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 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
 //
 // 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_CWISE_NULLARY_OP_H
 #define EIGEN_CWISE_NULLARY_OP_H
 
 namespace Eigen {
 
 namespace internal {
 template<typename NullaryOp, typename PlainObjectType>
 struct traits<CwiseNullaryOp<NullaryOp, PlainObjectType> > : traits<PlainObjectType>
 {
   enum {
     Flags = traits<PlainObjectType>::Flags & RowMajorBit
   };
 };
 
 } // namespace internal
 
 /** \class CwiseNullaryOp
   * \ingroup Core_Module
   *
   * \brief Generic expression of a matrix where all coefficients are defined by a functor
   *
   * \tparam NullaryOp template functor implementing the operator
   * \tparam PlainObjectType the underlying plain matrix/array type
   *
   * This class represents an expression of a generic nullary operator.
   * It is the return type of the Ones(), Zero(), Constant(), Identity() and Random() methods,
   * and most of the time this is the only way it is used.
   *
   * However, if you want to write a function returning such an expression, you
   * will need to use this class.
   *
   * The functor NullaryOp must expose one of the following method:
     <table class="manual">
     <tr            ><td>\c operator()() </td><td>if the procedural generation does not depend on the coefficient entries (e.g., random numbers)</td></tr>
     <tr class="alt"><td>\c operator()(Index i)</td><td>if the procedural generation makes sense for vectors only and that it depends on the coefficient index \c i (e.g., linspace) </td></tr>
     <tr            ><td>\c operator()(Index i,Index j)</td><td>if the procedural generation depends on the matrix coordinates \c i, \c j (e.g., to generate a checkerboard with 0 and 1)</td></tr>
     </table>
   * It is also possible to expose the last two operators if the generation makes sense for matrices but can be optimized for vectors.
   *
   * See DenseBase::NullaryExpr(Index,const CustomNullaryOp&) for an example binding
   * C++11 random number generators.
   *
   * A nullary expression can also be used to implement custom sophisticated matrix manipulations
   * that cannot be covered by the existing set of natively supported matrix manipulations.
   * See this \ref TopicCustomizing_NullaryExpr "page" for some examples and additional explanations
   * on the behavior of CwiseNullaryOp.
   *
   * \sa class CwiseUnaryOp, class CwiseBinaryOp, DenseBase::NullaryExpr
   */
 template<typename NullaryOp, typename PlainObjectType>
 class CwiseNullaryOp : public internal::dense_xpr_base< CwiseNullaryOp<NullaryOp, PlainObjectType> >::type, internal::no_assignment_operator
 {
   public:
 
     typedef typename internal::dense_xpr_base<CwiseNullaryOp>::type Base;
     EIGEN_DENSE_PUBLIC_INTERFACE(CwiseNullaryOp)
 
     EIGEN_DEVICE_FUNC
     CwiseNullaryOp(Index rows, Index cols, const NullaryOp& func = NullaryOp())
       : m_rows(rows), m_cols(cols), m_functor(func)
     {
       eigen_assert(rows >= 0
             && (RowsAtCompileTime == Dynamic || RowsAtCompileTime == rows)
             &&  cols >= 0
             && (ColsAtCompileTime == Dynamic || ColsAtCompileTime == cols));
     }
 
     EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE EIGEN_CONSTEXPR
     Index rows() const { return m_rows.value(); }
     EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE EIGEN_CONSTEXPR
     Index cols() const { return m_cols.value(); }
 
     /** \returns the functor representing the nullary operation */
     EIGEN_DEVICE_FUNC
     const NullaryOp& functor() const { return m_functor; }
 
   protected:
     const internal::variable_if_dynamic<Index, RowsAtCompileTime> m_rows;
     const internal::variable_if_dynamic<Index, ColsAtCompileTime> m_cols;
     const NullaryOp m_functor;
 };
 
 
 /** \returns an expression of a matrix defined by a custom functor \a func
   *
   * The parameters \a rows and \a cols are the number of rows and of columns of
   * the returned matrix. Must be compatible with this MatrixBase type.
   *
   * This variant is meant to be used for dynamic-size matrix types. For fixed-size types,
   * it is redundant to pass \a rows and \a cols as arguments, so Zero() should be used
   * instead.
   *
   * The template parameter \a CustomNullaryOp is the type of the functor.
   *
   * \sa class CwiseNullaryOp
   */
 template<typename Derived>
 template<typename CustomNullaryOp>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const CwiseNullaryOp<CustomNullaryOp,typename DenseBase<Derived>::PlainObject>
 #else
 const CwiseNullaryOp<CustomNullaryOp,PlainObject>
 #endif
 DenseBase<Derived>::NullaryExpr(Index rows, Index cols, const CustomNullaryOp& func)
 {
   return CwiseNullaryOp<CustomNullaryOp, PlainObject>(rows, cols, func);
 }
 
 /** \returns an expression of a matrix defined by a custom functor \a func
   *
   * The parameter \a size is the size of the returned vector.
   * Must be compatible with this MatrixBase type.
   *
   * \only_for_vectors
   *
   * This variant is meant to be used for dynamic-size vector types. For fixed-size types,
   * it is redundant to pass \a size as argument, so Zero() should be used
   * instead.
   *
   * The template parameter \a CustomNullaryOp is the type of the functor.
   *
   * Here is an example with C++11 random generators: \include random_cpp11.cpp
   * Output: \verbinclude random_cpp11.out
   *
   * \sa class CwiseNullaryOp
   */
 template<typename Derived>
 template<typename CustomNullaryOp>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const CwiseNullaryOp<CustomNullaryOp, typename DenseBase<Derived>::PlainObject>
 #else
 const CwiseNullaryOp<CustomNullaryOp, PlainObject>
 #endif
 DenseBase<Derived>::NullaryExpr(Index size, const CustomNullaryOp& func)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   if(RowsAtCompileTime == 1) return CwiseNullaryOp<CustomNullaryOp, PlainObject>(1, size, func);
   else return CwiseNullaryOp<CustomNullaryOp, PlainObject>(size, 1, func);
 }
 
 /** \returns an expression of a matrix defined by a custom functor \a func
   *
   * This variant is only for fixed-size DenseBase types. For dynamic-size types, you
   * need to use the variants taking size arguments.
   *
   * The template parameter \a CustomNullaryOp is the type of the functor.
   *
   * \sa class CwiseNullaryOp
   */
 template<typename Derived>
 template<typename CustomNullaryOp>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 const CwiseNullaryOp<CustomNullaryOp, typename DenseBase<Derived>::PlainObject>
 #else
 const CwiseNullaryOp<CustomNullaryOp, PlainObject>
 #endif
 DenseBase<Derived>::NullaryExpr(const CustomNullaryOp& func)
 {
   return CwiseNullaryOp<CustomNullaryOp, PlainObject>(RowsAtCompileTime, ColsAtCompileTime, func);
 }
 
 /** \returns an expression of a constant matrix of value \a value
   *
   * The parameters \a rows and \a cols are the number of rows and of columns of
   * the returned matrix. Must be compatible with this DenseBase type.
   *
   * This variant is meant to be used for dynamic-size matrix types. For fixed-size types,
   * it is redundant to pass \a rows and \a cols as arguments, so Zero() should be used
   * instead.
   *
   * The template parameter \a CustomNullaryOp is the type of the functor.
   *
   * \sa class CwiseNullaryOp
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::ConstantReturnType
 DenseBase<Derived>::Constant(Index rows, Index cols, const Scalar& value)
 {
   return DenseBase<Derived>::NullaryExpr(rows, cols, internal::scalar_constant_op<Scalar>(value));
 }
 
 /** \returns an expression of a constant matrix of value \a value
   *
   * The parameter \a size is the size of the returned vector.
   * Must be compatible with this DenseBase type.
   *
   * \only_for_vectors
   *
   * This variant is meant to be used for dynamic-size vector types. For fixed-size types,
   * it is redundant to pass \a size as argument, so Zero() should be used
   * instead.
   *
   * The template parameter \a CustomNullaryOp is the type of the functor.
   *
   * \sa class CwiseNullaryOp
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::ConstantReturnType
 DenseBase<Derived>::Constant(Index size, const Scalar& value)
 {
   return DenseBase<Derived>::NullaryExpr(size, internal::scalar_constant_op<Scalar>(value));
 }
 
 /** \returns an expression of a constant matrix of value \a value
   *
   * This variant is only for fixed-size DenseBase types. For dynamic-size types, you
   * need to use the variants taking size arguments.
   *
   * The template parameter \a CustomNullaryOp is the type of the functor.
   *
   * \sa class CwiseNullaryOp
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::ConstantReturnType
 DenseBase<Derived>::Constant(const Scalar& value)
 {
   EIGEN_STATIC_ASSERT_FIXED_SIZE(Derived)
   return DenseBase<Derived>::NullaryExpr(RowsAtCompileTime, ColsAtCompileTime, internal::scalar_constant_op<Scalar>(value));
 }
 
 /** \deprecated because of accuracy loss. In Eigen 3.3, it is an alias for LinSpaced(Index,const Scalar&,const Scalar&)
   *
   * \only_for_vectors
   *
   * Example: \include DenseBase_LinSpaced_seq_deprecated.cpp
   * Output: \verbinclude DenseBase_LinSpaced_seq_deprecated.out
   *
   * \sa LinSpaced(Index,const Scalar&, const Scalar&), setLinSpaced(Index,const Scalar&,const Scalar&)
   */
 template<typename Derived>
 EIGEN_DEPRECATED EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::RandomAccessLinSpacedReturnType
 DenseBase<Derived>::LinSpaced(Sequential_t, Index size, const Scalar& low, const Scalar& high)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return DenseBase<Derived>::NullaryExpr(size, internal::linspaced_op<Scalar>(low,high,size));
 }
 
 /** \deprecated because of accuracy loss. In Eigen 3.3, it is an alias for LinSpaced(const Scalar&,const Scalar&)
   *
   * \sa LinSpaced(const Scalar&, const Scalar&)
   */
 template<typename Derived>
 EIGEN_DEPRECATED EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::RandomAccessLinSpacedReturnType
 DenseBase<Derived>::LinSpaced(Sequential_t, const Scalar& low, const Scalar& high)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   EIGEN_STATIC_ASSERT_FIXED_SIZE(Derived)
   return DenseBase<Derived>::NullaryExpr(Derived::SizeAtCompileTime, internal::linspaced_op<Scalar>(low,high,Derived::SizeAtCompileTime));
 }
 
 /**
   * \brief Sets a linearly spaced vector.
   *
   * The function generates 'size' equally spaced values in the closed interval [low,high].
   * When size is set to 1, a vector of length 1 containing 'high' is returned.
   *
   * \only_for_vectors
   *
   * Example: \include DenseBase_LinSpaced.cpp
   * Output: \verbinclude DenseBase_LinSpaced.out
   *
   * For integer scalar types, an even spacing is possible if and only if the length of the range,
   * i.e., \c high-low is a scalar multiple of \c size-1, or if \c size is a scalar multiple of the
   * number of values \c high-low+1 (meaning each value can be repeated the same number of time).
   * If one of these two considions is not satisfied, then \c high is lowered to the largest value
   * satisfying one of this constraint.
   * Here are some examples:
   *
   * Example: \include DenseBase_LinSpacedInt.cpp
   * Output: \verbinclude DenseBase_LinSpacedInt.out
   *
   * \sa setLinSpaced(Index,const Scalar&,const Scalar&), CwiseNullaryOp
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::RandomAccessLinSpacedReturnType
 DenseBase<Derived>::LinSpaced(Index size, const Scalar& low, const Scalar& high)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return DenseBase<Derived>::NullaryExpr(size, internal::linspaced_op<Scalar>(low,high,size));
 }
 
 /**
   * \copydoc DenseBase::LinSpaced(Index, const Scalar&, const Scalar&)
   * Special version for fixed size types which does not require the size parameter.
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::RandomAccessLinSpacedReturnType
 DenseBase<Derived>::LinSpaced(const Scalar& low, const Scalar& high)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   EIGEN_STATIC_ASSERT_FIXED_SIZE(Derived)
   return DenseBase<Derived>::NullaryExpr(Derived::SizeAtCompileTime, internal::linspaced_op<Scalar>(low,high,Derived::SizeAtCompileTime));
 }
 
 /** \returns true if all coefficients in this matrix are approximately equal to \a val, to within precision \a prec */
 template<typename Derived>
 EIGEN_DEVICE_FUNC bool DenseBase<Derived>::isApproxToConstant
 (const Scalar& val, const RealScalar& prec) const
 {
   typename internal::nested_eval<Derived,1>::type self(derived());
   for(Index j = 0; j < cols(); ++j)
     for(Index i = 0; i < rows(); ++i)
       if(!internal::isApprox(self.coeff(i, j), val, prec))
         return false;
   return true;
 }
 
 /** This is just an alias for isApproxToConstant().
   *
   * \returns true if all coefficients in this matrix are approximately equal to \a value, to within precision \a prec */
 template<typename Derived>
 EIGEN_DEVICE_FUNC bool DenseBase<Derived>::isConstant
 (const Scalar& val, const RealScalar& prec) const
 {
   return isApproxToConstant(val, prec);
 }
 
 /** Alias for setConstant(): sets all coefficients in this expression to \a val.
   *
   * \sa setConstant(), Constant(), class CwiseNullaryOp
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void DenseBase<Derived>::fill(const Scalar& val)
 {
   setConstant(val);
 }
 
 /** Sets all coefficients in this expression to value \a val.
   *
   * \sa fill(), setConstant(Index,const Scalar&), setConstant(Index,Index,const Scalar&), setZero(), setOnes(), Constant(), class CwiseNullaryOp, setZero(), setOnes()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& DenseBase<Derived>::setConstant(const Scalar& val)
 {
   return derived() = Constant(rows(), cols(), val);
 }
 
 /** Resizes to the given \a size, and sets all coefficients in this expression to the given value \a val.
   *
   * \only_for_vectors
   *
   * Example: \include Matrix_setConstant_int.cpp
   * Output: \verbinclude Matrix_setConstant_int.out
   *
   * \sa MatrixBase::setConstant(const Scalar&), setConstant(Index,Index,const Scalar&), class CwiseNullaryOp, MatrixBase::Constant(const Scalar&)
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setConstant(Index size, const Scalar& val)
 {
   resize(size);
   return setConstant(val);
 }
 
 /** Resizes to the given size, and sets all coefficients in this expression to the given value \a val.
   *
   * \param rows the new number of rows
   * \param cols the new number of columns
   * \param val the value to which all coefficients are set
   *
   * Example: \include Matrix_setConstant_int_int.cpp
   * Output: \verbinclude Matrix_setConstant_int_int.out
   *
   * \sa MatrixBase::setConstant(const Scalar&), setConstant(Index,const Scalar&), class CwiseNullaryOp, MatrixBase::Constant(const Scalar&)
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setConstant(Index rows, Index cols, const Scalar& val)
 {
   resize(rows, cols);
   return setConstant(val);
 }
 
 /** Resizes to the given size, changing only the number of columns, and sets all
   * coefficients in this expression to the given value \a val. For the parameter
   * of type NoChange_t, just pass the special value \c NoChange.
   *
   * \sa MatrixBase::setConstant(const Scalar&), setConstant(Index,const Scalar&), class CwiseNullaryOp, MatrixBase::Constant(const Scalar&)
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setConstant(NoChange_t, Index cols, const Scalar& val)
 {
   return setConstant(rows(), cols, val);
 }
 
 /** Resizes to the given size, changing only the number of rows, and sets all
   * coefficients in this expression to the given value \a val. For the parameter
   * of type NoChange_t, just pass the special value \c NoChange.
   *
   * \sa MatrixBase::setConstant(const Scalar&), setConstant(Index,const Scalar&), class CwiseNullaryOp, MatrixBase::Constant(const Scalar&)
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setConstant(Index rows, NoChange_t, const Scalar& val)
 {
   return setConstant(rows, cols(), val);
 }
 
 
 /**
   * \brief Sets a linearly spaced vector.
   *
   * The function generates 'size' equally spaced values in the closed interval [low,high].
   * When size is set to 1, a vector of length 1 containing 'high' is returned.
   *
   * \only_for_vectors
   *
   * Example: \include DenseBase_setLinSpaced.cpp
   * Output: \verbinclude DenseBase_setLinSpaced.out
   *
   * For integer scalar types, do not miss the explanations on the definition
   * of \link LinSpaced(Index,const Scalar&,const Scalar&) even spacing \endlink.
   *
   * \sa LinSpaced(Index,const Scalar&,const Scalar&), CwiseNullaryOp
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& DenseBase<Derived>::setLinSpaced(Index newSize, const Scalar& low, const Scalar& high)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return derived() = Derived::NullaryExpr(newSize, internal::linspaced_op<Scalar>(low,high,newSize));
 }
 
 /**
   * \brief Sets a linearly spaced vector.
   *
   * The function fills \c *this with equally spaced values in the closed interval [low,high].
   * When size is set to 1, a vector of length 1 containing 'high' is returned.
   *
   * \only_for_vectors
   *
   * For integer scalar types, do not miss the explanations on the definition
   * of \link LinSpaced(Index,const Scalar&,const Scalar&) even spacing \endlink.
   *
   * \sa LinSpaced(Index,const Scalar&,const Scalar&), setLinSpaced(Index, const Scalar&, const Scalar&), CwiseNullaryOp
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& DenseBase<Derived>::setLinSpaced(const Scalar& low, const Scalar& high)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return setLinSpaced(size(), low, high);
 }
 
 // zero:
 
 /** \returns an expression of a zero matrix.
   *
   * The parameters \a rows and \a cols are the number of rows and of columns of
   * the returned matrix. Must be compatible with this MatrixBase type.
   *
   * This variant is meant to be used for dynamic-size matrix types. For fixed-size types,
   * it is redundant to pass \a rows and \a cols as arguments, so Zero() should be used
   * instead.
   *
   * Example: \include MatrixBase_zero_int_int.cpp
   * Output: \verbinclude MatrixBase_zero_int_int.out
   *
   * \sa Zero(), Zero(Index)
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::ConstantReturnType
 DenseBase<Derived>::Zero(Index rows, Index cols)
 {
   return Constant(rows, cols, Scalar(0));
 }
 
 /** \returns an expression of a zero vector.
   *
   * The parameter \a size is the size of the returned vector.
   * Must be compatible with this MatrixBase type.
   *
   * \only_for_vectors
   *
   * This variant is meant to be used for dynamic-size vector types. For fixed-size types,
   * it is redundant to pass \a size as argument, so Zero() should be used
   * instead.
   *
   * Example: \include MatrixBase_zero_int.cpp
   * Output: \verbinclude MatrixBase_zero_int.out
   *
   * \sa Zero(), Zero(Index,Index)
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::ConstantReturnType
 DenseBase<Derived>::Zero(Index size)
 {
   return Constant(size, Scalar(0));
 }
 
 /** \returns an expression of a fixed-size zero matrix or vector.
   *
   * This variant is only for fixed-size MatrixBase types. For dynamic-size types, you
   * need to use the variants taking size arguments.
   *
   * Example: \include MatrixBase_zero.cpp
   * Output: \verbinclude MatrixBase_zero.out
   *
   * \sa Zero(Index), Zero(Index,Index)
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::ConstantReturnType
 DenseBase<Derived>::Zero()
 {
   return Constant(Scalar(0));
 }
 
 /** \returns true if *this is approximately equal to the zero matrix,
   *          within the precision given by \a prec.
   *
   * Example: \include MatrixBase_isZero.cpp
   * Output: \verbinclude MatrixBase_isZero.out
   *
   * \sa class CwiseNullaryOp, Zero()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC bool DenseBase<Derived>::isZero(const RealScalar& prec) const
 {
   typename internal::nested_eval<Derived,1>::type self(derived());
   for(Index j = 0; j < cols(); ++j)
     for(Index i = 0; i < rows(); ++i)
       if(!internal::isMuchSmallerThan(self.coeff(i, j), static_cast<Scalar>(1), prec))
         return false;
   return true;
 }
 
 /** Sets all coefficients in this expression to zero.
   *
   * Example: \include MatrixBase_setZero.cpp
   * Output: \verbinclude MatrixBase_setZero.out
   *
   * \sa class CwiseNullaryOp, Zero()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& DenseBase<Derived>::setZero()
 {
   return setConstant(Scalar(0));
 }
 
 /** Resizes to the given \a size, and sets all coefficients in this expression to zero.
   *
   * \only_for_vectors
   *
   * Example: \include Matrix_setZero_int.cpp
   * Output: \verbinclude Matrix_setZero_int.out
   *
   * \sa DenseBase::setZero(), setZero(Index,Index), class CwiseNullaryOp, DenseBase::Zero()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setZero(Index newSize)
 {
   resize(newSize);
   return setConstant(Scalar(0));
 }
 
 /** Resizes to the given size, and sets all coefficients in this expression to zero.
   *
   * \param rows the new number of rows
   * \param cols the new number of columns
   *
   * Example: \include Matrix_setZero_int_int.cpp
   * Output: \verbinclude Matrix_setZero_int_int.out
   *
   * \sa DenseBase::setZero(), setZero(Index), class CwiseNullaryOp, DenseBase::Zero()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setZero(Index rows, Index cols)
 {
   resize(rows, cols);
   return setConstant(Scalar(0));
 }
 
 /** Resizes to the given size, changing only the number of columns, and sets all
   * coefficients in this expression to zero. For the parameter of type NoChange_t,
   * just pass the special value \c NoChange.
   *
   * \sa DenseBase::setZero(), setZero(Index), setZero(Index, Index), setZero(Index, NoChange_t), class CwiseNullaryOp, DenseBase::Zero()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setZero(NoChange_t, Index cols)
 {
   return setZero(rows(), cols);
 }
 
 /** Resizes to the given size, changing only the number of rows, and sets all
   * coefficients in this expression to zero. For the parameter of type NoChange_t,
   * just pass the special value \c NoChange.
   *
   * \sa DenseBase::setZero(), setZero(Index), setZero(Index, Index), setZero(NoChange_t, Index), class CwiseNullaryOp, DenseBase::Zero()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setZero(Index rows, NoChange_t)
 {
   return setZero(rows, cols());
 }
 
 // ones:
 
 /** \returns an expression of a matrix where all coefficients equal one.
   *
   * The parameters \a rows and \a cols are the number of rows and of columns of
   * the returned matrix. Must be compatible with this MatrixBase type.
   *
   * This variant is meant to be used for dynamic-size matrix types. For fixed-size types,
   * it is redundant to pass \a rows and \a cols as arguments, so Ones() should be used
   * instead.
   *
   * Example: \include MatrixBase_ones_int_int.cpp
   * Output: \verbinclude MatrixBase_ones_int_int.out
   *
   * \sa Ones(), Ones(Index), isOnes(), class Ones
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::ConstantReturnType
 DenseBase<Derived>::Ones(Index rows, Index cols)
 {
   return Constant(rows, cols, Scalar(1));
 }
 
 /** \returns an expression of a vector where all coefficients equal one.
   *
   * The parameter \a newSize is the size of the returned vector.
   * Must be compatible with this MatrixBase type.
   *
   * \only_for_vectors
   *
   * This variant is meant to be used for dynamic-size vector types. For fixed-size types,
   * it is redundant to pass \a size as argument, so Ones() should be used
   * instead.
   *
   * Example: \include MatrixBase_ones_int.cpp
   * Output: \verbinclude MatrixBase_ones_int.out
   *
   * \sa Ones(), Ones(Index,Index), isOnes(), class Ones
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::ConstantReturnType
 DenseBase<Derived>::Ones(Index newSize)
 {
   return Constant(newSize, Scalar(1));
 }
 
 /** \returns an expression of a fixed-size matrix or vector where all coefficients equal one.
   *
   * This variant is only for fixed-size MatrixBase types. For dynamic-size types, you
   * need to use the variants taking size arguments.
   *
   * Example: \include MatrixBase_ones.cpp
   * Output: \verbinclude MatrixBase_ones.out
   *
   * \sa Ones(Index), Ones(Index,Index), isOnes(), class Ones
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::ConstantReturnType
 DenseBase<Derived>::Ones()
 {
   return Constant(Scalar(1));
 }
 
 /** \returns true if *this is approximately equal to the matrix where all coefficients
   *          are equal to 1, within the precision given by \a prec.
   *
   * Example: \include MatrixBase_isOnes.cpp
   * Output: \verbinclude MatrixBase_isOnes.out
   *
   * \sa class CwiseNullaryOp, Ones()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC bool DenseBase<Derived>::isOnes
 (const RealScalar& prec) const
 {
   return isApproxToConstant(Scalar(1), prec);
 }
 
 /** Sets all coefficients in this expression to one.
   *
   * Example: \include MatrixBase_setOnes.cpp
   * Output: \verbinclude MatrixBase_setOnes.out
   *
   * \sa class CwiseNullaryOp, Ones()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& DenseBase<Derived>::setOnes()
 {
   return setConstant(Scalar(1));
 }
 
 /** Resizes to the given \a newSize, and sets all coefficients in this expression to one.
   *
   * \only_for_vectors
   *
   * Example: \include Matrix_setOnes_int.cpp
   * Output: \verbinclude Matrix_setOnes_int.out
   *
   * \sa MatrixBase::setOnes(), setOnes(Index,Index), class CwiseNullaryOp, MatrixBase::Ones()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setOnes(Index newSize)
 {
   resize(newSize);
   return setConstant(Scalar(1));
 }
 
 /** Resizes to the given size, and sets all coefficients in this expression to one.
   *
   * \param rows the new number of rows
   * \param cols the new number of columns
   *
   * Example: \include Matrix_setOnes_int_int.cpp
   * Output: \verbinclude Matrix_setOnes_int_int.out
   *
   * \sa MatrixBase::setOnes(), setOnes(Index), class CwiseNullaryOp, MatrixBase::Ones()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setOnes(Index rows, Index cols)
 {
   resize(rows, cols);
   return setConstant(Scalar(1));
 }
 
 /** Resizes to the given size, changing only the number of rows, and sets all
   * coefficients in this expression to one. For the parameter of type NoChange_t,
   * just pass the special value \c NoChange.
   *
  * \sa MatrixBase::setOnes(), setOnes(Index), setOnes(Index, Index), setOnes(NoChange_t, Index), class CwiseNullaryOp, MatrixBase::Ones()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setOnes(Index rows, NoChange_t)
 {
   return setOnes(rows, cols());
 }
 
 /** Resizes to the given size, changing only the number of columns, and sets all
   * coefficients in this expression to one. For the parameter of type NoChange_t,
   * just pass the special value \c NoChange.
   *
  * \sa MatrixBase::setOnes(), setOnes(Index), setOnes(Index, Index), setOnes(Index, NoChange_t) class CwiseNullaryOp, MatrixBase::Ones()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived&
 PlainObjectBase<Derived>::setOnes(NoChange_t, Index cols)
 {
   return setOnes(rows(), cols);
 }
 
 // Identity:
 
 /** \returns an expression of the identity matrix (not necessarily square).
   *
   * The parameters \a rows and \a cols are the number of rows and of columns of
   * the returned matrix. Must be compatible with this MatrixBase type.
   *
   * This variant is meant to be used for dynamic-size matrix types. For fixed-size types,
   * it is redundant to pass \a rows and \a cols as arguments, so Identity() should be used
   * instead.
   *
   * Example: \include MatrixBase_identity_int_int.cpp
   * Output: \verbinclude MatrixBase_identity_int_int.out
   *
   * \sa Identity(), setIdentity(), isIdentity()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::IdentityReturnType
 MatrixBase<Derived>::Identity(Index rows, Index cols)
 {
   return DenseBase<Derived>::NullaryExpr(rows, cols, internal::scalar_identity_op<Scalar>());
 }
 
 /** \returns an expression of the identity matrix (not necessarily square).
   *
   * This variant is only for fixed-size MatrixBase types. For dynamic-size types, you
   * need to use the variant taking size arguments.
   *
   * Example: \include MatrixBase_identity.cpp
   * Output: \verbinclude MatrixBase_identity.out
   *
   * \sa Identity(Index,Index), setIdentity(), isIdentity()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::IdentityReturnType
 MatrixBase<Derived>::Identity()
 {
   EIGEN_STATIC_ASSERT_FIXED_SIZE(Derived)
   return MatrixBase<Derived>::NullaryExpr(RowsAtCompileTime, ColsAtCompileTime, internal::scalar_identity_op<Scalar>());
 }
 
 /** \returns true if *this is approximately equal to the identity matrix
   *          (not necessarily square),
   *          within the precision given by \a prec.
   *
   * Example: \include MatrixBase_isIdentity.cpp
   * Output: \verbinclude MatrixBase_isIdentity.out
   *
   * \sa class CwiseNullaryOp, Identity(), Identity(Index,Index), setIdentity()
   */
 template<typename Derived>
 bool MatrixBase<Derived>::isIdentity
 (const RealScalar& prec) const
 {
   typename internal::nested_eval<Derived,1>::type self(derived());
   for(Index j = 0; j < cols(); ++j)
   {
     for(Index i = 0; i < rows(); ++i)
     {
       if(i == j)
       {
         if(!internal::isApprox(self.coeff(i, j), static_cast<Scalar>(1), prec))
           return false;
       }
       else
       {
         if(!internal::isMuchSmallerThan(self.coeff(i, j), static_cast<RealScalar>(1), prec))
           return false;
       }
     }
   }
   return true;
 }
 
 namespace internal {
 
 template<typename Derived, bool Big = (Derived::SizeAtCompileTime>=16)>
 struct setIdentity_impl
 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE Derived& run(Derived& m)
   {
     return m = Derived::Identity(m.rows(), m.cols());
   }
 };
 
 template<typename Derived>
 struct setIdentity_impl<Derived, true>
 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE Derived& run(Derived& m)
   {
     m.setZero();
     const Index size = numext::mini(m.rows(), m.cols());
     for(Index i = 0; i < size; ++i) m.coeffRef(i,i) = typename Derived::Scalar(1);
     return m;
   }
 };
 
 } // end namespace internal
 
 /** Writes the identity expression (not necessarily square) into *this.
   *
   * Example: \include MatrixBase_setIdentity.cpp
   * Output: \verbinclude MatrixBase_setIdentity.out
   *
   * \sa class CwiseNullaryOp, Identity(), Identity(Index,Index), isIdentity()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& MatrixBase<Derived>::setIdentity()
 {
   return internal::setIdentity_impl<Derived>::run(derived());
 }
 
 /** \brief Resizes to the given size, and writes the identity expression (not necessarily square) into *this.
   *
   * \param rows the new number of rows
   * \param cols the new number of columns
   *
   * Example: \include Matrix_setIdentity_int_int.cpp
   * Output: \verbinclude Matrix_setIdentity_int_int.out
   *
   * \sa MatrixBase::setIdentity(), class CwiseNullaryOp, MatrixBase::Identity()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& MatrixBase<Derived>::setIdentity(Index rows, Index cols)
 {
   derived().resize(rows, cols);
   return setIdentity();
 }
 
 /** \returns an expression of the i-th unit (basis) vector.
   *
   * \only_for_vectors
   *
   * \sa MatrixBase::Unit(Index), MatrixBase::UnitX(), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::BasisReturnType MatrixBase<Derived>::Unit(Index newSize, Index i)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return BasisReturnType(SquareMatrixType::Identity(newSize,newSize), i);
 }
 
 /** \returns an expression of the i-th unit (basis) vector.
   *
   * \only_for_vectors
   *
   * This variant is for fixed-size vector only.
   *
   * \sa MatrixBase::Unit(Index,Index), MatrixBase::UnitX(), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::BasisReturnType MatrixBase<Derived>::Unit(Index i)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
   return BasisReturnType(SquareMatrixType::Identity(),i);
 }
 
 /** \returns an expression of the X axis unit vector (1{,0}^*)
   *
   * \only_for_vectors
   *
   * \sa MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::BasisReturnType MatrixBase<Derived>::UnitX()
 { return Derived::Unit(0); }
 
 /** \returns an expression of the Y axis unit vector (0,1{,0}^*)
   *
   * \only_for_vectors
   *
   * \sa MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::BasisReturnType MatrixBase<Derived>::UnitY()
 { return Derived::Unit(1); }
 
 /** \returns an expression of the Z axis unit vector (0,0,1{,0}^*)
   *
   * \only_for_vectors
   *
   * \sa MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::BasisReturnType MatrixBase<Derived>::UnitZ()
 { return Derived::Unit(2); }
 
 /** \returns an expression of the W axis unit vector (0,0,0,1)
   *
   * \only_for_vectors
   *
   * \sa MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename MatrixBase<Derived>::BasisReturnType MatrixBase<Derived>::UnitW()
 { return Derived::Unit(3); }
 
 /** \brief Set the coefficients of \c *this to the i-th unit (basis) vector
   *
   * \param i index of the unique coefficient to be set to 1
   *
   * \only_for_vectors
   *
   * \sa MatrixBase::setIdentity(), class CwiseNullaryOp, MatrixBase::Unit(Index,Index)
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& MatrixBase<Derived>::setUnit(Index i)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived);
   eigen_assert(i<size());
   derived().setZero();
   derived().coeffRef(i) = Scalar(1);
   return derived();
 }
 
 /** \brief Resizes to the given \a newSize, and writes the i-th unit (basis) vector into *this.
   *
   * \param newSize the new size of the vector
   * \param i index of the unique coefficient to be set to 1
   *
   * \only_for_vectors
   *
   * \sa MatrixBase::setIdentity(), class CwiseNullaryOp, MatrixBase::Unit(Index,Index)
   */
 template<typename Derived>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& MatrixBase<Derived>::setUnit(Index newSize, Index i)
 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived);
   eigen_assert(i<newSize);
   derived().resize(newSize);
   return setUnit(i);
 }
 
 } // end namespace Eigen
 
 #endif // EIGEN_CWISE_NULLARY_OP_H

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