XprHelper.h

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2006-2008 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_XPRHELPER_H
#define EIGEN_XPRHELPER_H
 
#include "../InternalHeaderCheck.h"
 
namespace Eigen {
 
namespace internal {
 
 
// useful for unsigned / signed integer comparisons when idx is intended to be non-negative
template <typename IndexType>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename make_unsigned<IndexType>::type returnUnsignedIndexValue(
    const IndexType& idx) {
  EIGEN_STATIC_ASSERT((NumTraits<IndexType>::IsInteger), THIS FUNCTION IS FOR INTEGER TYPES)
  eigen_internal_assert(idx >= 0 && "Index value is negative and target type is unsigned");
  using UnsignedType = typename make_unsigned<IndexType>::type;
  return static_cast<UnsignedType>(idx);
}
 
template <typename IndexDest, typename IndexSrc, 
          bool IndexDestIsInteger = NumTraits<IndexDest>::IsInteger,
          bool IndexDestIsSigned = NumTraits<IndexDest>::IsSigned,
          bool IndexSrcIsInteger = NumTraits<IndexSrc>::IsInteger,
          bool IndexSrcIsSigned = NumTraits<IndexSrc>::IsSigned>
struct convert_index_impl {
  static inline EIGEN_DEVICE_FUNC IndexDest run(const IndexSrc& idx) {
    eigen_internal_assert(idx <= NumTraits<IndexDest>::highest() && "Index value is too big for target type");
    return static_cast<IndexDest>(idx);
  }
};
template <typename IndexDest, typename IndexSrc>
struct convert_index_impl<IndexDest, IndexSrc, true, true, true, false> {
  // IndexDest is a signed integer
  // IndexSrc is an unsigned integer
  static inline EIGEN_DEVICE_FUNC IndexDest run(const IndexSrc& idx) {
    eigen_internal_assert(idx <= returnUnsignedIndexValue(NumTraits<IndexDest>::highest()) &&
                          "Index value is too big for target type");
    return static_cast<IndexDest>(idx);
  }
};
template <typename IndexDest, typename IndexSrc>
struct convert_index_impl<IndexDest, IndexSrc, true, false, true, true> {
  // IndexDest is an unsigned integer
  // IndexSrc is a signed integer
  static inline EIGEN_DEVICE_FUNC IndexDest run(const IndexSrc& idx) {
    eigen_internal_assert(returnUnsignedIndexValue(idx) <= NumTraits<IndexDest>::highest() &&
                          "Index value is too big for target type");
    return static_cast<IndexDest>(idx);
  }
};
 
template <typename IndexDest, typename IndexSrc>
EIGEN_DEVICE_FUNC inline IndexDest convert_index(const IndexSrc& idx) {
  return convert_index_impl<IndexDest, IndexSrc>::run(idx);
}
 
// true if T can be considered as an integral index (i.e., and integral type or enum)
template<typename T> struct is_valid_index_type
{
  enum { value = internal::is_integral<T>::value || std::is_enum<T>::value
  };
};
 
// true if both types are not valid index types
template<typename RowIndices, typename ColIndices>
struct valid_indexed_view_overload {
  enum { value = !(internal::is_valid_index_type<RowIndices>::value && internal::is_valid_index_type<ColIndices>::value) };
};
 
// promote_scalar_arg is an helper used in operation between an expression and a scalar, like:
//    expression * scalar
// Its role is to determine how the type T of the scalar operand should be promoted given the scalar type ExprScalar of the given expression.
// The IsSupported template parameter must be provided by the caller as: internal::has_ReturnType<ScalarBinaryOpTraits<ExprScalar,T,op> >::value using the proper order for ExprScalar and T.
// Then the logic is as follows:
//  - if the operation is natively supported as defined by IsSupported, then the scalar type is not promoted, and T is returned.
//  - otherwise, NumTraits<ExprScalar>::Literal is returned if T is implicitly convertible to NumTraits<ExprScalar>::Literal AND that this does not imply a float to integer conversion.
//  - otherwise, ExprScalar is returned if T is implicitly convertible to ExprScalar AND that this does not imply a float to integer conversion.
//  - In all other cases, the promoted type is not defined, and the respective operation is thus invalid and not available (SFINAE).
template<typename ExprScalar,typename T, bool IsSupported>
struct promote_scalar_arg;
 
template<typename S,typename T>
struct promote_scalar_arg<S,T,true>
{
  typedef T type;
};
 
// Recursively check safe conversion to PromotedType, and then ExprScalar if they are different.
template<typename ExprScalar,typename T,typename PromotedType,
  bool ConvertibleToLiteral = internal::is_convertible<T,PromotedType>::value,
  bool IsSafe = NumTraits<T>::IsInteger || !NumTraits<PromotedType>::IsInteger>
struct promote_scalar_arg_unsupported;
 
// Start recursion with NumTraits<ExprScalar>::Literal
template<typename S,typename T>
struct promote_scalar_arg<S,T,false> : promote_scalar_arg_unsupported<S,T,typename NumTraits<S>::Literal> {};
 
// We found a match!
template<typename S,typename T, typename PromotedType>
struct promote_scalar_arg_unsupported<S,T,PromotedType,true,true>
{
  typedef PromotedType type;
};
 
// No match, but no real-to-integer issues, and ExprScalar and current PromotedType are different,
// so let's try to promote to ExprScalar
template<typename ExprScalar,typename T, typename PromotedType>
struct promote_scalar_arg_unsupported<ExprScalar,T,PromotedType,false,true>
   : promote_scalar_arg_unsupported<ExprScalar,T,ExprScalar>
{};
 
// Unsafe real-to-integer, let's stop.
template<typename S,typename T, typename PromotedType, bool ConvertibleToLiteral>
struct promote_scalar_arg_unsupported<S,T,PromotedType,ConvertibleToLiteral,false> {};
 
// T is not even convertible to ExprScalar, let's stop.
template<typename S,typename T>
struct promote_scalar_arg_unsupported<S,T,S,false,true> {};
 
//classes inheriting no_assignment_operator don't generate a default operator=.
class no_assignment_operator
{
  private:
    no_assignment_operator& operator=(const no_assignment_operator&);
  protected:
    EIGEN_DEFAULT_COPY_CONSTRUCTOR(no_assignment_operator)
    EIGEN_DEFAULT_EMPTY_CONSTRUCTOR_AND_DESTRUCTOR(no_assignment_operator)
};
 
template<typename I1, typename I2>
struct promote_index_type
{
  typedef std::conditional_t<(sizeof(I1)<sizeof(I2)), I2, I1> type;
};
 
template<typename T, int Value> class variable_if_dynamic
{
  public:
    EIGEN_DEFAULT_EMPTY_CONSTRUCTOR_AND_DESTRUCTOR(variable_if_dynamic)
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit variable_if_dynamic(T v) { EIGEN_ONLY_USED_FOR_DEBUG(v); eigen_assert(v == T(Value)); }
    EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE EIGEN_CONSTEXPR
    T value() { return T(Value); }
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE EIGEN_CONSTEXPR
    operator T() const { return T(Value); }
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
    void setValue(T v) const { EIGEN_ONLY_USED_FOR_DEBUG(v); eigen_assert(v == T(Value)); }
};
 
template<typename T> class variable_if_dynamic<T, Dynamic>
{
    T m_value;
  public:
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit variable_if_dynamic(T value = 0) EIGEN_NO_THROW : m_value(value) {}
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T value() const { return m_value; }
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE operator T() const { return m_value; }
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void setValue(T value) { m_value = value; }
};
 
template<typename T, int Value> class variable_if_dynamicindex
{
  public:
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit variable_if_dynamicindex(T v) { EIGEN_ONLY_USED_FOR_DEBUG(v); eigen_assert(v == T(Value)); }
    EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE EIGEN_CONSTEXPR
    T value() { return T(Value); }
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
    void setValue(T) {}
};
 
template<typename T> class variable_if_dynamicindex<T, DynamicIndex>
{
    T m_value;
    EIGEN_DEVICE_FUNC variable_if_dynamicindex() { eigen_assert(false); }
  public:
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit variable_if_dynamicindex(T value) : m_value(value) {}
    EIGEN_DEVICE_FUNC T EIGEN_STRONG_INLINE value() const { return m_value; }
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void setValue(T value) { m_value = value; }
};
 
template<typename T> struct functor_traits
{
  enum
  {
    Cost = 10,
    PacketAccess = false,
    IsRepeatable = false
  };
};
 
template<typename T> struct packet_traits;
 
template<typename T> struct unpacket_traits;
 
template<int Size, typename PacketType,
         bool Stop = Size==Dynamic || (Size%unpacket_traits<PacketType>::size)==0 || is_same<PacketType,typename unpacket_traits<PacketType>::half>::value>
struct find_best_packet_helper;
 
template< int Size, typename PacketType>
struct find_best_packet_helper<Size,PacketType,true>
{
  typedef PacketType type;
};
 
template<int Size, typename PacketType>
struct find_best_packet_helper<Size,PacketType,false>
{
  typedef typename find_best_packet_helper<Size,typename unpacket_traits<PacketType>::half>::type type;
};
 
template<typename T, int Size>
struct find_best_packet
{
  typedef typename find_best_packet_helper<Size,typename packet_traits<T>::type>::type type;
};
 
template <int Size, typename PacketType,
          bool Stop = (Size == unpacket_traits<PacketType>::size) ||
                      is_same<PacketType, typename unpacket_traits<PacketType>::half>::value>
struct find_packet_by_size_helper;
template <int Size, typename PacketType>
struct find_packet_by_size_helper<Size, PacketType, true> {
  using type = PacketType;
};
template <int Size, typename PacketType>
struct find_packet_by_size_helper<Size, PacketType, false> {
  using type = typename find_packet_by_size_helper<Size, typename unpacket_traits<PacketType>::half>::type;
};
 
template <typename T, int Size>
struct find_packet_by_size {
  using type = typename find_packet_by_size_helper<Size, typename packet_traits<T>::type>::type;
  static constexpr bool value = (Size == unpacket_traits<type>::size);
};
template <typename T>
struct find_packet_by_size<T, 1> {
  using type = typename unpacket_traits<T>::type;
  static constexpr bool value = (unpacket_traits<type>::size == 1);
};
 
#if EIGEN_MAX_STATIC_ALIGN_BYTES>0
constexpr inline int compute_default_alignment_helper(int ArrayBytes, int AlignmentBytes) {
  if((ArrayBytes % AlignmentBytes) == 0) {
    return AlignmentBytes;
  } else if (EIGEN_MIN_ALIGN_BYTES<AlignmentBytes) {
    return compute_default_alignment_helper(ArrayBytes, AlignmentBytes/2);
  } else {
    return 0;
  }
}
#else
// If static alignment is disabled, no need to bother.
// This also avoids a division by zero
constexpr inline int compute_default_alignment_helper(int ArrayBytes, int AlignmentBytes) {
  EIGEN_UNUSED_VARIABLE(ArrayBytes);
  EIGEN_UNUSED_VARIABLE(AlignmentBytes);
  return 0;
}
#endif
 
template<typename T, int Size> struct compute_default_alignment {
  enum { value = compute_default_alignment_helper(Size*sizeof(T), EIGEN_MAX_STATIC_ALIGN_BYTES) };
};
 
template<typename T> struct compute_default_alignment<T,Dynamic> {
  enum { value = EIGEN_MAX_ALIGN_BYTES };
};
 
template<typename Scalar_, int Rows_, int Cols_,
         int Options_ = AutoAlign |
                          ( (Rows_==1 && Cols_!=1) ? RowMajor
                          : (Cols_==1 && Rows_!=1) ? ColMajor
                          : EIGEN_DEFAULT_MATRIX_STORAGE_ORDER_OPTION ),
         int MaxRows_ = Rows_,
         int MaxCols_ = Cols_
> class make_proper_matrix_type
{
    enum {
      IsColVector = Cols_==1 && Rows_!=1,
      IsRowVector = Rows_==1 && Cols_!=1,
      Options = IsColVector ? (Options_ | ColMajor) & ~RowMajor
              : IsRowVector ? (Options_ | RowMajor) & ~ColMajor
              : Options_
    };
  public:
    typedef Matrix<Scalar_, Rows_, Cols_, Options, MaxRows_, MaxCols_> type;
};
 
constexpr inline unsigned compute_matrix_flags(int Options) {
  unsigned row_major_bit = Options&RowMajor ? RowMajorBit : 0;
  // FIXME currently we still have to handle DirectAccessBit at the expression level to handle DenseCoeffsBase<>
  // and then propagate this information to the evaluator's flags.
  // However, I (Gael) think that DirectAccessBit should only matter at the evaluation stage.
  return DirectAccessBit | LvalueBit | NestByRefBit | row_major_bit;
}
 
constexpr inline int size_at_compile_time(int rows, int cols) {
  if (rows == 0 || cols == 0) return 0;
  if (rows == Dynamic || cols == Dynamic) return Dynamic;
  return rows * cols;
}
 
template<typename XprType> struct size_of_xpr_at_compile_time
{
  enum { ret = size_at_compile_time(traits<XprType>::RowsAtCompileTime, traits<XprType>::ColsAtCompileTime) };
};
 
/* plain_matrix_type : the difference from eval is that plain_matrix_type is always a plain matrix type,
 * whereas eval is a const reference in the case of a matrix
 */
 
template<typename T, typename StorageKind = typename traits<T>::StorageKind> struct plain_matrix_type;
template<typename T, typename BaseClassType, int Flags> struct plain_matrix_type_dense;
template<typename T> struct plain_matrix_type<T,Dense>
{
  typedef typename plain_matrix_type_dense<T,typename traits<T>::XprKind, traits<T>::Flags>::type type;
};
template<typename T> struct plain_matrix_type<T,DiagonalShape>
{
  typedef typename T::PlainObject type;
};
 
template<typename T> struct plain_matrix_type<T,SkewSymmetricShape>
{
  typedef typename T::PlainObject type;
};
 
template<typename T, int Flags> struct plain_matrix_type_dense<T,MatrixXpr,Flags>
{
  typedef Matrix<typename traits<T>::Scalar,
                traits<T>::RowsAtCompileTime,
                traits<T>::ColsAtCompileTime,
                AutoAlign | (Flags&RowMajorBit ? RowMajor : ColMajor),
                traits<T>::MaxRowsAtCompileTime,
                traits<T>::MaxColsAtCompileTime
          > type;
};
 
template<typename T, int Flags> struct plain_matrix_type_dense<T,ArrayXpr,Flags>
{
  typedef Array<typename traits<T>::Scalar,
                traits<T>::RowsAtCompileTime,
                traits<T>::ColsAtCompileTime,
                AutoAlign | (Flags&RowMajorBit ? RowMajor : ColMajor),
                traits<T>::MaxRowsAtCompileTime,
                traits<T>::MaxColsAtCompileTime
          > type;
};
 
/* eval : the return type of eval(). For matrices, this is just a const reference
 * in order to avoid a useless copy
 */
 
template<typename T, typename StorageKind = typename traits<T>::StorageKind> struct eval;
 
template<typename T> struct eval<T,Dense>
{
  typedef typename plain_matrix_type<T>::type type;
//   typedef typename T::PlainObject type;
//   typedef T::Matrix<typename traits<T>::Scalar,
//                 traits<T>::RowsAtCompileTime,
//                 traits<T>::ColsAtCompileTime,
//                 AutoAlign | (traits<T>::Flags&RowMajorBit ? RowMajor : ColMajor),
//                 traits<T>::MaxRowsAtCompileTime,
//                 traits<T>::MaxColsAtCompileTime
//           > type;
};
 
template<typename T> struct eval<T,DiagonalShape>
{
  typedef typename plain_matrix_type<T>::type type;
};
 
template<typename T> struct eval<T,SkewSymmetricShape>
{
  typedef typename plain_matrix_type<T>::type type;
};
 
// for matrices, no need to evaluate, just use a const reference to avoid a useless copy
template<typename Scalar_, int Rows_, int Cols_, int Options_, int MaxRows_, int MaxCols_>
struct eval<Matrix<Scalar_, Rows_, Cols_, Options_, MaxRows_, MaxCols_>, Dense>
{
  typedef const Matrix<Scalar_, Rows_, Cols_, Options_, MaxRows_, MaxCols_>& type;
};
 
template<typename Scalar_, int Rows_, int Cols_, int Options_, int MaxRows_, int MaxCols_>
struct eval<Array<Scalar_, Rows_, Cols_, Options_, MaxRows_, MaxCols_>, Dense>
{
  typedef const Array<Scalar_, Rows_, Cols_, Options_, MaxRows_, MaxCols_>& type;
};
 
 
/* similar to plain_matrix_type, but using the evaluator's Flags */
template<typename T, typename StorageKind = typename traits<T>::StorageKind> struct plain_object_eval;
 
template<typename T>
struct plain_object_eval<T,Dense>
{
  typedef typename plain_matrix_type_dense<T,typename traits<T>::XprKind, evaluator<T>::Flags>::type type;
};
 
 
/* plain_matrix_type_column_major : same as plain_matrix_type but guaranteed to be column-major
 */
template<typename T> struct plain_matrix_type_column_major
{
  enum { Rows = traits<T>::RowsAtCompileTime,
         Cols = traits<T>::ColsAtCompileTime,
         MaxRows = traits<T>::MaxRowsAtCompileTime,
         MaxCols = traits<T>::MaxColsAtCompileTime
  };
  typedef Matrix<typename traits<T>::Scalar,
                Rows,
                Cols,
                (MaxRows==1&&MaxCols!=1) ? RowMajor : ColMajor,
                MaxRows,
                MaxCols
          > type;
};
 
/* plain_matrix_type_row_major : same as plain_matrix_type but guaranteed to be row-major
 */
template<typename T> struct plain_matrix_type_row_major
{
  enum { Rows = traits<T>::RowsAtCompileTime,
         Cols = traits<T>::ColsAtCompileTime,
         MaxRows = traits<T>::MaxRowsAtCompileTime,
         MaxCols = traits<T>::MaxColsAtCompileTime
  };
  typedef Matrix<typename traits<T>::Scalar,
                Rows,
                Cols,
                (MaxCols==1&&MaxRows!=1) ? ColMajor : RowMajor,
                MaxRows,
                MaxCols
          > type;
};
 
template <typename T>
struct ref_selector
{
  typedef std::conditional_t<
    bool(traits<T>::Flags & NestByRefBit),
    T const&,
    const T
  > type;
 
  typedef std::conditional_t<
    bool(traits<T>::Flags & NestByRefBit),
    T &,
    T
  > non_const_type;
};
 
template<typename T1, typename T2>
struct transfer_constness
{
  typedef std::conditional_t<
    bool(internal::is_const<T1>::value),
    add_const_on_value_type_t<T2>,
    T2
  > type;
};
 
 
// However, we still need a mechanism to detect whether an expression which is evaluated multiple time
// has to be evaluated into a temporary.
// That's the purpose of this new nested_eval helper:
template<typename T, int n, typename PlainObject = typename plain_object_eval<T>::type> struct nested_eval
{
  enum {
    ScalarReadCost = NumTraits<typename traits<T>::Scalar>::ReadCost,
    CoeffReadCost = evaluator<T>::CoeffReadCost,  // NOTE What if an evaluator evaluate itself into a temporary?
                                                  //      Then CoeffReadCost will be small (e.g., 1) but we still have to evaluate, especially if n>1.
                                                  //      This situation is already taken care by the EvalBeforeNestingBit flag, which is turned ON
                                                  //      for all evaluator creating a temporary. This flag is then propagated by the parent evaluators.
                                                  //      Another solution could be to count the number of temps?
    NAsInteger = n == Dynamic ? HugeCost : n,
    CostEval   = (NAsInteger+1) * ScalarReadCost + CoeffReadCost,
    CostNoEval = NAsInteger * CoeffReadCost,
    Evaluate = (int(evaluator<T>::Flags) & EvalBeforeNestingBit) || (int(CostEval) < int(CostNoEval))
  };
 
  typedef std::conditional_t<Evaluate, PlainObject, typename ref_selector<T>::type> type;
};
 
template<typename T>
EIGEN_DEVICE_FUNC
inline T* const_cast_ptr(const T* ptr)
{
  return const_cast<T*>(ptr);
}
 
template<typename Derived, typename XprKind = typename traits<Derived>::XprKind>
struct dense_xpr_base
{
  /* dense_xpr_base should only ever be used on dense expressions, thus falling either into the MatrixXpr or into the ArrayXpr cases */
};
 
template<typename Derived>
struct dense_xpr_base<Derived, MatrixXpr>
{
  typedef MatrixBase<Derived> type;
};
 
template<typename Derived>
struct dense_xpr_base<Derived, ArrayXpr>
{
  typedef ArrayBase<Derived> type;
};
 
template<typename Derived, typename XprKind = typename traits<Derived>::XprKind, typename StorageKind = typename traits<Derived>::StorageKind>
struct generic_xpr_base;
 
template<typename Derived, typename XprKind>
struct generic_xpr_base<Derived, XprKind, Dense>
{
  typedef typename dense_xpr_base<Derived,XprKind>::type type;
};
 
template<typename XprType, typename CastType> struct cast_return_type
{
  typedef typename XprType::Scalar CurrentScalarType;
  typedef remove_all_t<CastType> CastType_;
  typedef typename CastType_::Scalar NewScalarType;
  typedef std::conditional_t<is_same<CurrentScalarType,NewScalarType>::value,
                              const XprType&,CastType> type;
};
 
template <typename A, typename B> struct promote_storage_type;
 
template <typename A> struct promote_storage_type<A,A>
{
  typedef A ret;
};
template <typename A> struct promote_storage_type<A, const A>
{
  typedef A ret;
};
template <typename A> struct promote_storage_type<const A, A>
{
  typedef A ret;
};
 
template <typename A, typename B, typename Functor> struct cwise_promote_storage_type;
 
template <typename A, typename Functor>                   struct cwise_promote_storage_type<A,A,Functor>                                      { typedef A      ret; };
template <typename Functor>                               struct cwise_promote_storage_type<Dense,Dense,Functor>                              { typedef Dense  ret; };
template <typename A, typename Functor>                   struct cwise_promote_storage_type<A,Dense,Functor>                                  { typedef Dense  ret; };
template <typename B, typename Functor>                   struct cwise_promote_storage_type<Dense,B,Functor>                                  { typedef Dense  ret; };
template <typename Functor>                               struct cwise_promote_storage_type<Sparse,Dense,Functor>                             { typedef Sparse ret; };
template <typename Functor>                               struct cwise_promote_storage_type<Dense,Sparse,Functor>                             { typedef Sparse ret; };
 
template <typename LhsKind, typename RhsKind, int LhsOrder, int RhsOrder> struct cwise_promote_storage_order {
  enum { value = LhsOrder };
};
 
template <typename LhsKind, int LhsOrder, int RhsOrder>   struct cwise_promote_storage_order<LhsKind,Sparse,LhsOrder,RhsOrder>                { enum { value = RhsOrder }; };
template <typename RhsKind, int LhsOrder, int RhsOrder>   struct cwise_promote_storage_order<Sparse,RhsKind,LhsOrder,RhsOrder>                { enum { value = LhsOrder }; };
template <int Order>                                      struct cwise_promote_storage_order<Sparse,Sparse,Order,Order>                       { enum { value = Order }; };
 
 
template <typename A, typename B, int ProductTag> struct product_promote_storage_type;
 
template <typename A, int ProductTag> struct product_promote_storage_type<A,                  A,                  ProductTag> { typedef A     ret;};
template <int ProductTag>             struct product_promote_storage_type<Dense,              Dense,              ProductTag> { typedef Dense ret;};
template <typename A, int ProductTag> struct product_promote_storage_type<A,                  Dense,              ProductTag> { typedef Dense ret; };
template <typename B, int ProductTag> struct product_promote_storage_type<Dense,              B,                  ProductTag> { typedef Dense ret; };
 
template <typename A, int ProductTag> struct product_promote_storage_type<A,                  DiagonalShape,      ProductTag> { typedef A ret; };
template <typename B, int ProductTag> struct product_promote_storage_type<DiagonalShape,      B,                  ProductTag> { typedef B ret; };
template <int ProductTag>             struct product_promote_storage_type<Dense,              DiagonalShape,      ProductTag> { typedef Dense ret; };
template <int ProductTag>             struct product_promote_storage_type<DiagonalShape,      Dense,              ProductTag> { typedef Dense ret; };
 
template <typename A, int ProductTag> struct product_promote_storage_type<A,                  SkewSymmetricShape, ProductTag> { typedef A ret; };
template <typename B, int ProductTag> struct product_promote_storage_type<SkewSymmetricShape, B,                  ProductTag> { typedef B ret; };
template <int ProductTag>             struct product_promote_storage_type<Dense,              SkewSymmetricShape, ProductTag> { typedef Dense ret; };
template <int ProductTag>             struct product_promote_storage_type<SkewSymmetricShape, Dense,              ProductTag> { typedef Dense ret; };
template <int ProductTag>             struct product_promote_storage_type<SkewSymmetricShape, SkewSymmetricShape, ProductTag> { typedef Dense ret; };
 
template <typename A, int ProductTag> struct product_promote_storage_type<A,                  PermutationStorage, ProductTag> { typedef A ret; };
template <typename B, int ProductTag> struct product_promote_storage_type<PermutationStorage, B,                  ProductTag> { typedef B ret; };
template <int ProductTag>             struct product_promote_storage_type<Dense,              PermutationStorage, ProductTag> { typedef Dense ret; };
template <int ProductTag>             struct product_promote_storage_type<PermutationStorage, Dense,              ProductTag> { typedef Dense ret; };
 
template<typename ExpressionType, typename Scalar = typename ExpressionType::Scalar>
struct plain_row_type
{
  typedef Matrix<Scalar, 1, ExpressionType::ColsAtCompileTime,
                 int(ExpressionType::PlainObject::Options) | int(RowMajor), 1, ExpressionType::MaxColsAtCompileTime> MatrixRowType;
  typedef Array<Scalar, 1, ExpressionType::ColsAtCompileTime,
                 int(ExpressionType::PlainObject::Options) | int(RowMajor), 1, ExpressionType::MaxColsAtCompileTime> ArrayRowType;
 
  typedef std::conditional_t<
    is_same< typename traits<ExpressionType>::XprKind, MatrixXpr >::value,
    MatrixRowType,
    ArrayRowType
  > type;
};
 
template<typename ExpressionType, typename Scalar = typename ExpressionType::Scalar>
struct plain_col_type
{
  typedef Matrix<Scalar, ExpressionType::RowsAtCompileTime, 1,
                 ExpressionType::PlainObject::Options & ~RowMajor, ExpressionType::MaxRowsAtCompileTime, 1> MatrixColType;
  typedef Array<Scalar, ExpressionType::RowsAtCompileTime, 1,
                 ExpressionType::PlainObject::Options & ~RowMajor, ExpressionType::MaxRowsAtCompileTime, 1> ArrayColType;
 
  typedef std::conditional_t<
    is_same< typename traits<ExpressionType>::XprKind, MatrixXpr >::value,
    MatrixColType,
    ArrayColType
  > type;
};
 
template<typename ExpressionType, typename Scalar = typename ExpressionType::Scalar>
struct plain_diag_type
{
  enum { diag_size = internal::min_size_prefer_dynamic(ExpressionType::RowsAtCompileTime, ExpressionType::ColsAtCompileTime),
         max_diag_size = min_size_prefer_fixed(ExpressionType::MaxRowsAtCompileTime,
                                               ExpressionType::MaxColsAtCompileTime)
  };
  typedef Matrix<Scalar, diag_size, 1, ExpressionType::PlainObject::Options & ~RowMajor, max_diag_size, 1> MatrixDiagType;
  typedef Array<Scalar, diag_size, 1, ExpressionType::PlainObject::Options & ~RowMajor, max_diag_size, 1> ArrayDiagType;
 
  typedef std::conditional_t<
    is_same< typename traits<ExpressionType>::XprKind, MatrixXpr >::value,
    MatrixDiagType,
    ArrayDiagType
  > type;
};
 
template<typename Expr,typename Scalar = typename Expr::Scalar>
struct plain_constant_type
{
  enum { Options = (traits<Expr>::Flags&RowMajorBit)?RowMajor:0 };
 
  typedef Array<Scalar,  traits<Expr>::RowsAtCompileTime,   traits<Expr>::ColsAtCompileTime,
                Options, traits<Expr>::MaxRowsAtCompileTime,traits<Expr>::MaxColsAtCompileTime> array_type;
 
  typedef Matrix<Scalar,  traits<Expr>::RowsAtCompileTime,   traits<Expr>::ColsAtCompileTime,
                 Options, traits<Expr>::MaxRowsAtCompileTime,traits<Expr>::MaxColsAtCompileTime> matrix_type;
 
  typedef CwiseNullaryOp<scalar_constant_op<Scalar>, const std::conditional_t<is_same< typename traits<Expr>::XprKind, MatrixXpr >::value, matrix_type, array_type> > type;
};
 
template<typename ExpressionType>
struct is_lvalue
{
  enum { value = (!bool(is_const<ExpressionType>::value)) &&
                 bool(traits<ExpressionType>::Flags & LvalueBit) };
};
 
template<typename T> struct is_diagonal
{ enum { ret = false }; };
 
template<typename T> struct is_diagonal<DiagonalBase<T> >
{ enum { ret = true }; };
 
template<typename T> struct is_diagonal<DiagonalWrapper<T> >
{ enum { ret = true }; };
 
template<typename T, int S> struct is_diagonal<DiagonalMatrix<T,S> >
{ enum { ret = true }; };
 
 
template<typename T> struct is_identity
{ enum { value = false }; };
 
template<typename T> struct is_identity<CwiseNullaryOp<internal::scalar_identity_op<typename T::Scalar>, T> >
{ enum { value = true }; };
 
 
template<typename S1, typename S2> struct glue_shapes;
template<> struct glue_shapes<DenseShape,TriangularShape> { typedef TriangularShape type;  };
 
template<typename T1, typename T2>
struct possibly_same_dense {
  enum { value = has_direct_access<T1>::ret && has_direct_access<T2>::ret && is_same<typename T1::Scalar,typename T2::Scalar>::value };
};
 
template<typename T1, typename T2>
EIGEN_DEVICE_FUNC
bool is_same_dense(const T1 &mat1, const T2 &mat2, std::enable_if_t<possibly_same_dense<T1,T2>::value> * = 0)
{
  return (mat1.data()==mat2.data()) && (mat1.innerStride()==mat2.innerStride()) && (mat1.outerStride()==mat2.outerStride());
}
 
template<typename T1, typename T2>
EIGEN_DEVICE_FUNC
bool is_same_dense(const T1 &, const T2 &, std::enable_if_t<!possibly_same_dense<T1,T2>::value> * = 0)
{
  return false;
}
 
// Internal helper defining the cost of a scalar division for the type T.
// The default heuristic can be specialized for each scalar type and architecture.
template<typename T,bool Vectorized=false,typename EnableIf = void>
struct scalar_div_cost {
  enum { value = 8*NumTraits<T>::MulCost };
};
 
template<typename T,bool Vectorized>
struct scalar_div_cost<std::complex<T>, Vectorized> {
  enum { value = 2*scalar_div_cost<T>::value
               + 6*NumTraits<T>::MulCost
               + 3*NumTraits<T>::AddCost
  };
};
 
 
template<bool Vectorized>
struct scalar_div_cost<signed long,Vectorized, std::conditional_t<sizeof(long)==8,void,false_type>> { enum { value = 24 }; };
template<bool Vectorized>
struct scalar_div_cost<unsigned long,Vectorized, std::conditional_t<sizeof(long)==8,void,false_type>> { enum { value = 21 }; };
 
 
#ifdef EIGEN_DEBUG_ASSIGN
std::string demangle_traversal(int t)
{
  if(t==DefaultTraversal) return "DefaultTraversal";
  if(t==LinearTraversal) return "LinearTraversal";
  if(t==InnerVectorizedTraversal) return "InnerVectorizedTraversal";
  if(t==LinearVectorizedTraversal) return "LinearVectorizedTraversal";
  if(t==SliceVectorizedTraversal) return "SliceVectorizedTraversal";
  return "?";
}
std::string demangle_unrolling(int t)
{
  if(t==NoUnrolling) return "NoUnrolling";
  if(t==InnerUnrolling) return "InnerUnrolling";
  if(t==CompleteUnrolling) return "CompleteUnrolling";
  return "?";
}
std::string demangle_flags(int f)
{
  std::string res;
  if(f&RowMajorBit)                 res += " | RowMajor";
  if(f&PacketAccessBit)             res += " | Packet";
  if(f&LinearAccessBit)             res += " | Linear";
  if(f&LvalueBit)                   res += " | Lvalue";
  if(f&DirectAccessBit)             res += " | Direct";
  if(f&NestByRefBit)                res += " | NestByRef";
  if(f&NoPreferredStorageOrderBit)  res += " | NoPreferredStorageOrderBit";
 
  return res;
}
#endif
 
} // end namespace internal
 
 
template<typename ScalarA, typename ScalarB, typename BinaryOp=internal::scalar_product_op<ScalarA,ScalarB> >
struct ScalarBinaryOpTraits
#ifndef EIGEN_PARSED_BY_DOXYGEN
  // for backward compatibility, use the hints given by the (deprecated) internal::scalar_product_traits class.
  : internal::scalar_product_traits<ScalarA,ScalarB>
#endif // EIGEN_PARSED_BY_DOXYGEN
{};
 
template<typename T, typename BinaryOp>
struct ScalarBinaryOpTraits<T,T,BinaryOp>
{
  typedef T ReturnType;
};
 
template <typename T, typename BinaryOp>
struct ScalarBinaryOpTraits<T, typename NumTraits<std::enable_if_t<NumTraits<T>::IsComplex,T>>::Real, BinaryOp>
{
  typedef T ReturnType;
};
template <typename T, typename BinaryOp>
struct ScalarBinaryOpTraits<typename NumTraits<std::enable_if_t<NumTraits<T>::IsComplex,T>>::Real, T, BinaryOp>
{
  typedef T ReturnType;
};
 
// For Matrix * Permutation
template<typename T, typename BinaryOp>
struct ScalarBinaryOpTraits<T,void,BinaryOp>
{
  typedef T ReturnType;
};
 
// For Permutation * Matrix
template<typename T, typename BinaryOp>
struct ScalarBinaryOpTraits<void,T,BinaryOp>
{
  typedef T ReturnType;
};
 
// for Permutation*Permutation
template<typename BinaryOp>
struct ScalarBinaryOpTraits<void,void,BinaryOp>
{
  typedef void ReturnType;
};
 
// We require Lhs and Rhs to have "compatible" scalar types.
// It is tempting to always allow mixing different types but remember that this is often impossible in the vectorized paths.
// So allowing mixing different types gives very unexpected errors when enabling vectorization, when the user tries to
// add together a float matrix and a double matrix.
#define EIGEN_CHECK_BINARY_COMPATIBILIY(BINOP,LHS,RHS) \
  EIGEN_STATIC_ASSERT((Eigen::internal::has_ReturnType<ScalarBinaryOpTraits<LHS, RHS,BINOP> >::value), \
    YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY)
 
} // end namespace Eigen
 
#endif // EIGEN_XPRHELPER_H