namespace Eigen {

/** \page TopicNewExpressionType Adding a new expression type

<!--<span style="font-size:130%; color:red; font-weight: 900;"></span>-->
\warning
Disclaimer: this page is tailored to very advanced users who are not afraid of dealing with some %Eigen's internal aspects.
In most cases, a custom expression can be avoided by either using custom \ref MatrixBase::unaryExpr "unary" or \ref MatrixBase::binaryExpr "binary" functors,
while extremely complex matrix manipulations can be achieved by a nullary functors as described in the \ref TopicCustomizing_NullaryExpr "previous page".

This page describes with the help of an example how to implement a new
light-weight expression type in %Eigen. This consists of three parts:
the expression type itself, a traits class containing compile-time
information about the expression, and the evaluator class which is
used to evaluate the expression to a matrix.

\b TO \b DO: Write a page explaining the design, with details on
vectorization etc., and refer to that page here.


\eigenAutoToc

\section TopicSetting The setting

A circulant matrix is a matrix where each column is the same as the
column to the left, except that it is cyclically shifted downwards.
For example, here is a 4-by-4 circulant matrix:
\f[ \begin{bmatrix}
    1 & 8 & 4 & 2 \\
    2 & 1 & 8 & 4 \\
    4 & 2 & 1 & 8 \\
    8 & 4 & 2 & 1
\end{bmatrix} \f]
A circulant matrix is uniquely determined by its first column. We wish
to write a function \c makeCirculant which, given the first column,
returns an expression representing the circulant matrix.

For simplicity, we restrict the \c makeCirculant function to dense
matrices. It may make sense to also allow arrays, or sparse matrices,
but we will not do so here. We also do not want to support
vectorization.


\section TopicPreamble Getting started

We will present the file implementing the \c makeCirculant function
part by part. We start by including the appropriate header files and
forward declaring the expression class, which we will call
\c Circulant. The \c makeCirculant function will return an object of
this type. The class \c Circulant is in fact a class template; the
template argument \c ArgType refers to the type of the vector passed
to the \c makeCirculant function.

\include make_circulant.cpp.preamble


\section TopicTraits The traits class

For every expression class \c X, there should be a traits class
\c Traits<X> in the \c Eigen::internal namespace containing
information about \c X known as compile time.

As explained in \ref TopicSetting, we designed the \c Circulant
expression class to refer to dense matrices. The entries of the
circulant matrix have the same type as the entries of the vector
passed to the \c makeCirculant function. The type used to index the
entries is also the same. Again for simplicity, we will only return
column-major matrices. Finally, the circulant matrix is a square
matrix (number of rows equals number of columns), and the number of
rows equals the number of rows of the column vector passed to the
\c makeCirculant function. If this is a dynamic-size vector, then the
size of the circulant matrix is not known at compile-time.

This leads to the following code:

\include make_circulant.cpp.traits


\section TopicExpression The expression class

The next step is to define the expression class itself. In our case,
we want to inherit from \c MatrixBase in order to expose the interface
for dense matrices. In the constructor, we check that we are passed a
column vector (see \ref TopicAssertions) and we store the vector from
which we are going to build the circulant matrix in the member
variable \c m_arg. Finally, the expression class should compute the
size of the corresponding circulant matrix. As explained above, this
is a square matrix with as many columns as the vector used to
construct the matrix.

\b TO \b DO: What about the \c Nested typedef? It seems to be
necessary; is this only temporary?

\include make_circulant.cpp.expression


\section TopicEvaluator The evaluator

The last big fragment implements the evaluator for the \c Circulant
expression. The evaluator computes the entries of the circulant
matrix; this is done in the \c .coeff() member function. The entries
are computed by finding the corresponding entry of the vector from
which the circulant matrix is constructed. Getting this entry may
actually be non-trivial when the circulant matrix is constructed from
a vector which is given by a complicated expression, so we use the
evaluator which corresponds to the vector.

The \c CoeffReadCost constant records the cost of computing an entry
of the circulant matrix; we ignore the index computation and say that
this is the same as the cost of computing an entry of the vector from
which the circulant matrix is constructed.

In the constructor, we save the evaluator for the column vector which
defined the circulant matrix. We also save the size of that vector;
remember that we can query an expression object to find the size but
not the evaluator.

\include make_circulant.cpp.evaluator


\section TopicEntry The entry point

After all this, the \c makeCirculant function is very simple. It
simply creates an expression object and returns it.

\include make_circulant.cpp.entry


\section TopicMain A simple main function for testing

Finally, a short \c main function that shows how the \c makeCirculant
function can be called.

\include make_circulant.cpp.main

If all the fragments are combined, the following output is produced,
showing that the program works as expected:

\include make_circulant.out

*/
}
