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lines 8-312 of file: include/cppad/core/abs_normal_fun.hpp
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{xrst_begin abs_normal_fun}

Create An Abs-normal Representation of a Function
#################################################

Syntax
******
| *f* . ``abs_normal_fun`` ( *g* , *a* )

f
*
The object *f* has prototype

    ``const ADFun`` < *Base* >& *f*

It represents a function :math:`f : \B{R}^n \rightarrow \B{R}^m`.
We assume that the only non-smooth terms in the representation are
absolute value functions and use :math:`s \in \B{Z}_+`
to represent the number of these terms.

n
=
We use *n* to denote the dimension of the domain space for *f* .

m
=
We use *m* to denote the dimension of the range space for *f* .

s
=
We use *s* to denote the number of absolute value terms in *f* .

a
*
The object *a* has prototype

    ``ADFun`` < *Base* > *a*

The initial function representation in *a* is lost.
Upon return it represents the result of the absolute terms
:math:`a : \B{R}^n \rightarrow \B{R}^s`; see :math:`a(x)` defined below.
Note that *a* is constructed by copying *f*
and then changing the dependent variables. There may
be many calculations in this representation that are not necessary
and can be removed using

    *a* . ``optimize`` ()

This optimization is not done automatically by ``abs_normal_fun``
because it may take a significant amount of time.

zeta
====
Let :math:`\zeta_0 ( x )`
denote the argument for the first absolute value term in :math:`f(x)`,
:math:`\zeta_1 ( x , |\zeta_0 (x)| )` for the second term, and so on.

a(x)
====
For :math:`i = 0 , \ldots , {s-1}` define

.. math::

    a_i (x)
    =
    | \zeta_i ( x , a_0 (x) , \ldots , a_{i-1} (x ) ) |

This defines :math:`a : \B{R}^n \rightarrow \B{R}^s`.

g
*
The object *g* has prototype

    ``ADFun`` < *Base* > *g*

The initial function representation in *g* is lost.
Upon return it represents the smooth function
:math:`g : \B{R}^{n + s} \rightarrow  \B{R}^{m + s}` is defined by

.. math::

    g( x , u )
    =
    \left[ \begin{array}{c} y(x, u) \\ z(x, u) \end{array} \right]

were :math:`y(x, u)` and :math:`z(x, u)` are defined below.

z(x, u)
=======
Define the smooth function
:math:`z : \B{R}^{n + s} \rightarrow  \B{R}^s` by

.. math::

    z_i ( x , u ) = \zeta_i ( x , u_0 , \ldots , u_{i-1} )

Note that the partial of :math:`z_i` with respect to :math:`u_j` is zero
for :math:`j \geq i`.

y(x, u)
=======
There is a smooth function
:math:`y : \B{R}^{n + s} \rightarrow  \B{R}^m`
such that :math:`y( x , u ) = f(x)` whenever :math:`u = a(x)`.

Affine Approximation
********************
We define the affine approximations

.. math::
    :nowrap:

    \begin{eqnarray}
    y[ \hat{x} ]( x , u )
    & = &
    y ( \hat{x}, a( \hat{x} ) )
        + \partial_x y ( \hat{x}, a( \hat{x} ) ) ( x - \hat{x} )
        + \partial_u y ( \hat{x}, a( \hat{x} ) ) ( u - a( \hat{x} ) )
    \\
    z[ \hat{x} ]( x , u )
    & = &
    z ( \hat{x}, a( \hat{x} ) )
        + \partial_x z ( \hat{x}, a( \hat{x} ) ) ( x - \hat{x} )
        + \partial_u z ( \hat{x}, a( \hat{x} ) ) ( u - a( \hat{x} ) )
    \end{eqnarray}

It follows that

.. math::
    :nowrap:

    \begin{eqnarray}
    y( x , u )
    & = &
    y[ \hat{x} ]( x , u ) + o ( x - \hat{x}, u - a( \hat{x} ) )
    \\
    z( x , u )
    & = &
    z[ \hat{x} ]( x , u ) + o ( x - \hat{x}, u - a( \hat{x} ) )
    \end{eqnarray}

Abs-normal Approximation
************************

Approximating a(x)
==================
The function :math:`a(x)` is not smooth, but it is equal to
:math:`| z(x, u) |` when :math:`u = a(x)`.
Furthermore

.. math::

    z[ \hat{x} ]( x , u )
    =
    z ( \hat{x}, a( \hat{x} ) )
        + \partial_x z ( \hat{x}, a( \hat{x} ) ) ( x - \hat{x} )
        + \partial_u z ( \hat{x}, a( \hat{x} ) ) ( u - a( \hat{x} ) )

The partial of :math:`z_i` with respect to :math:`u_j` is zero
for :math:`j \geq i`. It follows that

.. math::

    z_i [ \hat{x} ]( x , u )
    =
    z_i ( \hat{x}, a( \hat{x} ) )
        + \partial_x z_i ( \hat{x}, a( \hat{x} ) ) ( x - \hat{x} )
        + \sum_{j < i} \partial_{u(j)}
            z_i ( \hat{x}, a( \hat{x} ) ) ( u_j - a_j ( \hat{x} ) )

Considering the case :math:`i = 0` we define

.. math::

    a_0 [ \hat{x} ]( x )
    =
    | z_0 [ \hat{x} ]( x , u ) |
    =
    \left|
        z_0 ( \hat{x}, a( \hat{x} ) )
        + \partial_x z_0 ( \hat{x}, a( \hat{x} ) ) ( x - \hat{x} )
    \right|

It follows that

.. math::

    a_0 (x) = a_0 [ \hat{x} ]( x ) + o ( x - \hat{x} )

In general, we define :math:`a_i [ \hat{x} ]` using
:math:`a_j [ \hat{x} ]` for :math:`j < i` as follows:

.. math::

    a_i [ \hat{x} ]( x )
    =
    \left |
        z_i ( \hat{x}, a( \hat{x} ) )
        + \partial_x z_i ( \hat{x}, a( \hat{x} ) ) ( x - \hat{x} )
        + \sum_{j < i} \partial_{u(j)}
            z_i ( \hat{x}, a( \hat{x} ) )
                ( a_j [ \hat{x} ] ( x )  - a_j ( \hat{x} ) )
    \right|

It follows that

.. math::

    a (x) = a[ \hat{x} ]( x ) + o ( x - \hat{x} )

Note that in the case where :math:`z(x, u)` and :math:`y(x, u)` are
affine,

.. math::

    a[ \hat{x} ]( x ) = a( x )

Approximating f(x)
==================

.. math::

    f(x)
    =
    y ( x , a(x ) )
    =
    y [ \hat{x} ] ( x , a[ \hat{x} ] ( x ) )
    + o( x - \hat{x} )

Correspondence to Literature
****************************
Using the notation
:math:`Z = \partial_x z(\hat{x}, \hat{u})`,
:math:`L = \partial_u z(\hat{x}, \hat{u})`,
:math:`J = \partial_x y(\hat{x}, \hat{u})`,
:math:`Y = \partial_u y(\hat{x}, \hat{u})`,
the approximation for :math:`z` and :math:`y` are

.. math::
    :nowrap:

    \begin{eqnarray}
    z[ \hat{x} ]( x , u )
    & = &
    z ( \hat{x}, a( \hat{x} ) ) + Z ( x - \hat{x} ) + L ( u - a( \hat{x} ) )
    \\
    y[ \hat{x} ]( x , u )
    & = &
    y ( \hat{x}, a( \hat{x} ) ) + J ( x - \hat{x} ) + Y ( u - a( \hat{x} ) )
    \end{eqnarray}

Moving the terms with :math:`\hat{x}` together, we have

.. math::
    :nowrap:

    \begin{eqnarray}
    z[ \hat{x} ]( x , u )
    & = &
    z ( \hat{x}, a( \hat{x} ) ) - Z \hat{x} - L a( \hat{x} )  + Z x + L u
    \\
    y[ \hat{x} ]( x , u )
    & = &
    y ( \hat{x}, a( \hat{x} ) ) - J \hat{x} - Y a( \hat{x} )  + J x + Y u
    \end{eqnarray}

Using the notation
:math:`c = z ( \hat{x}, \hat{u} ) - Z \hat{x} - L \hat{u}`,
:math:`b = y ( \hat{x}, \hat{u} ) - J \hat{x} - Y \hat{u}`,
we have

.. math::
    :nowrap:

    \begin{eqnarray}
    z[ \hat{x} ]( x , u ) & = & c + Z x + L u
    \\
    y[ \hat{x} ]( x , u ) & = & b + J x + Y u
    \end{eqnarray}

Considering the affine case, where the approximations are exact,
and choosing :math:`u = a(x) = |z(x, u)|`, we obtain

.. math::
    :nowrap:

    \begin{eqnarray}
    z( x , a(x ) ) & = & c + Z x + L |z( x , a(x ) )|
    \\
    y( x , a(x ) ) & = & b + J x + Y |z( x , a(x ) )|
    \end{eqnarray}

This is Equation (2) of the
:ref:`example_abs_normal@Reference` .
{xrst_toc_hidden
    example/abs_normal/abs_normal.xrst
}
Example
*******
The file :ref:`abs_get_started.cpp-name` contains
an example and test using this operation.
The section :ref:`example_abs_normal-name`
has a links to all the abs normal examples.

{xrst_end abs_normal_fun}