtheta: float

   :returns: **A** -- The transfer angle parameter.
   :rtype: float


.. py:function:: _y_at_psi(psi, r1_norm, r2_norm, A)

   Evaluates the value of y at given psi.

   :param psi: The free-parameter or independent variable.
   :type psi: float
   :param r1_norm: Initial position vector norm.
   :type r1_norm: float
   :param r2_norm: Final position vector norm.
   :type r2_norm: float
   :param A: The transfer angle parameter.
   :type A: float

   :returns: **y** -- Auxiliary variable.
   :rtype: float

   .. rubric:: Notes

   This is equation (7-59) simplified, similarly as made in [1].