cart-elc

BlockMethods.h (59020B)

---

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
 // Copyright (C) 2006-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
 //
 // This Source Code Form is subject to the terms of the Mozilla
 // Public License v. 2.0. If a copy of the MPL was not distributed
 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 
 /// \internal expression type of a column */
 typedef Block<Derived, internal::traits<Derived>::RowsAtCompileTime, 1, !IsRowMajor> ColXpr;
 typedef const Block<const Derived, internal::traits<Derived>::RowsAtCompileTime, 1, !IsRowMajor> ConstColXpr;
 /// \internal expression type of a row */
 typedef Block<Derived, 1, internal::traits<Derived>::ColsAtCompileTime, IsRowMajor> RowXpr;
 typedef const Block<const Derived, 1, internal::traits<Derived>::ColsAtCompileTime, IsRowMajor> ConstRowXpr;
 /// \internal expression type of a block of whole columns */
 typedef Block<Derived, internal::traits<Derived>::RowsAtCompileTime, Dynamic, !IsRowMajor> ColsBlockXpr;
 typedef const Block<const Derived, internal::traits<Derived>::RowsAtCompileTime, Dynamic, !IsRowMajor> ConstColsBlockXpr;
 /// \internal expression type of a block of whole rows */
 typedef Block<Derived, Dynamic, internal::traits<Derived>::ColsAtCompileTime, IsRowMajor> RowsBlockXpr;
 typedef const Block<const Derived, Dynamic, internal::traits<Derived>::ColsAtCompileTime, IsRowMajor> ConstRowsBlockXpr;
 /// \internal expression type of a block of whole columns */
 template<int N> struct NColsBlockXpr { typedef Block<Derived, internal::traits<Derived>::RowsAtCompileTime, N, !IsRowMajor> Type; };
 template<int N> struct ConstNColsBlockXpr { typedef const Block<const Derived, internal::traits<Derived>::RowsAtCompileTime, N, !IsRowMajor> Type; };
 /// \internal expression type of a block of whole rows */
 template<int N> struct NRowsBlockXpr { typedef Block<Derived, N, internal::traits<Derived>::ColsAtCompileTime, IsRowMajor> Type; };
 template<int N> struct ConstNRowsBlockXpr { typedef const Block<const Derived, N, internal::traits<Derived>::ColsAtCompileTime, IsRowMajor> Type; };
 /// \internal expression of a block */
 typedef Block<Derived> BlockXpr;
 typedef const Block<const Derived> ConstBlockXpr;
 /// \internal expression of a block of fixed sizes */
 template<int Rows, int Cols> struct FixedBlockXpr { typedef Block<Derived,Rows,Cols> Type; };
 template<int Rows, int Cols> struct ConstFixedBlockXpr { typedef Block<const Derived,Rows,Cols> Type; };
 
 typedef VectorBlock<Derived> SegmentReturnType;
 typedef const VectorBlock<const Derived> ConstSegmentReturnType;
 template<int Size> struct FixedSegmentReturnType { typedef VectorBlock<Derived, Size> Type; };
 template<int Size> struct ConstFixedSegmentReturnType { typedef const VectorBlock<const Derived, Size> Type; };
 
 /// \internal inner-vector
 typedef Block<Derived,IsRowMajor?1:Dynamic,IsRowMajor?Dynamic:1,true>       InnerVectorReturnType;
 typedef Block<const Derived,IsRowMajor?1:Dynamic,IsRowMajor?Dynamic:1,true> ConstInnerVectorReturnType;
 
 /// \internal set of inner-vectors
 typedef Block<Derived,Dynamic,Dynamic,true> InnerVectorsReturnType;
 typedef Block<const Derived,Dynamic,Dynamic,true> ConstInnerVectorsReturnType;
 
 #endif // not EIGEN_PARSED_BY_DOXYGEN
 
 /// \returns an expression of a block in \c *this with either dynamic or fixed sizes.
 ///
 /// \param  startRow  the first row in the block
 /// \param  startCol  the first column in the block
 /// \param  blockRows number of rows in the block, specified at either run-time or compile-time
 /// \param  blockCols number of columns in the block, specified at either run-time or compile-time
 /// \tparam NRowsType the type of the value handling the number of rows in the block, typically Index.
 /// \tparam NColsType the type of the value handling the number of columns in the block, typically Index.
 ///
 /// Example using runtime (aka dynamic) sizes: \include MatrixBase_block_int_int_int_int.cpp
 /// Output: \verbinclude MatrixBase_block_int_int_int_int.out
 ///
 /// \newin{3.4}:
 ///
 /// The number of rows \a blockRows and columns \a blockCols can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments. In the later case, \c n plays the role of a runtime fallback value in case \c N equals Eigen::Dynamic.
 /// Here is an example with a fixed number of rows \c NRows and dynamic number of columns \c cols:
 /// \code
 /// mat.block(i,j,fix<NRows>,cols)
 /// \endcode
 ///
 /// This function thus fully covers the features offered by the following overloads block<NRows,NCols>(Index, Index),
 /// and block<NRows,NCols>(Index, Index, Index, Index) that are thus obsolete. Indeed, this generic version avoids
 /// redundancy, it preserves the argument order, and prevents the need to rely on the template keyword in templated code.
 ///
 /// but with less redundancy and more consistency as it does not modify the argument order
 /// and seamlessly enable hybrid fixed/dynamic sizes.
 ///
 /// \note Even in the case that the returned expression has dynamic size, in the case
 /// when it is applied to a fixed-size matrix, it inherits a fixed maximal size,
 /// which means that evaluating it does not cause a dynamic memory allocation.
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa class Block, fix, fix<N>(int)
 ///
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 typename FixedBlockXpr<...,...>::Type
 #endif
 block(Index startRow, Index startCol, NRowsType blockRows, NColsType blockCols)
 {
   return typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type(
             derived(), startRow, startCol, internal::get_runtime_value(blockRows), internal::get_runtime_value(blockCols));
 }
 
 /// This is the const version of block(Index,Index,NRowsType,NColsType)
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 const typename ConstFixedBlockXpr<...,...>::Type
 #endif
 block(Index startRow, Index startCol, NRowsType blockRows, NColsType blockCols) const
 {
   return typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type(
             derived(), startRow, startCol, internal::get_runtime_value(blockRows), internal::get_runtime_value(blockCols));
 }
 
 
 
 /// \returns a expression of a top-right corner of \c *this with either dynamic or fixed sizes.
 ///
 /// \param cRows the number of rows in the corner
 /// \param cCols the number of columns in the corner
 /// \tparam NRowsType the type of the value handling the number of rows in the block, typically Index.
 /// \tparam NColsType the type of the value handling the number of columns in the block, typically Index.
 ///
 /// Example with dynamic sizes: \include MatrixBase_topRightCorner_int_int.cpp
 /// Output: \verbinclude MatrixBase_topRightCorner_int_int.out
 ///
 /// The number of rows \a blockRows and columns \a blockCols can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments. See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 typename FixedBlockXpr<...,...>::Type
 #endif
 topRightCorner(NRowsType cRows, NColsType cCols)
 {
   return typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, cols() - internal::get_runtime_value(cCols), internal::get_runtime_value(cRows), internal::get_runtime_value(cCols));
 }
 
 /// This is the const version of topRightCorner(NRowsType, NColsType).
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 const typename ConstFixedBlockXpr<...,...>::Type
 #endif
 topRightCorner(NRowsType cRows, NColsType cCols) const
 {
   return typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, cols() - internal::get_runtime_value(cCols), internal::get_runtime_value(cRows), internal::get_runtime_value(cCols));
 }
 
 /// \returns an expression of a fixed-size top-right corner of \c *this.
 ///
 /// \tparam CRows the number of rows in the corner
 /// \tparam CCols the number of columns in the corner
 ///
 /// Example: \include MatrixBase_template_int_int_topRightCorner.cpp
 /// Output: \verbinclude MatrixBase_template_int_int_topRightCorner.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa class Block, block<int,int>(Index,Index)
 ///
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedBlockXpr<CRows,CCols>::Type topRightCorner()
 {
   return typename FixedBlockXpr<CRows,CCols>::Type(derived(), 0, cols() - CCols);
 }
 
 /// This is the const version of topRightCorner<int, int>().
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<CRows,CCols>::Type topRightCorner() const
 {
   return typename ConstFixedBlockXpr<CRows,CCols>::Type(derived(), 0, cols() - CCols);
 }
 
 /// \returns an expression of a top-right corner of \c *this.
 ///
 /// \tparam CRows number of rows in corner as specified at compile-time
 /// \tparam CCols number of columns in corner as specified at compile-time
 /// \param  cRows number of rows in corner as specified at run-time
 /// \param  cCols number of columns in corner as specified at run-time
 ///
 /// This function is mainly useful for corners where the number of rows is specified at compile-time
 /// and the number of columns is specified at run-time, or vice versa. The compile-time and run-time
 /// information should not contradict. In other words, \a cRows should equal \a CRows unless
 /// \a CRows is \a Dynamic, and the same for the number of columns.
 ///
 /// Example: \include MatrixBase_template_int_int_topRightCorner_int_int.cpp
 /// Output: \verbinclude MatrixBase_template_int_int_topRightCorner_int_int.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa class Block
 ///
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedBlockXpr<CRows,CCols>::Type topRightCorner(Index cRows, Index cCols)
 {
   return typename FixedBlockXpr<CRows,CCols>::Type(derived(), 0, cols() - cCols, cRows, cCols);
 }
 
 /// This is the const version of topRightCorner<int, int>(Index, Index).
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<CRows,CCols>::Type topRightCorner(Index cRows, Index cCols) const
 {
   return typename ConstFixedBlockXpr<CRows,CCols>::Type(derived(), 0, cols() - cCols, cRows, cCols);
 }
 
 
 
 /// \returns an expression of a top-left corner of \c *this  with either dynamic or fixed sizes.
 ///
 /// \param cRows the number of rows in the corner
 /// \param cCols the number of columns in the corner
 /// \tparam NRowsType the type of the value handling the number of rows in the block, typically Index.
 /// \tparam NColsType the type of the value handling the number of columns in the block, typically Index.
 ///
 /// Example: \include MatrixBase_topLeftCorner_int_int.cpp
 /// Output: \verbinclude MatrixBase_topLeftCorner_int_int.out
 ///
 /// The number of rows \a blockRows and columns \a blockCols can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments. See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 typename FixedBlockXpr<...,...>::Type
 #endif
 topLeftCorner(NRowsType cRows, NColsType cCols)
 {
   return typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, 0, internal::get_runtime_value(cRows), internal::get_runtime_value(cCols));
 }
 
 /// This is the const version of topLeftCorner(Index, Index).
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 const typename ConstFixedBlockXpr<...,...>::Type
 #endif
 topLeftCorner(NRowsType cRows, NColsType cCols) const
 {
   return typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, 0, internal::get_runtime_value(cRows), internal::get_runtime_value(cCols));
 }
 
 /// \returns an expression of a fixed-size top-left corner of \c *this.
 ///
 /// The template parameters CRows and CCols are the number of rows and columns in the corner.
 ///
 /// Example: \include MatrixBase_template_int_int_topLeftCorner.cpp
 /// Output: \verbinclude MatrixBase_template_int_int_topLeftCorner.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedBlockXpr<CRows,CCols>::Type topLeftCorner()
 {
   return typename FixedBlockXpr<CRows,CCols>::Type(derived(), 0, 0);
 }
 
 /// This is the const version of topLeftCorner<int, int>().
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<CRows,CCols>::Type topLeftCorner() const
 {
   return typename ConstFixedBlockXpr<CRows,CCols>::Type(derived(), 0, 0);
 }
 
 /// \returns an expression of a top-left corner of \c *this.
 ///
 /// \tparam CRows number of rows in corner as specified at compile-time
 /// \tparam CCols number of columns in corner as specified at compile-time
 /// \param  cRows number of rows in corner as specified at run-time
 /// \param  cCols number of columns in corner as specified at run-time
 ///
 /// This function is mainly useful for corners where the number of rows is specified at compile-time
 /// and the number of columns is specified at run-time, or vice versa. The compile-time and run-time
 /// information should not contradict. In other words, \a cRows should equal \a CRows unless
 /// \a CRows is \a Dynamic, and the same for the number of columns.
 ///
 /// Example: \include MatrixBase_template_int_int_topLeftCorner_int_int.cpp
 /// Output: \verbinclude MatrixBase_template_int_int_topLeftCorner_int_int.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa class Block
 ///
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedBlockXpr<CRows,CCols>::Type topLeftCorner(Index cRows, Index cCols)
 {
   return typename FixedBlockXpr<CRows,CCols>::Type(derived(), 0, 0, cRows, cCols);
 }
 
 /// This is the const version of topLeftCorner<int, int>(Index, Index).
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<CRows,CCols>::Type topLeftCorner(Index cRows, Index cCols) const
 {
   return typename ConstFixedBlockXpr<CRows,CCols>::Type(derived(), 0, 0, cRows, cCols);
 }
 
 
 
 /// \returns an expression of a bottom-right corner of \c *this  with either dynamic or fixed sizes.
 ///
 /// \param cRows the number of rows in the corner
 /// \param cCols the number of columns in the corner
 /// \tparam NRowsType the type of the value handling the number of rows in the block, typically Index.
 /// \tparam NColsType the type of the value handling the number of columns in the block, typically Index.
 ///
 /// Example: \include MatrixBase_bottomRightCorner_int_int.cpp
 /// Output: \verbinclude MatrixBase_bottomRightCorner_int_int.out
 ///
 /// The number of rows \a blockRows and columns \a blockCols can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments. See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 typename FixedBlockXpr<...,...>::Type
 #endif
 bottomRightCorner(NRowsType cRows, NColsType cCols)
 {
   return typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
             (derived(), rows() - internal::get_runtime_value(cRows), cols() - internal::get_runtime_value(cCols),
                         internal::get_runtime_value(cRows), internal::get_runtime_value(cCols));
 }
 
 /// This is the const version of bottomRightCorner(NRowsType, NColsType).
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 const typename ConstFixedBlockXpr<...,...>::Type
 #endif
 bottomRightCorner(NRowsType cRows, NColsType cCols) const
 {
   return typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
             (derived(), rows() - internal::get_runtime_value(cRows), cols() - internal::get_runtime_value(cCols),
                         internal::get_runtime_value(cRows), internal::get_runtime_value(cCols));
 }
 
 /// \returns an expression of a fixed-size bottom-right corner of \c *this.
 ///
 /// The template parameters CRows and CCols are the number of rows and columns in the corner.
 ///
 /// Example: \include MatrixBase_template_int_int_bottomRightCorner.cpp
 /// Output: \verbinclude MatrixBase_template_int_int_bottomRightCorner.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedBlockXpr<CRows,CCols>::Type bottomRightCorner()
 {
   return typename FixedBlockXpr<CRows,CCols>::Type(derived(), rows() - CRows, cols() - CCols);
 }
 
 /// This is the const version of bottomRightCorner<int, int>().
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<CRows,CCols>::Type bottomRightCorner() const
 {
   return typename ConstFixedBlockXpr<CRows,CCols>::Type(derived(), rows() - CRows, cols() - CCols);
 }
 
 /// \returns an expression of a bottom-right corner of \c *this.
 ///
 /// \tparam CRows number of rows in corner as specified at compile-time
 /// \tparam CCols number of columns in corner as specified at compile-time
 /// \param  cRows number of rows in corner as specified at run-time
 /// \param  cCols number of columns in corner as specified at run-time
 ///
 /// This function is mainly useful for corners where the number of rows is specified at compile-time
 /// and the number of columns is specified at run-time, or vice versa. The compile-time and run-time
 /// information should not contradict. In other words, \a cRows should equal \a CRows unless
 /// \a CRows is \a Dynamic, and the same for the number of columns.
 ///
 /// Example: \include MatrixBase_template_int_int_bottomRightCorner_int_int.cpp
 /// Output: \verbinclude MatrixBase_template_int_int_bottomRightCorner_int_int.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa class Block
 ///
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedBlockXpr<CRows,CCols>::Type bottomRightCorner(Index cRows, Index cCols)
 {
   return typename FixedBlockXpr<CRows,CCols>::Type(derived(), rows() - cRows, cols() - cCols, cRows, cCols);
 }
 
 /// This is the const version of bottomRightCorner<int, int>(Index, Index).
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<CRows,CCols>::Type bottomRightCorner(Index cRows, Index cCols) const
 {
   return typename ConstFixedBlockXpr<CRows,CCols>::Type(derived(), rows() - cRows, cols() - cCols, cRows, cCols);
 }
 
 
 
 /// \returns an expression of a bottom-left corner of \c *this  with either dynamic or fixed sizes.
 ///
 /// \param cRows the number of rows in the corner
 /// \param cCols the number of columns in the corner
 /// \tparam NRowsType the type of the value handling the number of rows in the block, typically Index.
 /// \tparam NColsType the type of the value handling the number of columns in the block, typically Index.
 ///
 /// Example: \include MatrixBase_bottomLeftCorner_int_int.cpp
 /// Output: \verbinclude MatrixBase_bottomLeftCorner_int_int.out
 ///
 /// The number of rows \a blockRows and columns \a blockCols can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments. See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 typename FixedBlockXpr<...,...>::Type
 #endif
 bottomLeftCorner(NRowsType cRows, NColsType cCols)
 {
   return typename FixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
             (derived(), rows() - internal::get_runtime_value(cRows), 0,
                         internal::get_runtime_value(cRows), internal::get_runtime_value(cCols));
 }
 
 /// This is the const version of bottomLeftCorner(NRowsType, NColsType).
 template<typename NRowsType, typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
 #else
 typename ConstFixedBlockXpr<...,...>::Type
 #endif
 bottomLeftCorner(NRowsType cRows, NColsType cCols) const
 {
   return typename ConstFixedBlockXpr<internal::get_fixed_value<NRowsType>::value,internal::get_fixed_value<NColsType>::value>::Type
             (derived(), rows() - internal::get_runtime_value(cRows), 0,
                         internal::get_runtime_value(cRows), internal::get_runtime_value(cCols));
 }
 
 /// \returns an expression of a fixed-size bottom-left corner of \c *this.
 ///
 /// The template parameters CRows and CCols are the number of rows and columns in the corner.
 ///
 /// Example: \include MatrixBase_template_int_int_bottomLeftCorner.cpp
 /// Output: \verbinclude MatrixBase_template_int_int_bottomLeftCorner.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedBlockXpr<CRows,CCols>::Type bottomLeftCorner()
 {
   return typename FixedBlockXpr<CRows,CCols>::Type(derived(), rows() - CRows, 0);
 }
 
 /// This is the const version of bottomLeftCorner<int, int>().
 template<int CRows, int CCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<CRows,CCols>::Type bottomLeftCorner() const
 {
   return typename ConstFixedBlockXpr<CRows,CCols>::Type(derived(), rows() - CRows, 0);
 }
 
 /// \returns an expression of a bottom-left corner of \c *this.
 ///
 /// \tparam CRows number of rows in corner as specified at compile-time
 /// \tparam CCols number of columns in corner as specified at compile-time
 /// \param  cRows number of rows in corner as specified at run-time
 /// \param  cCols number of columns in corner as specified at run-time
 ///
 /// This function is mainly useful for corners where the number of rows is specified at compile-time
 /// and the number of columns is specified at run-time, or vice versa. The compile-time and run-time
 /// information should not contradict. In other words, \a cRows should equal \a CRows unless
 /// \a CRows is \a Dynamic, and the same for the number of columns.
 ///
 /// Example: \include MatrixBase_template_int_int_bottomLeftCorner_int_int.cpp
 /// Output: \verbinclude MatrixBase_template_int_int_bottomLeftCorner_int_int.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa class Block
 ///
 template<int CRows, int CCols>
 EIGEN_STRONG_INLINE
 typename FixedBlockXpr<CRows,CCols>::Type bottomLeftCorner(Index cRows, Index cCols)
 {
   return typename FixedBlockXpr<CRows,CCols>::Type(derived(), rows() - cRows, 0, cRows, cCols);
 }
 
 /// This is the const version of bottomLeftCorner<int, int>(Index, Index).
 template<int CRows, int CCols>
 EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<CRows,CCols>::Type bottomLeftCorner(Index cRows, Index cCols) const
 {
   return typename ConstFixedBlockXpr<CRows,CCols>::Type(derived(), rows() - cRows, 0, cRows, cCols);
 }
 
 
 
 /// \returns a block consisting of the top rows of \c *this.
 ///
 /// \param n the number of rows in the block
 /// \tparam NRowsType the type of the value handling the number of rows in the block, typically Index.
 ///
 /// Example: \include MatrixBase_topRows_int.cpp
 /// Output: \verbinclude MatrixBase_topRows_int.out
 ///
 /// The number of rows \a n can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments.
 /// See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(row-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NRowsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename NRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
 #else
 typename NRowsBlockXpr<...>::Type
 #endif
 topRows(NRowsType n)
 {
   return typename NRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
             (derived(), 0, 0, internal::get_runtime_value(n), cols());
 }
 
 /// This is the const version of topRows(NRowsType).
 template<typename NRowsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstNRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
 #else
 const typename ConstNRowsBlockXpr<...>::Type
 #endif
 topRows(NRowsType n) const
 {
   return typename ConstNRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
             (derived(), 0, 0, internal::get_runtime_value(n), cols());
 }
 
 /// \returns a block consisting of the top rows of \c *this.
 ///
 /// \tparam N the number of rows in the block as specified at compile-time
 /// \param n the number of rows in the block as specified at run-time
 ///
 /// The compile-time and run-time information should not contradict. In other words,
 /// \a n should equal \a N unless \a N is \a Dynamic.
 ///
 /// Example: \include MatrixBase_template_int_topRows.cpp
 /// Output: \verbinclude MatrixBase_template_int_topRows.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(row-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename NRowsBlockXpr<N>::Type topRows(Index n = N)
 {
   return typename NRowsBlockXpr<N>::Type(derived(), 0, 0, n, cols());
 }
 
 /// This is the const version of topRows<int>().
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename ConstNRowsBlockXpr<N>::Type topRows(Index n = N) const
 {
   return typename ConstNRowsBlockXpr<N>::Type(derived(), 0, 0, n, cols());
 }
 
 
 
 /// \returns a block consisting of the bottom rows of \c *this.
 ///
 /// \param n the number of rows in the block
 /// \tparam NRowsType the type of the value handling the number of rows in the block, typically Index.
 ///
 /// Example: \include MatrixBase_bottomRows_int.cpp
 /// Output: \verbinclude MatrixBase_bottomRows_int.out
 ///
 /// The number of rows \a n can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments.
 /// See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(row-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NRowsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename NRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
 #else
 typename NRowsBlockXpr<...>::Type
 #endif
 bottomRows(NRowsType n)
 {
   return typename NRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
             (derived(), rows() - internal::get_runtime_value(n), 0, internal::get_runtime_value(n), cols());
 }
 
 /// This is the const version of bottomRows(NRowsType).
 template<typename NRowsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstNRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
 #else
 const typename ConstNRowsBlockXpr<...>::Type
 #endif
 bottomRows(NRowsType n) const
 {
   return typename ConstNRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
             (derived(), rows() - internal::get_runtime_value(n), 0, internal::get_runtime_value(n), cols());
 }
 
 /// \returns a block consisting of the bottom rows of \c *this.
 ///
 /// \tparam N the number of rows in the block as specified at compile-time
 /// \param n the number of rows in the block as specified at run-time
 ///
 /// The compile-time and run-time information should not contradict. In other words,
 /// \a n should equal \a N unless \a N is \a Dynamic.
 ///
 /// Example: \include MatrixBase_template_int_bottomRows.cpp
 /// Output: \verbinclude MatrixBase_template_int_bottomRows.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(row-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename NRowsBlockXpr<N>::Type bottomRows(Index n = N)
 {
   return typename NRowsBlockXpr<N>::Type(derived(), rows() - n, 0, n, cols());
 }
 
 /// This is the const version of bottomRows<int>().
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename ConstNRowsBlockXpr<N>::Type bottomRows(Index n = N) const
 {
   return typename ConstNRowsBlockXpr<N>::Type(derived(), rows() - n, 0, n, cols());
 }
 
 
 
 /// \returns a block consisting of a range of rows of \c *this.
 ///
 /// \param startRow the index of the first row in the block
 /// \param n the number of rows in the block
 /// \tparam NRowsType the type of the value handling the number of rows in the block, typically Index.
 ///
 /// Example: \include DenseBase_middleRows_int.cpp
 /// Output: \verbinclude DenseBase_middleRows_int.out
 ///
 /// The number of rows \a n can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments.
 /// See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(row-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NRowsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename NRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
 #else
 typename NRowsBlockXpr<...>::Type
 #endif
 middleRows(Index startRow, NRowsType n)
 {
   return typename NRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
             (derived(), startRow, 0, internal::get_runtime_value(n), cols());
 }
 
 /// This is the const version of middleRows(Index,NRowsType).
 template<typename NRowsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstNRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
 #else
 const typename ConstNRowsBlockXpr<...>::Type
 #endif
 middleRows(Index startRow, NRowsType n) const
 {
   return typename ConstNRowsBlockXpr<internal::get_fixed_value<NRowsType>::value>::Type
             (derived(), startRow, 0, internal::get_runtime_value(n), cols());
 }
 
 /// \returns a block consisting of a range of rows of \c *this.
 ///
 /// \tparam N the number of rows in the block as specified at compile-time
 /// \param startRow the index of the first row in the block
 /// \param n the number of rows in the block as specified at run-time
 ///
 /// The compile-time and run-time information should not contradict. In other words,
 /// \a n should equal \a N unless \a N is \a Dynamic.
 ///
 /// Example: \include DenseBase_template_int_middleRows.cpp
 /// Output: \verbinclude DenseBase_template_int_middleRows.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(row-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename NRowsBlockXpr<N>::Type middleRows(Index startRow, Index n = N)
 {
   return typename NRowsBlockXpr<N>::Type(derived(), startRow, 0, n, cols());
 }
 
 /// This is the const version of middleRows<int>().
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename ConstNRowsBlockXpr<N>::Type middleRows(Index startRow, Index n = N) const
 {
   return typename ConstNRowsBlockXpr<N>::Type(derived(), startRow, 0, n, cols());
 }
 
 
 
 /// \returns a block consisting of the left columns of \c *this.
 ///
 /// \param n the number of columns in the block
 /// \tparam NColsType the type of the value handling the number of columns in the block, typically Index.
 ///
 /// Example: \include MatrixBase_leftCols_int.cpp
 /// Output: \verbinclude MatrixBase_leftCols_int.out
 ///
 /// The number of columns \a n can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments.
 /// See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(column-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename NColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
 #else
 typename NColsBlockXpr<...>::Type
 #endif
 leftCols(NColsType n)
 {
   return typename NColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, 0, rows(), internal::get_runtime_value(n));
 }
 
 /// This is the const version of leftCols(NColsType).
 template<typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstNColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
 #else
 const typename ConstNColsBlockXpr<...>::Type
 #endif
 leftCols(NColsType n) const
 {
   return typename ConstNColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, 0, rows(), internal::get_runtime_value(n));
 }
 
 /// \returns a block consisting of the left columns of \c *this.
 ///
 /// \tparam N the number of columns in the block as specified at compile-time
 /// \param n the number of columns in the block as specified at run-time
 ///
 /// The compile-time and run-time information should not contradict. In other words,
 /// \a n should equal \a N unless \a N is \a Dynamic.
 ///
 /// Example: \include MatrixBase_template_int_leftCols.cpp
 /// Output: \verbinclude MatrixBase_template_int_leftCols.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(column-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename NColsBlockXpr<N>::Type leftCols(Index n = N)
 {
   return typename NColsBlockXpr<N>::Type(derived(), 0, 0, rows(), n);
 }
 
 /// This is the const version of leftCols<int>().
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename ConstNColsBlockXpr<N>::Type leftCols(Index n = N) const
 {
   return typename ConstNColsBlockXpr<N>::Type(derived(), 0, 0, rows(), n);
 }
 
 
 
 /// \returns a block consisting of the right columns of \c *this.
 ///
 /// \param n the number of columns in the block
 /// \tparam NColsType the type of the value handling the number of columns in the block, typically Index.
 ///
 /// Example: \include MatrixBase_rightCols_int.cpp
 /// Output: \verbinclude MatrixBase_rightCols_int.out
 ///
 /// The number of columns \a n can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments.
 /// See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(column-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename NColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
 #else
 typename NColsBlockXpr<...>::Type
 #endif
 rightCols(NColsType n)
 {
   return typename NColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, cols() - internal::get_runtime_value(n), rows(), internal::get_runtime_value(n));
 }
 
 /// This is the const version of rightCols(NColsType).
 template<typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstNColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
 #else
 const typename ConstNColsBlockXpr<...>::Type
 #endif
 rightCols(NColsType n) const
 {
   return typename ConstNColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, cols() - internal::get_runtime_value(n), rows(), internal::get_runtime_value(n));
 }
 
 /// \returns a block consisting of the right columns of \c *this.
 ///
 /// \tparam N the number of columns in the block as specified at compile-time
 /// \param n the number of columns in the block as specified at run-time
 ///
 /// The compile-time and run-time information should not contradict. In other words,
 /// \a n should equal \a N unless \a N is \a Dynamic.
 ///
 /// Example: \include MatrixBase_template_int_rightCols.cpp
 /// Output: \verbinclude MatrixBase_template_int_rightCols.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(column-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename NColsBlockXpr<N>::Type rightCols(Index n = N)
 {
   return typename NColsBlockXpr<N>::Type(derived(), 0, cols() - n, rows(), n);
 }
 
 /// This is the const version of rightCols<int>().
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename ConstNColsBlockXpr<N>::Type rightCols(Index n = N) const
 {
   return typename ConstNColsBlockXpr<N>::Type(derived(), 0, cols() - n, rows(), n);
 }
 
 
 
 /// \returns a block consisting of a range of columns of \c *this.
 ///
 /// \param startCol the index of the first column in the block
 /// \param numCols the number of columns in the block
 /// \tparam NColsType the type of the value handling the number of columns in the block, typically Index.
 ///
 /// Example: \include DenseBase_middleCols_int.cpp
 /// Output: \verbinclude DenseBase_middleCols_int.out
 ///
 /// The number of columns \a n can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments.
 /// See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(column-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename NColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
 #else
 typename NColsBlockXpr<...>::Type
 #endif
 middleCols(Index startCol, NColsType numCols)
 {
   return typename NColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, startCol, rows(), internal::get_runtime_value(numCols));
 }
 
 /// This is the const version of middleCols(Index,NColsType).
 template<typename NColsType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstNColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
 #else
 const typename ConstNColsBlockXpr<...>::Type
 #endif
 middleCols(Index startCol, NColsType numCols) const
 {
   return typename ConstNColsBlockXpr<internal::get_fixed_value<NColsType>::value>::Type
             (derived(), 0, startCol, rows(), internal::get_runtime_value(numCols));
 }
 
 /// \returns a block consisting of a range of columns of \c *this.
 ///
 /// \tparam N the number of columns in the block as specified at compile-time
 /// \param startCol the index of the first column in the block
 /// \param n the number of columns in the block as specified at run-time
 ///
 /// The compile-time and run-time information should not contradict. In other words,
 /// \a n should equal \a N unless \a N is \a Dynamic.
 ///
 /// Example: \include DenseBase_template_int_middleCols.cpp
 /// Output: \verbinclude DenseBase_template_int_middleCols.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(column-major)
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename NColsBlockXpr<N>::Type middleCols(Index startCol, Index n = N)
 {
   return typename NColsBlockXpr<N>::Type(derived(), 0, startCol, rows(), n);
 }
 
 /// This is the const version of middleCols<int>().
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename ConstNColsBlockXpr<N>::Type middleCols(Index startCol, Index n = N) const
 {
   return typename ConstNColsBlockXpr<N>::Type(derived(), 0, startCol, rows(), n);
 }
 
 
 
 /// \returns a fixed-size expression of a block of \c *this.
 ///
 /// The template parameters \a NRows and \a NCols are the number of
 /// rows and columns in the block.
 ///
 /// \param startRow the first row in the block
 /// \param startCol the first column in the block
 ///
 /// Example: \include MatrixBase_block_int_int.cpp
 /// Output: \verbinclude MatrixBase_block_int_int.out
 ///
 /// \note The usage of of this overload is discouraged from %Eigen 3.4, better used the generic
 /// block(Index,Index,NRowsType,NColsType), here is the one-to-one equivalence:
 /// \code
 /// mat.template block<NRows,NCols>(i,j)  <-->  mat.block(i,j,fix<NRows>,fix<NCols>)
 /// \endcode
 ///
 /// \note since block is a templated member, the keyword template has to be used
 /// if the matrix type is also a template parameter: \code m.template block<3,3>(1,1); \endcode
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int NRows, int NCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedBlockXpr<NRows,NCols>::Type block(Index startRow, Index startCol)
 {
   return typename FixedBlockXpr<NRows,NCols>::Type(derived(), startRow, startCol);
 }
 
 /// This is the const version of block<>(Index, Index). */
 template<int NRows, int NCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<NRows,NCols>::Type block(Index startRow, Index startCol) const
 {
   return typename ConstFixedBlockXpr<NRows,NCols>::Type(derived(), startRow, startCol);
 }
 
 /// \returns an expression of a block of \c *this.
 ///
 /// \tparam NRows number of rows in block as specified at compile-time
 /// \tparam NCols number of columns in block as specified at compile-time
 /// \param  startRow  the first row in the block
 /// \param  startCol  the first column in the block
 /// \param  blockRows number of rows in block as specified at run-time
 /// \param  blockCols number of columns in block as specified at run-time
 ///
 /// This function is mainly useful for blocks where the number of rows is specified at compile-time
 /// and the number of columns is specified at run-time, or vice versa. The compile-time and run-time
 /// information should not contradict. In other words, \a blockRows should equal \a NRows unless
 /// \a NRows is \a Dynamic, and the same for the number of columns.
 ///
 /// Example: \include MatrixBase_template_int_int_block_int_int_int_int.cpp
 /// Output: \verbinclude MatrixBase_template_int_int_block_int_int_int_int.out
 ///
 /// \note The usage of of this overload is discouraged from %Eigen 3.4, better used the generic
 /// block(Index,Index,NRowsType,NColsType), here is the one-to-one complete equivalence:
 /// \code
 /// mat.template block<NRows,NCols>(i,j,rows,cols)     <-->  mat.block(i,j,fix<NRows>(rows),fix<NCols>(cols))
 /// \endcode
 /// If we known that, e.g., NRows==Dynamic and NCols!=Dynamic, then the equivalence becomes:
 /// \code
 /// mat.template block<Dynamic,NCols>(i,j,rows,NCols)  <-->  mat.block(i,j,rows,fix<NCols>)
 /// \endcode
 ///
 EIGEN_DOC_BLOCK_ADDONS_NOT_INNER_PANEL
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), class Block
 ///
 template<int NRows, int NCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedBlockXpr<NRows,NCols>::Type block(Index startRow, Index startCol,
                                                   Index blockRows, Index blockCols)
 {
   return typename FixedBlockXpr<NRows,NCols>::Type(derived(), startRow, startCol, blockRows, blockCols);
 }
 
 /// This is the const version of block<>(Index, Index, Index, Index).
 template<int NRows, int NCols>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const typename ConstFixedBlockXpr<NRows,NCols>::Type block(Index startRow, Index startCol,
                                                               Index blockRows, Index blockCols) const
 {
   return typename ConstFixedBlockXpr<NRows,NCols>::Type(derived(), startRow, startCol, blockRows, blockCols);
 }
 
 /// \returns an expression of the \a i-th column of \c *this. Note that the numbering starts at 0.
 ///
 /// Example: \include MatrixBase_col.cpp
 /// Output: \verbinclude MatrixBase_col.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(column-major)
 /**
   * \sa row(), class Block */
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 ColXpr col(Index i)
 {
   return ColXpr(derived(), i);
 }
 
 /// This is the const version of col().
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 ConstColXpr col(Index i) const
 {
   return ConstColXpr(derived(), i);
 }
 
 /// \returns an expression of the \a i-th row of \c *this. Note that the numbering starts at 0.
 ///
 /// Example: \include MatrixBase_row.cpp
 /// Output: \verbinclude MatrixBase_row.out
 ///
 EIGEN_DOC_BLOCK_ADDONS_INNER_PANEL_IF(row-major)
 /**
   * \sa col(), class Block */
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 RowXpr row(Index i)
 {
   return RowXpr(derived(), i);
 }
 
 /// This is the const version of row(). */
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 ConstRowXpr row(Index i) const
 {
   return ConstRowXpr(derived(), i);
 }
 
 /// \returns an expression of a segment (i.e. a vector block) in \c *this with either dynamic or fixed sizes.
 ///
 /// \only_for_vectors
 ///
 /// \param start the first coefficient in the segment
 /// \param n the number of coefficients in the segment
 /// \tparam NType the type of the value handling the number of coefficients in the segment, typically Index.
 ///
 /// Example: \include MatrixBase_segment_int_int.cpp
 /// Output: \verbinclude MatrixBase_segment_int_int.out
 ///
 /// The number of coefficients \a n can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments.
 /// See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 /// \note Even in the case that the returned expression has dynamic size, in the case
 /// when it is applied to a fixed-size vector, it inherits a fixed maximal size,
 /// which means that evaluating it does not cause a dynamic memory allocation.
 ///
 /// \sa block(Index,Index,NRowsType,NColsType), fix<N>, fix<N>(int), class Block
 ///
 template<typename NType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename FixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
 #else
 typename FixedSegmentReturnType<...>::Type
 #endif
 segment(Index start, NType n)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename FixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
             (derived(), start, internal::get_runtime_value(n));
 }
 
 
 /// This is the const version of segment(Index,NType).
 template<typename NType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstFixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
 #else
 const typename ConstFixedSegmentReturnType<...>::Type
 #endif
 segment(Index start, NType n) const
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename ConstFixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
             (derived(), start, internal::get_runtime_value(n));
 }
 
 /// \returns an expression of the first coefficients of \c *this with either dynamic or fixed sizes.
 ///
 /// \only_for_vectors
 ///
 /// \param n the number of coefficients in the segment
 /// \tparam NType the type of the value handling the number of coefficients in the segment, typically Index.
 ///
 /// Example: \include MatrixBase_start_int.cpp
 /// Output: \verbinclude MatrixBase_start_int.out
 ///
 /// The number of coefficients \a n can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments.
 /// See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 /// \note Even in the case that the returned expression has dynamic size, in the case
 /// when it is applied to a fixed-size vector, it inherits a fixed maximal size,
 /// which means that evaluating it does not cause a dynamic memory allocation.
 ///
 /// \sa class Block, block(Index,Index)
 ///
 template<typename NType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename FixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
 #else
 typename FixedSegmentReturnType<...>::Type
 #endif
 head(NType n)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename FixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
               (derived(), 0, internal::get_runtime_value(n));
 }
 
 /// This is the const version of head(NType).
 template<typename NType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstFixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
 #else
 const typename ConstFixedSegmentReturnType<...>::Type
 #endif
 head(NType n) const
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename ConstFixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
             (derived(), 0, internal::get_runtime_value(n));
 }
 
 /// \returns an expression of a last coefficients of \c *this with either dynamic or fixed sizes.
 ///
 /// \only_for_vectors
 ///
 /// \param n the number of coefficients in the segment
 /// \tparam NType the type of the value handling the number of coefficients in the segment, typically Index.
 ///
 /// Example: \include MatrixBase_end_int.cpp
 /// Output: \verbinclude MatrixBase_end_int.out
 ///
 /// The number of coefficients \a n can also be specified at compile-time by passing Eigen::fix<N>,
 /// or Eigen::fix<N>(n) as arguments.
 /// See \link block(Index,Index,NRowsType,NColsType) block() \endlink for the details.
 ///
 /// \note Even in the case that the returned expression has dynamic size, in the case
 /// when it is applied to a fixed-size vector, it inherits a fixed maximal size,
 /// which means that evaluating it does not cause a dynamic memory allocation.
 ///
 /// \sa class Block, block(Index,Index)
 ///
 template<typename NType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 typename FixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
 #else
 typename FixedSegmentReturnType<...>::Type
 #endif
 tail(NType n)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename FixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
             (derived(), this->size() - internal::get_runtime_value(n), internal::get_runtime_value(n));
 }
 
 /// This is the const version of tail(Index).
 template<typename NType>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const typename ConstFixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
 #else
 const typename ConstFixedSegmentReturnType<...>::Type
 #endif
 tail(NType n) const
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename ConstFixedSegmentReturnType<internal::get_fixed_value<NType>::value>::Type
             (derived(), this->size() - internal::get_runtime_value(n), internal::get_runtime_value(n));
 }
 
 /// \returns a fixed-size expression of a segment (i.e. a vector block) in \c *this
 ///
 /// \only_for_vectors
 ///
 /// \tparam N the number of coefficients in the segment as specified at compile-time
 /// \param start the index of the first element in the segment
 /// \param n the number of coefficients in the segment as specified at compile-time
 ///
 /// The compile-time and run-time information should not contradict. In other words,
 /// \a n should equal \a N unless \a N is \a Dynamic.
 ///
 /// Example: \include MatrixBase_template_int_segment.cpp
 /// Output: \verbinclude MatrixBase_template_int_segment.out
 ///
 /// \sa segment(Index,NType), class Block
 ///
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedSegmentReturnType<N>::Type segment(Index start, Index n = N)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename FixedSegmentReturnType<N>::Type(derived(), start, n);
 }
 
 /// This is the const version of segment<int>(Index).
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename ConstFixedSegmentReturnType<N>::Type segment(Index start, Index n = N) const
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename ConstFixedSegmentReturnType<N>::Type(derived(), start, n);
 }
 
 /// \returns a fixed-size expression of the first coefficients of \c *this.
 ///
 /// \only_for_vectors
 ///
 /// \tparam N the number of coefficients in the segment as specified at compile-time
 /// \param  n the number of coefficients in the segment as specified at run-time
 ///
 /// The compile-time and run-time information should not contradict. In other words,
 /// \a n should equal \a N unless \a N is \a Dynamic.
 ///
 /// Example: \include MatrixBase_template_int_start.cpp
 /// Output: \verbinclude MatrixBase_template_int_start.out
 ///
 /// \sa head(NType), class Block
 ///
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedSegmentReturnType<N>::Type head(Index n = N)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename FixedSegmentReturnType<N>::Type(derived(), 0, n);
 }
 
 /// This is the const version of head<int>().
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename ConstFixedSegmentReturnType<N>::Type head(Index n = N) const
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename ConstFixedSegmentReturnType<N>::Type(derived(), 0, n);
 }
 
 /// \returns a fixed-size expression of the last coefficients of \c *this.
 ///
 /// \only_for_vectors
 ///
 /// \tparam N the number of coefficients in the segment as specified at compile-time
 /// \param  n the number of coefficients in the segment as specified at run-time
 ///
 /// The compile-time and run-time information should not contradict. In other words,
 /// \a n should equal \a N unless \a N is \a Dynamic.
 ///
 /// Example: \include MatrixBase_template_int_end.cpp
 /// Output: \verbinclude MatrixBase_template_int_end.out
 ///
 /// \sa tail(NType), class Block
 ///
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename FixedSegmentReturnType<N>::Type tail(Index n = N)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename FixedSegmentReturnType<N>::Type(derived(), size() - n);
 }
 
 /// This is the const version of tail<int>.
 template<int N>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename ConstFixedSegmentReturnType<N>::Type tail(Index n = N) const
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return typename ConstFixedSegmentReturnType<N>::Type(derived(), size() - n);
 }
 
 /// \returns the \a outer -th column (resp. row) of the matrix \c *this if \c *this
 /// is col-major (resp. row-major).
 ///
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 InnerVectorReturnType innerVector(Index outer)
 { return InnerVectorReturnType(derived(), outer); }
 
 /// \returns the \a outer -th column (resp. row) of the matrix \c *this if \c *this
 /// is col-major (resp. row-major). Read-only.
 ///
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const ConstInnerVectorReturnType innerVector(Index outer) const
 { return ConstInnerVectorReturnType(derived(), outer); }
 
 /// \returns the \a outer -th column (resp. row) of the matrix \c *this if \c *this
 /// is col-major (resp. row-major).
 ///
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 InnerVectorsReturnType
 innerVectors(Index outerStart, Index outerSize)
 {
   return Block<Derived,Dynamic,Dynamic,true>(derived(),
                                              IsRowMajor ? outerStart : 0, IsRowMajor ? 0 : outerStart,
                                              IsRowMajor ? outerSize : rows(), IsRowMajor ? cols() : outerSize);
 
 }
 
 /// \returns the \a outer -th column (resp. row) of the matrix \c *this if \c *this
 /// is col-major (resp. row-major). Read-only.
 ///
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 const ConstInnerVectorsReturnType
 innerVectors(Index outerStart, Index outerSize) const
 {
   return Block<const Derived,Dynamic,Dynamic,true>(derived(),
                                                   IsRowMajor ? outerStart : 0, IsRowMajor ? 0 : outerStart,
                                                   IsRowMajor ? outerSize : rows(), IsRowMajor ? cols() : outerSize);
 
 }
 
 /** \returns the i-th subvector (column or vector) according to the \c Direction
   * \sa subVectors()
   */
 template<DirectionType Direction>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename internal::conditional<Direction==Vertical,ColXpr,RowXpr>::type
 subVector(Index i)
 {
   return typename internal::conditional<Direction==Vertical,ColXpr,RowXpr>::type(derived(),i);
 }
 
 /** This is the const version of subVector(Index) */
 template<DirectionType Direction>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 typename internal::conditional<Direction==Vertical,ConstColXpr,ConstRowXpr>::type
 subVector(Index i) const
 {
   return typename internal::conditional<Direction==Vertical,ConstColXpr,ConstRowXpr>::type(derived(),i);
 }
 
 /** \returns the number of subvectors (rows or columns) in the direction \c Direction
   * \sa subVector(Index)
   */
 template<DirectionType Direction>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE EIGEN_CONSTEXPR
 Index subVectors() const
 { return (Direction==Vertical)?cols():rows(); }