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lines 8-128 of file: include/cppad/example/atomic_four/lin_ode/for_type.hpp
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{xrst_begin atomic_four_lin_ode_for_type.hpp}

Atomic Linear ODE Forward Type Calculation: Example Implementation
##################################################################

Purpose
*******
The ``for_type`` routine overrides the virtual functions
used by the atomic_four base; see
:ref:`for_type<atomic_four_for_type-name>` .

Notation
********
We use the notation:
:ref:`atomic_four_lin_ode@call_id`
:ref:`atomic_four_lin_ode@r`
:ref:`atomic_four_lin_ode@pattern`
:ref:`atomic_four_lin_ode@transpose`
:ref:`atomic_four_lin_ode@pattern@nnz` ,
:ref:`atomic_four_lin_ode@pattern@row` ,
:ref:`atomic_four_lin_ode@pattern@col` ,
:ref:`atomic_four_lin_ode@x` ,
:ref:`atomic_four_lin_ode@x@n` ,
:ref:`atomic_four_lin_ode@x@A(x)` ,
:ref:`atomic_four_lin_ode@x@b(x)` ,
:ref:`atomic_four_lin_ode@y(x)` ,
:ref:`atomic_four_lin_ode@y(x)@m` ,
:ref:`atomic_four_lin_ode@vk(x)` ,
and the following additional notation:

T(s)
====
We use :math:`\R{T} ( s )` to denote the ad_type of a scalar value :math:`s`.
There are four possible
:ref:`ad_types<atomic_four_for_type@ad_type>` :
identical_zero, constant, dynamic, and variable in that order.

Theory
******
This routine must calculate the following value for
:math:`i = 0, \ldots, m-1`; see :ref:`atomic_four_lin_ode@y(x)@m` :

.. math::

   \R{T} [ y_i (x) ] = \max_k \R{T} [  v_i^k (x) ]

The type :math:`\R{T} [ v_i^0 (x) ] = \R{T}[ b_i (x) ]`.
This is easy to calculate given the type of the components of *x* ;
see :ref:`atomic_four_lin_ode@x@b(x)` .
Furthermore, for :math:`k > 0`

.. math::

   v_i^k (x)
   =
   \frac{r}{k} \sum_{j=0}^{m-1} A_{i,j} (x) v_j^{k-1} (x)

.. math::

   \R{T} [ v_i^k (x) ]
   =
   \max_j \R{T} [ A_{i,j} (x) v_j^{k-1} (x) ]

.. math::

   \R{T} [ A_{i,j} (x) v_j^k (x) ]
   =
   \left\{ \begin{array}{ll}
      \R{identical\_zero} &
      \R{if} A_{i,j} (x) \W{\R{or}} v_j^{k-1} (x) \W{\R{is}} \R{identical\_zero}
      \\
      \max\{ \R{T} [ A_{i,j} (x) ] \W{,} \R{T} [ v_j^{k-1} (x) ] \} &
      \R{otherwise}
   \end{array} \right.

If :math:`A_{i,j} (x)` is not in the sparsity
:ref:`atomic_four_lin_ode@pattern` , it is identically zero.
Furthermore we are allowing for the case where
:math:`A_{i,j} (x)` is in the pattern and it is identically zero; i.e.,
the sparsity pattern is not efficient as it could be.
The type :math:`\R{T} [ A_{i,j} (x) ]` for components in the sparsity pattern
is easy to calculate given the type of the components of *x* ;
see :ref:`atomic_four_lin_ode@x@A(x)` .
Suppose :math:`\ell` is such that for all :math:`i`

.. math::

   \R{T} [ v_i^\ell (x) ] \leq \max_{k < \ell} \R{T} [ v_i^k (x) ]

It follows that

.. math::

   \R{T} [ v_j^{\ell+1} (x) ]  = \max_j \R{T} [ A_{i,j} (x) v_j^\ell (x) ]

.. math::

   \R{T} [ v_j^{\ell+1} (x) ]
   \leq
   \max_{k < \ell} \max_j \R{T} [ A_{i,j} (x) v_j^k (x) ]

.. math::

   \R{T} [ v_j^{\ell+1} (x) ]
   \leq
   \max_{k < \ell} \R{T} [ v_i^k (x) ]

From this it is clear that

.. math::

   \R{T} [ y_i (x) ] = \max_{k < \ell} \R{T} [  v_i^k (x) ]

Source
******
{xrst_literal
   // BEGIN C++
   // END C++
}

{xrst_end atomic_four_lin_ode_for_type.hpp}