butools.moments.HankelMomsFromMoms
===================================

.. currentmodule:: butools.moments

.. np:function:: HankelMomsFromMoms

    .. list-table::
        :widths: 25 150

        * - Matlab:
          - :code:`hm = HankelMomsFromMoms(m)`
        * - Mathematica:
          - :code:`hm = HankelMomsFromMoms[m]`
        * - Python/Numpy:
          - :code:`hm = HankelMomsFromMoms(m)`
    
    Returns the Hankel moments given the raw moments.
    
    The raw moments are: :math:`m_i=E(\mathcal{X}^i)`
    
    The ith Hankel moment is the determinant of matrix 
    :math:`\Delta_{i/2}`, if i is even, 
    and it is the determinant of :math:`\Delta^{(1)}_{(i+1)/2}`, 
    if i is odd. For the definition of matrices :math:`\Delta`
    and :math:`\Delta^{(1)}` see [1]_.
       
    Parameters
    ----------
    m : vector of doubles
        The list of raw moments (starting with the first
        moment)
        
    Returns
    -------
    hm : vector of doubles
        The list of Hankel moments

    References
    ----------
    .. [1] http://en.wikipedia.org/wiki/Stieltjes_moment_problem

    Examples
    ========
    For Matlab:

    >>> M = [1.3, 2.4, 6.03, 20.5, 89.5, 474.9];
    >>> hmoms = HankelMomsFromMoms(M);
    >>> disp(hmoms);
              1.3         0.71        2.079       1.9973       13.841       44.916
    >>> moms = MomsFromHankelMoms(hmoms);
    >>> disp(moms);
              1.3          2.4         6.03         20.5         89.5        474.9
    >>> err = norm(moms-M);
    >>> disp(err);
        5.118e-13

    For Mathematica:

    >>> M = {1.3, 2.4, 6.03, 20.5, 89.5, 474.9};
    >>> hmoms = HankelMomsFromMoms[M];
    >>> Print[hmoms];
    {1.3, 0.7099999999999995, 2.079000000000002, 1.997299999999989, 13.841272999999994, 44.91574881000027}
    >>> moms = MomsFromHankelMoms[hmoms];
    >>> Print[moms];
    {1.3, 2.3999999999999995, 6.0299999999999985, 20.499999999999996, 89.50000000000006, 474.9000000000003}
    >>> err = Norm[moms-M];
    >>> Print[err];
    3.4578811769837537*^-13

    For Python/Numpy:

    >>> M = [1.3, 2.4, 6.03, 20.5, 89.5, 474.9]
    >>> hmoms = HankelMomsFromMoms(M)
    >>> print(hmoms)
    [1.3, 0.7099999999999995, 2.079000000000002, 1.997299999999989, 13.841272999999994, 44.91574881000027]
    >>> moms = MomsFromHankelMoms(hmoms)
    >>> print(moms)
    [1.3, 2.3999999999999995, 6.0299999999999985, 20.499999999999996, 89.500000000000057, 474.90000000000032]
    >>> err = la.norm(np.array(moms)-np.array(M))
    >>> print(err)
    3.4578811769837537e-13