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
 
// IWYU pragma: private
#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 };
};
 
// estimates the cost of lazily evaluating a generic functor by unwinding the expression
template <typename Xpr>
struct nested_functor_cost {
  static constexpr Index Cost = static_cast<Index>(functor_traits<Xpr>::Cost);
};
 
template <typename Scalar, int Rows, int Cols, int Options, int MaxRows, int MaxCols>
struct nested_functor_cost<Matrix<Scalar, Rows, Cols, Options, MaxRows, MaxCols>> {
  static constexpr Index Cost = 1;
};
 
template <typename Scalar, int Rows, int Cols, int Options, int MaxRows, int MaxCols>
struct nested_functor_cost<Array<Scalar, Rows, Cols, Options, MaxRows, MaxCols>> {
  static constexpr Index Cost = 1;
};
 
// TODO: assign a cost to the stride type?
template <typename PlainObjectType, int MapOptions, typename StrideType>
struct nested_functor_cost<Map<PlainObjectType, MapOptions, StrideType>> : nested_functor_cost<PlainObjectType> {};
 
template <typename Func, typename Xpr>
struct nested_functor_cost<CwiseUnaryOp<Func, Xpr>> {
  using XprCleaned = remove_all_t<Xpr>;
  using FuncCleaned = remove_all_t<Func>;
  static constexpr Index Cost = nested_functor_cost<FuncCleaned>::Cost + nested_functor_cost<XprCleaned>::Cost;
};
 
template <typename Func, typename Xpr>
struct nested_functor_cost<CwiseNullaryOp<Func, Xpr>> {
  using XprCleaned = remove_all_t<Xpr>;
  using FuncCleaned = remove_all_t<Func>;
  static constexpr Index Cost = nested_functor_cost<FuncCleaned>::Cost + nested_functor_cost<XprCleaned>::Cost;
};
 
template <typename Func, typename LhsXpr, typename RhsXpr>
struct nested_functor_cost<CwiseBinaryOp<Func, LhsXpr, RhsXpr>> {
  using LhsXprCleaned = remove_all_t<LhsXpr>;
  using RhsXprCleaned = remove_all_t<RhsXpr>;
  using FuncCleaned = remove_all_t<Func>;
  static constexpr Index Cost = nested_functor_cost<FuncCleaned>::Cost + nested_functor_cost<LhsXprCleaned>::Cost +
                                nested_functor_cost<RhsXprCleaned>::Cost;
};
 
template <typename Func, typename LhsXpr, typename MidXpr, typename RhsXpr>
struct nested_functor_cost<CwiseTernaryOp<Func, LhsXpr, MidXpr, RhsXpr>> {
  using LhsXprCleaned = remove_all_t<LhsXpr>;
  using MidXprCleaned = remove_all_t<MidXpr>;
  using RhsXprCleaned = remove_all_t<RhsXpr>;
  using FuncCleaned = remove_all_t<Func>;
  static constexpr Index Cost = nested_functor_cost<FuncCleaned>::Cost + nested_functor_cost<LhsXprCleaned>::Cost +
                                nested_functor_cost<MidXprCleaned>::Cost + nested_functor_cost<RhsXprCleaned>::Cost;
};
 
template <typename Xpr>
struct functor_cost {
  static constexpr Index Cost = plain_enum_max(nested_functor_cost<Xpr>::Cost, 1);
};
 
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 = int(NAsInteger) * int(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
 
template <typename XprType>
struct is_block_xpr : std::false_type {};
 
template <typename XprType, int BlockRows, int BlockCols, bool InnerPanel>
struct is_block_xpr<Block<XprType, BlockRows, BlockCols, InnerPanel>> : std::true_type {};
 
template <typename XprType, int BlockRows, int BlockCols, bool InnerPanel>
struct is_block_xpr<const Block<XprType, BlockRows, BlockCols, InnerPanel>> : std::true_type {};
 
// Helper utility for constructing non-recursive block expressions.
template <typename XprType>
struct block_xpr_helper {
  using BaseType = XprType;
 
  // For regular block expressions, simply forward along the InnerPanel argument,
  // which is set when calling row/column expressions.
  static constexpr bool is_inner_panel(bool inner_panel) { return inner_panel; }
 
  // Only enable non-const base function if XprType is not const (otherwise we get a duplicate definition).
  template <typename T = XprType, typename EnableIf = std::enable_if_t<!std::is_const<T>::value>>
  static EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE BaseType& base(XprType& xpr) {
    return xpr;
  }
  static EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE const BaseType& base(const XprType& xpr) { return xpr; }
  static constexpr EIGEN_ALWAYS_INLINE Index row(const XprType& /*xpr*/, Index r) { return r; }
  static constexpr EIGEN_ALWAYS_INLINE Index col(const XprType& /*xpr*/, Index c) { return c; }
};
 
template <typename XprType, int BlockRows, int BlockCols, bool InnerPanel>
struct block_xpr_helper<Block<XprType, BlockRows, BlockCols, InnerPanel>> {
  using BlockXprType = Block<XprType, BlockRows, BlockCols, InnerPanel>;
  // Recursive helper in case of explicit block-of-block expression.
  using NestedXprHelper = block_xpr_helper<XprType>;
  using BaseType = typename NestedXprHelper::BaseType;
 
  // For block-of-block expressions, we need to combine the InnerPannel trait
  // with that of the block subexpression.
  static constexpr bool is_inner_panel(bool inner_panel) { return InnerPanel && inner_panel; }
 
  // Only enable non-const base function if XprType is not const (otherwise we get a duplicates definition).
  template <typename T = XprType, typename EnableIf = std::enable_if_t<!std::is_const<T>::value>>
  static EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE BaseType& base(BlockXprType& xpr) {
    return NestedXprHelper::base(xpr.nestedExpression());
  }
  static EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE const BaseType& base(const BlockXprType& xpr) {
    return NestedXprHelper::base(xpr.nestedExpression());
  }
  static constexpr EIGEN_ALWAYS_INLINE Index row(const BlockXprType& xpr, Index r) {
    return xpr.startRow() + NestedXprHelper::row(xpr.nestedExpression(), r);
  }
  static constexpr EIGEN_ALWAYS_INLINE Index col(const BlockXprType& xpr, Index c) {
    return xpr.startCol() + NestedXprHelper::col(xpr.nestedExpression(), c);
  }
};
 
template <typename XprType, int BlockRows, int BlockCols, bool InnerPanel>
struct block_xpr_helper<const Block<XprType, BlockRows, BlockCols, InnerPanel>>
    : block_xpr_helper<Block<XprType, BlockRows, BlockCols, InnerPanel>> {};
 
template <typename XprType>
struct is_matrix_base_xpr : std::is_base_of<MatrixBase<remove_all_t<XprType>>, remove_all_t<XprType>> {};
 
template <typename XprType>
struct is_permutation_base_xpr : std::is_base_of<PermutationBase<remove_all_t<XprType>>, remove_all_t<XprType>> {};
 
}  // 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