namespace Eigen {

/** \page TutorialBlockOperations Tutorial page 4 - %Block operations
    \ingroup Tutorial

\li \b Previous: \ref TutorialArrayClass
\li \b Next: \ref TutorialAdvancedInitialization

This tutorial explains the essentials of Block operations together with many examples.

\b Table \b of \b contents
  - \ref TutorialBlockOperationsWhatIs
    - \ref TutorialBlockOperationsFixedAndDynamicSize
  - \ref TutorialBlockOperationsSyntax
     - \ref TutorialBlockOperationsSyntaxColumnRows
     - \ref TutorialBlockOperationsSyntaxCorners


\section TutorialBlockOperationsWhatIs What are Block operations?
Block operations are a set of functions that provide an easy way to access a set of coefficients 
inside a \b Matrix or \link ArrayBase Array \endlink. A typical example is accessing a single row or 
column within a given matrix, as well as extracting a sub-matrix from the latter.

Blocks are highly flexible and can be used both as \b rvalues and \b lvalues in expressions, simplifying 
the task of writing combined expressions with Eigen.

\subsection TutorialBlockOperationsFixedAndDynamicSize Block operations and compile-time optimizations
As said earlier, a block operation is a way of accessing a group of coefficients inside a Matrix or 
Array object. Eigen considers two different cases in order to provide compile-time optimization for 
block operations, depending on whether the the size of the block to be accessed is known at compile time or not. 

To deal with these two situations, for each type of block operation Eigen provides a default version that
is able to work with run-time dependant block sizes and another one for block operations whose block size is
known at compile-time.

Even though both functions can be applied to fixed-size objects, it is advisable to use special block operations
in this case, allowing Eigen to perform more optimizations at compile-time.

\section TutorialBlockOperationsUsing Using block operations
Block operations are implemented such that they are easy to use and combine with operators and other 
matrices or arrays.

The most general block operation in Eigen is called \link DenseBase::block() .block() \endlink.
This function returns a block of size <tt>(p,q)</tt> whose origin is at <tt>(i,j)</tt> by using 
the following syntax:

<table class="tutorial_code" align="center">
<tr><td align="center">\b Block \b operation</td>
<td align="center">Default \b version</td>
<td align="center">Optimized version when the<br>size is known at compile time</td></tr>
<tr><td>Block of length <tt>(p,q)</tt>, starting at <tt>(i,j)</tt></td>
    <td>\code
MatrixXf m;
std::cout << m.block(i,j,p,q);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.block<p,q>(i,j);\endcode </td>
</tr>
</table>

Therefore, if we want to print the values of a block inside a matrix we can simply write:
<table class="tutorial_code"><tr><td>
\include Tutorial_BlockOperations_print_block.cpp
</td>
<td>
Output:
\verbinclude Tutorial_BlockOperations_print_block.out
</td></tr></table>


In the previous example the \link DenseBase::block() .block() \endlink function was employed 
to read the values inside matrix \p m . Blocks can also be used to perform operations and 
assignments within matrices or arrays of different size:

<table class="tutorial_code"><tr><td>
\include Tutorial_BlockOperations_block_assignment.cpp
</td>
<td>
Output:
\verbinclude Tutorial_BlockOperations_block_assignment.out
</td></tr></table>


Blocks can also be combined with matrices and arrays to create more complex expressions:

\code
  MatrixXf m(3,3), n(2,2);
  MatrixXf p(3,3);
  
  m.block(0,0,2,2) = m.block(0,0,2,2) * n + p.block(1,1,2,2);
\endcode

It is important to point out that \link DenseBase::block() .block() \endlink is the 
general case for a block operation but there are many other useful block operations, 
as described in the next section.

\section TutorialBlockOperationsSyntax Block operation syntax
The following tables show a summary of Eigen's block operations and how they are applied to 
fixed- and dynamic-sized Eigen objects.

\subsection TutorialBlockOperationsSyntaxColumnRows Columns and rows
Other extremely useful block operations are \link DenseBase::col() .col() \endlink and 
\link DenseBase::row() .row() \endlink which provide access to a 
specific row or column. This is a special case in the sense that the syntax for fixed- and 
dynamic-sized objects is exactly the same:

<table class="tutorial_code" align="center">
<tr><td align="center">\b Block \b operation</td>
<td align="center">Default version</td>
<td align="center">Optimized version when the<br>size is known at compile time</td></tr>
<tr><td>i<sup>th</sup> row
                    \link DenseBase::row() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.row(i);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.row(i);\endcode </td>
</tr>
<tr><td>j<sup>th</sup> column
                    \link DenseBase::col() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.col(j);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.col(j);\endcode </td>
</tr>
</table>

A simple example demonstrating these feature follows:

<table class="tutorial_code"><tr><td>
C++ code:
\include Tutorial_BlockOperations_colrow.cpp
</td>
<td>
Output:
\include Tutorial_BlockOperations_colrow.out
</td></tr></table>


\b NOTE: the argument for \p col() and \p row() is the index of the column or row to be accessed, 
starting at 0. Therefore, \p col(0) will access the first column and \p col(1) the second one.


\subsection TutorialBlockOperationsSyntaxCorners Corner-related operations
<table class="tutorial_code" align="center">
<tr><td align="center">\b Block \b operation</td>
<td align="center">Default version</td>
<td align="center">Optimized version when the<br>size is known at compile time</td></tr>
<tr><td>Top-left p by q block \link DenseBase::topLeftCorner() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.topLeftCorner(p,q);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.topLeftCorner<p,q>();\endcode </td>
</tr>
<tr><td>Bottom-left p by q block
              \link DenseBase::bottomLeftCorner() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.bottomLeftCorner(p,q);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.bottomLeftCorner<p,q>();\endcode </td>
</tr>
<tr><td>Top-right p by q block
              \link DenseBase::topRightCorner() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.topRightCorner(p,q);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.topRightCorner<p,q>();\endcode </td>
</tr>
<tr><td>Bottom-right p by q block
               \link DenseBase::bottomRightCorner() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.bottomRightCorner(p,q);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.bottomRightCorner<p,q>();\endcode </td>
</tr>
<tr><td>Block containing the first q rows
                   \link DenseBase::topRows() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.topRows(q);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.topRows<q>();\endcode </td>
</tr>
<tr><td>Block containing the last q rows
                    \link DenseBase::bottomRows() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.bottomRows(q);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.bottomRows<q>();\endcode </td>
</tr>
<tr><td>Block containing the first p columns
                    \link DenseBase::leftCols() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.leftCols(p);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.leftCols<p>();\endcode </td>
</tr>
<tr><td>Block containing the last q columns
                    \link DenseBase::rightCols() * \endlink</td>
    <td>\code
MatrixXf m;
std::cout << m.rightCols(q);\endcode </td>
    <td>\code 
Matrix3f m;
std::cout << m.rightCols<q>();\endcode </td>
</tr>
</table>


Here is a simple example showing the power of the operations presented above:

<table class="tutorial_code"><tr><td>
C++ code:
\include Tutorial_BlockOperations_corner.cpp
</td>
<td>
Output:
\include Tutorial_BlockOperations_corner.out
</td></tr></table>








\subsection TutorialBlockOperationsSyntaxVectors Block operations for vectors
Eigen also provides a set of block operations designed specifically for vectors:

<table class="tutorial_code" align="center">
<tr><td align="center">\b Block \b operation</td>
<td align="center">Default version</td>
<td align="center">Optimized version when the<br>size is known at compile time</td></tr>
<tr><td>Block containing the first \p n elements 
                    \link DenseBase::head() * \endlink</td>
    <td>\code
VectorXf v;
std::cout << v.head(n);\endcode </td>
    <td>\code 
Vector3f v;
std::cout << v.head<n>();\endcode </td>
</tr>
<tr><td>Block containing the last \p n elements
                    \link DenseBase::tail() * \endlink</td>
    <td>\code
VectorXf v;
std::cout << v.tail(n);\endcode </td>
    <td>\code 
Vector3f m;
std::cout << v.tail<n>();\endcode </td>
</tr>
<tr><td>Block containing \p n elements, starting at position \p i
                    \link DenseBase::segment() * \endlink</td>
    <td>\code
VectorXf v;
std::cout << v.segment(i,n);\endcode </td>
    <td>\code 
Vector3f m;
std::cout << v.segment<n>(i);\endcode </td>
</tr>
</table>


An example is presented below:
<table class="tutorial_code"><tr><td>
C++ code:
\include Tutorial_BlockOperations_vector.cpp
</td>
<td>
Output:
\include Tutorial_BlockOperations_vector.out
</td></tr></table>

\li \b Next: \ref TutorialAdvancedInitialization

*/

}