.. _testing:

**************************************************
Testing the implementation of the contact function
**************************************************

This chapter describes how our implementation of the contact function is
tested. The source of the unit tests can be found in the file
``src/test_pw85.c``. Note that the tests described here are repeated over a
large set of tests case, including very flat and very slender sheroids, for
various relative orientations and center-to-center distances.

In the present chapter, we assume that the two ellipsoids (their matrices ``Q₁``
and ``Q₂`` are given), as well as their center-to-center radius vector
``r₁₂``. Then, a call to :cpp:func:`pw85::contact_function` delivers an estimate
of ``λ`` and ``μ²``.

We first assert that ``μ₁²`` and ``μ₂²`` as defined by Eq. :ref:`(3)
<optimization-eq-3>` in chapter :ref:`optimization` are close to the value
returned by :cpp:func:`pw85::contact_function`. For all the cases considered here,
this is true up to a relative error of ``10⁻¹⁰``.

We also check that ``f’(λ) = 0``, up to an absolute error of ``Δλf”(λ)`` where
``Δλ`` is the absolute tolerance on ``λ`` for the stopping criterion of the
Brent iterations, as defined by the constant :cpp:var:`pw85::lambda_atol`.