lines 162-229 of file: xrst/algorithm.xrst

{xrst_begin llsq_obj}

Linear Least Squares Objective
##############################

Function
********

.. math::

   y(x) = \frac{1}{2} \sum_{i=0}^{n-1} \left(
      s_i - x_0 - x_1 t_i - x_2 t_i^2 - \cdots
   \right)^2

where *s* and *t* in :math:`{\bf R}^n` are given by:

.. math::

   t_i & = -1 + i * 2 / (n - 1)
   \\
   s_i & = \mathrm{sign} ( t_i )
   =
   \cases{
      -1 & if $t_i < 0$ \\
       0 & if $t_i = 0$ \\
      +1 & if $t_i > 0$
   }


Vector Operations
*****************
The calculation of this objective can be vectorized
with respect to the index *j* above.
The intent is to enable AD packages that have vector operators
to use them reduce memory and speed up the calculation of derivatives.

option
******
This algorithm uses the ``n_arg`` and ``n_other`` options; see below:

n_arg
=====
This is the size of the vector *x* above .
There is an assert checking that *n_arg* > 0.

n_other
=======
This is the size of the vector *t* above .
There is an assert checking that *n_other* > 0.

Implementation
**************
:ref:`cpp_llsq_obj-name` , :ref:`py_llsq_obj-name` .

Derivative
**********
The partial derivative of :math:`y(x)` with respect to :math:`x_j` is

.. math::

   \frac{ \partial y(x)} { \partial x_j }
   =
   - \sum_{i=0}^{n-1} \left(
      s_i - x_0 - x_1 t_i - x_2 t_i^2 - \cdots
   \right) t_i^j


{xrst_end llsq_obj}