Eigen/src/Core/CwiseNullaryOp.h

// 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

// IWYU pragma: private
#include "./InternalHeaderCheck.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));
}

template <typename Derived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::RandomAccessEqualSpacedReturnType
DenseBase<Derived>::EqualSpaced(Index size, const Scalar& low, const Scalar& step) {
  EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
  return DenseBase<Derived>::NullaryExpr(size, internal::equalspaced_op<Scalar>(low, step));
}

template <typename Derived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename DenseBase<Derived>::RandomAccessEqualSpacedReturnType
DenseBase<Derived>::EqualSpaced(const Scalar& low, const Scalar& step) {
  EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
  return DenseBase<Derived>::NullaryExpr(Derived::SizeAtCompileTime, internal::equalspaced_op<Scalar>(low, step));
}

/** \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);
}

template <typename Derived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& DenseBase<Derived>::setEqualSpaced(Index newSize, const Scalar& low,
                                                                                  const Scalar& step) {
  EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
  return derived() = Derived::NullaryExpr(newSize, internal::equalspaced_op<Scalar>(low, step));
}
template <typename Derived>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Derived& DenseBase<Derived>::setEqualSpaced(const Scalar& low,
                                                                                  const Scalar& step) {
  EIGEN_STATIC_ASSERT_VECTOR_ONLY(Derived)
  return setEqualSpaced(size(), low, step);
}

// 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