gls_matrix.h

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/*! \file
  \brief  The GlsMatrix class.
 
  \par Copyright Information
 
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*/
#ifndef GLS_MATRIX_H
#define GLS_MATRIX_H
 
#include <cstring>
#include <iostream>
#include <math.h>
#include <memory>
 
namespace disti
{
//----------------------------------------------------------------------------
/**
* The GlsMatrix class provides support for an n x n square matrix of any 
* numeric type. Most basic matrix operations are supported.  Internally,
* the matrix terms are stored contiguously in column-major order so that the
* matrix data can be passed to OpenGL matrix operations.  However, instances
* of this class can be treated as row-major matrices since the API performs
* the conversion to column-major ordering for the user.
*/
//----------------------------------------------------------------------------
template<class Type, int DIM = 4>
class GlsMatrix
{
public:
    typedef Type CStyleMatrix[ DIM ][ DIM ]; ///< Typedef for a 4x4 matrix stored as a 2D array.
 
    //------------------------------------------------------------------------
    /** 
    * Default constructor. The new matrix will be the identity matrix.
    */
    //------------------------------------------------------------------------
    GlsMatrix();
 
    //------------------------------------------------------------------------
    /** 
    * Constructor. This will transpose m so that the matrix is stored in
    * column-major order, but a caller can pass one in that is in row-major
    * order, which is more intuitive.
    *
    * \param m  a row-major, two-dimensional array to initialize the matrix 
    *           with.
    */
    //------------------------------------------------------------------------
    GlsMatrix( const CStyleMatrix m );
 
    //------------------------------------------------------------------------
    /** 
    * Constructor.
    *
    * \param m  a column-major, one-dimensional array containing all values
    *           to initialize the matrix with. 
    */
    //------------------------------------------------------------------------
    GlsMatrix( const Type* m );
 
    //------------------------------------------------------------------------
    /** 
    * Copy Constructor.
    *
    * \param m  matrix to copy. 
    */
    //------------------------------------------------------------------------
    GlsMatrix( const GlsMatrix<Type, DIM>& m );
 
    //------------------------------------------------------------------------
    /** 
    * Destructor.
    */
    //------------------------------------------------------------------------
    virtual ~GlsMatrix();
 
    //------------------------------------------------------------------------
    /** 
    * Dump the contents of the matrix to standard output.
    */
    //------------------------------------------------------------------------
    void Dump() const;
 
    //------------------------------------------------------------------------
    /** 
    * Zero out the matrix so that all terms are equal to zero.
    */
    //------------------------------------------------------------------------
    inline void Clear() { memset( _data, 0, sizeof( Type ) * DIM * DIM ); }
 
    /// Provides access to the raw column-major matrix data as a one dimensional
    /// array in a format suitable for Open GL matrix operations.
    /// \return A pointer to the underlying matrix array.
    inline Type* Data() { return _data; }
 
    /// Provides access to the raw column-major matrix data as a one dimensional
    /// array in a format suitable for Open GL matrix operations.
    /// \return A const pointer to the underlying matrix array.
    inline const Type* Data() const { return _data; }
 
    //------------------------------------------------------------------------
    /** 
    * Set this matrix to the identity matrix. The identity matrix is 
    * I = [Aij], where Aii = 1 and Aij = 0 for i != j. Thus, for the identity 
    * matrix, the terms off the main diagonal are all zero, and the terms on 
    * the main diagonal are all one.
    */
    //------------------------------------------------------------------------
    void MakeIdentity();
 
    //------------------------------------------------------------------------
    /** 
    * Calculates the determinant of the matrix.
    *
    * \return the calculated value of the determinant.
    */
    //------------------------------------------------------------------------
    Type Determinant() const;
 
    //------------------------------------------------------------------------
    /** 
    * Calculates the inverse of this matrix. The inverse of matrix A is B if
    * AB = BA = I, where I is the identity matrix. The inverse of a matrix is
    * the notion corresponding to the reciprocal of a nonzero real number.
    *
    * \return the inverse of this matrix.
    */
    //------------------------------------------------------------------------
    GlsMatrix<Type, DIM> Inverse() const;
 
    //------------------------------------------------------------------------
    /** 
    * Sets this matrix equal to its inverse.
    */
    //------------------------------------------------------------------------
    void Invert();
 
    //------------------------------------------------------------------------
    /** 
    * Calculates the transpose of this matrix.  The transpose of a matrix
    * is obtained by interchanging its rows and columns.
    *
    * \return the transpose of this matrix.
    */
    //------------------------------------------------------------------------
    GlsMatrix<Type, DIM> Transposition() const;
 
    //------------------------------------------------------------------------
    /** 
    * Sets this matrix equal to its transpose.  
    */
    //------------------------------------------------------------------------
    void Transpose();
 
    //------------------------------------------------------------------------
    /** 
    * Determines if this is a diagonal matrix.  An n x n matrix A = [Aij] is
    * called a diagonal matrix if Aij = 0 for i != j. Thus, for a diagonal
    * matrix, the terms off the main diagonal are all zero.
    *
    * \return true if the matrix is a diagonal matrix, false otherwise.
    */
    //------------------------------------------------------------------------
    bool IsDiagonal() const;
 
    //------------------------------------------------------------------------
    /** 
    * Determines if this matrix is equal to the identity matrix. The 
    * identity matrix is I = [Aij], where Aii = 1 and Aij = 0 for i != j.
    * Thus, for the identity matrix, the terms off the main diagonal are all
    * zero, and the terms on the main diagonal are all one.
    *
    * \return true if the matrix is the identity matrix, false otherwise.
    */
    //------------------------------------------------------------------------
    bool IsIdentity() const;
 
    //------------------------------------------------------------------------
    /** 
    * Determines if this is a scalar matrix.  A scalar matrix is a diagonal
    * matrix whose diagonal terms are equal.
    *
    * \return true if the matrix is a scalar matrix, false otherwise.
    */
    //------------------------------------------------------------------------
    bool IsScalar() const;
 
    //------------------------------------------------------------------------
    /** 
    * Determines if this matrix is singular, or non-invertible.  A singular
    * matrix has no inverse.
    *
    * \return true if the matrix is singular, false otherwise.
    */
    //------------------------------------------------------------------------
    bool IsSingular() const;
 
    //------------------------------------------------------------------------
    /** 
    * Determines if this matrix is symmetric. A matrix is symmetric if it is
    * equal to its transpose, i.e. A = A'.
    *
    * \return true if the matrix is symmetric, false otherwise.
    */
    //------------------------------------------------------------------------
    bool IsSymmetric() const;
 
    //------------------------------------------------------------------------
    /** 
    * Determines if this matrix is skew symmetric. A matrix is skew symmetric
    * if its transpose is equal to its negative, i.e. -A = A'.
    *
    * \return true if the matrix is skew symmetric, false otherwise.
    */
    //------------------------------------------------------------------------
    bool IsSkewSymmetric() const;
 
    //------------------------------------------------------------------------
    /** 
    * Determines if this matrix is lower triangular. An n x n matrix A = [Aij]
    * is lower triangular if Aij = 0 for i < j.
    *
    * \return true if the matrix is lower triangular, false otherwise.
    */
    //------------------------------------------------------------------------
    bool IsLowerTriangular() const;
 
    //------------------------------------------------------------------------
    /** 
    * Determines if this matrix is upper triangular. An n x n matrix A = [Aij]
    * is upper triangular if Aij = 0 for i > j.
    *
    * \return true if the matrix is upper triangular, false otherwise.
    */
    //------------------------------------------------------------------------
    bool IsUpperTriangular() const;
 
    //------------------------------------------------------------------------
    // Operators
    //------------------------------------------------------------------------
 
    /// Subscript operator (zero-based, no bounds checking).
    /// \param row The row of the element to retrieve.
    /// \param col The column of the element to retrieve.
    /// \return A reference to the element at that position.
    Type& operator()( unsigned row, unsigned col );
 
    /// Subscript operator (zero-based, no bounds checking).
    /// \param row The row of the element to retrieve.
    /// \param col The column of the element to retrieve.
    /// \return The value of the element at that position.
    Type operator()( unsigned row, unsigned col ) const;
 
    // Equality operator
    /// \param rhs The matrix to compare.
    /// \return Whether or not the matrices are equal.
    bool operator==( const GlsMatrix<Type, DIM>& rhs ) const;
 
    // Inequality operator
    /// \param rhs The matrix to compare.
    /// \return whether or not the matrices are not equal.
    bool operator!=( const GlsMatrix<Type, DIM>& rhs ) const;
 
    /// Assignment operator
    /// \param m The matrix to copy from.
    /// \return The resulting matrix (this).
    GlsMatrix<Type, DIM>& operator=( const GlsMatrix<Type, DIM>& m );
 
    /// Assignment operator
    /// \param m The matrix to copy from.
    /// \return The resulting matrix (this).
    GlsMatrix<Type, DIM>& operator=( const CStyleMatrix m );
 
    /// Unary + operator
    /// \return This matrix.
    GlsMatrix<Type, DIM> operator+() const { return *this; }
 
    /// Unary - operator
    /// \return Matrix with all elements multiplied by -1.
    GlsMatrix<Type, DIM> operator-() const;
 
    /// Combined assignment += operator
    /// \param rhs Matrix to add.
    /// \return The resulting matrix (this).
    GlsMatrix<Type, DIM>& operator+=( const GlsMatrix<Type, DIM>& rhs );
 
    /// Combined assignment -= operator
    /// \param rhs Matrix to subtract.
    /// \return The resulting matrix (this).
    GlsMatrix<Type, DIM>& operator-=( const GlsMatrix<Type, DIM>& rhs );
 
    /// Combined assignment *= operator
    /// \param rhs Matrix to multiply by.
    /// \return The resulting matrix (this).
    GlsMatrix<Type, DIM>& operator*=( const GlsMatrix<Type, DIM>& rhs );
 
    /// Combined assignment *= operator
    /// \param scalar Value to multiply by.
    /// \return The resulting matrix (this).
    GlsMatrix<Type, DIM>& operator*=( Type scalar );
 
    /// Combined assignment /= operator
    /// \param scalar Value to divide by.
    /// \return The resulting matrix (this).
    GlsMatrix<Type, DIM>& operator/=( Type scalar );
 
    // Mathematical + operator
    /// \param rhs The matrix to add.
    /// \return The resulting matrix.
    GlsMatrix<Type, DIM> operator+( const GlsMatrix<Type, DIM>& rhs ) const;
 
    // Mathematical - operator
    /// \param rhs The matrix to subtract.
    /// \return The resulting matrix.
    GlsMatrix<Type, DIM> operator-( const GlsMatrix<Type, DIM>& rhs ) const;
 
    // Mathematical * operator
    /// \param rhs The matrix to multiply by.
    /// \return The resulting matrix.
    GlsMatrix<Type, DIM> operator*( const GlsMatrix<Type, DIM>& rhs ) const;
 
    // Mathematical * operator
    /// \param scalar The value to multiply by.
    /// \return The resulting matrix.
    GlsMatrix<Type, DIM> operator*( Type scalar ) const;
 
    // Mathematical / operator
    /// \param scalar The value to divide by.
    /// \return The resulting matrix.
    GlsMatrix<Type, DIM> operator/( Type scalar ) const;
 
protected:
    /** A very tiny number, close to zero. Used to compare floats */
    static const double PRECISION;
 
    /**
    * This is the actual matrix data stored in column-major ordering in
    * a one-dimensional array of the appropriate size to store all matrix
    * terms. The data is stored in this manner so that it can be passed
    * to OpenGL matrix operations, which expect matrices to be ordered
    * this way.
    */
    Type _data[ DIM * DIM ];
 
    /** 
    * A column index array. Each element in this array "points" 
    * to a column of the matrix within the above data. By using this index,
    * we can address matrix terms using two dimensions, column and row, 
    * without incurring a multiplication. Eg. matrix[0] points to a column
    * within data, which is the address of the first term in the column.
    * We can further address into the column and access a particular row
    * by adding another dimension, eg. matrix[0][1].  The indices must
    * be set up at construction time.
    */
    Type* _matrix[ DIM ];
 
    //------------------------------------------------------------------------
    /** 
    * This should be called from all constructors, as it initializes the
    * matrix column index so that matrix elements can be addressed in two
    * dimensions, but a multiplication is not incurred when doing so.
    */
    //------------------------------------------------------------------------
    void Initialize();
 
    /// Pivoting is used in conjunction with Gauss-Jordan reduction to find
    /// the inverse of a matrix and in Gaussian Elimination to find the
    /// determinant of a matrix.  It involves the elementary row operation of
    /// interchanging rows (or columns in our column-major matrix).  Only the
    /// row passed in and rows below it are considered.  The row passed in
    /// will be swapped with another row below it by selecting the desired
    /// pivot element.  The pivot is the largest term (in magnitude) in
    /// the column equal to row for any of the considered rows.  This will
    /// always put the largest (in magnitude) term from the column on the
    /// diagonal of the matrix.
    /// \param row is the current row to pivot about. Rows above this are not
    ///        considered during pivoting, as these rows should already be in
    ///        their proper form.
    /// \return the row number index that was interchanged, if the pivot is different to the incoming row
    ///         0 if the pivot was successful on the incoming row
    ///        -1 if no pivot is possible
    int Pivot( unsigned row );
 
    /// Decides if the two values are close together determined by the constant PRECISION.
    /// The formula is: lv - PRECISION < rv < lv + PRECISION
    /// \param lv The left value.
    /// \param rv The right value.
    /// \return Whether or not the two values are close together.
    bool closeValues( Type lv, Type rv ) const;
};
 
//============================================================================
//
//                         CLASS IMPLEMENTATION
//
//============================================================================
 
//----------------------------------------------------------------------------
template<class Type, int DIM>
const double GlsMatrix<Type, DIM>::PRECISION = 1e-7;
 
//----------------------------------------------------------------------------
template<class Type, int DIM>
GlsMatrix<Type, DIM>::GlsMatrix()
{
    Initialize();
    MakeIdentity();
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
GlsMatrix<Type, DIM>::GlsMatrix( const CStyleMatrix m )
{
    Initialize();
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            _matrix[ col ][ row ] = m[ row ][ col ];
        }
    }
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
GlsMatrix<Type, DIM>::GlsMatrix( const Type* m )
{
    Initialize();
    memcpy( _data, m, sizeof( Type ) * DIM * DIM );
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
GlsMatrix<Type, DIM>::GlsMatrix( const GlsMatrix<Type, DIM>& m )
{
    Initialize();
    memcpy( _data, m._data, sizeof( Type ) * DIM * DIM );
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
GlsMatrix<Type, DIM>::~GlsMatrix()
{
}
 
/// Mathematical * operator
/// \param rhs The matrix to multiply by.
/// \return The resulting matrix.
template<>
inline GlsMatrix<double, 4>
GlsMatrix<double, 4>::operator*( const GlsMatrix<double, 4>& rhs ) const
{
    GlsMatrix<double, 4> result;
 
    // Same as regular, just unroll the loop
    result._matrix[ 0 ][ 0 ] = _matrix[ 0 ][ 0 ] * rhs._matrix[ 0 ][ 0 ] + _matrix[ 1 ][ 0 ] * rhs._matrix[ 0 ][ 1 ] + _matrix[ 2 ][ 0 ] * rhs._matrix[ 0 ][ 2 ] + _matrix[ 3 ][ 0 ] * rhs._matrix[ 0 ][ 3 ];
    result._matrix[ 1 ][ 0 ] = _matrix[ 0 ][ 0 ] * rhs._matrix[ 1 ][ 0 ] + _matrix[ 1 ][ 0 ] * rhs._matrix[ 1 ][ 1 ] + _matrix[ 2 ][ 0 ] * rhs._matrix[ 1 ][ 2 ] + _matrix[ 3 ][ 0 ] * rhs._matrix[ 1 ][ 3 ];
    result._matrix[ 2 ][ 0 ] = _matrix[ 0 ][ 0 ] * rhs._matrix[ 2 ][ 0 ] + _matrix[ 1 ][ 0 ] * rhs._matrix[ 2 ][ 1 ] + _matrix[ 2 ][ 0 ] * rhs._matrix[ 2 ][ 2 ] + _matrix[ 3 ][ 0 ] * rhs._matrix[ 2 ][ 3 ];
    result._matrix[ 3 ][ 0 ] = _matrix[ 0 ][ 0 ] * rhs._matrix[ 3 ][ 0 ] + _matrix[ 1 ][ 0 ] * rhs._matrix[ 3 ][ 1 ] + _matrix[ 2 ][ 0 ] * rhs._matrix[ 3 ][ 2 ] + _matrix[ 3 ][ 0 ] * rhs._matrix[ 3 ][ 3 ];
    result._matrix[ 0 ][ 1 ] = _matrix[ 0 ][ 1 ] * rhs._matrix[ 0 ][ 0 ] + _matrix[ 1 ][ 1 ] * rhs._matrix[ 0 ][ 1 ] + _matrix[ 2 ][ 1 ] * rhs._matrix[ 0 ][ 2 ] + _matrix[ 3 ][ 1 ] * rhs._matrix[ 0 ][ 3 ];
    result._matrix[ 1 ][ 1 ] = _matrix[ 0 ][ 1 ] * rhs._matrix[ 1 ][ 0 ] + _matrix[ 1 ][ 1 ] * rhs._matrix[ 1 ][ 1 ] + _matrix[ 2 ][ 1 ] * rhs._matrix[ 1 ][ 2 ] + _matrix[ 3 ][ 1 ] * rhs._matrix[ 1 ][ 3 ];
    result._matrix[ 2 ][ 1 ] = _matrix[ 0 ][ 1 ] * rhs._matrix[ 2 ][ 0 ] + _matrix[ 1 ][ 1 ] * rhs._matrix[ 2 ][ 1 ] + _matrix[ 2 ][ 1 ] * rhs._matrix[ 2 ][ 2 ] + _matrix[ 3 ][ 1 ] * rhs._matrix[ 2 ][ 3 ];
    result._matrix[ 3 ][ 1 ] = _matrix[ 0 ][ 1 ] * rhs._matrix[ 3 ][ 0 ] + _matrix[ 1 ][ 1 ] * rhs._matrix[ 3 ][ 1 ] + _matrix[ 2 ][ 1 ] * rhs._matrix[ 3 ][ 2 ] + _matrix[ 3 ][ 1 ] * rhs._matrix[ 3 ][ 3 ];
    result._matrix[ 0 ][ 2 ] = _matrix[ 0 ][ 2 ] * rhs._matrix[ 0 ][ 0 ] + _matrix[ 1 ][ 2 ] * rhs._matrix[ 0 ][ 1 ] + _matrix[ 2 ][ 2 ] * rhs._matrix[ 0 ][ 2 ] + _matrix[ 3 ][ 2 ] * rhs._matrix[ 0 ][ 3 ];
    result._matrix[ 1 ][ 2 ] = _matrix[ 0 ][ 2 ] * rhs._matrix[ 1 ][ 0 ] + _matrix[ 1 ][ 2 ] * rhs._matrix[ 1 ][ 1 ] + _matrix[ 2 ][ 2 ] * rhs._matrix[ 1 ][ 2 ] + _matrix[ 3 ][ 2 ] * rhs._matrix[ 1 ][ 3 ];
    result._matrix[ 2 ][ 2 ] = _matrix[ 0 ][ 2 ] * rhs._matrix[ 2 ][ 0 ] + _matrix[ 1 ][ 2 ] * rhs._matrix[ 2 ][ 1 ] + _matrix[ 2 ][ 2 ] * rhs._matrix[ 2 ][ 2 ] + _matrix[ 3 ][ 2 ] * rhs._matrix[ 2 ][ 3 ];
    result._matrix[ 3 ][ 2 ] = _matrix[ 0 ][ 2 ] * rhs._matrix[ 3 ][ 0 ] + _matrix[ 1 ][ 2 ] * rhs._matrix[ 3 ][ 1 ] + _matrix[ 2 ][ 2 ] * rhs._matrix[ 3 ][ 2 ] + _matrix[ 3 ][ 2 ] * rhs._matrix[ 3 ][ 3 ];
    result._matrix[ 0 ][ 3 ] = _matrix[ 0 ][ 3 ] * rhs._matrix[ 0 ][ 0 ] + _matrix[ 1 ][ 3 ] * rhs._matrix[ 0 ][ 1 ] + _matrix[ 2 ][ 3 ] * rhs._matrix[ 0 ][ 2 ] + _matrix[ 3 ][ 3 ] * rhs._matrix[ 0 ][ 3 ];
    result._matrix[ 1 ][ 3 ] = _matrix[ 0 ][ 3 ] * rhs._matrix[ 1 ][ 0 ] + _matrix[ 1 ][ 3 ] * rhs._matrix[ 1 ][ 1 ] + _matrix[ 2 ][ 3 ] * rhs._matrix[ 1 ][ 2 ] + _matrix[ 3 ][ 3 ] * rhs._matrix[ 1 ][ 3 ];
    result._matrix[ 2 ][ 3 ] = _matrix[ 0 ][ 3 ] * rhs._matrix[ 2 ][ 0 ] + _matrix[ 1 ][ 3 ] * rhs._matrix[ 2 ][ 1 ] + _matrix[ 2 ][ 3 ] * rhs._matrix[ 2 ][ 2 ] + _matrix[ 3 ][ 3 ] * rhs._matrix[ 2 ][ 3 ];
    result._matrix[ 3 ][ 3 ] = _matrix[ 0 ][ 3 ] * rhs._matrix[ 3 ][ 0 ] + _matrix[ 1 ][ 3 ] * rhs._matrix[ 3 ][ 1 ] + _matrix[ 2 ][ 3 ] * rhs._matrix[ 3 ][ 2 ] + _matrix[ 3 ][ 3 ] * rhs._matrix[ 3 ][ 3 ];
 
    return result;
}
 
/// Mathematical * operator
/// \param rhs The matrix to multiply by.
/// \return The resulting matrix.
template<>
inline GlsMatrix<float, 4>
GlsMatrix<float, 4>::operator*( const GlsMatrix<float, 4>& rhs ) const
{
    GlsMatrix<float, 4> result;
 
    // Same as regular, just unroll the loop
    result._matrix[ 0 ][ 0 ] = _matrix[ 0 ][ 0 ] * rhs._matrix[ 0 ][ 0 ] + _matrix[ 1 ][ 0 ] * rhs._matrix[ 0 ][ 1 ] + _matrix[ 2 ][ 0 ] * rhs._matrix[ 0 ][ 2 ] + _matrix[ 3 ][ 0 ] * rhs._matrix[ 0 ][ 3 ];
    result._matrix[ 1 ][ 0 ] = _matrix[ 0 ][ 0 ] * rhs._matrix[ 1 ][ 0 ] + _matrix[ 1 ][ 0 ] * rhs._matrix[ 1 ][ 1 ] + _matrix[ 2 ][ 0 ] * rhs._matrix[ 1 ][ 2 ] + _matrix[ 3 ][ 0 ] * rhs._matrix[ 1 ][ 3 ];
    result._matrix[ 2 ][ 0 ] = _matrix[ 0 ][ 0 ] * rhs._matrix[ 2 ][ 0 ] + _matrix[ 1 ][ 0 ] * rhs._matrix[ 2 ][ 1 ] + _matrix[ 2 ][ 0 ] * rhs._matrix[ 2 ][ 2 ] + _matrix[ 3 ][ 0 ] * rhs._matrix[ 2 ][ 3 ];
    result._matrix[ 3 ][ 0 ] = _matrix[ 0 ][ 0 ] * rhs._matrix[ 3 ][ 0 ] + _matrix[ 1 ][ 0 ] * rhs._matrix[ 3 ][ 1 ] + _matrix[ 2 ][ 0 ] * rhs._matrix[ 3 ][ 2 ] + _matrix[ 3 ][ 0 ] * rhs._matrix[ 3 ][ 3 ];
    result._matrix[ 0 ][ 1 ] = _matrix[ 0 ][ 1 ] * rhs._matrix[ 0 ][ 0 ] + _matrix[ 1 ][ 1 ] * rhs._matrix[ 0 ][ 1 ] + _matrix[ 2 ][ 1 ] * rhs._matrix[ 0 ][ 2 ] + _matrix[ 3 ][ 1 ] * rhs._matrix[ 0 ][ 3 ];
    result._matrix[ 1 ][ 1 ] = _matrix[ 0 ][ 1 ] * rhs._matrix[ 1 ][ 0 ] + _matrix[ 1 ][ 1 ] * rhs._matrix[ 1 ][ 1 ] + _matrix[ 2 ][ 1 ] * rhs._matrix[ 1 ][ 2 ] + _matrix[ 3 ][ 1 ] * rhs._matrix[ 1 ][ 3 ];
    result._matrix[ 2 ][ 1 ] = _matrix[ 0 ][ 1 ] * rhs._matrix[ 2 ][ 0 ] + _matrix[ 1 ][ 1 ] * rhs._matrix[ 2 ][ 1 ] + _matrix[ 2 ][ 1 ] * rhs._matrix[ 2 ][ 2 ] + _matrix[ 3 ][ 1 ] * rhs._matrix[ 2 ][ 3 ];
    result._matrix[ 3 ][ 1 ] = _matrix[ 0 ][ 1 ] * rhs._matrix[ 3 ][ 0 ] + _matrix[ 1 ][ 1 ] * rhs._matrix[ 3 ][ 1 ] + _matrix[ 2 ][ 1 ] * rhs._matrix[ 3 ][ 2 ] + _matrix[ 3 ][ 1 ] * rhs._matrix[ 3 ][ 3 ];
    result._matrix[ 0 ][ 2 ] = _matrix[ 0 ][ 2 ] * rhs._matrix[ 0 ][ 0 ] + _matrix[ 1 ][ 2 ] * rhs._matrix[ 0 ][ 1 ] + _matrix[ 2 ][ 2 ] * rhs._matrix[ 0 ][ 2 ] + _matrix[ 3 ][ 2 ] * rhs._matrix[ 0 ][ 3 ];
    result._matrix[ 1 ][ 2 ] = _matrix[ 0 ][ 2 ] * rhs._matrix[ 1 ][ 0 ] + _matrix[ 1 ][ 2 ] * rhs._matrix[ 1 ][ 1 ] + _matrix[ 2 ][ 2 ] * rhs._matrix[ 1 ][ 2 ] + _matrix[ 3 ][ 2 ] * rhs._matrix[ 1 ][ 3 ];
    result._matrix[ 2 ][ 2 ] = _matrix[ 0 ][ 2 ] * rhs._matrix[ 2 ][ 0 ] + _matrix[ 1 ][ 2 ] * rhs._matrix[ 2 ][ 1 ] + _matrix[ 2 ][ 2 ] * rhs._matrix[ 2 ][ 2 ] + _matrix[ 3 ][ 2 ] * rhs._matrix[ 2 ][ 3 ];
    result._matrix[ 3 ][ 2 ] = _matrix[ 0 ][ 2 ] * rhs._matrix[ 3 ][ 0 ] + _matrix[ 1 ][ 2 ] * rhs._matrix[ 3 ][ 1 ] + _matrix[ 2 ][ 2 ] * rhs._matrix[ 3 ][ 2 ] + _matrix[ 3 ][ 2 ] * rhs._matrix[ 3 ][ 3 ];
    result._matrix[ 0 ][ 3 ] = _matrix[ 0 ][ 3 ] * rhs._matrix[ 0 ][ 0 ] + _matrix[ 1 ][ 3 ] * rhs._matrix[ 0 ][ 1 ] + _matrix[ 2 ][ 3 ] * rhs._matrix[ 0 ][ 2 ] + _matrix[ 3 ][ 3 ] * rhs._matrix[ 0 ][ 3 ];
    result._matrix[ 1 ][ 3 ] = _matrix[ 0 ][ 3 ] * rhs._matrix[ 1 ][ 0 ] + _matrix[ 1 ][ 3 ] * rhs._matrix[ 1 ][ 1 ] + _matrix[ 2 ][ 3 ] * rhs._matrix[ 1 ][ 2 ] + _matrix[ 3 ][ 3 ] * rhs._matrix[ 1 ][ 3 ];
    result._matrix[ 2 ][ 3 ] = _matrix[ 0 ][ 3 ] * rhs._matrix[ 2 ][ 0 ] + _matrix[ 1 ][ 3 ] * rhs._matrix[ 2 ][ 1 ] + _matrix[ 2 ][ 3 ] * rhs._matrix[ 2 ][ 2 ] + _matrix[ 3 ][ 3 ] * rhs._matrix[ 2 ][ 3 ];
    result._matrix[ 3 ][ 3 ] = _matrix[ 0 ][ 3 ] * rhs._matrix[ 3 ][ 0 ] + _matrix[ 1 ][ 3 ] * rhs._matrix[ 3 ][ 1 ] + _matrix[ 2 ][ 3 ] * rhs._matrix[ 3 ][ 2 ] + _matrix[ 3 ][ 3 ] * rhs._matrix[ 3 ][ 3 ];
 
    return result;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>
GlsMatrix<Type, DIM>::operator*( const GlsMatrix<Type, DIM>& rhs ) const
{
    GlsMatrix<Type, DIM> result;
    memset( result._data, 0, sizeof( Type ) * DIM * DIM );
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            for( int i = 0; i < DIM; ++i )
            {
                result._matrix[ col ][ row ] += _matrix[ i ][ row ] * rhs._matrix[ col ][ i ];
            }
        }
    }
    return result;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM> GlsMatrix<Type, DIM>::operator*( Type scalar ) const
{
    GlsMatrix<Type, DIM> result;
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            result._matrix[ col ][ row ] = _matrix[ col ][ row ] * scalar;
        }
    }
    return result;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
void GlsMatrix<Type, DIM>::Dump() const
{
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            std::cout.width( 11 );
            std::cout << _matrix[ col ][ row ] << ' ';
        }
        std::cout << std::endl;
    }
    std::cout << std::endl;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline void GlsMatrix<Type, DIM>::MakeIdentity()
{
    Clear();
    for( int i = 0; i < DIM; ++i )
    {
        _matrix[ i ][ i ] = Type( 1 );
    }
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
Type GlsMatrix<Type, DIM>::Determinant() const
{
    // In this algorithm we use gaussian elimination with pivoting to reduce
    // the matrix to an upper triangular form.  The determinant of an
    // upper triangular matrix is the product of the terms on the diagonal.
    // To perform the elimination, we will use two types of elementary row
    // operations on the matrix.
    // 1) Interchanging rows of the matrix.
    // 2) Add c times row i of the matrix to row j where i != j.
    //
    // The first operation of interchanging rows prevents numerical
    // instability, and the second allows us to zero out terms below
    // the diagonal by choosing c appropriately.
 
    Type pivotFactor;
    Type det = Type( 1 );
 
    // Make a temporary copy of the matrix so the original is not destroyed
    // during gaussian elimination and pivoting
    GlsMatrix<Type, DIM> temp( *this );
 
    // Loop through each row in the matrix
    for( int k = 0; k < DIM; ++k )
    {
        // Perform pivoting
        int index = temp.Pivot( k );
 
        // If pivot returns a -1, then a zero exists on the diagonal
        // so the determinant will be zero.
        if( index == -1 )
            return 0;
        // If pivot returns a number other than zero, rows have been
        // interchanged, so we need to change the sign of the determinant
        // due to the theorem that states "If matrix B results from A by
        // interchanging two rows (columns) of A, then det(B) = - det(A).
        if( index != 0 )
            det = -det;
 
        // The determinant will be the product of the diagonal terms. The
        // row that the [k][k] term is in already has the proper form
        // for an upper triangular matrix, so go ahead and use it in the det.
        det *= temp._matrix[ k ][ k ];
 
        // Since we are using the diagonal element from row k, column k, we
        // need all other terms below this element, in column k, to be
        // zero. We do this by an elementary operation (2) above to each
        // row below this diagonal element.
        for( int i = k + 1; i < DIM; ++i )
        {
            // Choose the pivot factor that would zero out the [i][k]term
            // when multiplied by the kth row and subtracted from the
            // ith row.
            pivotFactor = temp._matrix[ i ][ k ] / temp._matrix[ k ][ k ];
 
            // Perform the operation on each term in the row to the right
            // of the pivot. The terms to the left are assumed to be zero
            // and we never reference them again so no need to waste time
            // processing them.
            for( int j = k + 1; j < DIM; ++j )
            {
                temp._matrix[ i ][ j ] -= pivotFactor * temp._matrix[ k ][ j ];
            }
        }
    }
    return det;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
GlsMatrix<Type, DIM> GlsMatrix<Type, DIM>::Inverse() const
{
    // In this algorithm we use Gauss-Jordan reduction with pivoting to
    // transform the original matrix to the identity matrix, all the while
    // performing the same operations on what started out as the identity
    // matrix.  What started out as the identity matrix will become the
    // inverse of the original matrix. This is because A*Inv(A) = I and
    // performing the same operations on either side of the equation to
    // reduce A to the identity matrix, the equation becomes I*Inv(A) = Inv(A).
    // To perform the elimination, we will use two types of elementary row
    // operations on the matrix.
    // 1) Interchanging rows of the matrix.
    // 2) Multiply row i of the matrix by c, where c != 0.
    // 2) Add c times row i of the matrix to row j where i != j.
    //
    // The first operation of interchanging rows prevents numerical
    // instability. The second allows us to put 1's on the diagonal, by
    // dividing the row by the value of the term on the diagonal.
    // and the third to zero out terms that are not on the diagonal by
    // choosing c appropriately.
 
    Type a1_inv, a2;
 
    // Make a temporary copy of the matrix so the original is not destroyed
    // during Gauss-Jordan reduction and pivoting
    GlsMatrix<Type, DIM> temp( *this );
 
    // This starts out as the identity matrix. Every operation we perform
    // on temp, we will also do to this matrix to maintain the equality in
    // the equation we are starting out with: A*Inv(A) = I
    GlsMatrix<Type, DIM> inverse;
 
    // Loop through each row in the matrix
    for( int k = 0; k < DIM; ++k )
    {
        // Perform pivoting
        int index = temp.Pivot( k );
 
        // If pivot returns a -1, then the matrix is singular or
        // non-invertible. TBD: WHAT SHOULD WE DO HERE?
        if( index == -1 )
            return inverse;
 
        // If pivoting resulted in the swapping of rows, then
        // we need to do the same thing to the inverse matrix
        if( index != 0 )
        {
            // Swap rows
            Type   rowptr[ DIM ];
            size_t byteCount = sizeof( Type ) * DIM;
            memcpy( rowptr, inverse._matrix[ k ], byteCount );
            memcpy( inverse._matrix[ k ], inverse._matrix[ index ], byteCount );
            memcpy( inverse._matrix[ index ], rowptr, byteCount );
        }
 
        // Divide the current row by the term on the diagonal so we
        // get a 1 on the diagonal. Remember, do the same operation
        // to the inverse matrix.
        a1_inv = ( (Type)1.0 ) / temp._matrix[ k ][ k ];
        for( int j = 0; j < DIM; ++j )
        {
            temp._matrix[ k ][ j ] *= a1_inv;
            inverse._matrix[ k ][ j ] *= a1_inv;
        }
 
        // Now, loop through each row, skipping the current
        // one (k).
        // Choose the pivot factor (a2) that would zero out the [i][k]term
        // when multiplied by the kth row and subtracted from the
        // ith row.  Since the kth row has a 1 in the diagonal term,
        // the factor should be the [i][k] term itself.  This loop results
        // in zeros in the kth column, except for the diagonal term [k][k].
        for( int i = 0; i < DIM; ++i )
        {
            if( i != k )
            {
                a2 = temp._matrix[ i ][ k ];
                for( int j = 0; j < DIM; ++j )
                {
                    temp._matrix[ i ][ j ] -= a2 * temp._matrix[ k ][ j ];
                    inverse._matrix[ i ][ j ] -= a2 * inverse._matrix[ k ][ j ];
                }
            }
        }
    }
    return inverse;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline void GlsMatrix<Type, DIM>::Invert()
{
    *this = Inverse();
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM> GlsMatrix<Type, DIM>::Transposition() const
{
    // Interchange rows with columns
    GlsMatrix<Type, DIM> result;
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            result._matrix[ col ][ row ] = _matrix[ row ][ col ];
        }
    }
    return result;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline void GlsMatrix<Type, DIM>::Transpose()
{
    GlsMatrix<Type, DIM> result;
    result = Transposition();
    *this  = result;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::IsDiagonal() const
{
    // Look for the first term off the diagonal (where row != column) that
    // is not equal to zero. If we find one, the matrix is not a diagonal.
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            if( col != row && !closeValues( _matrix[ col ][ row ], Type( 0 ) ) )
                return false;
        }
    }
    return true;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::IsIdentity() const
{
    // First do a quick check to see if it is an un-touched identity matrix
    // This allows for zero tolerences, but should be much faster than
    // doing all the tolerance checking every time.
    static const GlsMatrix referenceIdentity; // Defaults to identity.
    if( 0 == memcmp( _data, referenceIdentity._data, sizeof( Type ) * DIM * DIM ) )
        return true;
 
    // If the matrix is scalar, it is diagonal and all terms on the diagonal
    // are equal. Make sure they are equal to 1 to be the identity matrix.
    if( IsScalar() && closeValues( _matrix[ 0 ][ 0 ], Type( 1 ) ) )
        return true;
    else
        return false;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::IsScalar() const
{
    // If the matrix is scalar, it is diagonal and all terms on the diagonal
    // are equal.
    if( !IsDiagonal() )
        return false;
 
    Type val = _matrix[ 0 ][ 0 ];
    for( int i = 0; i < DIM; ++i )
    {
        if( !closeValues( _matrix[ i ][ i ], val ) )
            return false;
    }
    return true;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::IsSingular() const
{
    return Determinant() == Type( 0 );
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::IsSymmetric() const
{
    return Transposition() == *this;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::IsSkewSymmetric() const
{
    return Transposition() == -( *this );
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::IsLowerTriangular() const
{
    // An n x n matrix A = [Aij] is lower triangular if Aij = 0 for i < j.
    for( int row = 0; row < DIM - 1; ++row )
    {
        for( int col = row + 1; col < DIM; ++col )
        {
            if( !closeValues( _matrix[ col ][ row ], Type( 0 ) ) )
                return false;
        }
    }
    return true;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::IsUpperTriangular() const
{
    // An n x n matrix A = [Aij] is upper triangular if Aij = 0 for i > j.
    for( int row = 1; row < DIM; ++row )
    {
        for( int col = 0; col < row; ++col )
        {
            if( !closeValues( _matrix[ col ][ row ], Type( 0 ) ) )
                return false;
        }
    }
    return true;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline Type& GlsMatrix<Type, DIM>::operator()( unsigned row, unsigned col )
{
    // transpose the row and column since the matrix is in column
    // major order but we want users to be able to use row-major ordering.
    return _matrix[ col ][ row ];
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline Type GlsMatrix<Type, DIM>::operator()( unsigned row, unsigned col ) const
{
    // transpose the row and column since the matrix is in column
    // major order but we want users to be able to use row-major ordering.
    return _matrix[ col ][ row ];
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::operator==( const GlsMatrix<Type, DIM>& rhs ) const
{
    static const int SIZE = DIM * DIM;
    for( int i = 0; i < SIZE; ++i )
    {
        if( !closeValues( _data[ i ], rhs._data[ i ] ) )
            return false;
    }
    return true;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::operator!=( const GlsMatrix<Type, DIM>& rhs ) const
{
    return !( *this == rhs );
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>&
GlsMatrix<Type, DIM>::operator=( const GlsMatrix<Type, DIM>& m )
{
    if( this == &m )
        return *this;
 
    memcpy( _data, m._data, sizeof( Type ) * DIM * DIM );
 
    return *this;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>&
GlsMatrix<Type, DIM>::operator=( const CStyleMatrix m )
{
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            _matrix[ col ][ row ] = m[ row ][ col ];
        }
    }
 
    return *this;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM> GlsMatrix<Type, DIM>::operator-() const
{
    GlsMatrix<Type, DIM> temp;
 
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            temp._matrix[ col ][ row ] = -_matrix[ col ][ row ];
        }
    }
 
    return temp;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>&
GlsMatrix<Type, DIM>::operator+=( const GlsMatrix<Type, DIM>& rhs )
{
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            _matrix[ col ][ row ] += rhs._matrix[ col ][ row ];
        }
    }
    return *this;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>&
GlsMatrix<Type, DIM>::operator-=( const GlsMatrix<Type, DIM>& rhs )
{
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            _matrix[ col ][ row ] -= rhs._matrix[ col ][ row ];
        }
    }
    return *this;
}
 
/// Combined assignment *= operator
/// \param rhs The matrix to multiply by.
/// \return The resulting matrix (this).
template<>
inline GlsMatrix<double, 4>&
GlsMatrix<double, 4>::operator*=( const GlsMatrix<double, 4>& rhs )
{
    return *this = *this * rhs; // faster because it calls the specialized operator*
}
 
/// Combined assignment *= operator
/// \param rhs The matrix to multiply by.
/// \return The resulting matrix (this).
template<>
inline GlsMatrix<float, 4>&
GlsMatrix<float, 4>::operator*=( const GlsMatrix<float, 4>& rhs )
{
    return *this = *this * rhs; // faster because it calls the specialized operator*
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>&
GlsMatrix<Type, DIM>::operator*=( const GlsMatrix<Type, DIM>& rhs )
{
    GlsMatrix<Type, DIM> result;
    memset( result._data, 0, sizeof( Type ) * DIM * DIM );
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            for( int i = 0; i < DIM; ++i )
            {
                result._matrix[ col ][ row ] += _matrix[ i ][ row ] * rhs._matrix[ col ][ i ];
            }
        }
    }
    *this = result;
    return *this;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>&
GlsMatrix<Type, DIM>::operator*=( Type scalar )
{
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            _matrix[ col ][ row ] *= scalar;
        }
    }
    return *this;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>&
GlsMatrix<Type, DIM>::operator/=( Type scalar )
{
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            _matrix[ col ][ row ] /= scalar;
        }
    }
    return *this;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>
GlsMatrix<Type, DIM>::operator+( const GlsMatrix<Type, DIM>& rhs ) const
{
    GlsMatrix<Type, DIM> result;
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            result._matrix[ col ][ row ] = _matrix[ col ][ row ] + rhs._matrix[ col ][ row ];
        }
    }
    return result;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM>
GlsMatrix<Type, DIM>::operator-( const GlsMatrix<Type, DIM>& rhs ) const
{
    GlsMatrix<Type, DIM> result;
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            result._matrix[ col ][ row ] = _matrix[ col ][ row ] - rhs._matrix[ col ][ row ];
        }
    }
    return result;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline GlsMatrix<Type, DIM> GlsMatrix<Type, DIM>::operator/( Type scalar ) const
{
    GlsMatrix<Type, DIM> result;
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            result._matrix[ col ][ row ] = _matrix[ col ][ row ] / scalar;
        }
    }
    return result;
}
 
/// Stream out operator
/// \param ostrm The stream to write to.
/// \param m The value to write to the stream.
/// \return The stream in its current state.
template<class Type, int DIM>
inline std::ostream& operator<<( std::ostream& ostrm, const GlsMatrix<Type, DIM>& m )
{
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            ostrm.width( 11 );
            ostrm << m( row, col ) << ' ';
        }
    }
    return ostrm;
}
 
/// Stream in operator
/// \param istrm The stream to read from.
/// \param m The returned value read from the stream.
/// \return The stream in its current state.
template<class Type, int DIM>
inline std::istream& operator>>( std::istream& istrm, GlsMatrix<Type, DIM>& m )
{
    Type x;
    for( int row = 0; row < DIM; ++row )
    {
        for( int col = 0; col < DIM; ++col )
        {
            istrm >> x;
            m( row, col ) = x;
        }
    }
    return istrm;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline void GlsMatrix<Type, DIM>::Initialize()
{
    // Set each element in the matrix index to point to a column of the
    // _matrix data.
    for( int i = 0, j = 0; i < DIM; ++i, j += DIM )
    {
        _matrix[ i ] = &( _data[ j ] );
    }
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
int GlsMatrix<Type, DIM>::Pivot( unsigned row )
{
    int    pivotRow;
    int    k    = int( row );
    double amax = -1;
    double temp;
 
    // Go through each element in the column below the diagonal element
    // [row][row] to find the largest term (in magnitude). Save which row
    // has the largest term in the column.
    for( unsigned i = row; i < unsigned( DIM ); ++i )
    {
        if( ( temp = fabs( _matrix[ i ][ row ] ) ) > amax && temp != 0.0 )
        {
            amax = temp;
            k    = i;
        }
    }
 
    if( _matrix[ k ][ row ] == Type( 0 ) )
    {
        // If the largest term is zero, then return a -1 so the caller knows
        // we cannot pivot.
        pivotRow = -1;
    }
    else if( k != int( row ) )
    {
        // If k is not equal to the passed in row, it means another row
        // has the largest term in it, so lets interchange rows.
        // Return the row that was interchanged to let the caller know.
        Type   rowptr[ DIM ];
        size_t byteCount = sizeof( Type ) * DIM;
        memcpy( rowptr, _matrix[ k ], byteCount );
        memcpy( _matrix[ k ], _matrix[ row ], byteCount );
        memcpy( _matrix[ row ], rowptr, byteCount );
        pivotRow = k;
    }
    else
    {
        // If k is equal to the passed in row, it means current row
        // has the largest term in it, so lets not interchange any rows.
        // return 0 to let the caller know.
        pivotRow = 0;
    }
 
    return pivotRow;
}
//----------------------------------------------------------------------------
template<class Type, int DIM>
inline bool GlsMatrix<Type, DIM>::closeValues( Type lv, Type rv ) const
{
    return ( lv - PRECISION ) <= rv && rv <= ( lv + PRECISION );
}
 
} // namespace disti
 
#endif