SquareMatrix4.h

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/*
 * Copyright (c) Meta Platforms, Inc. and affiliates.
 *
 * This source code is licensed under the MIT license found in the
 * LICENSE file in the root directory of this source tree.
 */
 
#ifndef META_OCEAN_MATH_SQUARE_MATRIX_4_H
#define META_OCEAN_MATH_SQUARE_MATRIX_4_H
 
#include "ocean/math/Math.h"
#include "ocean/math/Equation.h"
#include "ocean/math/Vector2.h"
#include "ocean/math/Vector3.h"
#include "ocean/math/Vector4.h"
 
#include <vector>
 
namespace Ocean
{
 
// Forward declaration.
template <typename T> class AnyCameraT;
 
// Forward declaration.
template <typename T> class HomogenousMatrixT4;
 
// Forward declaration.
template <typename T> class SquareMatrixT3;
 
// Forward declaration.
template <typename T> class SquareMatrixT4;
 
/**
 * Definition of the SquareMatrix4 object, depending on the OCEAN_MATH_USE_SINGLE_PRECISION either with single or double precision float data type.
 * @see SquareMatrixT4
 * @ingroup math
 */
using SquareMatrix4 = SquareMatrixT4<Scalar>;
 
/**
 * Instantiation of the SquareMatrixT4 template class using a double precision float data type.
 * @see SquareMatrixT4
 * @ingroup math
 */
using SquareMatrixD4 = SquareMatrixT4<double>;
 
/**
 * Instantiation of the SquareMatrixT4 template class using a double precision float data type.
 * @see SquareMatrixT4
 * @ingroup math
 */
using SquareMatrixF4 = SquareMatrixT4<float>;
 
/**
 * Definition of a typename alias for vectors with SquareMatrixT4 objects.
 * @see SquareMatrixT4
 * @ingroup math
 */
template <typename T>
using SquareMatricesT4 = std::vector<SquareMatrixT4<T>>;
 
/**
 * Definition of a vector holding SquareMatrix4 objects.
 * @see SquareMatrix4
 * @ingroup math
 */
using SquareMatrices4 = std::vector<SquareMatrix4>;
 
/**
 * This class implements a 4x4 square matrix.
 * The values are stored in a column aligned order with indices:
 * <pre>
 * | 0  4  8  12 |
 * | 1  5  9  13 |
 * | 2  6  10 14 |
 * | 3  7  11 15 |
 * </pre>
 * @tparam T Data type of matrix elements
 * @see SquareMatrix4, SquareMatrixF4, SquareMatrixD4.
 * @ingroup math
 */
template <typename T>
class SquareMatrixT4
{
    template <typename U> friend class SquareMatrixT4;
 
    public:
 
        /**
         * Definition of the used data type.
         */
        using Type = T;
 
    public:
 
        /**
         * Creates a new SquareMatrixT4 object with undefined elements.
         * Beware: This matrix is neither a zero nor an entity matrix!
         */
        inline SquareMatrixT4();
 
        /**
         * Copy constructor.
         * @param matrix The matrix to copy
         */
        SquareMatrixT4(const SquareMatrixT4<T>& matrix) = default;
 
        /**
         * Copy constructor for a matrix with difference element data type than T.
         * @param matrix The matrix to copy
         * @tparam U The element data type of the second matrix
         */
        template <typename U>
        inline explicit SquareMatrixT4(const SquareMatrixT4<U>& matrix);
 
        /**
         * Creates a new SquareMatrixT4 object.
         * @param setToIdentity Determines whether a entity matrix will be created
         */
        explicit SquareMatrixT4(const bool setToIdentity);
 
        /**
         * Creates a new SquareMatrixT4 object by an array of at least sixteen elements of float type U.
         * @param arrayValues The sixteen matrix elements defining the new matrix, must be valid
         * @tparam U The floating point type of the given elements
         */
        template <typename U>
        explicit SquareMatrixT4(const U* arrayValues);
 
        /**
         * Creates a new SquareMatrixT4 object by an array of at least sixteen elements.
         * @param arrayValues The sixteen matrix elements defining the new matrix, must be valid
         */
        explicit SquareMatrixT4(const T* arrayValues);
 
        /**
         * Creates a new SquareMatrixT4 object by an array of at least sixteen elements of float type U.
         * @param arrayValues The sixteen matrix elements defining the new matrix, must be valid
         * @param valuesRowAligned True, if the given values are stored in a row aligned order; False, if the values are stored in a column aligned order (which is the default case for this matrix)
         * @tparam U The floating point type of the given elements
         */
        template <typename U>
        SquareMatrixT4(const U* arrayValues, const bool valuesRowAligned);
 
        /**
         * Creates a new SquareMatrixT4 object by an array of at least sixteen elements.
         * @param valuesRowAligned True, if the given values are stored in a row aligned order; False, if the values are stored in a column aligned order (which is the default case for this matrix)
         * @param arrayValues The sixteen matrix elements defining the new matrix, must be valid
         */
        SquareMatrixT4(const T* arrayValues, const bool valuesRowAligned);
 
        /**
         * Creates a new SquareMatrixT4 object by given transformation matrix.
         * @param transformation The transformation matrix to copy
         */
        explicit SquareMatrixT4(const HomogenousMatrixT4<T>& transformation);
 
        /**
         * Creates a new SquareMatrixT4 object by given 3x3 sub matrix.
         * The lower and right elements of the 4x4 square matrix are set to zero.
         * @param subMatrix 3x3 subMatrix defining the upper left elements of the new matrix
         */
        explicit SquareMatrixT4(const SquareMatrixT3<T>& subMatrix);
 
        /**
         * Creates a new SquareMatrixT4 object by a given diagonal vector.
         * @param diagonal The diagonal vector for the new matrix
         */
        explicit SquareMatrixT4(const VectorT4<T>& diagonal);
 
        /**
         * Returns the transposed of this matrix.
         * @return Transposed matrix
         */
        [[nodiscard]] SquareMatrixT4<T> transposed() const;
 
        /**
         * Transposes the matrix.
         */
        void transpose();
 
        /**
         * Returns the inverted matrix of this matrix.
         * This matrix must not be singular.<br>
         * Beware: This function does not throw an exception if the matrix cannot be inverted.<br>
         * Thus, ensure that the matrix is invertible before calling this function.<br>
         * Even better: avoid the usage of this function and call invert() instead.<br>
         * In case, this matrix is not invertible, this matrix will be returned instead.
         * @return The inverted matrix
         * @see invert(), isSingular().
         */
        [[nodiscard]] SquareMatrixT4<T> inverted() const;
 
        /**
         * Inverts this matrix in place.
         * @return True, if the matrix is not singular and could be inverted
         * @see inverted().
         */
        bool invert();
 
        /**
         * Inverts the matrix and returns the result.
         * @param invertedMatrix The resulting inverted matrix
         * @return True, if the matrix is not singular and could be inverted
         * @see inverted().
         */
        bool invert(SquareMatrixT4<T>& invertedMatrix) const;
 
        /**
         * Returns the determinant of the matrix.
         * @return Matrix determinant
         */
        T determinant() const;
 
        /**
         * Returns the trace of the matrix which is the sum of the diagonal elements.
         * @return Trace of the matrix
         */
        T trace() const;
 
        /**
         * Sets the matrix to the identity matrix.
         */
        inline void toIdentity();
 
        /**
         * Sets the matrix to a zero matrix.
         * @see isNull();
         */
        inline void toNull();
 
        /**
         * Returns whether this matrix is a null matrix.
         * @return True, if so
         */
        bool isNull() const;
 
        /**
         * Returns whether this matrix is the identity matrix.
         * @return True, if so
         */
        bool isIdentity() const;
 
        /**
         * Returns whether this matrix is singular (and thus cannot be inverted).
         * A matrix is singular if the determinant of a matrix is 0.<br>
         * @return True, if so
         */
        inline bool isSingular() const;
 
        /**
         * Returns whether this matrix is symmetric.
         * @param epsilon The epsilon threshold to be used, with range [0, infinity)
         * @return True, if so
         */
        bool isSymmetric(const T epsilon = NumericT<T>::eps()) const;
 
        /**
         * Returns whether two matrices are almost identical up to a specified epsilon.
         * @param matrix Second matrix that will be checked
         * @param eps The epsilon threshold to be used, with range [0, infinity)
         * @return True, if so
         */
        inline bool isEqual(const SquareMatrixT4<T>& matrix, const T eps = NumericT<T>::eps()) const;
 
        /**
         * Performs an eigen value analysis.
         * @param eigenValues The resulting four eigen values, must be valid
         * @param eigenVectors The resulting four eigen vectors, must be valid
         * @return True, if succeeded
         */
        bool eigenSystem(T* eigenValues, VectorT4<T>* eigenVectors) const;
 
        /**
         * Returns a pointer to the internal values.
         * @return Pointer to the internal values
         */
        inline const T* data() const;
 
        /**
         * Returns a pointer to the internal values.
         * @return Pointer to the internal values
         */
        inline T* data();
 
        /**
         * Copies the elements of this matrix to an array with floating point values of type U.
         * @param arrayValues Array with 16 floating point values of type U receiving the elements of this matrix
         * @tparam U Floating point type
         */
        template <typename U>
        void copyElements(U* arrayValues) const;
 
        /**
         * Copies the elements of this matrix to an array with floating point values.
         * @param arrayValues Array with floating point values receiving the elements of this matrix
         */
        void copyElements(T* arrayValues) const;
 
        /**
         * Default assign operator.
         * @return Reference to this object
         */
        SquareMatrixT4<T>& operator=(const SquareMatrixT4<T>&) = default;
 
        /**
         * Returns whether two matrices are identical up to a small epsilon.
         * @param matrix Right operand
         * @return True, if so
         */
        bool operator==(const SquareMatrixT4<T>& matrix) const;
 
        /**
         * Returns whether two matrices are not identical up to a small epsilon.
         * @param matrix Right operand
         * @return True, if so
         */
        inline bool operator!=(const SquareMatrixT4<T>& matrix) const;
 
        /**
         * Adds two matrices.
         * @param matrix Right operand
         * @return Sum matrix
         */
        SquareMatrixT4<T> operator+(const SquareMatrixT4<T>& matrix) const;
 
        /**
         * Adds and assigns two matrices.
         * @param matrix Right operand
         * @return Reference to this object
         */
        SquareMatrixT4<T>& operator+=(const SquareMatrixT4<T>& matrix);
 
        /**
         * Subtracts two matrices.
         * @param matrix Right operand
         * @return Difference matrix
         */
        SquareMatrixT4<T> operator-(const SquareMatrixT4<T>& matrix) const;
 
        /**
         * Subtracts and assigns two matrices.
         * @param matrix Right operand
         * @return Reference to this object
         */
        SquareMatrixT4<T>& operator-=(const SquareMatrixT4<T>& matrix);
 
        /**
         * Returns the negative matrix of this matrix (all matrix elements are multiplied by -1).
         * @return Resulting negative matrix
         */
        inline SquareMatrixT4<T> operator-() const;
 
        /**
         * Multiplies two matrices.
         * @param matrix Right operand
         * @return Product matrix
         */
        OCEAN_FORCE_INLINE SquareMatrixT4<T> operator*(const SquareMatrixT4<T>& matrix) const;
 
        /**
         * Multiplies two matrices.
         * @param matrix Right operand
         * @return Product matrix
         */
        OCEAN_FORCE_INLINE SquareMatrixT4<T> operator*(const HomogenousMatrixT4<T>& matrix) const;
 
        /**
         * Multiplies and assigns two matrices.
         * @param matrix Right operand
         * @return Reference to this object
         */
        OCEAN_FORCE_INLINE SquareMatrixT4<T>& operator*=(const SquareMatrixT4<T>& matrix);
 
        /**
         * Multiplies and assigns two matrices.
         * @param matrix Right operand
         * @return Reference to this object
         */
        OCEAN_FORCE_INLINE SquareMatrixT4<T>& operator*=(const HomogenousMatrixT4<T>& matrix);
 
        /**
         * Multiply operator for a 3D vector.
         * The 3D vector is interpreted as a 4D vector with fourth component equal to 1.<br>
         * The final result will be de-homogenizated to provide a 3D vector result.<br>
         * Beware the dot product of the last row with the vector must not be zero!
         * @param vector Right operand
         * @return Resulting 3D vector
         */
        OCEAN_FORCE_INLINE VectorT3<T> operator*(const VectorT3<T>& vector) const;
 
        /**
         * Multiply operator for a 4D vector.
         * @param vector Right operand
         * @return Resulting 4D vector
         */
        OCEAN_FORCE_INLINE VectorT4<T> operator*(const VectorT4<T>& vector) const;
 
        /**
         * Multiplies this matrix with a scalar value.
         * @param value Right operand
         * @return Resulting matrix
         */
        SquareMatrixT4<T> operator*(const T value) const;
 
        /**
         * Multiplies and assigns this matrix with a scalar value.
         * @param value right operand
         * @return Reference to this object
         */
        SquareMatrixT4<T>& operator*=(const T value);
 
        /**
         * Element operator.
         * Beware: No range check will be done!
         * @param index The index of the element to return [0, 15]
         * @return Specified element
         */
        inline T operator[](const unsigned int index) const;
 
        /**
         * Element operator.
         * Beware: No range check will be done!
         * @param index The index of the element to return [0, 15]
         * @return Specified element
         */
        inline T& operator[](const unsigned int index);
 
        /**
         * Element operator.
         * Beware: No range check will be done!
         * @param row The row of the element to return [0, 3]
         * @param column The column of the element to return [0, 3]
         * @return Specified element
         */
        inline T operator()(const unsigned int row, const unsigned int column) const;
 
        /**
         * Element operator.
         * Beware: No range check will be done!
         * @param row The row of the element to return [0, 3]
         * @param column The column of the element to return [0, 3]
         * @return Specified element
         */
        inline T& operator()(const unsigned int row, const unsigned int column);
 
        /**
         * Element operator.
         * Beware: No range check will be done!
         * @param index The index of the element to return [0, 15]
         * @return Specified element
         */
        inline T operator()(const unsigned int index) const;
 
        /**
         * Element operator.
         * Beware: No range check will be done!
         * @param index The index of the element to return [0, 15]
         * @return Specified element
         */
        inline T& operator()(const unsigned int index);
 
        /**
         * Access operator.
         * @return Pointer to the internal values
         */
        inline const T* operator()() const;
 
        /**
         * Access operator.
         * @return Pointer to the internal values
         */
        inline T* operator()();
 
        /**
         * Hash function.
         * @param matrix The matrix for which the hash value will be determined
         * @return The resulting hash value
         */
        inline size_t operator()(const SquareMatrixT4<T>& matrix) const;
 
        /**
         * Returns the number of elements this matrix has.
         * @return The number of elements, always 16
         */
        static inline size_t elements();
 
        /**
         * Multiplies several 4D vectors with a given matrix.
         * @param matrix The matrix to be used for multiplication, may be nullptr if number is 0
         * @param vectors The input vectors that will be multiplied, may be nullptr if number is 0
         * @param results The resulting output (multiplied/transformed) vectors, with same number as the provided input vectors
         * @param number The number of provided vectors (input and output), with range [0, infinity)
         */
        static void multiply(const SquareMatrixT4<T>& matrix, const VectorT4<T>* vectors, VectorT4<T>* results, const size_t number);
 
        /**
         * Creates a projection matrix defined by the horizontal field of view, the aspect ratio and the near and far clipping plane.
         * @param fovX Horizontal field of view, in radian and a range of [0, Pi]
         * @param aspectRatio View aspect ratio which is width divided by the height of the projection window
         * @param nearDistance Positive distance to the near clipping plane
         * @param farDistance Positive distance to the far clipping lane
         * @return Specified projection matrix
         */
        static SquareMatrixT4<T> projectionMatrix(const T fovX, const T aspectRatio, const T nearDistance, const T farDistance);
 
        /**
         * Creates a projection matrix defined by a camera profile of a pinhole camera and the near and far clipping plane.
         * @param anyCamera The camera profile of a pinhole camera, without distortion parameters, must be valid
         * @param nearDistance Positive distance to the near clipping plane
         * @param farDistance Positive distance to the far clipping lane
         * @return Specified projection matrix
         */
        static SquareMatrixT4<T> projectionMatrix(const AnyCameraT<T>& anyCamera, const T nearDistance, const T farDistance);
 
        /**
         * Creates a projection matrix defined by an asymmetric viewing frustum.
         * The shape of the frustum is defined by the rectangle on the near plane.<br>
         * Afterwards, the field of view is defined by the (positive) distance to the near clipping plane.<br>
         * Followed by the (positive) far clipping plane to determine the entire frustum.
         * @param left Position of the left border of the rectangle on the near plane
         * @param right Position of the right border of the rectangle on the near plane
         * @param top Position of the top border of the rectangle on the near plane
         * @param bottom Position of the bottom border of the rectangle on the near plane
         * @param nearDistance Positive distance to the near clipping plane
         * @param farDistance Positive distance to the far clipping plane
         * @return Specified frustum projection matrix
         */
        static SquareMatrixT4<T> frustumMatrix(const T left, const T right, const T top, const T bottom, const T nearDistance, const T farDistance);
 
        /**
         * Creates a project matrix defined by an asymmetric viewing frustum.
         * The shape of the frustum is defined by the rectangle on the near plane.<br>
         * The viewing position is defined by the given view matrix while the near plane is expected to lie in the origin of the coordinate system.
         * @param width The width of the near plane, with range (0, infinity)
         * @param height The height of the near plane, with range (0, infinity)
         * @param viewingMatrix Viewing matrix transforming point defined in the camera coordinate system into points defined in the world coordinate system, must be invertible
         * @param nearDistance Positive distance to the near clipping plane, with range (0, infinity)
         * @param farDistance Positive distance to the far clipping plane, with range (nearDistance, infinity)
         * @return Specified frustum projection matrix
         */
        static SquareMatrixT4<T> frustumMatrix(const T width, const T height, const HomogenousMatrixT4<T>& viewingMatrix, const T nearDistance, const T farDistance);
 
        /**
         * Converts matrices with specific data type to matrices with different data type.
         * @param matrices The matrices to convert
         * @return The converted matrices
         * @tparam U The element data type of the matrices to convert
         */
        template <typename U>
        static inline std::vector< SquareMatrixT4<T> > matrices2matrices(const std::vector< SquareMatrixT4<U> >& matrices);
 
        /**
         * Converts matrices with specific data type to matrices with different data type.
         * @param matrices The matrices to convert
         * @param size The number of matrices to convert
         * @return The converted matrices
         * @tparam U The element data type of the matrices to convert
         */
        template <typename U>
        static inline std::vector< SquareMatrixT4<T> > matrices2matrices(const SquareMatrixT4<U>* matrices, const size_t size);
 
    protected:
 
        /**
         * Swaps two rows of this matrix.
         * @param row0 The index of the first row ,with range [0, 3]
         * @param row1 The index of the second row, with range [0, 3]
         */
        void swapRows(const unsigned int row0, const unsigned int row1);
 
        /**
         * Multiplies a row with a scalar value.
         * @param row The index of the row to multiply, with range [0, 3]
         * @param scalar The scalar to multiply
         */
        void multiplyRow(const unsigned int row, const T scalar);
 
        /**
         * Multiplies elements from a specific row with a scalar and adds them to another row.
         * @param targetRow The index of the target row, with range [0, 3]
         * @param sourceRow The index of the source row, with range [0, 3]
         * @param scalar The scalar to multiply the source elements with
         */
        void addRows(const unsigned int targetRow, const unsigned int sourceRow, const T scalar);
 
    protected:
 
        /// The sixteen values of the matrix.
        T values[16];
};
 
template <typename T>
inline SquareMatrixT4<T>::SquareMatrixT4()
{
    // nothing to do here
}
 
template <typename T>
template <typename U>
inline SquareMatrixT4<T>::SquareMatrixT4(const SquareMatrixT4<U>& matrix)
{
    for (unsigned int n = 0u; n < 16u; ++n)
    {
        values[n] = T(matrix.values[n]);
    }
}
 
template <typename T>
SquareMatrixT4<T>::SquareMatrixT4(const bool setToIdentity)
{
    if (setToIdentity)
    {
        toIdentity();
    }
    else
    {
        toNull();
    }
}
 
template <typename T>
template <typename U>
SquareMatrixT4<T>::SquareMatrixT4(const U* arrayValues)
{
    ocean_assert(arrayValues);
 
    for (unsigned int n = 0u; n < 16u; ++n)
    {
        values[n] = T(arrayValues[n]);
    }
}
 
template <typename T>
SquareMatrixT4<T>::SquareMatrixT4(const T* arrayValues)
{
    ocean_assert(arrayValues);
    memcpy(values, arrayValues, sizeof(T) * 16);
}
 
template <typename T>
template <typename U>
SquareMatrixT4<T>::SquareMatrixT4(const U* arrayValues, const bool valuesRowAligned)
{
    ocean_assert(arrayValues);
 
    if (valuesRowAligned)
    {
        values[ 0] = T(arrayValues[ 0]);
        values[ 1] = T(arrayValues[ 4]);
        values[ 2] = T(arrayValues[ 8]);
        values[ 3] = T(arrayValues[12]);
        values[ 4] = T(arrayValues[ 1]);
        values[ 5] = T(arrayValues[ 5]);
        values[ 6] = T(arrayValues[ 9]);
        values[ 7] = T(arrayValues[13]);
        values[ 8] = T(arrayValues[ 2]);
        values[ 9] = T(arrayValues[ 6]);
        values[10] = T(arrayValues[10]);
        values[11] = T(arrayValues[14]);
        values[12] = T(arrayValues[ 3]);
        values[13] = T(arrayValues[ 7]);
        values[14] = T(arrayValues[11]);
        values[15] = T(arrayValues[15]);
    }
    else
    {
        for (unsigned int n = 0u; n < 16u; ++n)
        {
            values[n] = T(arrayValues[n]);
        }
    }
}
 
template <typename T>
SquareMatrixT4<T>::SquareMatrixT4(const T* arrayValues, const bool valuesRowAligned)
{
    ocean_assert(arrayValues);
 
    if (valuesRowAligned)
    {
        values[ 0] = arrayValues[ 0];
        values[ 1] = arrayValues[ 4];
        values[ 2] = arrayValues[ 8];
        values[ 3] = arrayValues[12];
        values[ 4] = arrayValues[ 1];
        values[ 5] = arrayValues[ 5];
        values[ 6] = arrayValues[ 9];
        values[ 7] = arrayValues[13];
        values[ 8] = arrayValues[ 2];
        values[ 9] = arrayValues[ 6];
        values[10] = arrayValues[10];
        values[11] = arrayValues[14];
        values[12] = arrayValues[ 3];
        values[13] = arrayValues[ 7];
        values[14] = arrayValues[11];
        values[15] = arrayValues[15];
    }
    else
    {
        memcpy(values, arrayValues, sizeof(T) * 16);
    }
}
 
template <typename T>
SquareMatrixT4<T>::SquareMatrixT4(const HomogenousMatrixT4<T>& transformation)
{
    memcpy(values, transformation(), sizeof(T) * 16);
}
 
template <typename T>
SquareMatrixT4<T>::SquareMatrixT4(const SquareMatrixT3<T>& subMatrix)
{
    values[ 3] = T(0.0);
    values[ 7] = T(0.0);
    values[11] = T(0.0);
    values[12] = T(0.0);
    values[13] = T(0.0);
    values[14] = T(0.0);
    values[15] = T(0.0);
 
    memcpy(values, subMatrix(), sizeof(T) * 3); // Set values[0] - values[2]
    memcpy(values + 4, subMatrix() + 3, sizeof(T) * 3); // Set values[4] - values[6]
    memcpy(values + 8, subMatrix() + 6, sizeof(T) * 3); // Set values[8] - values[10]
}
 
template <typename T>
SquareMatrixT4<T>::SquareMatrixT4(const VectorT4<T>& diagonal)
{
    values[ 0] = diagonal[0];
    values[ 1] = T(0.0);
    values[ 2] = T(0.0);
    values[ 3] = T(0.0);
    values[ 4] = T(0.0);
    values[ 5] = diagonal[1];
    values[ 6] = T(0.0);
    values[ 7] = T(0.0);
    values[ 8] = T(0.0);
    values[ 9] = T(0.0);
    values[10] = diagonal[2];
    values[11] = T(0.0);
    values[12] = T(0.0);
    values[13] = T(0.0);
    values[14] = T(0.0);
    values[15] = diagonal[3];
}
 
template <typename T>
inline const T* SquareMatrixT4<T>::data() const
{
    return values;
}
 
template <typename T>
inline T* SquareMatrixT4<T>::data()
{
    return values;
}
 
template <typename T>
SquareMatrixT4<T> SquareMatrixT4<T>::transposed() const
{
    SquareMatrixT4<T> result(*this);
 
    result.values[1] = values[4];
    result.values[4] = values[1];
 
    result.values[2] = values[8];
    result.values[8] = values[2];
 
    result.values[3] = values[12];
    result.values[12] = values[3];
 
    result.values[6] = values[9];
    result.values[9] = values[6];
 
    result.values[7] = values[13];
    result.values[13] = values[7];
 
    result.values[11] = values[14];
    result.values[14] = values[11];
 
    return result;
}
 
template <typename T>
void SquareMatrixT4<T>::transpose()
{
    SquareMatrixT4<T> tmp(*this);
 
    values[4] = tmp.values[1];
    values[1] = tmp.values[4];
 
    values[8] = tmp.values[2];
    values[2] = tmp.values[8];
 
    values[12] = tmp.values[3];
    values[3] = tmp.values[12];
 
    values[9] = tmp.values[6];
    values[6] = tmp.values[9];
 
    values[13] = tmp.values[7];
    values[7] = tmp.values[13];
 
    values[14] = tmp.values[11];
    values[11] = tmp.values[14];
}
 
template <typename T>
SquareMatrixT4<T> SquareMatrixT4<T>::inverted() const
{
    SquareMatrixT4 invertedMatrix;
 
    if (!invert(invertedMatrix))
    {
        ocean_assert(false && "Could not invert matrix.");
        return *this;
    }
 
    return invertedMatrix;
}
 
template <typename T>
bool SquareMatrixT4<T>::invert()
{
    SquareMatrixT4<T> invertedMatrix;
 
    if (!invert(invertedMatrix))
    {
        return false;
    }
 
    *this = invertedMatrix;
 
    return true;
}
 
template <typename T>
bool SquareMatrixT4<T>::invert(SquareMatrixT4<T>& invertedMatrix) const
{
    // implements the Gauss-Jordon Elimination
 
    SquareMatrixT4<T> source(*this);
    invertedMatrix.toIdentity();
 
    for (unsigned int col = 0; col < 4u; ++col)
    {
        // find largest absolute value in the col-th column,
        // to remove zeros from the main diagonal and to provide numerical stability
        T absolute = 0.0;
        unsigned int selectedRow = 0u;
 
        for (unsigned int row = col; row < 4u; ++row)
        {
            const T value = NumericT<T>::abs(source(row, col));
            if (absolute < value)
            {
                absolute = value;
                selectedRow = row;
            }
        }
 
        // if there was no greater absolute value than 0 this matrix is singular
 
        if (NumericT<T>::isEqualEps(absolute))
        {
            return false;
        }
 
        // exchange the two rows
        if (selectedRow != col)
        {
            source.swapRows(col, selectedRow);
            invertedMatrix.swapRows(col, selectedRow);
        }
 
        // now the element at (col, col) will be 1.0
        if (NumericT<T>::isNotEqual(source(col, col), T(1.0)))
        {
            const T divisor = T(1.0) / source(col, col);
            ocean_assert(divisor != T(0.0));
 
            source.multiplyRow(col, divisor);
            invertedMatrix.multiplyRow(col, divisor);
        }
 
        // clear each entry above and below the selected row and column to zero
        for (unsigned int row = 0; row < 4u; ++row)
        {
            if (row != col)
            {
                const T value = -source(row, col);
 
                source.addRows(row, col, value);
                invertedMatrix.addRows(row, col, value);
            }
        }
    }
 
    return true;
}
 
template <typename T>
T SquareMatrixT4<T>::determinant() const
{
    const T v6_15 = values[6] * values[15];
    const T v10_15 = values[10] * values[15];
    const T v11_14 = values[11] * values[14];
    const T v7_10 = values[7] * values[10];
    const T v9_14 = values[9] * values[14];
    const T v6_13 = values[6] * values[13];
    const T v2_13 = values[2] * values[13];
    const T v2_9 = values[2] * values[9];
    const T v3_10 = values[3] * values[10];
    const T v2_5 = values[2] * values[5];
 
    return values[0] * (values[5] * v10_15 - values[13] * v7_10
                        + v9_14 * values[7] - values[5] * v11_14
                        + v6_13 * values[11] - values[9] * v6_15)
        - values[4] * (v9_14 * values[3] - values[1] * v11_14
                        + v2_13 * values[11] - v2_9 * values[15]
                        + values[1] * v10_15 - values[13] * v3_10)
        + values[8] * (values[1] * v6_15 - v6_13 * values[3]
                        + values[5] * values[14] * values[3] - values[1] * values[14] * values[7]
                        + v2_13 * values[7] - v2_5 * values[15])
        - values[12] * (values[1] * values[6] * values[11] - values[9] * values[6] * values[3]
                        + values[5] * v3_10 - values[1] * v7_10
                        + v2_9 * values[7] - v2_5 * values[11]);
}
 
template <typename T>
T SquareMatrixT4<T>::trace() const
{
    return values[0] + values[5] + values[10] + values[15];
}
 
template <typename T>
inline void SquareMatrixT4<T>::toIdentity()
{
    values[ 0] = T(1.0);
    values[ 1] = T(0.0);
    values[ 2] = T(0.0);
    values[ 3] = T(0.0);
    values[ 4] = T(0.0);
    values[ 5] = T(1.0);
    values[ 6] = T(0.0);
    values[ 7] = T(0.0);
    values[ 8] = T(0.0);
    values[ 9] = T(0.0);
    values[10] = T(1.0);
    values[11] = T(0.0);
    values[12] = T(0.0);
    values[13] = T(0.0);
    values[14] = T(0.0);
    values[15] = T(1.0);
}
 
template <typename T>
inline void SquareMatrixT4<T>::toNull()
{
    for (unsigned int n = 0u; n < 16u; ++n)
    {
        values[n] = T(0.0);
    }
}
 
template <typename T>
bool SquareMatrixT4<T>::isIdentity() const
{
    return NumericT<T>::isEqual(values[0], 1) && NumericT<T>::isEqualEps(values[1]) && NumericT<T>::isEqualEps(values[2]) && NumericT<T>::isEqualEps(values[3])
                && NumericT<T>::isEqualEps(values[4]) && NumericT<T>::isEqual(values[5], 1) && NumericT<T>::isEqualEps(values[6]) && NumericT<T>::isEqualEps(values[7])
                && NumericT<T>::isEqualEps(values[8]) && NumericT<T>::isEqualEps(values[9]) && NumericT<T>::isEqual(values[10], 1) && NumericT<T>::isEqualEps(values[11])
                && NumericT<T>::isEqualEps(values[12]) && NumericT<T>::isEqualEps(values[13]) && NumericT<T>::isEqualEps(values[14]) && NumericT<T>::isEqual(values[15], 1);
}
 
template <typename T>
bool SquareMatrixT4<T>::isNull() const
{
    return NumericT<T>::isEqualEps(values[0]) && NumericT<T>::isEqualEps(values[1]) && NumericT<T>::isEqualEps(values[2]) && NumericT<T>::isEqualEps(values[3])
                && NumericT<T>::isEqualEps(values[4]) && NumericT<T>::isEqualEps(values[5]) && NumericT<T>::isEqualEps(values[6]) && NumericT<T>::isEqualEps(values[7])
                && NumericT<T>::isEqualEps(values[8]) && NumericT<T>::isEqualEps(values[9]) && NumericT<T>::isEqualEps(values[10]) && NumericT<T>::isEqualEps(values[11])
                && NumericT<T>::isEqualEps(values[12]) && NumericT<T>::isEqualEps(values[13]) && NumericT<T>::isEqualEps(values[14]) && NumericT<T>::isEqualEps(values[15]);
}
 
template <typename T>
inline bool SquareMatrixT4<T>::isSingular() const
{
    return NumericT<T>::isEqualEps(determinant());
}
 
template <typename T>
bool SquareMatrixT4<T>::isSymmetric(const T epsilon) const
{
    ocean_assert(epsilon >= T(0));
 
    return NumericT<T>::isEqual(values[1], values[4], epsilon) && NumericT<T>::isEqual(values[2], values[8], epsilon) && NumericT<T>::isEqual(values[3], values[12], epsilon)
            && NumericT<T>::isEqual(values[6], values[9], epsilon) && NumericT<T>::isEqual(values[7], values[13], epsilon) && NumericT<T>::isEqual(values[11], values[14], epsilon);
}
 
template <typename T>
inline bool SquareMatrixT4<T>::isEqual(const SquareMatrixT4<T>& matrix, const T eps) const
{
    return NumericT<T>::isEqual(values[0], matrix.values[0], eps) && NumericT<T>::isEqual(values[1], matrix.values[1], eps)
            && NumericT<T>::isEqual(values[2], matrix.values[2], eps) && NumericT<T>::isEqual(values[3], matrix.values[3], eps)
            && NumericT<T>::isEqual(values[4], matrix.values[4], eps) && NumericT<T>::isEqual(values[5], matrix.values[5], eps)
            && NumericT<T>::isEqual(values[6], matrix.values[6], eps) && NumericT<T>::isEqual(values[7], matrix.values[7], eps)
            && NumericT<T>::isEqual(values[8], matrix.values[8], eps) && NumericT<T>::isEqual(values[9], matrix.values[9], eps)
            && NumericT<T>::isEqual(values[10], matrix.values[10], eps) && NumericT<T>::isEqual(values[11], matrix.values[11], eps)
            && NumericT<T>::isEqual(values[12], matrix.values[12], eps) && NumericT<T>::isEqual(values[13], matrix.values[13], eps)
            && NumericT<T>::isEqual(values[14], matrix.values[14], eps) && NumericT<T>::isEqual(values[15], matrix.values[15], eps);
}
 
template <typename T>
bool SquareMatrixT4<T>::eigenSystem(T* eigenValues, VectorT4<T>* eigenVectors) const
{
    ocean_assert(eigenValues != nullptr && eigenVectors != nullptr);
 
    /**
     * <pre>
     * Computation of the characteristic polynomial
     *
     *     [ a b c d ]
     * A = [ e f g h ]
     *     [ i j k l ]
     *     [ m n o p ]
     *
     *             [ a-x   b    c    d  ]
     * A - x * E = [  e   f-x   g    h  ]
     *             [  i    j   k-x   l  ]
     *             [  m    n    o   p-x ]
     *
     * Polynomial = Det|A - x * E| = 0
     *            = x^4 + (-a - f - k - p)x^3 + (-be + af - ci - gj + ak + fk - dm - hn - lo + ap + fp + kp)x^2
     *               + (cfi - bgi - cej + agj + bek - afk + dfm - bhm + dkm - clm - den + ahn + hkn - gln - dio - hjo + alo + flo + bep - afp + cip + gjp - akp - fkp)x
     *               + (dgjm - chjm - dfkm + bhkm + cflm - bglm)
     *               + (-dgin + chin + dekn - ahkn - celn + agln)
     *               + (dfio - bhio - dejo + ahjo + belo - aflo)
     *               + (-cfip + bgip + cejp - agjp - bekp + afkp)
     *            = a1x^4 + a2x^3 + a3x^2 + a4x + a5 = 0
     * </pre>
     */
 
    const T a = values[0];
    const T e = values[1];
    const T i = values[2];
    const T m = values[3];
 
    const T b = values[4];
    const T f = values[5];
    const T j = values[6];
    const T n = values[7];
 
    const T c = values[8];
    const T g = values[9];
    const T k = values[10];
    const T o = values[11];
 
    const T d = values[12];
    const T h = values[13];
    const T l = values[14];
    const T p = values[15];
 
    const T a1 = 1.0;
    const T a2 = -a - f - k - p;
    const T a3 = -b * e + a * f - c * i - g * j + a * k + f * k - d * m - h * n - l * o + a * p + f * p + k * p;
    const T a4 = c * f * i - b * g * i - c * e * j + a * g * j + b * e * k - a * f * k + d * f * m - b * h * m
                        + d * k * m - c * l * m - d * e * n + a * h * n + h * k * n - g * l * n - d * i * o - h * j * o
                        + a * l * o + f * l * o + b * e * p - a * f * p + c * i * p + g * j * p - a * k * p - f * k * p;
    const T a5 = d * g * j * m - c * h * j * m - d * f * k * m + b * h * k * m + c * f * l * m - b * g * l * m
                        - d * g * i * n + c * h * i * n + d * e * k * n - a * h * k * n - c * e * l * n + a * g * l * n
                        + d * f * i * o - b * h * i * o - d * e * j * o + a * h * j * o + b * e * l * o - a * f * l * o
                        - c * f * i * p + b * g * i * p + c * e * j * p - a * g * j * p - b * e * k * p + a * f * k * p;
 
    // Solve the quartic equation to find eigenvalues
 
    double x[4];
    const unsigned int solutions = EquationT<double>::solveQuartic(double(a1), double(a2), double(a3), double(a4), double(a5), x);
 
    if (solutions != 4u)
    {
        return false;
    }
 
    for (unsigned int iEigen = 0u; iEigen < solutions && iEigen < 4u; ++iEigen)
    {
        eigenValues[iEigen] = T(x[iEigen]);
    }
 
    for (unsigned int iEigen = solutions; iEigen < 4u; ++iEigen)
    {
        eigenValues[iEigen] = T(0);
    }
 
    Utilities::sortHighestToFront4(eigenValues[0], eigenValues[1], eigenValues[2], eigenValues[3]);
 
    /**
     * <pre>
     * Determination of the eigen vectors (vx, vy, vz, vw):
     *             [ a-x   b    c    d  ]   [ vx ]
     * A - x * I = [  e   f-x   g    h  ] * [ vy ] = 0
     *             [  i    j   k-x   l  ]   [ vz ]
     *             [  m    n    o   p-x ]   [ vw ]
     * </pre>
     */
 
    for (unsigned int nSolution = 0u; nSolution < solutions; ++nSolution)
    {
        const T lambda = eigenValues[nSolution];
 
        // create matrix (A - x*I) as rows
 
        VectorT4<T> rows[4];
        rows[0] = VectorT4<T>(a - lambda, b, c, d);
        rows[1] = VectorT4<T>(e, f - lambda, g, h);
        rows[2] = VectorT4<T>(i, j, k - lambda, l);
        rows[3] = VectorT4<T>(m, n, o, p - lambda);
 
        // Find eigenvector by solving (A - lambda*I) * v = 0
        // We'll try setting each component to 1 and solving for the others
 
        VectorT4<T> eigenvector(T(0), T(0), T(0), T(0));
        T bestResidual = NumericT<T>::maxValue();
 
        for (unsigned int freeIndex = 0u; freeIndex < 4u; ++freeIndex)
        {
            // Try setting component freeIndex to 1
            VectorT4<T> candidate(T(0), T(0), T(0), T(0));
            candidate[freeIndex] = T(1);
 
            // Find the 3 rows with largest coefficients at freeIndex position
            // (these will give us the most stable equations)
            unsigned int rowIndices[4] = {0, 1, 2, 3};
            T rowScores[4];
 
            for (unsigned int rowNum = 0u; rowNum < 4u; ++rowNum)
            {
                rowScores[rowNum] = NumericT<T>::abs(rows[rowNum][freeIndex]);
            }
 
            // Sort rows by score (descending)
            for (unsigned int row = 0u; row < 3u; ++row)
            {
                for (unsigned int row2 = row + 1u; row2 < 4u; ++row2)
                {
                    if (rowScores[row2] > rowScores[row])
                    {
                        std::swap(rowScores[row], rowScores[row2]);
                        std::swap(rowIndices[row], rowIndices[row2]);
                    }
                }
            }
 
            // Build 3x3 system from the 3 best rows
            // System: coeffMatrix * unknowns = rhs
            T coeffMatrix[3][3];
            T rhs[3];
            unsigned int unknownIndices[3];
            unsigned int idx = 0u;
 
            for (unsigned int col = 0u; col < 4u; ++col)
            {
                if (col != freeIndex)
                {
                    unknownIndices[idx++] = col;
                }
            }
 
            for (unsigned int rowIdx = 0u; rowIdx < 3u; ++rowIdx)
            {
                const VectorT4<T>& row = rows[rowIndices[rowIdx]];
                rhs[rowIdx] = -row[freeIndex];
 
                for (unsigned int colIdx = 0u; colIdx < 3u; ++colIdx)
                {
                    coeffMatrix[rowIdx][colIdx] = row[unknownIndices[colIdx]];
                }
            }
 
            // Solve 3x3 system using Gaussian elimination
            T augmented[3][4];
            T rowScales[3];
 
            for (unsigned int rowIdx = 0u; rowIdx < 3u; ++rowIdx)
            {
                // Copy coefficients
                for (unsigned int colIdx = 0u; colIdx < 3u; ++colIdx)
                {
                    augmented[rowIdx][colIdx] = coeffMatrix[rowIdx][colIdx];
                }
                augmented[rowIdx][3] = rhs[rowIdx];
 
                // Compute row scale for better numerical conditioning
                T maxRowVal = T(0);
                for (unsigned int colIdx = 0u; colIdx < 4u; ++colIdx)
                {
                    maxRowVal = std::max(maxRowVal, NumericT<T>::abs(augmented[rowIdx][colIdx]));
                }
 
                rowScales[rowIdx] = maxRowVal;
 
                // Scale the row if possible (improves conditioning)
                if (maxRowVal > NumericT<T>::weakEps())
                {
                    const T scale = T(1) / maxRowVal;
                    for (unsigned int colIdx = 0u; colIdx < 4u; ++colIdx)
                    {
                        augmented[rowIdx][colIdx] *= scale;
                    }
                }
            }
 
            // Forward elimination with partial pivoting
            for (unsigned int pivot = 0u; pivot < 3u; ++pivot)
            {
                // Find row with largest pivot element
                unsigned int maxRow = pivot;
                T maxVal = NumericT<T>::abs(augmented[pivot][pivot]);
 
                for (unsigned int rowIdx = pivot + 1u; rowIdx < 3u; ++rowIdx)
                {
                    const T val = NumericT<T>::abs(augmented[rowIdx][pivot]);
                    if (val > maxVal)
                    {
                        maxVal = val;
                        maxRow = rowIdx;
                    }
                }
 
                // Swap rows if needed
                if (maxRow != pivot)
                {
                    for (unsigned int colIdx = 0u; colIdx < 4u; ++colIdx)
                    {
                        std::swap(augmented[pivot][colIdx], augmented[maxRow][colIdx]);
                    }
                }
 
                // Use generous threshold (system may be near-singular as we're finding null space)
                const T pivotThreshold = std::is_same<T, float>::value ? (NumericT<T>::weakEps() * T(10)) : NumericT<T>::weakEps();
 
                // Skip elimination if pivot is too small, but continue (system is expected to be singular)
                if (NumericT<T>::abs(augmented[pivot][pivot]) < pivotThreshold)
                {
                    continue;
                }
 
                // Eliminate below pivot
                for (unsigned int rowIdx = pivot + 1u; rowIdx < 3u; ++rowIdx)
                {
                    const T factor = augmented[rowIdx][pivot] / augmented[pivot][pivot];
                    for (unsigned int colIdx = pivot; colIdx < 4u; ++colIdx)
                    {
                        augmented[rowIdx][colIdx] -= factor * augmented[pivot][colIdx];
                    }
                }
            }
 
            // Back substitution
            T solution[3] = {T(0), T(0), T(0)};
 
            for (int rowIdx = 2; rowIdx >= 0; --rowIdx)
            {
                T sum = augmented[rowIdx][3];
 
                for (unsigned int colIdx = rowIdx + 1u; colIdx < 3u; ++colIdx)
                {
                    sum -= augmented[rowIdx][colIdx] * solution[colIdx];
                }
 
                const T pivotThreshold = std::is_same<T, float>::value ? (NumericT<T>::weakEps() * T(10)) : NumericT<T>::weakEps();
                if (NumericT<T>::abs(augmented[rowIdx][rowIdx]) > pivotThreshold)
                {
                    solution[rowIdx] = sum / augmented[rowIdx][rowIdx];
                }
                // else: leave solution[rowIdx] as 0 (free variable in under-determined system)
            }
 
            // Build candidate eigenvector
            for (unsigned int idx2 = 0u; idx2 < 3u; ++idx2)
            {
                candidate[unknownIndices[idx2]] = solution[idx2];
            }
 
            // Evaluate residual: ||A*v - lambda*v||
            const VectorT4<T> Av = (*this) * candidate;
            const VectorT4<T> lambdaV = candidate * lambda;
            const T residual = (Av - lambdaV).length();
 
            if (residual < bestResidual)
            {
                bestResidual = residual;
                eigenvector = candidate;
            }
        }
 
        // Normalize the eigenvector
        if (!eigenvector.normalize())
        {
            // Normalization failed - use standard basis vector
            eigenvector = VectorT4<T>(T(0), T(0), T(0), T(0));
            eigenvector[nSolution % 4u] = T(1);
        }
 
        eigenVectors[nSolution] = eigenvector;
    }
 
    return true;
}
 
template <typename T>
template <typename U>
void SquareMatrixT4<T>::copyElements(U* arrayValues) const
{
    ocean_assert(arrayValues != nullptr);
 
    for (unsigned int n = 0u; n < 16u; ++n)
    {
        arrayValues[n] = U(values[n]);
    }
}
 
template <typename T>
void SquareMatrixT4<T>::copyElements(T* arrayValues) const
{
    ocean_assert(arrayValues != nullptr);
 
    memcpy(arrayValues, values, sizeof(T) * 16);
}
 
template <typename T>
inline bool SquareMatrixT4<T>::operator!=(const SquareMatrixT4<T>& matrix) const
{
    return !(*this == matrix);
}
 
template <typename T>
inline SquareMatrixT4<T>& SquareMatrixT4<T>::operator*=(const SquareMatrixT4<T>& matrix)
{
    *this = *this * matrix;
    return *this;
}
 
template <typename T>
inline SquareMatrixT4<T>& SquareMatrixT4<T>::operator*=(const HomogenousMatrixT4<T>& matrix)
{
    *this = *this * matrix;
    return *this;
}
 
template <typename T>
inline T SquareMatrixT4<T>::operator[](const unsigned int index) const
{
    ocean_assert(index < 16u);
    return values[index];
}
 
template <typename T>
inline T& SquareMatrixT4<T>::operator[](const unsigned int index)
{
    ocean_assert(index < 16u);
    return values[index];
}
 
template <typename T>
inline T SquareMatrixT4<T>::operator()(const unsigned int row, const unsigned int column) const
{
    ocean_assert(row < 4u && column < 4u);
    return values[(column << 2) + row]; // values[(column * 4) + row];
}
 
template <typename T>
inline T& SquareMatrixT4<T>::operator()(const unsigned int row, const unsigned int column)
{
    ocean_assert(row < 4u && column < 4u);
    return values[(column << 2) + row]; // values[(column * 4) + row];
}
 
template <typename T>
inline T SquareMatrixT4<T>::operator()(const unsigned int index) const
{
    ocean_assert(index < 16u);
    return values[index];
}
 
template <typename T>
inline T& SquareMatrixT4<T>::operator()(const unsigned int index)
{
    ocean_assert(index < 16u);
    return values[index];
}
 
template <typename T>
inline const T* SquareMatrixT4<T>::operator()() const
{
    return values;
}
 
template <typename T>
inline T* SquareMatrixT4<T>::operator()()
{
    return values;
}
 
template <typename T>
inline size_t SquareMatrixT4<T>::operator()(const SquareMatrixT4<T>& matrix) const
{
    size_t seed = std::hash<T>{}(matrix.values[0]);
 
    for (unsigned int n = 1u; n < 16u; ++n)
    {
        seed ^= std::hash<T>{}(matrix.values[n]) + 0x9e3779b9 + (seed << 6) + (seed >> 2);
    }
 
    return seed;
}
 
template <typename T>
inline size_t SquareMatrixT4<T>::elements()
{
    return 16;
}
 
template <typename T>
bool SquareMatrixT4<T>::operator==(const SquareMatrixT4<T>& matrix) const
{
    return isEqual(matrix);
}
 
template <typename T>
SquareMatrixT4<T> SquareMatrixT4<T>::operator+(const SquareMatrixT4<T>& matrix) const
{
    SquareMatrixT4<T> result(*this);
 
    result += matrix;
 
    return result;
}
 
template <typename T>
SquareMatrixT4<T>& SquareMatrixT4<T>::operator+=(const SquareMatrixT4<T>& matrix)
{
    for (unsigned int n = 0u; n < 16u; ++n)
    {
        values[n] += matrix.values[n];
    }
 
    return *this;
}
 
template <typename T>
SquareMatrixT4<T> SquareMatrixT4<T>::operator-(const SquareMatrixT4<T>& matrix) const
{
    SquareMatrixT4<T> result(*this);
 
    result -= matrix;
 
    return result;
}
 
template <typename T>
SquareMatrixT4<T>& SquareMatrixT4<T>::operator-=(const SquareMatrixT4<T>& matrix)
{
    for (unsigned int n = 0u; n < 16u; ++n)
    {
        values[n] -= matrix.values[n];
    }
 
    return *this;
}
 
template <typename T>
inline SquareMatrixT4<T> SquareMatrixT4<T>::operator-() const
{
    SquareMatrixT4<T> result;
 
    for (unsigned int n = 0u; n < 16u; ++n)
    {
        result.values[n] = -values[n];
    }
 
    return result;
}
 
template <typename T>
OCEAN_FORCE_INLINE SquareMatrixT4<T> SquareMatrixT4<T>::operator*(const SquareMatrixT4<T>& matrix) const
{
    SquareMatrixT4<T> result;
 
    result.values[0] = values[0] * matrix.values[0] + values[4] * matrix.values[1] + values[8] * matrix.values[2] + values[12] * matrix.values[3];
    result.values[1] = values[1] * matrix.values[0] + values[5] * matrix.values[1] + values[9] * matrix.values[2] + values[13] * matrix.values[3];
    result.values[2] = values[2] * matrix.values[0] + values[6] * matrix.values[1] + values[10] * matrix.values[2] + values[14] * matrix.values[3];
    result.values[3] = values[3] * matrix.values[0] + values[7] * matrix.values[1] + values[11] * matrix.values[2] + values[15] * matrix.values[3];
 
    result.values[4] = values[0] * matrix.values[4] + values[4] * matrix.values[5] + values[8] * matrix.values[6] + values[12] * matrix.values[7];
    result.values[5] = values[1] * matrix.values[4] + values[5] * matrix.values[5] + values[9] * matrix.values[6] + values[13] * matrix.values[7];
    result.values[6] = values[2] * matrix.values[4] + values[6] * matrix.values[5] + values[10] * matrix.values[6] + values[14] * matrix.values[7];
    result.values[7] = values[3] * matrix.values[4] + values[7] * matrix.values[5] + values[11] * matrix.values[6] + values[15] * matrix.values[7];
 
    result.values[8] = values[0] * matrix.values[8] + values[4] * matrix.values[9] + values[8] * matrix.values[10] + values[12] * matrix.values[11];
    result.values[9] = values[1] * matrix.values[8] + values[5] * matrix.values[9] + values[9] * matrix.values[10] + values[13] * matrix.values[11];
    result.values[10] = values[2] * matrix.values[8] + values[6] * matrix.values[9] + values[10] * matrix.values[10] + values[14] * matrix.values[11];
    result.values[11] = values[3] * matrix.values[8] + values[7] * matrix.values[9] + values[11] * matrix.values[10] + values[15] * matrix.values[11];
 
    result.values[12] = values[0] * matrix.values[12] + values[4] * matrix.values[13] + values[8] * matrix.values[14] + values[12] * matrix.values[15];
    result.values[13] = values[1] * matrix.values[12] + values[5] * matrix.values[13] + values[9] * matrix.values[14] + values[13] * matrix.values[15];
    result.values[14] = values[2] * matrix.values[12] + values[6] * matrix.values[13] + values[10] * matrix.values[14] + values[14] * matrix.values[15];
    result.values[15] = values[3] * matrix.values[12] + values[7] * matrix.values[13] + values[11] * matrix.values[14] + values[15] * matrix.values[15];
 
    return result;
}
 
#if defined(OCEAN_HARDWARE_AVX_VERSION) && OCEAN_HARDWARE_AVX_VERSION >= 10
 
template <>
OCEAN_FORCE_INLINE SquareMatrixT4<double> SquareMatrixT4<double>::operator*(const SquareMatrixT4<double>& matrix) const
{
    // the following code uses the following AVX instructions, and needs AVX1 or higher
 
    // AVX1:
    // _mm256_broadcast_sd
    // _mm256_loadu_pd
    // _mm256_mul_pd
    // _mm256_add_pd
    // _mm256_storeu_pd
 
    // we use the same strategy as we apply for matrix-vector multiplication
    // further, here we interpret the right matrix as 4 vectors
 
    // we load the columns of the left matrix
    __m256d c0 = _mm256_loadu_pd(values +  0);
    __m256d c1 = _mm256_loadu_pd(values +  4);
    __m256d c2 = _mm256_loadu_pd(values +  8);
    __m256d c3 = _mm256_loadu_pd(values + 12);
 
 
    // we determine the first vector of the resulting matrix
    __m256d v0 = _mm256_broadcast_sd(matrix.data() + 0);
    __m256d r0 = _mm256_mul_pd(c0, v0);
 
    __m256d v1 = _mm256_broadcast_sd(matrix.data() + 1);
    __m256d r1 = _mm256_mul_pd(c1, v1);
 
    r0 = _mm256_add_pd(r0, r1);
 
    __m256d v2 = _mm256_broadcast_sd(matrix.data() + 2);
    __m256d r2 = _mm256_mul_pd(c2, v2);
 
    r0 = _mm256_add_pd(r0, r2);
 
    __m256d v3 = _mm256_broadcast_sd(matrix.data() + 3);
    __m256d r3 = _mm256_mul_pd(c3, v3);
 
    r0 = _mm256_add_pd(r0, r3);
 
    SquareMatrixT4<double> result;
 
    _mm256_storeu_pd(result.data(), r0);
 
 
    // we determine the second vector of the resulting matrix
    __m256d v4 = _mm256_broadcast_sd(matrix.data() + 4);
    __m256d r4 = _mm256_mul_pd(c0, v4);
 
    __m256d v5 = _mm256_broadcast_sd(matrix.data() + 5);
    __m256d r5 = _mm256_mul_pd(c1, v5);
 
    r4 = _mm256_add_pd(r4, r5);
 
    __m256d v6 = _mm256_broadcast_sd(matrix.data() + 6);
    __m256d r6 = _mm256_mul_pd(c2, v6);
 
    r4 = _mm256_add_pd(r4, r6);
 
    __m256d v7 = _mm256_broadcast_sd(matrix.data() + 7);
    __m256d r7 = _mm256_mul_pd(c3, v7);
 
    r4 = _mm256_add_pd(r4, r7);
 
    _mm256_storeu_pd(result.data() + 4, r4);
 
 
    // we determine the third vector of the resulting matrix
    __m256d v8 = _mm256_broadcast_sd(matrix.data() + 8);
    __m256d r8 = _mm256_mul_pd(c0, v8);
 
    __m256d v9 = _mm256_broadcast_sd(matrix.data() + 9);
    __m256d r9 = _mm256_mul_pd(c1, v9);
 
    r8 = _mm256_add_pd(r8, r9);
 
    __m256d v10 = _mm256_broadcast_sd(matrix.data() + 10);
    __m256d r10 = _mm256_mul_pd(c2, v10);
 
    r8 = _mm256_add_pd(r8, r10);
 
    __m256d v11 = _mm256_broadcast_sd(matrix.data() + 11);
    __m256d r11 = _mm256_mul_pd(c3, v11);
 
    r8 = _mm256_add_pd(r8, r11);
 
    _mm256_storeu_pd(result.data() + 8, r8);
 
 
    // we determine the forth vector of the resulting matrix
    __m256d v12 = _mm256_broadcast_sd(matrix.data() + 12);
    __m256d r12 = _mm256_mul_pd(c0, v12);
 
    __m256d v13 = _mm256_broadcast_sd(matrix.data() + 13);
    __m256d r13 = _mm256_mul_pd(c1, v13);
 
    r12 = _mm256_add_pd(r12, r13);
 
    __m256d v14 = _mm256_broadcast_sd(matrix.data() + 14);
    __m256d r14 = _mm256_mul_pd(c2, v14);
 
    r12 = _mm256_add_pd(r12, r14);
 
    __m256d v15 = _mm256_broadcast_sd(matrix.data() + 15);
    __m256d r15 = _mm256_mul_pd(c3, v15);
 
    r12 = _mm256_add_pd(r12, r15);
 
    _mm256_storeu_pd(result.data() + 12, r12);
 
    return result;
}
 
template <>
OCEAN_FORCE_INLINE SquareMatrixT4<float> SquareMatrixT4<float>::operator*(const SquareMatrixT4<float>& matrix) const
{
    // the following code uses the following AVX instructions, and needs AVX1 or higher
 
    // we use the same strategy as we apply for matrix-vector multiplication
    // further, here we interpret the right matrix as 4 vectors, and combine two vectors into one 256 bit register
 
    // we load the four columns of the left matrix
    __m256 c0 = _mm256_broadcast_ps((const __m128*)values + 0);
    __m256 c1 = _mm256_broadcast_ps((const __m128*)values + 1);
    __m256 c2 = _mm256_broadcast_ps((const __m128*)values + 2);
    __m256 c3 = _mm256_broadcast_ps((const __m128*)values + 3);
 
    __m256 m01 = _mm256_loadu_ps(matrix.data() + 0);
    __m256 m23 = _mm256_loadu_ps(matrix.data() + 8);
 
 
    // we determine the first two vectors of the resulting matrix
    __m256 v0_4 = _mm256_permute_ps(m01, 0x00);
    __m256 r0 = _mm256_mul_ps(c0, v0_4);
 
#ifdef OCEAN_COMPILER_MSC
    ocean_assert(NumericF::isEqual(r0.m256_f32[0], values[0] * matrix[0]));
    ocean_assert(NumericF::isEqual(r0.m256_f32[1], values[1] * matrix[0]));
    ocean_assert(NumericF::isEqual(r0.m256_f32[2], values[2] * matrix[0]));
    ocean_assert(NumericF::isEqual(r0.m256_f32[3], values[3] * matrix[0]));
 
    ocean_assert(NumericF::isEqual(r0.m256_f32[4], values[0] * matrix[4]));
    ocean_assert(NumericF::isEqual(r0.m256_f32[5], values[1] * matrix[4]));
    ocean_assert(NumericF::isEqual(r0.m256_f32[6], values[2] * matrix[4]));
    ocean_assert(NumericF::isEqual(r0.m256_f32[7], values[3] * matrix[4]));
#endif
 
    __m256 v1_5 = _mm256_permute_ps(m01, 0x55);
    __m256 r1 = _mm256_mul_ps(c1, v1_5);
 
#ifdef OCEAN_COMPILER_MSC
    ocean_assert(NumericF::isEqual(r1.m256_f32[0], values[ 4] * matrix[1]));
    ocean_assert(NumericF::isEqual(r1.m256_f32[1], values[ 5] * matrix[1]));
    ocean_assert(NumericF::isEqual(r1.m256_f32[2], values[ 6] * matrix[1]));
    ocean_assert(NumericF::isEqual(r1.m256_f32[3], values[ 7] * matrix[1]));
 
    ocean_assert(NumericF::isEqual(r1.m256_f32[4], values[4] * matrix[5]));
    ocean_assert(NumericF::isEqual(r1.m256_f32[5], values[5] * matrix[5]));
    ocean_assert(NumericF::isEqual(r1.m256_f32[6], values[6] * matrix[5]));
    ocean_assert(NumericF::isEqual(r1.m256_f32[7], values[7] * matrix[5]));
#endif
 
    r0 = _mm256_add_ps(r0, r1);
 
 
    __m256 v2_6 = _mm256_permute_ps(m01, 0xAA);
    __m256 r2 = _mm256_mul_ps(c2, v2_6);
 
    r0 = _mm256_add_ps(r0, r2);
 
 
    __m256 v3_7 = _mm256_permute_ps(m01, 0xFF);
    __m256 r3 = _mm256_mul_ps(c3, v3_7);
 
    r0 = _mm256_add_ps(r0, r3);
 
    SquareMatrixT4<float> result;
 
    _mm256_storeu_ps(result.data(), r0);
 
 
 
    // we determine the last two vectors of the resulting matrix
    __m256 v8_12 = _mm256_permute_ps(m23, 0x00);
    __m256 r8 = _mm256_mul_ps(c0, v8_12);
 
 
    __m256 v9_13 = _mm256_permute_ps(m23, 0x55);
    __m256 r9 = _mm256_mul_ps(c1, v9_13);
 
    r8 = _mm256_add_ps(r8, r9);
 
 
    __m256 v10_14 = _mm256_permute_ps(m23, 0xAA);
    __m256 r10 = _mm256_mul_ps(c2, v10_14);
 
    r8 = _mm256_add_ps(r8, r10);
 
 
    __m256 v11_15 = _mm256_permute_ps(m23, 0xFF);
    __m256 r11 = _mm256_mul_ps(c3, v11_15);
 
    r8 = _mm256_add_ps(r8, r11);
 
    _mm256_storeu_ps(result.data() + 8, r8);
 
    return result;
}
 
#else // OCEAN_HARDWARE_AVX_VERSION < 10
 
#if defined(OCEAN_HARDWARE_SSE_VERSION) && OCEAN_HARDWARE_SSE_VERSION >= 10
 
template <>
OCEAN_FORCE_INLINE SquareMatrixT4<float> SquareMatrixT4<float>::operator*(const SquareMatrixT4<float>& matrix) const
{
    // the following code uses the following SSE instructions, and needs SSE1 or higher
 
    // SSE1:
    // _mm_load1_ps
    // _mm_loadu_ps
    // _mm_mul_ps
    // _mm_add_ps
    // _mm_storeu_ps
 
    // we use the same strategy as we apply for matrix-vector multiplication
    // further, here we interpret the right matrix as 4 vectors
 
    // we load the columns of the left matrix
    __m128 c0 = _mm_loadu_ps(values +  0);
    __m128 c1 = _mm_loadu_ps(values +  4);
    __m128 c2 = _mm_loadu_ps(values +  8);
    __m128 c3 = _mm_loadu_ps(values + 12);
 
 
    // we determine the first vector of the resulting matrix
    __m128 v0 = _mm_load1_ps(matrix.data() + 0);
    __m128 r0 = _mm_mul_ps(c0, v0);
 
    __m128 v1 = _mm_load1_ps(matrix.data() + 1);
    __m128 r1 = _mm_mul_ps(c1, v1);
 
    r0 = _mm_add_ps(r0, r1);
 
    __m128 v2 = _mm_load1_ps(matrix.data() + 2);
    __m128 r2 = _mm_mul_ps(c2, v2);
 
    r0 = _mm_add_ps(r0, r2);
 
    __m128 v3 = _mm_load1_ps(matrix.data() + 3);
    __m128 r3 = _mm_mul_ps(c3, v3);
 
    r0 = _mm_add_ps(r0, r3);
 
    SquareMatrixT4<float> result;
 
    _mm_storeu_ps(result.data(), r0);
 
 
    // we determine the second vector of the resulting matrix
    __m128 v4 = _mm_load1_ps(matrix.data() + 4);
    __m128 r4 = _mm_mul_ps(c0, v4);
 
    __m128 v5 = _mm_load1_ps(matrix.data() + 5);
    __m128 r5 = _mm_mul_ps(c1, v5);
 
    r4 = _mm_add_ps(r4, r5);
 
    __m128 v6 = _mm_load1_ps(matrix.data() + 6);
    __m128 r6 = _mm_mul_ps(c2, v6);
 
    r4 = _mm_add_ps(r4, r6);
 
    __m128 v7 = _mm_load1_ps(matrix.data() + 7);
    __m128 r7 = _mm_mul_ps(c3, v7);
 
    r4 = _mm_add_ps(r4, r7);
 
    _mm_storeu_ps(result.data() + 4, r4);
 
 
    // we determine the third vector of the resulting matrix
    __m128 v8 = _mm_load1_ps(matrix.data() + 8);
    __m128 r8 = _mm_mul_ps(c0, v8);
 
    __m128 v9 = _mm_load1_ps(matrix.data() + 9);
    __m128 r9 = _mm_mul_ps(c1, v9);
 
    r8 = _mm_add_ps(r8, r9);
 
    __m128 v10 = _mm_load1_ps(matrix.data() + 10);
    __m128 r10 = _mm_mul_ps(c2, v10);
 
    r8 = _mm_add_ps(r8, r10);
 
    __m128 v11 = _mm_load1_ps(matrix.data() + 11);
    __m128 r11 = _mm_mul_ps(c3, v11);
 
    r8 = _mm_add_ps(r8, r11);
 
    _mm_storeu_ps(result.data() + 8, r8);
 
 
    // we determine the forth vector of the resulting matrix
    __m128 v12 = _mm_load1_ps(matrix.data() + 12);
    __m128 r12 = _mm_mul_ps(c0, v12);
 
    __m128 v13 = _mm_load1_ps(matrix.data() + 13);
    __m128 r13 = _mm_mul_ps(c1, v13);
 
    r12 = _mm_add_ps(r12, r13);
 
    __m128 v14 = _mm_load1_ps(matrix.data() + 14);
    __m128 r14 = _mm_mul_ps(c2, v14);
 
    r12 = _mm_add_ps(r12, r14);
 
    __m128 v15 = _mm_load1_ps(matrix.data() + 15);
    __m128 r15 = _mm_mul_ps(c3, v15);
 
    r12 = _mm_add_ps(r12, r15);
 
    _mm_storeu_ps(result.data() + 12, r12);
 
    return result;
}
 
#endif // OCEAN_HARDWARE_SSE_VERSION >= 10
 
#endif // OCEAN_HARDWARE_AVX_VERSION >= 10
 
template <typename T>
OCEAN_FORCE_INLINE SquareMatrixT4<T> SquareMatrixT4<T>::operator*(const HomogenousMatrixT4<T>& matrix) const
{
    SquareMatrixT4<T> result;
 
    result.values[0] = values[0] * matrix[0] + values[4] * matrix[1] + values[8] * matrix[2]; // + values[12] * matrix[3];
    result.values[1] = values[1] * matrix[0] + values[5] * matrix[1] + values[9] * matrix[2]; // + values[13] * matrix[3];
    result.values[2] = values[2] * matrix[0] + values[6] * matrix[1] + values[10] * matrix[2]; // + values[14] * matrix[3];
    result.values[3] = values[3] * matrix[0] + values[7] * matrix[1] + values[11] * matrix[2]; // + values[15] * matrix[3];
 
    result.values[4] = values[0] * matrix[4] + values[4] * matrix[5] + values[8] * matrix[6]; // + values[12] * matrix[7];
    result.values[5] = values[1] * matrix[4] + values[5] * matrix[5] + values[9] * matrix[6]; // + values[13] * matrix[7];
    result.values[6] = values[2] * matrix[4] + values[6] * matrix[5] + values[10] * matrix[6]; // + values[14] * matrix[7];
    result.values[7] = values[3] * matrix[4] + values[7] * matrix[5] + values[11] * matrix[6]; // + values[15] * matrix[7];
 
    result.values[8] = values[0] * matrix[8] + values[4] * matrix[9] + values[8] * matrix[10]; // + values[12] * matrix[11];
    result.values[9] = values[1] * matrix[8] + values[5] * matrix[9] + values[9] * matrix[10]; // + values[13] * matrix[11];
    result.values[10] = values[2] * matrix[8] + values[6] * matrix[9] + values[10] * matrix[10]; // + values[14] * matrix[11];
    result.values[11] = values[3] * matrix[8] + values[7] * matrix[9] + values[11] * matrix[10]; // + values[15] * matrix[11];
 
    result.values[12] = values[0] * matrix[12] + values[4] * matrix[13] + values[8] * matrix[14] + values[12]; // * matrix[15];
    result.values[13] = values[1] * matrix[12] + values[5] * matrix[13] + values[9] * matrix[14] + values[13]; // * matrix[15];
    result.values[14] = values[2] * matrix[12] + values[6] * matrix[13] + values[10] * matrix[14] + values[14]; // * matrix[15];
    result.values[15] = values[3] * matrix[12] + values[7] * matrix[13] + values[11] * matrix[14] + values[15]; // * matrix[15];
 
    return result;
}
 
template <typename T>
OCEAN_FORCE_INLINE VectorT3<T> SquareMatrixT4<T>::operator*(const VectorT3<T>& vector) const
{
    const T w = values[3] * vector[0] + values[7] * vector[1] + values[11] * vector[2] + values[15];
    ocean_assert(NumericT<T>::isNotEqualEps(w) && "Division by zero!");
 
    const T factor = 1 / w;
 
    return VectorT3<T>((values[0] * vector[0] + values[4] * vector[1] + values[8] * vector[2] + values[12]) * factor,
                    (values[1] * vector[0] + values[5] * vector[1] + values[9] * vector[2] + values[13]) * factor,
                    (values[2] * vector[0] + values[6] * vector[1] + values[10] * vector[2] + values[14]) * factor);
}
 
template <typename T>
OCEAN_FORCE_INLINE VectorT4<T> SquareMatrixT4<T>::operator*(const VectorT4<T>& vector) const
{
    return VectorT4<T>(values[0] * vector[0] + values[4] * vector[1] + values[8] * vector[2] + values[12] * vector[3],
                        values[1] * vector[0] + values[5] * vector[1] + values[9] * vector[2] + values[13] * vector[3],
                        values[2] * vector[0] + values[6] * vector[1] + values[10] * vector[2] + values[14] * vector[3],
                        values[3] * vector[0] + values[7] * vector[1] + values[11] * vector[2] + values[15] * vector[3]);
}
 
#if defined(OCEAN_HARDWARE_AVX_VERSION) && OCEAN_HARDWARE_AVX_VERSION >= 10
 
#ifdef OCEAN_USE_SLOWER_IMPLEMENTATION
 
// we keep following implementation inside 'OCEAN_USE_SLOWER_IMPLEMENTATION' showing an alternative (which is slower)
 
template <>
OCEAN_FORCE_INLINE VectorT4<double> SquareMatrixT4<double>::operator*(const VectorT4<double>& vector) const
{
    // the following code uses the following AVX instructions, and needs AVX1 or higher
 
    // AVX1:
    // _mm256_loadu_pd
    // _mm256_mul_pd
    // _mm256_shuffle_pd
    // _mm256_hadd_pd
    // _mm256_permute2f128_pd
    // _mm256_storeu_pd
 
    // first we load the first four rows of the matrix
    __m256d row0 = _mm256_loadu_pd(values + 0);
    __m256d row1 = _mm256_loadu_pd(values + 4);
    __m256d row2 = _mm256_loadu_pd(values + 8);
    __m256d row3 = _mm256_loadu_pd(values + 12);
 
    // we load the four values of the vector
    __m256d v = _mm256_loadu_pd(vector.data());
 
    // first, we transpose the 4x4 matrix
 
    // A E I M     A B C D
    // B F J N     E F G H
    // C G K O     I J K L
    // D H L P     M N O P
 
    // A B C D,     E F G H     ->    A E C G
    __m256d temp0 = _mm256_shuffle_pd(row0, row1, 0x00); // 0x00 = 0000 0000
    // A B C D,     E F G H     ->    B F D H
    __m256d temp2 = _mm256_shuffle_pd(row0, row1, 0x0F); // 0x0F = 0000 1111
    // I J K L,     M N O P     ->    I M K O
    __m256d temp1 = _mm256_shuffle_pd(row2, row3, 0x00);
    // I J K L,     M N O P     ->    J N L P
    __m256d temp3 = _mm256_shuffle_pd(row2, row3, 0x0F);
 
    // A E C G      I M K O     ->    A E I M
    row0 = _mm256_permute2f128_pd(temp0, temp1, 0x20); // 0x20 = 0010 0000
    // B F D H      J N L P     ->    B F J N
    row1 = _mm256_permute2f128_pd(temp2, temp3, 0x20);
    // A E C G      I M K O     ->    C G K O
    row2 = _mm256_permute2f128_pd(temp0, temp1, 0x31); // 0x31 = 0011 0001
    // B F D H      J N L P     ->    D H L P
    row3 = _mm256_permute2f128_pd(temp2, temp3, 0x31);
 
#ifdef OCEAN_COMPILER_MSC
    ocean_assert(row0.m256d_f64[0] == values[0] && row0.m256d_f64[1] == values[4] && row0.m256d_f64[2] == values[ 8] && row0.m256d_f64[3] == values[12]);
    ocean_assert(row1.m256d_f64[0] == values[1] && row1.m256d_f64[1] == values[5] && row1.m256d_f64[2] == values[ 9] && row1.m256d_f64[3] == values[13]);
    ocean_assert(row2.m256d_f64[0] == values[2] && row2.m256d_f64[1] == values[6] && row2.m256d_f64[2] == values[10] && row2.m256d_f64[3] == values[14]);
    ocean_assert(row3.m256d_f64[0] == values[3] && row3.m256d_f64[1] == values[7] && row3.m256d_f64[2] == values[11] && row3.m256d_f64[3] == values[15]);
#endif
 
    // unfortunately the AVX does not offer _mm256_dp_pd (the determination of the dot product) so we have to do it on our own
 
    __m256d r0v = _mm256_mul_pd(row0, v);
    __m256d r1v = _mm256_mul_pd(row1, v);
    __m256d r2v = _mm256_mul_pd(row2, v);
    __m256d r3v = _mm256_mul_pd(row3, v);
 
    // we sum both multiplication results horizontally (at least two neighboring products)
    __m256d sum_interleaved_r0_r1 = _mm256_hadd_pd(r0v, r1v);
    __m256d sum_interleaved_r2_r3 = _mm256_hadd_pd(r2v, r3v);
 
    // now we reorder the interleaved sums
    __m256d sum_first = _mm256_permute2f128_pd(sum_interleaved_r0_r1, sum_interleaved_r2_r3, 0x20); //  0x20 = 0010 0000
    __m256d sum_second = _mm256_permute2f128_pd(sum_interleaved_r0_r1, sum_interleaved_r2_r3, 0x31); // 0x31 = 0011 0001
 
    // we finally add both reordered sums
    __m256d sum = _mm256_add_pd(sum_first, sum_second);
 
    VectorT4<double> result;
    _mm256_storeu_pd(result.data(), sum);
 
    ocean_assert(NumericD::isEqual(result[0], values[0] * vector[0] + values[4] * vector[1] + values[ 8] * vector[2] + values[12] * vector[3], NumericD::eps() * 100));
    ocean_assert(NumericD::isEqual(result[1], values[1] * vector[0] + values[5] * vector[1] + values[ 9] * vector[2] + values[13] * vector[3], NumericD::eps() * 100));
    ocean_assert(NumericD::isEqual(result[2], values[2] * vector[0] + values[6] * vector[1] + values[10] * vector[2] + values[14] * vector[3], NumericD::eps() * 100));
    ocean_assert(NumericD::isEqual(result[3], values[3] * vector[0] + values[7] * vector[1] + values[11] * vector[2] + values[15] * vector[3], NumericD::eps() * 100));
 
    return result;
}
 
#else // OCEAN_USE_SLOWER_IMPLEMENTATION
 
template <>
OCEAN_FORCE_INLINE VectorT4<double> SquareMatrixT4<double>::operator*(const VectorT4<double>& vector) const
{
    // the following code uses the following AVX instructions, and needs AVX1 or higher
 
    // AVX1:
    // _mm256_broadcast_sd
    // _mm256_loadu_pd
    // _mm256_mul_pd
    // _mm256_add_pd
    // _mm256_storeu_pd
 
    // we use the same strategy as for the 32 bit float values
 
    // first we load the first vector element in all 64bit elements of the 256 bit register, so that we receive [a, a, a, a]
    __m256d v0 = _mm256_broadcast_sd(vector.data() + 0);
 
    // now we load the first column to receive: [A, B, C, D]
    __m256d c0 = _mm256_loadu_pd(values + 0);
 
    // now we multiply the 256 bit register [A, B, C, D] * [a, a, a, a] = [Aa, Ba, Ca, Da]
    __m256d r0 = _mm256_mul_pd(c0, v0);
 
 
    // now we proceed with the second column
    __m256d v1 = _mm256_broadcast_sd(vector.data() + 1);
    __m256d c1 = _mm256_loadu_pd(values + 4);
    __m256d r1 = _mm256_mul_pd(c1, v1);
 
    // and we sum the result of the first column with the result of the second column
    r0 = _mm256_add_pd(r0, r1);
 
 
    // now we proceed with the third column
    __m256d v2 = _mm256_broadcast_sd(vector.data() + 2);
    __m256d c2 = _mm256_loadu_pd(values + 8);
    __m256d r2 = _mm256_mul_pd(c2, v2);
 
    // we sum the results
    r0 = _mm256_add_pd(r0, r2);
 
 
    // now we proceed with the fourth column
    __m256d v3 = _mm256_broadcast_sd(vector.data() + 3);
    __m256d c3 = _mm256_loadu_pd(values + 12);
    __m256d r3 = _mm256_mul_pd(c3, v3);
 
    // we sum the results
    r0 = _mm256_add_pd(r0, r3);
 
    // and finally we store the results back to the vector
    VectorT4<double> result;
 
    _mm256_storeu_pd(result.data(), r0);
 
    return result;
}
 
#endif // OCEAN_USE_SLOWER_IMPLEMENTATION
 
#else // OCEAN_HARDWARE_AVX_VERSION
 
#if defined(OCEAN_HARDWARE_SSE_VERSION) && OCEAN_HARDWARE_SSE_VERSION >= 20
 
template <>
OCEAN_FORCE_INLINE VectorT4<double> SquareMatrixT4<double>::operator*(const VectorT4<double>& vector) const
{
    // the following code uses the following SSE instructions, and needs SSE2 or higher
 
    // SSE2:
    // _mm_load1_pd
    // _mm_loadu_pd
    // _mm_add_pd
    // _mm_storeu_pd
    // _mm_mul_pd
 
    // we use the following strategy:
    // the values of the matrix are column aligned so that we normally would need to transpose the matrix before we can apply simple SIMD instructions
    // however, we do not transpose the matrix (we avoid the shuffle instructions) and instead multiply the matrix column-wise:
    // finally we sum the four columns and have the result, compared to the transpose-based approach this approach is approx. two times faster
    //
    // A E I M     a     Aa  +  Eb  +  Ic  +  Md
    // B F J N     b     Ba  +  Fb  +  Jc  +  Nd
    // C G K O  *  c  =  Ca  +  Gb  +  Kc  +  Od
    // D H L P     d     Da  +  Hb  +  Lc  +  Pd
 
    // first we load the first vector element in both 64 elements of the 128 bit register, so that we receive [a, a]
    __m128d v0 = _mm_load1_pd(vector.data() + 0);
 
    // now we load the first column to receive: [A, B]  and  [C, D]
    __m128d c0a = _mm_loadu_pd(values + 0);
    __m128d c0b = _mm_loadu_pd(values + 2);
 
    // now we multiply both 128 bit registers by: [A, B] * [a, a] = [Aa, Ba]  and  [C, D] * [a, a] = [Ca, Da]
    __m128d r0a = _mm_mul_pd(c0a, v0);
    __m128d r0b = _mm_mul_pd(c0b, v0);
 
 
    // now we proceed with the second column
    __m128d v1 = _mm_load1_pd(vector.data() + 1);
 
    __m128d c1a = _mm_loadu_pd(values + 4);
    __m128d c1b = _mm_loadu_pd(values + 6);
 
    __m128d r1a = _mm_mul_pd(c1a, v1);
    __m128d r1b = _mm_mul_pd(c1b, v1);
 
    // and we sum the result of the first column with the result of the second column
    r0a = _mm_add_pd(r0a, r1a);
    r0b = _mm_add_pd(r0b, r1b);
 
 
    // now we proceed with the third column
    __m128d v2 = _mm_load1_pd(vector.data() + 2);
 
    __m128d c2a = _mm_loadu_pd(values + 8);
    __m128d c2b = _mm_loadu_pd(values + 10);
 
    __m128d r2a = _mm_mul_pd(c2a, v2);
    __m128d r2b = _mm_mul_pd(c2b, v2);
 
    // we sum the results
    r0a = _mm_add_pd(r0a, r2a);
    r0b = _mm_add_pd(r0b, r2b);
 
 
    // now we proceed with the fourth column
    __m128d v3 = _mm_load1_pd(vector.data() + 3);
 
    __m128d c3a = _mm_loadu_pd(values + 12);
    __m128d c3b = _mm_loadu_pd(values + 14);
 
    __m128d r3a = _mm_mul_pd(c3a, v3);
    __m128d r3b = _mm_mul_pd(c3b, v3);
 
    // we sum the results
    r0a = _mm_add_pd(r0a, r3a);
    r0b = _mm_add_pd(r0b, r3b);
 
    // and finally we store the results back to the vector
    VectorT4<double> result;
 
    _mm_storeu_pd(result.data() + 0, r0a);
    _mm_storeu_pd(result.data() + 2, r0b);
 
    return result;
}
 
#endif // OCEAN_HARDWARE_SSE_VERSION >= 20
 
#endif // OCEAN_HARDWARE_AVX_VERSION >= 10
 
#ifdef OCEAN_USE_SLOWER_IMPLEMENTATION
 
// we keep following implementation inside 'OCEAN_USE_SLOWER_IMPLEMENTATION' showing an alternative (which is slower)
 
#if defined(OCEAN_HARDWARE_SSE_VERSION) && OCEAN_HARDWARE_SSE_VERSION >= 41
 
template <>
OCEAN_FORCE_INLINE VectorT4<float> SquareMatrixT4<float>::operator*(const VectorT4<float>& vector) const
{
    // the following code uses the following SSE instructions, and needs SSE 4.1 or higher
 
    // SSE:
    // _mm_loadu_ps
    // _mm_shuffle_ps
    // _mm_or_ps
    // _mm_storeu_ps
 
    // SSE4.1:
    // _mm_dp_ps
 
    // first we load the first four rows of the matrix
    __m128 row0 = _mm_loadu_ps(values + 0);
    __m128 row1 = _mm_loadu_ps(values + 4);
    __m128 row2 = _mm_loadu_ps(values + 8);
    __m128 row3 = _mm_loadu_ps(values + 12);
 
    // we load the four values of the vector
    __m128 v = _mm_loadu_ps(vector.data());
 
    // first, we transpose the 4x4 matrix
 
    // A E I M     A B C D
    // B F J N     E F G H
    // C G K O     I J K L
    // D H L P     M N O P
 
    // A B C D,     E F G H     ->    A B E F
    __m128 temp0 = _mm_shuffle_ps(row0, row1, 0x44); // 0x44 = 0100 0100
    // A B C D,     E F G H     ->    C D G H
    __m128 temp2 = _mm_shuffle_ps(row0, row1, 0xEE); // 0xEE = 1110 1110
    // I J K L,     M N O P     ->    I J M N
    __m128 temp1 = _mm_shuffle_ps(row2, row3, 0x44);
    // I J K L,     M N O P     ->    K L O P
    __m128 temp3 = _mm_shuffle_ps(row2, row3, 0xEE);
 
    // A B E F,     I J M N     ->    A E I M
    row0 = _mm_shuffle_ps(temp0, temp1, 0x88); // 0x88 = 1000 1000
    // A B E F,     I J M N     ->    B F J N
    row1 = _mm_shuffle_ps(temp0, temp1, 0xDD); // 0xDD = 1101 1101
    // C D G H,     K L O P     ->    C G K O
    row2 = _mm_shuffle_ps(temp2, temp3, 0x88);
    // C D G H,     K L O P     ->    D H L P
    row3 = _mm_shuffle_ps(temp2, temp3, 0xDD);
 
#ifdef OCEAN_COMPILER_MSC
    ocean_assert(row0.m128_f32[0] == values[0] && row0.m128_f32[1] == values[4] && row0.m128_f32[2] == values[ 8] && row0.m128_f32[3] == values[12]);
    ocean_assert(row1.m128_f32[0] == values[1] && row1.m128_f32[1] == values[5] && row1.m128_f32[2] == values[ 9] && row1.m128_f32[3] == values[13]);
    ocean_assert(row2.m128_f32[0] == values[2] && row2.m128_f32[1] == values[6] && row2.m128_f32[2] == values[10] && row2.m128_f32[3] == values[14]);
    ocean_assert(row3.m128_f32[0] == values[3] && row3.m128_f32[1] == values[7] && row3.m128_f32[2] == values[11] && row3.m128_f32[3] == values[15]);
#endif
 
    // we determine the dot product between the first row and the vector and store the result in the first float bin
    row0 = _mm_dp_ps(row0, v, 0xF1); // 0xF1 = 1111 0001
 
    // we determine the dot product between the second row and the vector and store the result in the second float bin
    row1 = _mm_dp_ps(row1, v, 0xF2); // 0xF2 = 1111 0010
    row2 = _mm_dp_ps(row2, v, 0xF4); // 0xF4 = 1111 0100
    row3 = _mm_dp_ps(row3, v, 0xF8); // 0xF8 = 1111 1000
 
    // now we blend the results by applying the bit-wise or operator
    __m128 result01 = _mm_or_ps(row0, row1);
    __m128 result23 = _mm_or_ps(row2, row3);
    __m128 result03 = _mm_or_ps(result01, result23);
 
    VectorT4<float> result;
    _mm_storeu_ps(result.data(), result03);
 
    return result;
}
 
#endif // OCEAN_HARDWARE_SSE_VERSION >= 41
 
#else // OCEAN_USE_SLOWER_IMPLEMENTATION
 
#if defined(OCEAN_HARDWARE_SSE_VERSION) && OCEAN_HARDWARE_SSE_VERSION >= 10
 
template <>
OCEAN_FORCE_INLINE VectorT4<float> SquareMatrixT4<float>::operator*(const VectorT4<float>& vector) const
{
    // the following code uses the following SSE instructions, and needs SSE1 or higher
 
    // SSE1:
    // _mm_load1_ps
    // _mm_loadu_ps
    // _mm_mul_ps
    // _mm_add_ps
    // _mm_storeu_ps
 
 
    // we use the following strategy:
    // the values of the matrix are column aligned so that we normally would need to transpose the matrix before we can apply simple SIMD instructions
    // however, we do not transpose the matrix (we avoid the shuffle instructions) and instead multiply the matrix column-wise:
    // finally we sum the four columns and have the result, compared to the transpose-based approach this approach is approx. two times faster
    //
    // A E I M     a     Aa  +  Eb  +  Ic  +  Md
    // B F J N     b     Ba  +  Fb  +  Jc  +  Nd
    // C G K O  *  c  =  Ca  +  Gb  +  Kc  +  Od
    // D H L P     d     Da  +  Hb  +  Lc  +  Pd
 
    // first we load the first vector element in all 32bit elements of the 128 bit register, so that we receive [a, a, a, a]
    __m128 v0 = _mm_load1_ps(vector.data() + 0);
 
    // now we load the first column to receive: [A, B, C, D]
    __m128 c0 = _mm_loadu_ps(values + 0);
 
    // now we multiply the 128 bit register [A, B, C, D] * [a, a, a, a] = [Aa, Ba, Ca, Da]
    __m128 r0 = _mm_mul_ps(c0, v0);
 
 
    // now we proceed with the second column
    __m128 v1 = _mm_load1_ps(vector.data() + 1);
    __m128 c1 = _mm_loadu_ps(values + 4);
    __m128 r1 = _mm_mul_ps(c1, v1);
 
    // and we sum the result of the first column with the result of the second column
    r0 = _mm_add_ps(r0, r1);
 
 
    // now we proceed with the third column
    __m128 v2 = _mm_load1_ps(vector.data() + 2);
    __m128 c2 = _mm_loadu_ps(values + 8);
    __m128 r2 = _mm_mul_ps(c2, v2);
 
    // we sum the results
    r0 = _mm_add_ps(r0, r2);
 
 
    // now we proceed with the fourth column
    __m128 v3 = _mm_load1_ps(vector.data() + 3);
    __m128 c3 = _mm_loadu_ps(values + 12);
    __m128 r3 = _mm_mul_ps(c3, v3);
 
    // we sum the results
    r0 = _mm_add_ps(r0, r3);
 
    // and finally we store the results back to the vector
    VectorT4<float> result;
 
    _mm_storeu_ps(result.data(), r0);
 
    return result;
}
 
#endif // OCEAN_HARDWARE_SSE_VERSION >= 10
 
#endif // OCEAN_USE_SLOWER_IMPLEMENTATION
 
#if defined(OCEAN_HARDWARE_NEON_VERSION) && OCEAN_HARDWARE_NEON_VERSION >= 10
 
#ifdef __aarch64__
 
template <>
OCEAN_FORCE_INLINE VectorT4<double> SquareMatrixT4<double>::operator*(const VectorT4<double>& vector) const
{
    // the following NEON code is almost identical to the SSE implementation
 
    // we use the following strategy:
    // the values of the matrix are column aligned so that we normally would need to transpose the matrix before we can apply simple SIMD instructions
    // however, we do not transpose the matrix (we avoid the shuffle instructions) and instead multiply the matrix column-wise:
    // finally we sum the four columns and have the result, compared to the transpose-based approach this approach is approx. two times faster
    //
    // A E I M     a     Aa  +  Eb  +  Ic  +  Md
    // B F J N     b     Ba  +  Fb  +  Jc  +  Nd
    // C G K O  *  c  =  Ca  +  Gb  +  Kc  +  Od
    // D H L P     d     Da  +  Hb  +  Lc  +  Pd
 
    // first we load the first vector element in both 64 elements of the 128 bit register, so that we receive [a, a]
    float64x2_t v0 = vld1q_dup_f64(vector.data() + 0);
 
    // now we load the first column to receive: [A, B]  and  [C, D]
    float64x2_t c0a = vld1q_f64(values + 0);
    float64x2_t c0b = vld1q_f64(values + 2);
 
    // now we multiply both 128 bit registers by: [A, B] * [a, a] = [Aa, Ba]  and  [C, D] * [a, a] = [Ca, Da]
    float64x2_t r0a = vmulq_f64(c0a, v0);
    float64x2_t r0b = vmulq_f64(c0b, v0);
 
 
    // now we proceed with the second column
    float64x2_t v1 = vld1q_dup_f64(vector.data() + 1);
 
    float64x2_t c1a = vld1q_f64(values + 4);
    float64x2_t c1b = vld1q_f64(values + 6);
 
    float64x2_t r1a = vmulq_f64(c1a, v1);
    float64x2_t r1b = vmulq_f64(c1b, v1);
 
    // and we sum the result of the first column with the result of the second column
    r0a = vaddq_f64(r0a, r1a);
    r0b = vaddq_f64(r0b, r1b);
 
 
    // now we proceed with the third column
    float64x2_t v2 = vld1q_dup_f64(vector.data() + 2);
 
    float64x2_t c2a = vld1q_f64(values + 8);
    float64x2_t c2b = vld1q_f64(values + 10);
 
    float64x2_t r2a = vmulq_f64(c2a, v2);
    float64x2_t r2b = vmulq_f64(c2b, v2);
 
    // we sum the results
    r0a = vaddq_f64(r0a, r2a);
    r0b = vaddq_f64(r0b, r2b);
 
 
    // now we proceed with the fourth column
    float64x2_t v3 = vld1q_dup_f64(vector.data() + 3);
 
    float64x2_t c3a = vld1q_f64(values + 12);
    float64x2_t c3b = vld1q_f64(values + 14);
 
    float64x2_t r3a = vmulq_f64(c3a, v3);
    float64x2_t r3b = vmulq_f64(c3b, v3);
 
    // we sum the results
    r0a = vaddq_f64(r0a, r3a);
    r0b = vaddq_f64(r0b, r3b);
 
    // and finally we store the results back to the vector
    VectorT4<double> result;
 
    vst1q_f64(result.data() + 0, r0a);
    vst1q_f64(result.data() + 2, r0b);
 
    return result;
}
 
#endif // __aarch64__
 
template <>
OCEAN_FORCE_INLINE VectorT4<float> SquareMatrixT4<float>::operator*(const VectorT4<float>& vector) const
{
    // the following NEON code is almost identical to the SSE implementation
 
    // we use the following strategy:
    // the values of the matrix are column aligned so that we normally would need to transpose the matrix before we can apply simple SIMD instructions
    // however, we do not transpose the matrix (we avoid the shuffle instructions) and instead multiply the matrix column-wise:
    // finally we sum the four columns and have the result, compared to the transpose-based approach this approach is approx. two times faster
    //
    // A E I M     a     Aa  +  Eb  +  Ic  +  Md
    // B F J N     b     Ba  +  Fb  +  Jc  +  Nd
    // C G K O  *  c  =  Ca  +  Gb  +  Kc  +  Od
    // D H L P     d     Da  +  Hb  +  Lc  +  Pd
 
    // first we load the first vector element in all 32bit elements of the 128 bit register, so that we receive [a, a, a, a]
    float32x4_t v0 = vld1q_dup_f32(vector.data() + 0);
 
    // now we load the first column to receive: [A, B, C, D]
    float32x4_t c0 = vld1q_f32(values + 0);
 
    // now we multiply the 128 bit register [A, B, C, D] * [a, a, a, a] = [Aa, Ba, Ca, Da]
    float32x4_t r0 = vmulq_f32(c0, v0);
 
 
    // now we proceed with the second column
    float32x4_t v1 = vld1q_dup_f32(vector.data() + 1);
    float32x4_t c1 = vld1q_f32(values + 4);
    float32x4_t r1 = vmulq_f32(c1, v1);
 
    // and we sum the result of the first column with the result of the second column
    r0 = vaddq_f32(r0, r1);
 
 
    // now we proceed with the third column
    float32x4_t v2 = vld1q_dup_f32(vector.data() + 2);
    float32x4_t c2 = vld1q_f32(values + 8);
    float32x4_t r2 = vmulq_f32(c2, v2);
 
    // we sum the results
    r0 = vaddq_f32(r0, r2);
 
 
    // now we proceed with the fourth column
    float32x4_t v3 = vld1q_dup_f32(vector.data() + 3);
    float32x4_t c3 = vld1q_f32(values + 12);
    float32x4_t r3 = vmulq_f32(c3, v3);
 
    // we sum the results
    r0 = vaddq_f32(r0, r3);
 
    // and finally we store the results back to the vector
    VectorT4<float> result;
 
    vst1q_f32(result.data(), r0);
 
    return result;
}
 
#endif
 
template <typename T>
SquareMatrixT4<T> SquareMatrixT4<T>::operator*(const T value) const
{
    SquareMatrixT4<T> result(*this);
 
    result.values[0] *= value;
    result.values[1] *= value;
    result.values[2] *= value;
    result.values[3] *= value;
    result.values[4] *= value;
    result.values[5] *= value;
    result.values[6] *= value;
    result.values[7] *= value;
    result.values[8] *= value;
    result.values[9] *= value;
    result.values[10] *= value;
    result.values[11] *= value;
    result.values[12] *= value;
    result.values[13] *= value;
    result.values[14] *= value;
    result.values[15] *= value;
 
    return result;
}
 
template <typename T>
SquareMatrixT4<T>& SquareMatrixT4<T>::operator*=(const T value)
{
    values[0] *= value;
    values[1] *= value;
    values[2] *= value;
    values[3] *= value;
    values[4] *= value;
    values[5] *= value;
    values[6] *= value;
    values[7] *= value;
    values[8] *= value;
    values[9] *= value;
    values[10] *= value;
    values[11] *= value;
    values[12] *= value;
    values[13] *= value;
    values[14] *= value;
    values[15] *= value;
 
    return *this;
}
 
template <typename T>
SquareMatrixT4<T> SquareMatrixT4<T>::projectionMatrix(const T fovX, const T aspectRatio, const T nearDistance, const T farDistance)
{
    /*
     * <pre>
     * Creates the following frustum projection matrix.
     *
     *  --------------------------------------------------
     * |   t/a          0             0              0     |
     * |    0           t             0              0     |
     * |    0           0        (f+n)/(n-f)    -2fn/(n-f) |
     * |    0           0            -1              0     |
     *  --------------------------------------------------
     *
     * With: t = 1 / tan (fovY / 2), a = aspectRatio, n = nearDistance, f = farDistance
     * </pre>
     */
 
    ocean_assert(fovX > 0 && fovX < NumericT<T>::pi());
    ocean_assert(aspectRatio > 0);
    ocean_assert(nearDistance > 0);
    ocean_assert(nearDistance < farDistance);
 
    const T fovY = T(2.0) * NumericT<T>::atan(NumericT<T>::tan(T(0.5) * fovX) / aspectRatio);
 
    SquareMatrixT4<T> matrix(false);
    ocean_assert(matrix(1, 0) == 0 && matrix(2, 0) == 0 && matrix(3, 0) == 0);
    ocean_assert(matrix(0, 1) == 0 && matrix(2, 1) == 0 && matrix(3, 1) == 0);
    ocean_assert(matrix(0, 2) == 0 && matrix(1, 2) == 0);
    ocean_assert(matrix(0, 3) == 0 && matrix(1, 3) == 0 && matrix(3, 3) == 0);
 
    ocean_assert(NumericT<T>::isNotEqual(farDistance, nearDistance));
 
    const T f = T(1.0) / NumericT<T>::tan(fovY * T(0.5));
    const T factor = T(1.0) / (nearDistance - farDistance);
 
    matrix(0, 0) = f / aspectRatio;
    matrix(1, 1) = f;
    matrix(2, 2) = (farDistance + nearDistance) * factor;
    matrix(3, 2) = -T(1.0);
    matrix(2, 3) = (T(2.0) * farDistance * nearDistance) * factor;
 
    return matrix;
}
 
template <typename T>
SquareMatrixT4<T> SquareMatrixT4<T>::projectionMatrix(const AnyCameraT<T>& anyCamera, const T nearDistance, const T farDistance)
{
    /*
     * <pre>
     * Creates the following frustum projection matrix.
     *
     *  --------------------------------------------------
     * |   Fx           0             mx              0    |
     * |    0          Fy             my              0    |
     * |    0           0        (f+n)/(n-f)    -2fn/(n-f) |
     * |    0           0            -1              0     |
     *  --------------------------------------------------
     *
     * n = nearDistance, f = farDistance
     * </pre>
     */
 
    ocean_assert(anyCamera.isValid());
 
    ocean_assert(nearDistance > 0);
    ocean_assert(nearDistance < farDistance);
 
    const T fxPixel = anyCamera.focalLengthX();
    const T fyPixel = anyCamera.focalLengthY();
    ocean_assert(fxPixel > T(1) && fyPixel > T(1));
 
    const T mxPixel = anyCamera.principalPointX();
    const T myPixel = anyCamera.principalPointY();
 
    const T width_2 = T(anyCamera.width()) / T(2);
    const T height_2 = T(anyCamera.height()) / T(2);
 
    ocean_assert(NumericT<T>::isNotEqualEps(width_2));
    ocean_assert(NumericT<T>::isNotEqualEps(height_2));
 
    const T fx = fxPixel / width_2;
    const T fy = fyPixel / height_2;
 
    const T mx = (mxPixel - width_2) / width_2; // principal point with range [-1, 1]
    const T my = (myPixel - height_2) / height_2;
 
    const T factor = T(1.0) / (nearDistance - farDistance);
 
    SquareMatrixT4<T> matrix(false);
 
    matrix(0, 0) = fx;
    matrix(1, 1) = fy;
    matrix(0, 2) = -mx;
    matrix(1, 2) = my;
    matrix(2, 2) = (farDistance + nearDistance) * factor;
    matrix(3, 2) = -T(1);
    matrix(2, 3) = (T(2) * farDistance * nearDistance) * factor;
 
    return matrix;
}
 
template <typename T>
SquareMatrixT4<T> SquareMatrixT4<T>::frustumMatrix(const T left, const T right, const T top, const T bottom, const T nearDistance, const T farDistance)
{
    /**
     * <pre>
     * Creates the following frustum projection matrix:
     *
     *  --------------------------------------------------
     * | 2n/(r-l)       0       (r+l)/(r-l)         0     |
     * |    0        2n/(t-b)   (t+b)/(t-b)         0     |
     * |    0           0      -(f+n)/(f-n)    -2fn/(f-n) |
     * |    0           0           -1              0     |
     *  --------------------------------------------------
     * </pre>
     */
 
    ocean_assert(NumericT<T>::isNotEqual(left, right));
    ocean_assert(NumericT<T>::isNotEqual(top, bottom));
    ocean_assert(NumericT<T>::isNotEqual(nearDistance, farDistance));
 
    SquareMatrixT4<T> matrix(false);
    ocean_assert(matrix(1, 0) == 0 && matrix(2, 0) == 0 && matrix(3, 0) == 0);
    ocean_assert(matrix(0, 1) == 0 && matrix(2, 1) == 0 && matrix(3, 1) == 0);
    ocean_assert(matrix(0, 3) == 0 && matrix(1, 3) == 0 && matrix(3, 3) == 0);
 
    const T rightLeft = T(1.0) / (right - left);
    const T near2 = nearDistance * T(2.0);
 
    matrix(0, 0) = near2 * rightLeft;
    matrix(0, 2) = (right + left) * rightLeft;
 
    const T topBottom = T(1.0) / (top - bottom);
 
    matrix(1, 1) = near2 * topBottom;
    matrix(1, 2) = (top + bottom) * topBottom;
 
    const T farNear = T(1.0) / (farDistance - nearDistance);
 
    matrix(2, 2) = -(farDistance + nearDistance) * farNear;
    matrix(2, 3) = -T(2.0) * farDistance * nearDistance * farNear;
 
    matrix(3, 2) = -T(1.0);
 
    return matrix;
}
 
template <typename T>
SquareMatrixT4<T> SquareMatrixT4<T>::frustumMatrix(const T width, const T height, const HomogenousMatrixT4<T>& viewingMatrix, const T nearDistance, const T farDistance)
{
    ocean_assert(width > NumericT<T>::eps() && height > NumericT<T>::eps());
    ocean_assert(nearDistance >= NumericT<T>::eps() && farDistance > nearDistance);
 
    const T planeDistance = NumericT<T>::abs(viewingMatrix.translation().z());
    ocean_assert(viewingMatrix.isValid() && NumericT<T>::isNotEqualEps(planeDistance));
 
    const HomogenousMatrixT4<T> inversedViewingMatrix(viewingMatrix.inverted());
 
    const VectorT3<T> leftTop(width * -T(0.5), height * T(0.5), 0);
    const VectorT3<T> rightBottom(width * T(0.5), height * -T(0.5), 0);
 
    const VectorT3<T> leftTopInCamera(inversedViewingMatrix * leftTop);
    const VectorT3<T> rightBottomInCamera(inversedViewingMatrix * rightBottom);
 
    const T factor = nearDistance / planeDistance;
 
    return frustumMatrix(factor * leftTopInCamera.x(), factor * rightBottomInCamera.x(), factor * leftTopInCamera.y(), factor * rightBottomInCamera.y(), nearDistance, farDistance);
}
 
template <typename T>
void SquareMatrixT4<T>::multiply(const SquareMatrixT4<T>& matrix, const VectorT4<T>* vectors, VectorT4<T>* results, const size_t number)
{
    ocean_assert((vectors && results) || number == 0);
 
    for (size_t n = 0; n < number; ++n)
    {
        results[n] = matrix * vectors[n];
    }
}
 
#if defined(OCEAN_HARDWARE_AVX_VERSION) && OCEAN_HARDWARE_AVX_VERSION >= 10
 
template <>
inline void SquareMatrixT4<double>::multiply(const SquareMatrixT4<double>& matrix, const VectorT4<double>* vectors, VectorT4<double>* results, const size_t number)
{
    // the following code uses the following AVX instructions, and needs AVX1 or higher
 
    // AVX1:
    // _mm256_broadcast_sd
    // _mm256_loadu_pd
    // _mm256_mul_pd
    // _mm256_add_pd
    // _mm256_storeu_pd
 
    // we use the same strategy as for the 32 bit float values
 
    __m256d c0 = _mm256_loadu_pd(matrix.values + 0);
    __m256d c1 = _mm256_loadu_pd(matrix.values + 4);
    __m256d c2 = _mm256_loadu_pd(matrix.values + 8);
    __m256d c3 = _mm256_loadu_pd(matrix.values + 12);
 
    for (size_t n = 0; n < number; ++n)
    {
        __m256d v0 = _mm256_broadcast_sd(vectors[n].data() + 0);
        __m256d r0 = _mm256_mul_pd(c0, v0);
 
        __m256d v1 = _mm256_broadcast_sd(vectors[n].data() + 1);
        __m256d r1 = _mm256_mul_pd(c1, v1);
 
        r0 = _mm256_add_pd(r0, r1);
 
        __m256d v2 = _mm256_broadcast_sd(vectors[n].data() + 2);
        __m256d r2 = _mm256_mul_pd(c2, v2);
 
        r0 = _mm256_add_pd(r0, r2);
 
        __m256d v3 = _mm256_broadcast_sd(vectors[n].data() + 3);
        __m256d r3 = _mm256_mul_pd(c3, v3);
 
        r0 = _mm256_add_pd(r0, r3);
 
        _mm256_storeu_pd(results[n].data(), r0);
    }
}
 
#else // OCEAN_HARDWARE_AVX_VERSION
 
#ifdef OCEAN_USE_SLOWER_IMPLEMENTATION
 
// we keep following implementation inside 'OCEAN_USE_SLOWER_IMPLEMENTATION' showing an alternative (which is slower)
 
#if defined(OCEAN_HARDWARE_SSE_VERSION) && OCEAN_HARDWARE_SSE_VERSION >= 41
 
template <>
inline void SquareMatrixT4<float>::multiply(const SquareMatrixT4<float>& matrix, const VectorT4<float>* vectors, VectorT4<float>* results, const size_t number)
{
    // the following code uses the following SSE instructions, and needs SSE 4.1 or higher
 
    // SSE:
    // _mm_loadu_ps
    // _mm_shuffle_ps
    // _mm_or_ps
    // _mm_storeu_ps
 
    // SSE4.1:
    // _mm_dp_ps
 
    // first we load the four rows of the matrix
    __m128 row0 = _mm_loadu_ps(matrix.values + 0);
    __m128 row1 = _mm_loadu_ps(matrix.values + 4);
    __m128 row2 = _mm_loadu_ps(matrix.values + 8);
    __m128 row3 = _mm_loadu_ps(matrix.values + 12);
 
    // first, we transpose the 4x4 matrix
 
    // A E I M     A B C D
    // B F J N     E F G H
    // C G K O     I J K L
    // D H L P     M N O P
 
    // A B C D,     E F G H     ->    A B E F
    __m128 temp0 = _mm_shuffle_ps(row0, row1, 0x44); // 0x44 = 0100 0100
    // A B C D,     E F G H     ->    C D G H
    __m128 temp2 = _mm_shuffle_ps(row0, row1, 0xEE); // 0xEE = 1110 1110
    // I J K L,     M N O P     ->    I J M N
    __m128 temp1 = _mm_shuffle_ps(row2, row3, 0x44);
    // I J K L,     M N O P     ->    K L O P
    __m128 temp3 = _mm_shuffle_ps(row2, row3, 0xEE);
 
    // A B E F,     I J M N     ->    A E I M
    row0 = _mm_shuffle_ps(temp0, temp1, 0x88); // 0x88 = 1000 1000
    // A B E F,     I J M N     ->    B F J N
    row1 = _mm_shuffle_ps(temp0, temp1, 0xDD); // 0xDD = 1101 1101
    // C D G H,     K L O P     ->    C G K O
    row2 = _mm_shuffle_ps(temp2, temp3, 0x88);
    // C D G H,     K L O P     ->    D H L P
    row3 = _mm_shuffle_ps(temp2, temp3, 0xDD);
 
#ifdef OCEAN_COMPILER_MSC
    ocean_assert(row0.m128_f32[0] == matrix.values[0] && row0.m128_f32[1] == matrix.values[4] && row0.m128_f32[2] == matrix.values[ 8] && row0.m128_f32[3] == matrix.values[12]);
    ocean_assert(row1.m128_f32[0] == matrix.values[1] && row1.m128_f32[1] == matrix.values[5] && row1.m128_f32[2] == matrix.values[ 9] && row1.m128_f32[3] == matrix.values[13]);
    ocean_assert(row2.m128_f32[0] == matrix.values[2] && row2.m128_f32[1] == matrix.values[6] && row2.m128_f32[2] == matrix.values[10] && row2.m128_f32[3] == matrix.values[14]);
    ocean_assert(row3.m128_f32[0] == matrix.values[3] && row3.m128_f32[1] == matrix.values[7] && row3.m128_f32[2] == matrix.values[11] && row3.m128_f32[3] == matrix.values[15]);
#endif
 
    for (size_t n = 0u; n < number; ++n)
    {
        // we load the four values of the vector
        __m128 v = _mm_loadu_ps(vectors[n].data());
 
        // we determine the dot product between the first row and the vector and store the result in the first float bin
        __m128 dot0 = _mm_dp_ps(row0, v, 0xF1); // 0xF1 = 1111 0001
 
        // we determine the dot product between the second row and the vector and store the result in the second float bin
        __m128 dot1 = _mm_dp_ps(row1, v, 0xF2); // 0xF2 = 1111 0010
        __m128 dot2 = _mm_dp_ps(row2, v, 0xF4); // 0xF4 = 1111 0100
        __m128 dot3 = _mm_dp_ps(row3, v, 0xF8); // 0xF8 = 1111 1000
 
        // now we blend the results by applying the bit-wise or operator
        __m128 result01 = _mm_or_ps(dot0, dot1);
        __m128 result23 = _mm_or_ps(dot2, dot3);
        __m128 result03 = _mm_or_ps(result01, result23);
 
        _mm_storeu_ps(results[n].data(), result03);
    }
}
 
#endif // OCEAN_HARDWARE_SSE_VERSION >= 41
 
#else // OCEAN_USE_SLOWER_IMPLEMENTATION
 
#if defined(OCEAN_HARDWARE_SSE_VERSION) && OCEAN_HARDWARE_SSE_VERSION >= 20
 
template <>
inline void SquareMatrixT4<double>::multiply(const SquareMatrixT4<double>& matrix, const VectorT4<double>* vectors, VectorT4<double>* results, const size_t number)
{
    // the following code uses the following SSE instructions, and needs SSE2 or higher
 
    // SSE2:
    // _mm_load1_pd
    // _mm_loadu_pd
    // _mm_add_pd
    // _mm_storeu_pd
    // _mm_mul_pd
 
    // now we load the four columns
    __m128d c0a = _mm_loadu_pd(matrix.values + 0);
    __m128d c0b = _mm_loadu_pd(matrix.values + 2);
    __m128d c1a = _mm_loadu_pd(matrix.values + 4);
    __m128d c1b = _mm_loadu_pd(matrix.values + 6);
    __m128d c2a = _mm_loadu_pd(matrix.values + 8);
    __m128d c2b = _mm_loadu_pd(matrix.values + 10);
    __m128d c3a = _mm_loadu_pd(matrix.values + 12);
    __m128d c3b = _mm_loadu_pd(matrix.values + 14);
 
    for (size_t n = 0u; n < number; ++n)
    {
        __m128d v0 = _mm_load1_pd(vectors[n].data() + 0);
        __m128d v1 = _mm_load1_pd(vectors[n].data() + 1);
        __m128d v2 = _mm_load1_pd(vectors[n].data() + 2);
        __m128d v3 = _mm_load1_pd(vectors[n].data() + 3);
 
        // first column
        __m128d r0a = _mm_mul_pd(c0a, v0);
        __m128d r0b = _mm_mul_pd(c0b, v0);
 
        // second column
        __m128d r1a = _mm_mul_pd(c1a, v1);
        __m128d r1b = _mm_mul_pd(c1b, v1);
 
        r0a = _mm_add_pd(r0a, r1a);
        r0b = _mm_add_pd(r0b, r1b);
 
        // third column
        __m128d r2a = _mm_mul_pd(c2a, v2);
        __m128d r2b = _mm_mul_pd(c2b, v2);
        r0a = _mm_add_pd(r0a, r2a);
        r0b = _mm_add_pd(r0b, r2b);
 
        // fourth column
        __m128d r3a = _mm_mul_pd(c3a, v3);
        __m128d r3b = _mm_mul_pd(c3b, v3);
        r0a = _mm_add_pd(r0a, r3a);
        r0b = _mm_add_pd(r0b, r3b);
 
        _mm_storeu_pd(results[n].data() + 0, r0a);
        _mm_storeu_pd(results[n].data() + 2, r0b);
    }
}
 
#endif // OCEAN_HARDWARE_SSE_VERSION >= 20
 
#if defined(OCEAN_HARDWARE_SSE_VERSION) && OCEAN_HARDWARE_SSE_VERSION >= 10
 
template <>
inline void SquareMatrixT4<float>::multiply(const SquareMatrixT4<float>& matrix, const VectorT4<float>* vectors, VectorT4<float>* results, const size_t number)
{
    // the following code uses the following SSE instructions, and needs SSE1 or higher
 
    // SSE1:
    // _mm_load1_ps
    // _mm_loadu_ps
    // _mm_mul_ps
    // _mm_add_ps
    // _mm_storeu_ps
 
    // now we load the four columns
    __m128 c0 = _mm_loadu_ps(matrix.values + 0);
    __m128 c1 = _mm_loadu_ps(matrix.values + 4);
    __m128 c2 = _mm_loadu_ps(matrix.values + 8);
    __m128 c3 = _mm_loadu_ps(matrix.values + 12);
 
    for (size_t n = 0u; n < number; ++n)
    {
        __m128 v0 = _mm_load1_ps(vectors[n].data() + 0);
        __m128 v1 = _mm_load1_ps(vectors[n].data() + 1);
        __m128 v2 = _mm_load1_ps(vectors[n].data() + 2);
        __m128 v3 = _mm_load1_ps(vectors[n].data() + 3);
 
        // first column
        __m128 r0 = _mm_mul_ps(c0, v0);
 
        // second column
        __m128 r1 = _mm_mul_ps(c1, v1);
 
        r0 = _mm_add_ps(r0, r1);
 
        // third column
        __m128 r2 = _mm_mul_ps(c2, v2);
        r0 = _mm_add_ps(r0, r2);
 
        // fourth column
        __m128 r3 = _mm_mul_ps(c3, v3);
        r0 = _mm_add_ps(r0, r3);
 
        _mm_storeu_ps(results[n].data(), r0);
    }
}
 
#endif // OCEAN_HARDWARE_SSE_VERSION >= 10
 
#endif // OCEAN_USE_SLOWER_IMPLEMENTATION
 
#endif // OCEAN_HARDWARE_AVX_VERSION
 
#if defined(OCEAN_HARDWARE_NEON_VERSION) && OCEAN_HARDWARE_NEON_VERSION >= 10
 
#ifdef __aarch64__
 
template <>
inline void SquareMatrixT4<double>::multiply(const SquareMatrixT4<double>& matrix, const VectorT4<double>* vectors, VectorT4<double>* results, const size_t number)
{
    // the following NEON code is almost identical to the SSE implementation
 
    // now we load the four columns
    float64x2_t c0a = vld1q_f64(matrix.values + 0);
    float64x2_t c0b = vld1q_f64(matrix.values + 2);
    float64x2_t c1a = vld1q_f64(matrix.values + 4);
    float64x2_t c1b = vld1q_f64(matrix.values + 6);
    float64x2_t c2a = vld1q_f64(matrix.values + 8);
    float64x2_t c2b = vld1q_f64(matrix.values + 10);
    float64x2_t c3a = vld1q_f64(matrix.values + 12);
    float64x2_t c3b = vld1q_f64(matrix.values + 14);
 
    for (size_t n = 0u; n < number; ++n)
    {
        float64x2_t v0 = vld1q_dup_f64(vectors[n].data() + 0);
        float64x2_t v1 = vld1q_dup_f64(vectors[n].data() + 1);
        float64x2_t v2 = vld1q_dup_f64(vectors[n].data() + 2);
        float64x2_t v3 = vld1q_dup_f64(vectors[n].data() + 3);
 
        float64x2_t r0a = vmulq_f64(c0a, v0);
        float64x2_t r0b = vmulq_f64(c0b, v0);
 
        float64x2_t r1a = vmulq_f64(c1a, v1);
        float64x2_t r1b = vmulq_f64(c1b, v1);
 
        r0a = vaddq_f64(r0a, r1a);
        r0b = vaddq_f64(r0b, r1b);
 
        float64x2_t r2a = vmulq_f64(c2a, v2);
        float64x2_t r2b = vmulq_f64(c2b, v2);
 
        r0a = vaddq_f64(r0a, r2a);
        r0b = vaddq_f64(r0b, r2b);
 
        float64x2_t r3a = vmulq_f64(c3a, v3);
        float64x2_t r3b = vmulq_f64(c3b, v3);
 
        r0a = vaddq_f64(r0a, r3a);
        r0b = vaddq_f64(r0b, r3b);
 
        vst1q_f64(results[n].data() + 0, r0a);
        vst1q_f64(results[n].data() + 2, r0b);
    }
}
 
#endif // __aarch64__
 
template <>
inline void SquareMatrixT4<float>::multiply(const SquareMatrixT4<float>& matrix, const VectorT4<float>* vectors, VectorT4<float>* results, const size_t number)
{
    // the following NEON code is almost identical to the SSE implementation
 
    // now we load the four columns
    float32x4_t c0 = vld1q_f32(matrix.values + 0);
    float32x4_t c1 = vld1q_f32(matrix.values + 4);
    float32x4_t c2 = vld1q_f32(matrix.values + 8);
    float32x4_t c3 = vld1q_f32(matrix.values + 12);
 
    for (size_t n = 0u; n < number; ++n)
    {
        float32x4_t v0 = vld1q_dup_f32(vectors[n].data() + 0);
        float32x4_t v1 = vld1q_dup_f32(vectors[n].data() + 1);
        float32x4_t v2 = vld1q_dup_f32(vectors[n].data() + 2);
        float32x4_t v3 = vld1q_dup_f32(vectors[n].data() + 3);
 
        // first column
        float32x4_t r0 = vmulq_f32(c0, v0);
 
        // second column
        float32x4_t r1 = vmulq_f32(c1, v1);
 
        r0 = vaddq_f32(r0, r1);
 
        // third column
        float32x4_t r2 = vmulq_f32(c2, v2);
 
        r0 = vaddq_f32(r0, r2);
 
        // fourth column
        float32x4_t r3 = vmulq_f32(c3, v3);
 
        r0 = vaddq_f32(r0, r3);
 
        vst1q_f32(results[n].data(), r0);
    }
}
 
#endif
 
template <typename T>
template <typename U>
inline std::vector< SquareMatrixT4<T> > SquareMatrixT4<T>::matrices2matrices(const std::vector< SquareMatrixT4<U> >& matrices)
{
    std::vector< SquareMatrixT4<T> > result;
    result.reserve(matrices.size());
 
    for (typename std::vector< SquareMatrixT4<U> >::const_iterator i = matrices.begin(); i != matrices.end(); ++i)
    {
        result.push_back(SquareMatrixT4<T>(*i));
    }
 
    return result;
}
 
template <>
template <>
inline std::vector< SquareMatrixT4<float> > SquareMatrixT4<float>::matrices2matrices(const std::vector< SquareMatrixT4<float> >& matrices)
{
    return matrices;
}
 
template <>
template <>
inline std::vector< SquareMatrixT4<double> > SquareMatrixT4<double>::matrices2matrices(const std::vector< SquareMatrixT4<double> >& matrices)
{
    return matrices;
}
 
template <typename T>
template <typename U>
inline std::vector< SquareMatrixT4<T> > SquareMatrixT4<T>::matrices2matrices(const SquareMatrixT4<U>* matrices, const size_t size)
{
    std::vector< SquareMatrixT4<T> > result;
    result.reserve(size);
 
    for (size_t n = 0; n < size; ++n)
    {
        result.push_back(SquareMatrixT4<T>(matrices[n]));
    }
 
    return result;
}
 
template <typename T>
void SquareMatrixT4<T>::swapRows(const unsigned int row0, const unsigned int row1)
{
    ocean_assert(row0 < 4u && row1 < 4u);
 
    if (row0 == row1)
    {
        return;
    }
 
    T* first = values + row0;
    T* second = values + row1;
 
    T tmp = *first;
    *first = *second;
    *second = tmp;
 
    first += 4;
    second += 4;
    tmp = *first;
    *first = *second;
    *second = tmp;
 
    first += 4;
    second += 4;
    tmp = *first;
    *first = *second;
    *second = tmp;
 
    first += 4;
    second += 4;
    tmp = *first;
    *first = *second;
    *second = tmp;
}
 
template <typename T>
void SquareMatrixT4<T>::multiplyRow(const unsigned int row, const T scalar)
{
    ocean_assert(row < 4u);
 
    T* element = values + row;
 
    *element *= scalar;
    element += 4;
    *element *= scalar;
    element += 4;
    *element *= scalar;
    element += 4;
    *element *= scalar;
}
 
template <typename T>
void SquareMatrixT4<T>::addRows(const unsigned int targetRow, unsigned int const sourceRow, const T scalar)
{
    ocean_assert(targetRow < 4u && sourceRow < 4u);
    ocean_assert(targetRow != sourceRow);
 
    T* target = values + targetRow;
    T* source = values + sourceRow;
 
    *target += *source * scalar;
 
    target += 4;
    source += 4;
    *target += *source * scalar;
 
    target += 4;
    source += 4;
    *target += *source * scalar;
 
    target += 4;
    source += 4;
    *target += *source * scalar;
}
 
template <typename T>
std::ostream& operator<<(std::ostream& stream, const SquareMatrixT4<T>& matrix)
{
    stream << "|" << matrix(0, 0) << ", " << matrix(0, 1) << ", " << matrix(0, 2) << ", " << matrix(0, 3) << "|" << std::endl;
    stream << "|" << matrix(1, 0) << ", " << matrix(1, 1) << ", " << matrix(1, 2) << ", " << matrix(1, 3) << "|" << std::endl;
    stream << "|" << matrix(2, 0) << ", " << matrix(2, 1) << ", " << matrix(2, 2) << ", " << matrix(2, 3) << "|" << std::endl;
    stream << "|" << matrix(3, 0) << ", " << matrix(3, 1) << ", " << matrix(3, 2) << ", " << matrix(3, 3) << "|";
 
    return stream;
}
 
template <bool tActive, typename T>
MessageObject<tActive>& operator<<(MessageObject<tActive>& messageObject, const SquareMatrixT4<T>& matrix)
{
    return messageObject << "|" << matrix(0, 0) << ", " << matrix(0, 1) << ", " << matrix(0, 2) << ", " << matrix(0, 3) << "|\n|"
                            << matrix(1, 0) << ", " << matrix(1, 1) << ", " << matrix(1, 2) << ", " << matrix(1, 3) << "|\n|"
                            << matrix(2, 0) << ", " << matrix(2, 1) << ", " << matrix(2, 2) << ", " << matrix(2, 3) << "|\n|"
                            << matrix(3, 0) << ", " << matrix(3, 1) << ", " << matrix(3, 2) << ", " << matrix(3, 3) << "|";
}
 
template <bool tActive, typename T>
MessageObject<tActive>& operator<<(MessageObject<tActive>&& messageObject, const SquareMatrixT4<T>& matrix)
{
    return messageObject << "|" << matrix(0, 0) << ", " << matrix(0, 1) << ", " << matrix(0, 2) << ", " << matrix(0, 3) << "|\n|"
                            << matrix(1, 0) << ", " << matrix(1, 1) << ", " << matrix(1, 2) << ", " << matrix(1, 3) << "|\n|"
                            << matrix(2, 0) << ", " << matrix(2, 1) << ", " << matrix(2, 2) << ", " << matrix(2, 3) << "|\n|"
                            << matrix(3, 0) << ", " << matrix(3, 1) << ", " << matrix(3, 2) << ", " << matrix(3, 3) << "|";
}
 
}
 
#endif // META_OCEAN_MATH_SQUARE_MATRIX_4_H