magnum/src/Magnum/Math/RectangularMatrix.h

#ifndef Magnum_Math_RectangularMatrix_h
#define Magnum_Math_RectangularMatrix_h
/*
    This file is part of Magnum.

    Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
                2020, 2021, 2022, 2023 Vladimír Vondruš <mosra@centrum.cz>

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    copy of this software and associated documentation files (the "Software"),
    to deal in the Software without restriction, including without limitation
    the rights to use, copy, modify, merge, publish, distribute, sublicense,
    and/or sell copies of the Software, and to permit persons to whom the
    Software is furnished to do so, subject to the following conditions:

    The above copyright notice and this permission notice shall be included
    in all copies or substantial portions of the Software.

    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
    THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
    LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
    FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
    DEALINGS IN THE SOFTWARE.
*/

/** @file
 * @brief Class @ref Magnum::Math::RectangularMatrix, alias @ref Magnum::Math::Matrix2x3, @ref Magnum::Math::Matrix3x2, @ref Magnum::Math::Matrix2x4, @ref Magnum::Math::Matrix4x2, @ref Magnum::Math::Matrix3x4, @ref Magnum::Math::Matrix4x3
 */

/* std::declval() is said to be in <utility> but libstdc++, libc++ and MSVC STL
   all have it directly in <type_traits> because it just makes sense */
#include <type_traits>

#include "Magnum/Math/Vector.h"

namespace Magnum { namespace Math {

namespace Implementation {
    template<std::size_t, std::size_t, class, class> struct RectangularMatrixConverter;

    template<std::size_t cols, std::size_t rows, std::size_t otherCols, std::size_t otherRows, class T, std::size_t col, std::size_t ...row> constexpr Vector<rows, T> valueOrZeroVector(Containers::Implementation::Sequence<row...>, const RectangularMatrix<otherCols, otherRows, T>& other) {
        return {(col < otherCols && row < otherRows ? other[col][row] : T{0})...};
    }

    template<std::size_t cols, std::size_t rows, std::size_t otherCols, std::size_t otherRows, class T, std::size_t col> constexpr Vector<rows, T> valueOrZeroVector(const RectangularMatrix<otherCols, otherRows, T>& other) {
        return valueOrZeroVector<cols, rows, otherCols, otherRows, T, col>(typename Containers::Implementation::GenerateSequence<rows>::Type{}, other);
    }

    template<std::size_t cols, std::size_t rows, std::size_t otherCols, std::size_t otherRows, class T, std::size_t col, std::size_t ...row> constexpr Vector<rows, T> valueOrIdentityVector(Containers::Implementation::Sequence<row...>, const RectangularMatrix<otherCols, otherRows, T>& other, T value) {
        return {(col < otherCols && row < otherRows ? other[col][row] :
            col == row ? value : T{0})...};
    }

    template<std::size_t cols, std::size_t rows, std::size_t otherCols, std::size_t otherRows, class T, std::size_t col> constexpr Vector<rows, T> valueOrIdentityVector(const RectangularMatrix<otherCols, otherRows, T>& other, T value) {
        return valueOrIdentityVector<cols, rows, otherCols, otherRows, T, col>(typename Containers::Implementation::GenerateSequence<rows>::Type{}, other, value);
    }
}

/**
@brief Rectangular matrix
@tparam cols    Column count
@tparam rows    Row count
@tparam T       Underlying data type

See @ref matrix-vector for brief introduction. See also @ref Matrix (square),
@ref Matrix3, @ref Matrix4 and @ref Vector.

The data are stored in column-major order, to reflect that, all indices in
math formulas are in reverse order (i.e. @f$ \boldsymbol A_{ji} @f$ instead
of @f$ \boldsymbol A_{ij} @f$).
@see @ref Matrix2x1, @ref Matrix2x3, @ref Matrix2x4, @ref Matrix3x1,
    @ref Matrix3x2, @ref Matrix3x4, @ref Matrix4x1, @ref Matrix4x2,
    @ref Matrix4x3, @ref Magnum::Matrix2x1, @ref Magnum::Matrix2x1d,
    @ref Magnum::Matrix2x3, @ref Magnum::Matrix2x3d, @ref Magnum::Matrix2x3h,
    @ref Magnum::Matrix2x3b, @ref Magnum::Matrix2x3s, @ref Magnum::Matrix2x4,
    @ref Magnum::Matrix2x4d, @ref Magnum::Matrix2x4h, @ref Magnum::Matrix2x4b,
    @ref Magnum::Matrix2x4s, @ref Magnum::Matrix3x1, @ref Magnum::Matrix3x1d,
    @ref Magnum::Matrix3x2, @ref Magnum::Matrix3x2d, @ref Magnum::Matrix3x2h,
    @ref Magnum::Matrix3x2b, @ref Magnum::Matrix3x2s, @ref Magnum::Matrix3x4,
    @ref Magnum::Matrix3x4d, @ref Magnum::Matrix3x4h, @ref Magnum::Matrix3x4b,
    @ref Magnum::Matrix3x4s, @ref Magnum::Matrix4x1, @ref Magnum::Matrix4x1,
    @ref Magnum::Matrix4x1d, @ref Magnum::Matrix4x2, @ref Magnum::Matrix4x2d,
    @ref Magnum::Matrix4x2h, @ref Magnum::Matrix4x2b, @ref Magnum::Matrix4x2s,
    @ref Magnum::Matrix4x3, @ref Magnum::Matrix4x3d, @ref Magnum::Matrix4x3h,
    @ref Magnum::Matrix4x3b, @ref Magnum::Matrix4x3s
*/
template<std::size_t cols, std::size_t rows, class T> class RectangularMatrix {
    static_assert(cols != 0 && rows != 0, "RectangularMatrix cannot have zero elements");

    template<std::size_t, std::size_t, class> friend class RectangularMatrix;

    public:
        typedef T Type;         /**< @brief Underlying data type */

        enum: std::size_t {
            Cols = cols,        /**< Matrix column count */
            Rows = rows,        /**< Matrix row count */

            /**
             * Size of matrix diagonal
             * @see @ref fromDiagonal(), @ref diagonal()
             */
            DiagonalSize = (cols < rows ? cols : rows)
        };

        /**
         * @brief Matrix from an array
         * @return Reference to the data as if it was matrix, thus doesn't
         *      perform any copying.
         *
         * Use with caution, the function doesn't check whether the array is
         * long enough. If possible, prefer to use the
         * @ref RectangularMatrix(const T(&)[cols_][rows_]) constructor.
         */
        static RectangularMatrix<cols, rows, T>& from(T* data) {
            return *reinterpret_cast<RectangularMatrix<cols, rows, T>*>(data);
        }
        /** @overload */
        static const RectangularMatrix<cols, rows, T>& from(const T* data) {
            return *reinterpret_cast<const RectangularMatrix<cols, rows, T>*>(data);
        }

        /**
         * @brief Construct a matrix from a vector
         *
         * Rolls the vector into matrix, i.e. first `rows` elements of the
         * vector will make first column of resulting matrix.
         * @see @ref toVector()
         */
        static RectangularMatrix<cols, rows, T> fromVector(const Vector<cols*rows, T>& vector) {
            return *reinterpret_cast<const RectangularMatrix<cols, rows, T>*>(vector.data());
        }

        /**
         * @brief Construct a diagonal matrix
         *
         * @see @ref diagonal()
         */
        constexpr static RectangularMatrix<cols, rows, T> fromDiagonal(const Vector<DiagonalSize, T>& diagonal) noexcept {
            return RectangularMatrix(typename Containers::Implementation::GenerateSequence<cols>::Type{}, diagonal);
        }

        /**
         * @brief Default constructor
         *
         * Equivalent to @ref RectangularMatrix(ZeroInitT).
         */
        constexpr /*implicit*/ RectangularMatrix() noexcept: RectangularMatrix<cols, rows, T>{typename Containers::Implementation::GenerateSequence<cols>::Type{}, ZeroInit} {}

        /**
         * @brief Construct a zero-filled matrix
         *
         * @see @ref RectangularMatrix(IdentityInitT, T)
         */
        constexpr explicit RectangularMatrix(ZeroInitT) noexcept: RectangularMatrix<cols, rows, T>{typename Containers::Implementation::GenerateSequence<cols>::Type{}, ZeroInit} {}

        /**
         * @brief Construct an identity matrix
         * @m_since_latest
         *
         * For non-square matrices, the diagonal has @ref DiagonalSize
         * elements. The @p value allows you to specify a value on diagonal.
         * @see @ref RectangularMatrix(ZeroInitT), @ref fromDiagonal()
         */
        constexpr explicit RectangularMatrix(IdentityInitT, T value = T(1)) noexcept: RectangularMatrix<cols, rows, T>{typename Containers::Implementation::GenerateSequence<DiagonalSize>::Type{}, Vector<DiagonalSize, T>(value)} {}

        /** @brief Construct without initializing the contents */
        explicit RectangularMatrix(Magnum::NoInitT) noexcept: RectangularMatrix<cols, rows, T>{typename Containers::Implementation::GenerateSequence<cols>::Type{}, Magnum::NoInit} {}

        /** @brief Construct from column vectors */
        template<class ...U> constexpr /*implicit*/ RectangularMatrix(const Vector<rows, T>& first, const U&... next) noexcept: _data{first, next...} {
            static_assert(sizeof...(next)+1 == cols, "Improper number of arguments passed to RectangularMatrix constructor");
        }

        /** @brief Construct with one value for all components */
        constexpr explicit RectangularMatrix(T value) noexcept: RectangularMatrix{typename Containers::Implementation::GenerateSequence<cols>::Type{}, value} {}

        /**
         * @brief Construct from a fixed-size array
         * @m_since_latest
         *
         * Use @ref Vector::from(T*) "from(const T*)" to reinterpret an
         * arbitrary pointer to a matrix.
         */
        #if !defined(CORRADE_TARGET_GCC) || defined(CORRADE_TARGET_CLANG) || __GNUC__ >= 5
        template<std::size_t cols_, std::size_t rows_> constexpr explicit RectangularMatrix(const T(&data)[cols_][rows_]) noexcept: RectangularMatrix{typename Containers::Implementation::GenerateSequence<cols_>::Type{}, data} {
            static_assert(cols_ == cols && rows_ == rows, "wrong number of initializers");
        }
        #else
        /* GCC 4.8 isn't able to figure out the size on its own. Which means
           there we use the type-provided size and lose the check for element
           count, but at least it compiles. */
        constexpr explicit RectangularMatrix(const T(&data)[cols][rows]) noexcept: RectangularMatrix{typename Containers::Implementation::GenerateSequence<cols>::Type{}, data} {}
        #endif

        /**
         * @brief Construct from a matrix of a different type
         *
         * Performs only default casting on the values, no rounding or
         * anything else. Example usage:
         *
         * @snippet Math.cpp RectangularMatrix-conversion
         */
        template<class U> constexpr explicit RectangularMatrix(const RectangularMatrix<cols, rows, U>& other) noexcept: RectangularMatrix(typename Containers::Implementation::GenerateSequence<cols>::Type{}, other) {}

        /**
         * @brief Construct by slicing or expanding a matrix of different size, leaving the rest at zero
         * @m_since_latest
         *
         * If the other matrix has less columns or rows, the corresponding
         * vectors and components are set to zeros.
         */
        template<std::size_t otherCols, std::size_t otherRows> constexpr explicit RectangularMatrix(ZeroInitT, const RectangularMatrix<otherCols, otherRows, T>& other) noexcept: RectangularMatrix<cols, rows, T>{ZeroInit, typename Containers::Implementation::GenerateSequence<cols>::Type{}, other} {}

        /**
         * @brief Construct by slicing or expanding a matrix of different size, leaving the rest at identity
         * @m_since_latest
         *
         * If the other matrix has less columns or rows, the corresponding
         * vectors and components are set to either zeros or @p value on the
         * diagonal.
         */
        template<std::size_t otherCols, std::size_t otherRows> constexpr explicit RectangularMatrix(IdentityInitT, const RectangularMatrix<otherCols, otherRows, T>& other, T value = T(1)) noexcept: RectangularMatrix<cols, rows, T>{IdentityInit, typename Containers::Implementation::GenerateSequence<cols>::Type{}, other, value} {}

        /**
         * @brief Construct by slicing or expanding a matrix of different size
         * @m_since_latest
         *
         * Equivalent to @ref RectangularMatrix(ZeroInitT, const RectangularMatrix<otherCols, otherRows, T>&).
         * Note that this default is different from @ref Matrix, where it's
         * equivalent to the @ref IdentityInit variant instead.
         */
        template<std::size_t otherCols, std::size_t otherRows> constexpr explicit RectangularMatrix(const RectangularMatrix<otherCols, otherRows, T>& other) noexcept: RectangularMatrix<cols, rows, T>{ZeroInit, typename Containers::Implementation::GenerateSequence<cols>::Type{}, other} {}

        /** @brief Construct a matrix from external representation */
        template<class U, class = decltype(Implementation::RectangularMatrixConverter<cols, rows, T, U>::from(std::declval<U>()))> constexpr explicit RectangularMatrix(const U& other): RectangularMatrix(Implementation::RectangularMatrixConverter<cols, rows, T, U>::from(other)) {}

        /** @brief Convert the matrix to external representation */
        template<class U, class = decltype(Implementation::RectangularMatrixConverter<cols, rows, T, U>::to(std::declval<RectangularMatrix<cols, rows, T>>()))> constexpr explicit operator U() const {
            return Implementation::RectangularMatrixConverter<cols, rows, T, U>::to(*this);
        }

        /**
         * @brief Raw data
         *
         * Contrary to what Doxygen shows, returns reference to an
         * one-dimensional fixed-size array of @cpp cols*rows @ce elements,
         * i.e. @cpp T(&)[cols*rows] @ce.
         * @see @ref operator[]()
         * @todoc Fix once there's a possibility to patch the signature in a
         *      post-processing step (https://github.com/mosra/m.css/issues/56)
         */
        #ifdef DOXYGEN_GENERATING_OUTPUT
        T* data();
        const T* data() const; /**< @overload */
        #else
        auto data() -> T(&)[cols*rows] {
            return reinterpret_cast<T(&)[cols*rows]>(_data);
        }
        /* Can't really be constexpr anymore, the only other solution is having
           an union with `T _rawData[cols*rows]` and return that, but in a
           constexpr context it'd mean we'd have to initialize it instead of
           `_data` in the constructor ... and then operator[]() could not be
           constexpr anymore. I can't think of a practical use case where
           constexpr data() would be needed and operator[]() could not be used
           instead. Having a statically sized array returned is a far more
           useful property than constexpr, so that wins. */
        auto data() const -> const T(&)[cols*rows] {
            return reinterpret_cast<const T(&)[cols*rows]>(_data);
        }
        #endif

        /**
         * @brief Column at given position
         *
         * Particular elements can be accessed using @ref Vector::operator[](),
         * e.g.:
         *
         * @snippet Math.cpp RectangularMatrix-access
         *
         * @see @ref row(), @ref data()
         */
        Vector<rows, T>& operator[](std::size_t col) { return _data[col]; }
        /* returns const& so [][] operations are also constexpr */
        constexpr const Vector<rows, T>& operator[](std::size_t col) const { return _data[col]; } /**< @overload */

        /**
         * @brief Row at given position
         *
         * Consider using @ref transposed() when accessing rows frequently, as
         * this is slower than accessing columns due to the way the matrix is
         * stored.
         * @see @ref setRow(), @ref operator[]()
         */
        Vector<cols, T> row(std::size_t row) const;

        /**
         * @brief Set matrix row
         *
         * Consider using @ref transposed() when accessing rows frequently, as
         * this is slower than accessing columns due to the way the matrix is
         * stored.
         * @see @ref row(), @ref operator[]()
         */
        void setRow(std::size_t row, const Vector<cols, T>& data);

        /** @brief Equality comparison */
        bool operator==(const RectangularMatrix<cols, rows, T>& other) const {
            for(std::size_t i = 0; i != cols; ++i)
                if(_data[i] != other._data[i]) return false;

            return true;
        }

        /**
         * @brief Non-equality comparison
         *
         * @see @ref Vector::operator<(), @ref Vector::operator<=(),
         *      @ref Vector::operator>=(), @ref Vector::operator>()
         */
        bool operator!=(const RectangularMatrix<cols, rows, T>& other) const {
            return !operator==(other);
        }

        /**
         * @brief Component-wise less than
         *
         * Calls @ref Vector::operator<() on @ref toVector().
         */
        BitVector<cols*rows> operator<(const RectangularMatrix<cols, rows, T>& other) const {
            return toVector() < other.toVector();
        }

        /**
         * @brief Component-wise less than or equal
         *
         * Calls @ref Vector::operator<=() on @ref toVector().
         */
        BitVector<cols*rows> operator<=(const RectangularMatrix<cols, rows, T>& other) const {
            return toVector() <= other.toVector();
        }

        /**
         * @brief Component-wise greater than or equal
         *
         * Calls @ref Vector::operator>=() on @ref toVector().
         */
        BitVector<cols*rows> operator>=(const RectangularMatrix<cols, rows, T>& other) const {
            return toVector() >= other.toVector();
        }

        /**
         * @brief Component-wise greater than
         *
         * Calls @ref Vector::operator>() on @ref toVector().
         */
        BitVector<cols*rows> operator>(const RectangularMatrix<cols, rows, T>& other) const {
            return toVector() > other.toVector();
        }

        /**
         * @brief Promotion
         * @m_since_latest
         *
         * Returns the value as-is.
         */
        RectangularMatrix<cols, rows, T> operator+() const { return *this; }

        /**
         * @brief Negated matrix
         *
         * The computation is done column-wise. @f[
         *      \boldsymbol B_j = -\boldsymbol A_j
         * @f]
         */
        RectangularMatrix<cols, rows, T> operator-() const;

        /**
         * @brief Add and assign a matrix
         *
         * The computation is done column-wise in-place. @f[
         *      \boldsymbol A_j = \boldsymbol A_j + \boldsymbol B_j
         * @f]
         */
        RectangularMatrix<cols, rows, T>& operator+=(const RectangularMatrix<cols, rows, T>& other) {
            for(std::size_t i = 0; i != cols; ++i)
                _data[i] += other._data[i];

            return *this;
        }

        /**
         * @brief Add a matrix
         *
         * @see @ref operator+=()
         */
        RectangularMatrix<cols, rows, T> operator+(const RectangularMatrix<cols, rows, T>& other) const {
            return RectangularMatrix<cols, rows, T>(*this)+=other;
        }

        /**
         * @brief Subtract and assign a matrix
         *
         * The computation is done column-wise in-place. @f[
         *      \boldsymbol A_j = \boldsymbol A_j - \boldsymbol B_j
         * @f]
         */
        RectangularMatrix<cols, rows, T>& operator-=(const RectangularMatrix<cols, rows, T>& other) {
            for(std::size_t i = 0; i != cols; ++i)
                _data[i] -= other._data[i];

            return *this;
        }

        /**
         * @brief Subtract a matrix
         *
         * @see @ref operator-=()
         */
        RectangularMatrix<cols, rows, T> operator-(const RectangularMatrix<cols, rows, T>& other) const {
            return RectangularMatrix<cols, rows, T>(*this)-=other;
        }

        /**
         * @brief Multiply with a scalar and assign
         *
         * The computation is done column-wise in-place. @f[
         *      \boldsymbol A_j = a \boldsymbol A_j
         * @f]
         */
        RectangularMatrix<cols, rows, T>& operator*=(T scalar) {
            for(std::size_t i = 0; i != cols; ++i)
                _data[i] *= scalar;

            return *this;
        }

        /**
         * @brief Multiply with a scalar
         *
         * @see @ref operator*=(T), @ref operator*(T, const RectangularMatrix<cols, rows, T>&)
         */
        RectangularMatrix<cols, rows, T> operator*(T scalar) const {
            return RectangularMatrix<cols, rows, T>(*this) *= scalar;
        }

        /**
         * @brief Multiply a scalar with a matrix
         *
         * Same as @ref operator*(T) const.
         */
        friend RectangularMatrix<cols, rows, T> operator*(
            #ifdef DOXYGEN_GENERATING_OUTPUT
            T
            #else
            typename std::common_type<T>::type
            #endif
            scalar, const RectangularMatrix<cols, rows, T>& matrix)
        {
            return matrix*scalar;
        }

        /**
         * @brief Divide with a scalar and assign
         *
         * The computation is done column-wise in-place. @f[
         *      \boldsymbol A_j = \frac{\boldsymbol A_j} a
         * @f]
         */
        RectangularMatrix<cols, rows, T>& operator/=(T scalar) {
            for(std::size_t i = 0; i != cols; ++i)
                _data[i] /= scalar;

            return *this;
        }

        /**
         * @brief Divide with a scalar
         *
         * @see @ref operator/=(T),
         *      @ref operator/(T, const RectangularMatrix<cols, rows, T>&)
         */
        RectangularMatrix<cols, rows, T> operator/(T scalar) const {
            return RectangularMatrix<cols, rows, T>(*this) /= scalar;
        }

        /**
         * @brief Divide a matrix with a scalar and invert
         *
         * The computation is done column-wise. @f[
         *      \boldsymbol B_j = \frac a {\boldsymbol A_j}
         * @f]
         * @see @ref operator/(T) const
         */
        friend RectangularMatrix<cols, rows, T> operator/(
            #ifdef DOXYGEN_GENERATING_OUTPUT
            T
            #else
            typename std::common_type<T>::type
            #endif
            scalar, const RectangularMatrix<cols, rows, T>& matrix)
        {
            RectangularMatrix<cols, rows, T> out{Magnum::NoInit};

            for(std::size_t i = 0; i != cols; ++i)
                out._data[i] = scalar/matrix._data[i];

            return out;
        }

        /**
         * @brief Multiply a matrix
         *
         * @f[
         *      (\boldsymbol {AB})_{ji} = \sum_{k=0}^{m-1} \boldsymbol A_{ki} \boldsymbol B_{jk}
         * @f]
         * @m_keyword{outerProduct(),GLSL outerProduct(),}
         */
        template<std::size_t size> RectangularMatrix<size, rows, T> operator*(const RectangularMatrix<size, cols, T>& other) const;

        /**
         * @brief Multiply a vector
         *
         * Internally the same as multiplying with a one-column matrix, but
         * returns a vector instead of a one-column matrix. @f[
         *      (\boldsymbol {Aa})_i = \sum_{k=0}^{m-1} \boldsymbol A_{ki} \boldsymbol a_k
         * @f]
         *
         * Vectors are treated as columns. An equivalent operation is
         * multiplying a transposed matrix with an one-row matrix from the left
         * side instead:
         *
         * @snippet Math.cpp RectangularMatrix-multiply-vector
         */
        Vector<rows, T> operator*(const Vector<cols, T>& other) const {
            return operator*(RectangularMatrix<1, cols, T>(other))[0];
        }

        /**
         * @brief Transposed matrix
         *
         * @f[
         *      \boldsymbol{A}^T_ij = \boldsymbol{A}_ji
         * @f]
         * @see @ref row(), @ref flippedCols(), @ref flippedRows()
         * @m_keyword{transpose(),GLSL transpose(),}
         */
        RectangularMatrix<rows, cols, T> transposed() const;

        /**
         * @brief Matrix with flipped cols
         *
         * The order of columns is reversed.
         * @see @ref transposed(), @ref flippedRows(), @ref Vector::flipped()
         */
        constexpr RectangularMatrix<cols, rows, T> flippedCols() const {
            return flippedColsInternal(typename Containers::Implementation::GenerateSequence<cols>::Type{});
        }

        /**
         * @brief Matrix with flipped rows
         *
         * The order of rows is reversed.
         * @see @ref transposed(), @ref flippedCols(), @ref Vector::flipped()
         */
        constexpr RectangularMatrix<cols, rows, T> flippedRows() const {
            return flippedRowsInternal(typename Containers::Implementation::GenerateSequence<cols>::Type{});
        }

        /**
         * @brief Values on diagonal
         *
         * @see @ref fromDiagonal()
         */
        constexpr Vector<DiagonalSize, T> diagonal() const {
            /* NVCC (from CUDA) has problems compiling this function under
               Windows when there's a separate definition due to DiagonalSize
               (see https://github.com/mosra/magnum/issues/345 for details) */
            return diagonalInternal(typename Containers::Implementation::GenerateSequence<DiagonalSize>::Type{});
        }

        /**
         * @brief Convert matrix to vector
         *
         * Returns the matrix unrolled into one large vector, i.e. first column
         * of the matrix will make first `rows` elements of resulting vector.
         * Useful for performing vector operations with the matrix (e.g.
         * summing the elements etc.).
         * @see @ref fromVector()
         */
        Vector<rows*cols, T> toVector() const {
            return *reinterpret_cast<const Vector<rows*cols, T>*>(data());
        }

    private:
        /* These two needed to access _data to speed up debug builds,
           Matrix::ij() needs access to different Matrix sizes */
        template<std::size_t, class> friend class Matrix;
        template<std::size_t, class> friend struct Implementation::MatrixDeterminant;

        /* Implementation for RectangularMatrix<cols, rows, T>::RectangularMatrix(T) and Matrix<size, T>(T) */
        /* MSVC 2015 can't handle {} here */
        template<std::size_t ...sequence> constexpr explicit RectangularMatrix(Containers::Implementation::Sequence<sequence...>, T value) noexcept: _data{Vector<rows, T>((static_cast<void>(sequence), value))...} {}

        /* Implementation for RectangularMatrix<cols, rows, T>::fromDiagonal() and RectangularMatrix<cols, rows, T>(IdentityInitT, T) */
        template<std::size_t ...sequence> constexpr explicit RectangularMatrix(Containers::Implementation::Sequence<sequence...>, const Vector<DiagonalSize, T>& diagonal);

        /* Implementation for RectangularMatrix<cols, rows, T>::RectangularMatrix(const T(&data)[cols_][rows_]).
           Can't use Vector<rows, T>{} here because MSVC 2015 chokes on it. */
        template<std::size_t rows_, std::size_t ...sequence> constexpr explicit RectangularMatrix(Containers::Implementation::Sequence<sequence...>, const T(&data)[sizeof...(sequence)][rows_]) noexcept: _data{Vector<rows, T>(data[sequence])...} {}

        /* Implementation for RectangularMatrix<cols, rows, T>::RectangularMatrix(const RectangularMatrix<cols, rows, U>&) */
        template<class U, std::size_t ...sequence> constexpr explicit RectangularMatrix(Containers::Implementation::Sequence<sequence...>, const RectangularMatrix<cols, rows, U>& matrix) noexcept: _data{Vector<rows, T>(matrix[sequence])...} {}

        /* Implementation for RectangularMatrix<cols, rows, T>::RectangularMatrix(ZeroInitT, const RectangularMatrix<otherCols, otherRows, T>&) */
        template<std::size_t otherCols, std::size_t otherRows, std::size_t ...col> constexpr explicit RectangularMatrix(ZeroInitT, Containers::Implementation::Sequence<col...>, const RectangularMatrix<otherCols, otherRows, T>& other) noexcept: RectangularMatrix<cols, rows, T>{Implementation::valueOrZeroVector<cols, rows, otherCols, otherRows, T, col>(other)...} {}

        /* Implementation for RectangularMatrix<cols, rows, T>::RectangularMatrix(IdentityInitT, const RectangularMatrix<otherCols, otherRows, T>&) */
        template<std::size_t otherCols, std::size_t otherRows, std::size_t ...col> constexpr explicit RectangularMatrix(IdentityInitT, Containers::Implementation::Sequence<col...>, const RectangularMatrix<otherCols, otherRows, T>& other, T value) noexcept: RectangularMatrix<cols, rows, T>{Implementation::valueOrIdentityVector<cols, rows, otherCols, otherRows, T, col>(other, value)...} {}

        /* Implementation for RectangularMatrix<cols, rows, T>::RectangularMatrix(ZeroInitT) and RectangularMatrix<cols, rows, T>::RectangularMatrix(NoInitT) */
        /* MSVC 2015 can't handle {} here */
        template<class U, std::size_t ...sequence> constexpr explicit RectangularMatrix(Containers::Implementation::Sequence<sequence...>, U) noexcept: _data{Vector<rows, T>((static_cast<void>(sequence), U{typename U::Init{}}))...} {}

        template<std::size_t ...sequence> constexpr RectangularMatrix<cols, rows, T> flippedColsInternal(Containers::Implementation::Sequence<sequence...>) const {
            return {_data[cols - 1 - sequence]...};
        }

        template<std::size_t ...sequence> constexpr RectangularMatrix<cols, rows, T> flippedRowsInternal(Containers::Implementation::Sequence<sequence...>) const {
            return {_data[sequence].flipped()...};
        }

        template<std::size_t ...sequence> constexpr Vector<DiagonalSize, T> diagonalInternal(Containers::Implementation::Sequence<sequence...>) const;

        Vector<rows, T> _data[cols];
};

/**
@brief Matrix with 2 columns and 1 row
@m_since_latest

Convenience alternative to @cpp RectangularMatrix<2, 1, T> @ce. See
@ref RectangularMatrix for more information. There's no 1-column 2-row matrix typedef, use @ref Vector2 instead.
@see @ref Magnum::Matrix2x1, @ref Magnum::Matrix2x1d
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix2x1 = RectangularMatrix<2, 1, T>;
#endif

/**
@brief Matrix with 2 columns and 3 rows

Convenience alternative to @cpp RectangularMatrix<2, 3, T> @ce. See
@ref RectangularMatrix for more information.
@see @ref Magnum::Matrix2x3, @ref Magnum::Matrix2x3d, @ref Magnum::Matrix2x3h,
    @ref Magnum::Matrix2x3b, @ref Magnum::Matrix2x3s
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix2x3 = RectangularMatrix<2, 3, T>;
#endif

/**
@brief Matrix with 2 columns and 4 rows

Convenience alternative to @cpp RectangularMatrix<2, 4, T> @ce. See
@ref RectangularMatrix for more information.
@see @ref Magnum::Matrix2x4, @ref Magnum::Matrix2x4d, @ref Magnum::Matrix2x4h,
    @ref Magnum::Matrix2x4b, @ref Magnum::Matrix2x4s
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix2x4 = RectangularMatrix<2, 4, T>;
#endif

/**
@brief Matrix with 3 columns and 1 row
@m_since_latest

Convenience alternative to @cpp RectangularMatrix<3, 1, T> @ce. See
@ref RectangularMatrix for more information. There's no 1-column 3-row matrix typedef, use @ref Vector3 instead.
@see @ref Magnum::Matrix3x1, @ref Magnum::Matrix3x1d
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix3x1 = RectangularMatrix<3, 1, T>;
#endif

/**
@brief Matrix with 3 columns and 2 rows

Convenience alternative to @cpp RectangularMatrix<3, 2, T> @ce. See
@ref RectangularMatrix for more information.
@see @ref Magnum::Matrix3x2, @ref Magnum::Matrix3x2d, @ref Magnum::Matrix3x2h,
    @ref Magnum::Matrix3x2b, @ref Magnum::Matrix3x2s
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix3x2 = RectangularMatrix<3, 2, T>;
#endif

/**
@brief Matrix with 3 columns and 4 rows

Convenience alternative to @cpp RectangularMatrix<3, 4, T> @ce. See
@ref RectangularMatrix for more information.
@see @ref Magnum::Matrix3x4, @ref Magnum::Matrix3x4d, @ref Magnum::Matrix3x4h,
    @ref Magnum::Matrix3x4b, @ref Magnum::Matrix3x4s
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix3x4 = RectangularMatrix<3, 4, T>;
#endif

/**
@brief Matrix with 4 columns and 1 row
@m_since_latest

Convenience alternative to @cpp RectangularMatrix<4, 1, T> @ce. See
@ref RectangularMatrix for more information. There's no 1-column 4-row matrix typedef, use @ref Vector4 instead.
@see @ref Magnum::Matrix4x1, @ref Magnum::Matrix4x1d
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix4x1 = RectangularMatrix<4, 1, T>;
#endif

/**
@brief Matrix with 4 columns and 2 rows

Convenience alternative to @cpp RectangularMatrix<4, 2, T> @ce. See
@ref RectangularMatrix for more information.
@see @ref Magnum::Matrix4x2, @ref Magnum::Matrix4x2d, @ref Magnum::Matrix4x2h,
    @ref Magnum::Matrix4x2b, @ref Magnum::Matrix4x2s
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix4x2 = RectangularMatrix<4, 2, T>;
#endif

/**
@brief Matrix with 4 columns and 3 rows

Convenience alternative to @cpp RectangularMatrix<4, 3, T> @ce. See
@ref RectangularMatrix for more information.
@see @ref Magnum::Matrix4x3, @ref Magnum::Matrix4x3d, @ref Magnum::Matrix4x3h,
    @ref Magnum::Matrix4x3b, @ref Magnum::Matrix4x3s
*/
#ifndef CORRADE_MSVC2015_COMPATIBILITY /* Multiple definitions still broken */
template<class T> using Matrix4x3 = RectangularMatrix<4, 3, T>;
#endif

/** @relates RectangularMatrix
@brief Multiply a vector with a rectangular matrix

Internally the same as multiplying one-column matrix with one-row matrix. @f[
    (\boldsymbol {aA})_{ji} = \boldsymbol a_i \boldsymbol A_j
@f]
@see @ref RectangularMatrix::operator*(const RectangularMatrix<size, cols, T>&) const
*/
template<std::size_t size, std::size_t cols, class T> inline RectangularMatrix<cols, size, T> operator*(const Vector<size, T>& vector, const RectangularMatrix<cols, 1, T>& matrix) {
    return RectangularMatrix<1, size, T>(vector)*matrix;
}

#ifndef CORRADE_SINGLES_NO_DEBUG
/** @debugoperator{RectangularMatrix} */
template<std::size_t cols, std::size_t rows, class T> Debug& operator<<(Debug& debug, const Magnum::Math::RectangularMatrix<cols, rows, T>& value) {
    /** @todo might make sense to propagate the flags also, for hex value
        printing etc */
    const bool packed = debug.immediateFlags() >= Debug::Flag::Packed;
    debug << (packed ? "{" : "Matrix(") << Debug::nospace;
    for(std::size_t row = 0; row != rows; ++row) {
        if(row != 0) debug << Debug::nospace << (packed ? ",\n" : ",\n      ");
        for(std::size_t col = 0; col != cols; ++col) {
            if(col != 0) debug << Debug::nospace << ",";
            debug << value[col][row];
        }
    }
    return debug << Debug::nospace << (packed ? "}" : ")");
}

/* Explicit instantiation for commonly used types */
#ifndef DOXYGEN_GENERATING_OUTPUT
/* Square matrices */
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<2, 2, Float>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<3, 3, Float>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<4, 4, Float>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<2, 2, Double>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<3, 3, Double>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<4, 4, Double>&);

/* Rectangular matrices */
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<2, 3, Float>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<3, 2, Float>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<2, 4, Float>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<4, 2, Float>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<3, 4, Float>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<4, 3, Float>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<2, 3, Double>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<3, 2, Double>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<2, 4, Double>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<4, 2, Double>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<3, 4, Double>&);
extern template MAGNUM_EXPORT Debug& operator<<(Debug&, const RectangularMatrix<4, 3, Double>&);
#endif
#endif

#ifndef DOXYGEN_GENERATING_OUTPUT
#define MAGNUM_RECTANGULARMATRIX_SUBCLASS_IMPLEMENTATION(cols, rows, ...)   \
    static __VA_ARGS__& from(T* data) {                                     \
        return *reinterpret_cast<__VA_ARGS__*>(data);                       \
    }                                                                       \
    static const __VA_ARGS__& from(const T* data) {                         \
        return *reinterpret_cast<const __VA_ARGS__*>(data);                 \
    }                                                                       \
    constexpr static __VA_ARGS__ fromDiagonal(const Vector<Math::RectangularMatrix<cols, rows, T>::DiagonalSize, T>& diagonal) { \
        return Math::RectangularMatrix<cols, rows, T>::fromDiagonal(diagonal); \
    }                                                                       \
                                                                            \
    __VA_ARGS__ operator+() const {                                         \
        return Math::RectangularMatrix<cols, rows, T>::operator+();         \
    }                                                                       \
    __VA_ARGS__ operator-() const {                                         \
        return Math::RectangularMatrix<cols, rows, T>::operator-();         \
    }                                                                       \
    __VA_ARGS__& operator+=(const Math::RectangularMatrix<cols, rows, T>& other) { \
        Math::RectangularMatrix<cols, rows, T>::operator+=(other);          \
        return *this;                                                       \
    }                                                                       \
    __VA_ARGS__ operator+(const Math::RectangularMatrix<cols, rows, T>& other) const { \
        return Math::RectangularMatrix<cols, rows, T>::operator+(other);    \
    }                                                                       \
    __VA_ARGS__& operator-=(const Math::RectangularMatrix<cols, rows, T>& other) { \
        Math::RectangularMatrix<cols, rows, T>::operator-=(other);          \
        return *this;                                                       \
    }                                                                       \
    __VA_ARGS__ operator-(const Math::RectangularMatrix<cols, rows, T>& other) const { \
        return Math::RectangularMatrix<cols, rows, T>::operator-(other);    \
    }                                                                       \
    __VA_ARGS__& operator*=(T number) {                                     \
        Math::RectangularMatrix<cols, rows, T>::operator*=(number);         \
        return *this;                                                       \
    }                                                                       \
    __VA_ARGS__ operator*(T number) const {                                 \
        return Math::RectangularMatrix<cols, rows, T>::operator*(number);   \
    }                                                                       \
    __VA_ARGS__& operator/=(T number) {                                     \
        Math::RectangularMatrix<cols, rows, T>::operator/=(number);         \
        return *this;                                                       \
    }                                                                       \
    __VA_ARGS__ operator/(T number) const {                                 \
        return Math::RectangularMatrix<cols, rows, T>::operator/(number);   \
    }                                                                       \
    constexpr __VA_ARGS__ flippedCols() const {                             \
        return Math::RectangularMatrix<cols, rows, T>::flippedCols();       \
    }                                                                       \
    constexpr __VA_ARGS__ flippedRows() const {                             \
        return Math::RectangularMatrix<cols, rows, T>::flippedRows();       \
    }                                                                       \

/* Unlike with Vector, these are kept non-member and non-friend as it'd mean
   too many macro combinations otherwise */

#define MAGNUM_MATRIX_OPERATOR_IMPLEMENTATION(...)                          \
    template<std::size_t size, class T> inline __VA_ARGS__ operator*(typename std::common_type<T>::type number, const __VA_ARGS__& matrix) { \
        return number*static_cast<const Math::RectangularMatrix<size, size, T>&>(matrix); \
    }                                                                       \
    template<std::size_t size, class T> inline __VA_ARGS__ operator/(typename std::common_type<T>::type number, const __VA_ARGS__& matrix) { \
        return number/static_cast<const Math::RectangularMatrix<size, size, T>&>(matrix); \
    }                                                                       \
    template<std::size_t size, class T> inline __VA_ARGS__ operator*(const Vector<size, T>& vector, const RectangularMatrix<size, 1, T>& matrix) { \
        return Math::RectangularMatrix<1, size, T>(vector)*matrix;          \
    }

#define MAGNUM_MATRIXn_OPERATOR_IMPLEMENTATION(size, Type)                  \
    template<class T> inline Type<T> operator*(typename std::common_type<T>::type number, const Type<T>& matrix) { \
        return number*static_cast<const Math::RectangularMatrix<size, size, T>&>(matrix); \
    }                                                                       \
    template<class T> inline Type<T> operator/(typename std::common_type<T>::type number, const Type<T>& matrix) { \
        return number/static_cast<const Math::RectangularMatrix<size, size, T>&>(matrix); \
    }                                                                       \
    template<class T> inline Type<T> operator*(const Vector<size, T>& vector, const RectangularMatrix<size, 1, T>& matrix) { \
        return Math::RectangularMatrix<1, size, T>(vector)*matrix;          \
    }
#endif

namespace Implementation {
    template<std::size_t rows, std::size_t i, class T, std::size_t ...sequence> constexpr Vector<rows, T> diagonalMatrixColumn2(Containers::Implementation::Sequence<sequence...>, const T& number) {
        return {(sequence == i ? number : T(0))...};
    }
    template<std::size_t rows, std::size_t i, class T> constexpr Vector<rows, T> diagonalMatrixColumn(const T& number) {
        return diagonalMatrixColumn2<rows, i, T>(typename Containers::Implementation::GenerateSequence<rows>::Type{}, number);
    }
}

template<std::size_t cols, std::size_t rows, class T> template<std::size_t ...sequence> constexpr RectangularMatrix<cols, rows, T>::RectangularMatrix(Containers::Implementation::Sequence<sequence...>, const Vector<DiagonalSize, T>& diagonal): _data{Implementation::diagonalMatrixColumn<rows, sequence>(sequence < DiagonalSize ? diagonal[sequence] : T{})...} {}

template<std::size_t cols, std::size_t rows, class T> inline Vector<cols, T> RectangularMatrix<cols, rows, T>::row(std::size_t row) const {
    Vector<cols, T> out;

    /* Using ._data[] instead of [] to avoid function call indirection
       on debug builds (saves a lot, yet doesn't obfuscate too much) */
    for(std::size_t i = 0; i != cols; ++i)
        out[i] = _data[i]._data[row];

    return out;
}

template<std::size_t cols, std::size_t rows, class T> inline void RectangularMatrix<cols, rows, T>::setRow(std::size_t row, const Vector<cols, T>& data) {
    /* Using ._data[] instead of [] to avoid function call indirection
       on debug builds (saves a lot, yet doesn't obfuscate too much) */
    for(std::size_t i = 0; i != cols; ++i)
        _data[i]._data[row] = data._data[i];
}

template<std::size_t cols, std::size_t rows, class T> inline RectangularMatrix<cols, rows, T> RectangularMatrix<cols, rows, T>::operator-() const {
    RectangularMatrix<cols, rows, T> out;

    for(std::size_t i = 0; i != cols; ++i)
        out._data[i] = -_data[i];

    return out;
}

template<std::size_t cols, std::size_t rows, class T> template<std::size_t size> inline RectangularMatrix<size, rows, T> RectangularMatrix<cols, rows, T>::operator*(const RectangularMatrix<size, cols, T>& other) const {
    RectangularMatrix<size, rows, T> out{ZeroInit};

    /* Using ._data[] instead of [] to avoid function call indirection
       on debug builds (saves a lot, yet doesn't obfuscate too much) */
    for(std::size_t col = 0; col != size; ++col)
        for(std::size_t row = 0; row != rows; ++row)
            for(std::size_t pos = 0; pos != cols; ++pos)
                out._data[col]._data[row] += _data[pos]._data[row]*other._data[col]._data[pos];

    return out;
}

template<std::size_t cols, std::size_t rows, class T> inline RectangularMatrix<rows, cols, T> RectangularMatrix<cols, rows, T>::transposed() const {
    RectangularMatrix<rows, cols, T> out{Magnum::NoInit};

    /* Using ._data[] instead of [] to avoid function call indirection
       on debug builds (saves a lot, yet doesn't obfuscate too much) */
    for(std::size_t col = 0; col != cols; ++col)
        for(std::size_t row = 0; row != rows; ++row)
            out._data[row]._data[col] = _data[col]._data[row];

    return out;
}

#ifndef DOXYGEN_GENERATING_OUTPUT
template<std::size_t cols, std::size_t rows, class T> template<std::size_t ...sequence> constexpr auto RectangularMatrix<cols, rows, T>::diagonalInternal(Containers::Implementation::Sequence<sequence...>) const -> Vector<DiagonalSize, T> {
    return {_data[sequence][sequence]...};
}
#endif

#ifndef MAGNUM_NO_MATH_STRICT_WEAK_ORDERING
namespace Implementation {

template<std::size_t cols, std::size_t rows, class T> struct StrictWeakOrdering<RectangularMatrix<cols, rows, T>> {
    bool operator()(const RectangularMatrix<cols, rows, T>& a, const RectangularMatrix<cols, rows, T>& b) const {
        StrictWeakOrdering<Vector<rows, T>> o;
        for(std::size_t i = 0; i < cols; ++i) {
            if(o(a[i], b[i]))
                return true;
            if(o(b[i], a[i]))
                return false;
        }

        return false; /* a and b are equivalent */
    }
};

}
#endif

}}

#endif