%feature("docstring") OT::AdaptiveStieltjesAlgorithm
"AdaptiveStieltjes algorithm used to build the orthonormal basis.

The algorithm builds a polynomial basis orthonormal with respect to a specific distribution.

Parameters
----------
measure : :class:`~openturns.Distribution`
    A measure for which the orthonormal polynomial basis is built. 

See also
--------
OrthonormalizationAlgorithm

Notes
-----
It implements an adaptive Stieltjes algorithm that builds the polynomial family
orthonormal with respect to the distribution *measure*, using the
:class:`~openturns.GaussKronrod` adaptive integration method to compute the
following dot-products: :math:`R_n=\langle Q_n, Q_n\rangle` and
:math:`S_n=\langle xQ_n, Q_n\rangle` where :math:`Q_n` is the monic polynomial
associated to the orthonormal polynomial :math:`P_n`, needed to compute the
coefficients of the three-terms recurrence relation that defines :math:`P_n`
(see :class:`~openturns.OrthogonalUnivariatePolynomialFamily`):

.. math::

    a_n = 1/\sqrt{\beta_{n+1}}\quad b_n = -\alpha_n/sqrt{\beta_{n+1}}\quad c_n = -\sqrt{\beta_n/\beta_{n+1}}

where :math:`\alpha_n = S_n / R_n` and :math:`\beta_n  = R_n / R_{n-1}, n>0, \beta_0 = 0`.
"
