namespace Eigen {

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

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

This tutorial page explains the essentials of block operations.
A block is a rectangular part of a matrix or array. Blocks expressions can be used both
as rvalues and as lvalues. As usual with Eigen expressions, this abstraction has zero runtime cost
provided that you let your compiler optimize.

\b Table \b of \b contents
  - \ref TutorialBlockOperationsUsing
  - \ref TutorialBlockOperationsSyntaxColumnRows
  - \ref TutorialBlockOperationsSyntaxCorners
  - \ref TutorialBlockOperationsSyntaxVectors


\section TutorialBlockOperationsUsing Using block operations

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>.
There are two versions, whose syntax is as follows:

<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 of size <tt>(p,q)</tt>, starting at <tt>(i,j)</tt></td>
    <td>\code
matrix.block(i,j,p,q);\endcode </td>
    <td>\code 
matrix.block<p,q>(i,j);\endcode </td>
</tr>
</table>

The default version is a method which takes four arguments. It can always be used. The optimized version
takes two template arguments (the size of the block) and two normal arguments (the position of the block).
It can only be used if the size of the block is known at compile time, but it may be faster than the
non-optimized version, especially if the size of the block is small. Both versions can be used on fixed-size
and dynamic-size matrices and arrays.

The following program uses the default and optimized versions to print the values of several blocks inside a
matrix.

<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 above example the \link DenseBase::block() .block() \endlink function was employed 
to read the values inside matrix \p m . However, blocks can also be used as lvalues, meaning that you can
assign to a block. 

This is illustrated in the following example, which uses arrays instead of matrices.  The coefficients of the
5-by-5 array \c n are first all set to 0.6, but then the 3-by-3 block in the middle is set to the values in 
\c m . The penultimate line shows that blocks can be combined with matrices and arrays to create more complex
expressions. Blocks of an array are an array expression, and thus the multiplication here is coefficient-wise
multiplication.

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

The \link DenseBase::block() .block() \endlink method is used for general block operations, but there are
other methods for special cases. These are described in the rest of this page.


\section TutorialBlockOperationsSyntaxColumnRows Columns and rows

Individual columns and rows are special cases of blocks. Eigen provides methods to easily access them:
\link DenseBase::col() .col() \endlink and \link DenseBase::row() .row()\endlink. There is no syntax variant
for an optimized version.

<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
matrix.row(i);\endcode </td>
    <td>\code 
matrix.row(i);\endcode </td>
</tr>
<tr><td>j<sup>th</sup> column
                    \link DenseBase::col() * \endlink</td>
    <td>\code
matrix.col(j);\endcode </td>
    <td>\code 
matrix.col(j);\endcode </td>
</tr>
</table>

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.

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


\section TutorialBlockOperationsSyntaxCorners Corner-related operations

Eigen also provides special methods for blocks that are flushed against one of the corners or sides of a
matrix or array. For instance, \link DenseBase::topLeftCorner() .topLeftCorner() \endlink can be used to refer
to a block in the top-left corner of a matrix. Use <tt>matrix.topLeftCorner(p,q)</tt> to access the block
consisting of the coefficients <tt>matrix(i,j)</tt> with \c i &lt; \c p and \c j &lt; \c q. As an other
example, blocks consisting of whole rows flushed against the top side of the matrix can be accessed by
\link DenseBase::topRows() .topRows() \endlink. 

The different possibilities are summarized in the following table:

<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
matrix.topLeftCorner(p,q);\endcode </td>
    <td>\code 
matrix.topLeftCorner<p,q>();\endcode </td>
</tr>
<tr><td>Bottom-left p by q block
              \link DenseBase::bottomLeftCorner() * \endlink</td>
    <td>\code
matrix.bottomLeftCorner(p,q);\endcode </td>
    <td>\code 
matrix.bottomLeftCorner<p,q>();\endcode </td>
</tr>
<tr><td>Top-right p by q block
              \link DenseBase::topRightCorner() * \endlink</td>
    <td>\code
matrix.topRightCorner(p,q);\endcode </td>
    <td>\code 
matrix.topRightCorner<p,q>();\endcode </td>
</tr>
<tr><td>Bottom-right p by q block
               \link DenseBase::bottomRightCorner() * \endlink</td>
    <td>\code
matrix.bottomRightCorner(p,q);\endcode </td>
    <td>\code 
matrix.bottomRightCorner<p,q>();\endcode </td>
</tr>
<tr><td>%Block containing the first q rows
                   \link DenseBase::topRows() * \endlink</td>
    <td>\code
matrix.topRows(q);\endcode </td>
    <td>\code 
matrix.topRows<q>();\endcode </td>
</tr>
<tr><td>%Block containing the last q rows
                    \link DenseBase::bottomRows() * \endlink</td>
    <td>\code
matrix.bottomRows(q);\endcode </td>
    <td>\code 
matrix.bottomRows<q>();\endcode </td>
</tr>
<tr><td>%Block containing the first p columns
                    \link DenseBase::leftCols() * \endlink</td>
    <td>\code
matrix.leftCols(p);\endcode </td>
    <td>\code 
matrix.leftCols<p>();\endcode </td>
</tr>
<tr><td>%Block containing the last q columns
                    \link DenseBase::rightCols() * \endlink</td>
    <td>\code
matrix.rightCols(q);\endcode </td>
    <td>\code 
matrix.rightCols<q>();\endcode </td>
</tr>
</table>

Here is a simple example illustrating the use of the operations presented above:

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


\section TutorialBlockOperationsSyntaxVectors Block operations for vectors

Eigen also provides a set of block operations designed specifically for vectors and one-dimensional arrays:

<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
vector.head(n);\endcode </td>
    <td>\code 
vector.head<n>();\endcode </td>
</tr>
<tr><td>%Block containing the last \p n elements
                    \link DenseBase::tail() * \endlink</td>
    <td>\code
vector.tail(n);\endcode </td>
    <td>\code 
vector.tail<n>();\endcode </td>
</tr>
<tr><td>%Block containing \p n elements, starting at position \p i
                    \link DenseBase::segment() * \endlink</td>
    <td>\code
vector.segment(i,n);\endcode </td>
    <td>\code 
vector.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:
\verbinclude Tutorial_BlockOperations_vector.out
</td></tr></table>

\li \b Next: \ref TutorialAdvancedInitialization

*/

}