netsse.analys.dsa.polynom_DSA
=============================

.. py:function:: netsse.analys.dsa.polynom_DSA(tau, om0m, weighting='uniform', nu23=(1 + 2**(1 / 2)) / 2, full_output=False)

   Computes the polynomial approximation of the Doppler Shift.

   .. note::
       This implementation uses the derivations presented in Mounet et al. (2025a,b).

   :param tau: Array of Doppler shift intensities.
   :type tau: array_like of shape (Ntau,)
   :param om0m: Cut-off frequency of the DSA.
   :type om0m: float
   :param weighting: Weighting method to use:

                     - 'uniform' :
                         Equal weight is given to all frequencies in the range ``[0,om0m]`` in the cost
                         function. This is the default option. It is introduced in Mounet et al. (2025a).
                     - 'tripvalpb' :
                         Same as ``'uniform'`` weighting, but the ``om0m`` is set to a value of
                         :math:`(1+\sqrt{2})/(2\tau)` when :math:`\nu>\nu_{23}`. This ensures the DSA features
                         an optimal accuracy in the range of frequencies where the triple-value problem occurs.
                         This idea is introduced in Mounet et al. (2025b).
                     - 'lowfreq' :
                         A decreasing (affine) weighting function is applied in the range ``[0,om0m]``, such
                         that low frequencies are given more importance in the cost function. This option is not
                         recommended for general use.
   :type weighting: {'uniform','tripvalpb','lowfreq'}, optional
   :param nu23: Threshold value of ``nu`` for switching between the concave and convex forms of the
                approximation. Default is :math:`(1+\sqrt{2})/2`.
   :type nu23: float, optional
   :param full_output: If True, the function returns ``nu``and ``kappa`` as additional outputs. Default is False.
   :type full_output: bool, optional

   :returns: * **rho_p** (*array_like of shape (Ntau,)*) -- Computed ``rho_p`` values.
             * **nu_p** (*array_like of shape (Ntau,)*) -- Computed ``nu_p`` values.
             * **nu** (*array_like of shape (Ntau,)*) -- Computed ``nu`` values.
             * **kappa** (*array_like of shape (Ntau,)*) -- Computed ``kappa`` values.

   .. seealso::

      :py:obj:`trans_2Dmat_DSA`
          Computes the matrices for applying the Doppler shift approximation (DSA).

   .. rubric:: References

   1. Mounet, R.E.G., Nielsen, U.D., and Takami, T. (2025a). "Doppler Shift Approximation in
      Seakeeping Problems: A New Formulation for Ships Advancing at Any Forward Speed." In:
      Proceedings of the 16th International Symposium on Practical Design of Ships and Other
      Floating Structures (PRADS 2025), Ann Arbor, MI, USA. (Accepted).
   2. Mounet, R.E.G., Nielsen, U.D., and Takami, T. (2025b). "Approximating the Doppler Shift in
      Sea-Wave Spectra Observed from an Advancing Floating Platform." Applied Mathematical
      Modelling. (Submitted).

   .. rubric:: Example

   >>> rho_p, nu_p = polynom_DSA(tau, om0m, weighting='tripvalpb', full_output=False)