GMatrixSymmetric.cpp

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/***************************************************************************
 *                GMatrixSymmetric.cpp - Symmetric matrix class            *
 * ----------------------------------------------------------------------- *
 *  copyright (C) 2006-2021 by Juergen Knoedlseder                         *
 * ----------------------------------------------------------------------- *
 *                                                                         *
 *  This program is free software: you can redistribute it and/or modify   *
 *  it under the terms of the GNU General Public License as published by   *
 *  the Free Software Foundation, either version 3 of the License, or      *
 *  (at your option) any later version.                                    *
 *                                                                         *
 *  This program is distributed in the hope that it will be useful,        *
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of         *
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the          *
 *  GNU General Public License for more details.                           *
 *                                                                         *
 *  You should have received a copy of the GNU General Public License      *
 *  along with this program.  If not, see <http://www.gnu.org/licenses/>.  *
 *                                                                         *
 ***************************************************************************/
/**
 * @file GMatrixSymmetric.cpp
 * @brief Symmetric matrix class implementation
 * @author Juergen Knoedlseder
 */
 
/* __ Includes ___________________________________________________________ */
#ifdef HAVE_CONFIG_H
#include <config.h>
#endif
#include <cmath>
#include "GTools.hpp"
#include "GException.hpp"
#include "GVector.hpp"
#include "GMatrix.hpp"
#include "GMatrixSparse.hpp"
#include "GMatrixSymmetric.hpp"
 
/* __ Method name definitions ____________________________________________ */
#define G_CONSTRUCTOR        "GMatrixSymmetric::GMatrixSymmetric(int&, int&)"
#define G_MATRIX               "GMatrixSymmetric::GMatrixSymmetric(GMatrix&)"
#define G_SPARSEMATRIX   "GMatrixSymmetric::GMatrixSymmetric(GSparseMatrix&)"
#define G_OP_ADD            "GMatrixSymmetric::operator+=(GMatrixSymmetric&)"
#define G_OP_SUB            "GMatrixSymmetric::operator-=(GMatrixSymmetric&)"
#define G_OP_MUL_VEC                  "GMatrixSymmetric::operator*(GVector&)"
#define G_OP_MAT_MUL        "GMatrixSymmetric::operator*=(GMatrixSymmetric&)"
#define G_AT                               "GMatrixSymmetric::at(int&, int&)"
#define G_EXTRACT_ROW                           "GMatrixSymmetric::row(int&)"
#define G_SET_ROW                     "GMatrixSymmetric::row(int&, GVector&)"
#define G_EXTRACT_COLUMN                     "GMatrixSymmetric::column(int&)"
#define G_SET_COLUMN               "GMatrixSymmetric::column(int&, GVector&)"
#define G_ADD_TO_ROW           "GMatrixSymmetric::add_to_row(int&, GVector&)"
#define G_ADD_TO_COLUMN     "GMatrixSymmetric::add_to_column(int&, GVector&)"
#define G_CHOL_DECOMP            "GMatrixSymmetric::cholesky_decompose(int&)"
#define G_CHOL_SOLVE      "GMatrixSymmetric::cholesky_solver(GVector&, int&)"
#define G_CHOL_INVERT               "GMatrixSymmetric::cholesky_invert(int&)"
#define G_COPY_MEMBERS    "GMatrixSymmetric::copy_members(GMatrixSymmetric&)"
#define G_ALLOC_MEMBERS         "GMatrixSymmetric::alloc_members(int&, int&)"
 
 
/*==========================================================================
 =                                                                         =
 =                         Constructors/destructors                        =
 =                                                                         =
 ==========================================================================*/
 
/***********************************************************************//**
 * @brief Void constructor
 *
 * Constructs an empty matrix. An empty matrix has zero rows and columns.
 ***************************************************************************/
GMatrixSymmetric::GMatrixSymmetric(void) : GMatrixBase()
{
    // Initialise class members for clean destruction
    init_members();
 
    // Return
    return;
}
 
/***********************************************************************//**
 * @brief Matrix constructor
 *
 * @param[in] rows Number of rows [>=0].
 * @param[in] columns Number of columns [>=0].
 *
 * @exception GException::invalid_argument
 *            Number of rows or columns is negative.
 *
 * Constructs a matrix with the specified number of @p rows and @p columns.
 ***************************************************************************/
GMatrixSymmetric::GMatrixSymmetric(const int& rows, const int& columns) :
                  GMatrixBase()
{
    // Compile option: raise an exception if number of rows or columns is
    // negative
    #if defined(G_RANGE_CHECK)
    if (rows < 0) {
        std::string msg = "Number of rows "+gammalib::str(rows)+" is negative. "
                          "Please specify a non-negative number of rows.";
        throw GException::invalid_argument(G_CONSTRUCTOR, msg);
    }
    if (columns < 0) {
        std::string msg = "Number of columns "+gammalib::str(columns)+" is "
                          "negative. Please specify a non-negative number of "
                          "columns.";
        throw GException::invalid_argument(G_CONSTRUCTOR, msg);
    }
    #endif
 
    // Initialise class members for clean destruction
    init_members();
 
    // Allocate matrix memory
    alloc_members(rows, columns);
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief GMatrix to GMatrixSymmetric storage class convertor
 *
 * @param[in] matrix General matrix (GMatrix).
 *
 * @exception GException::invalid_argument
 *            Matrix is not symmetric.
 *
 * Converts a general matrix into the symmetric storage class. If the input
 * matrix is not symmetric, an exception is thrown.
 ***************************************************************************/
GMatrixSymmetric::GMatrixSymmetric(const GMatrix& matrix)
{
    // Initialise class members for clean destruction
    init_members();
 
    // Allocate matrix memory
    alloc_members(matrix.rows(), matrix.columns());
 
    // Fill matrix
    for (int col = 0; col < m_cols; ++col) {
        for (int row = col; row < m_rows; ++row) {
            double value_ll = matrix(row,col);
            double value_ur = matrix(col,row);
            if (value_ll != value_ur) {
                std::string msg = "Value "+gammalib::str(value_ll)+" of "
                                  "matrix element at (row,column)=("+
                                  gammalib::str(row)+","+
                                  gammalib::str(col)+") differs from value "+
                                  gammalib::str(value_ur)+" of "
                                  "matrix element at (row,column)=("+
                                  gammalib::str(col)+","+
                                  gammalib::str(row)+"). Please specify a "
                                  "symmetric matrix.";
                throw GException::invalid_argument(G_MATRIX, msg);
            }
            (*this)(row, col) = matrix(row, col);
        }
    }
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief GMatrixSparse to GMatrixSymmetric storage class convertor
 *
 * @param[in] matrix Sparse matrix (GMatrixSparse).
 *
 * @exception GException::invalid_argument
 *            Sparse matrix is not symmetric.
 *
 * Converts a sparse matrix into the symmetric storage class. If the input
 * matrix is not symmetric, an exception is thrown.
 ***************************************************************************/
GMatrixSymmetric::GMatrixSymmetric(const GMatrixSparse& matrix)
{
    // Initialise class members for clean destruction
    init_members();
 
    // Allocate matrix memory
    alloc_members(matrix.rows(), matrix.columns());
 
    // Fill matrix
    for (int col = 0; col < m_cols; ++col) {
        for (int row = col; row < m_rows; ++row) {
            double value_ll = matrix(row,col);
            double value_ur = matrix(col,row);
            if (value_ll != value_ur) {
                std::string msg = "Value "+gammalib::str(value_ll)+" of "
                                  "matrix element at (row,column)=("+
                                  gammalib::str(row)+","+
                                  gammalib::str(col)+") differs from value "+
                                  gammalib::str(value_ur)+" of "
                                  "matrix element at (row,column)=("+
                                  gammalib::str(col)+","+
                                  gammalib::str(row)+"). Please specify a "
                                  "symmetric matrix.";
                throw GException::invalid_argument(G_MATRIX, msg);
            }
            (*this)(row, col) = matrix(row, col);
        }
    }
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Copy constructor
 *
 * @param[in] matrix Matrix.
 ***************************************************************************/
GMatrixSymmetric::GMatrixSymmetric(const GMatrixSymmetric& matrix) :
                  GMatrixBase(matrix)
{
    // Initialise private members for clean destruction
    init_members();
 
    // Copy members
    copy_members(matrix);
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Destructor
 ***************************************************************************/
GMatrixSymmetric::~GMatrixSymmetric(void)
{
    // Free members
    free_members();
 
    // Return
    return;
}
 
 
/*==========================================================================
 =                                                                         =
 =                               Operators                                 =
 =                                                                         =
 ==========================================================================*/
 
/***********************************************************************//**
 * @brief Matrix assignment operator
 *
 * @param[in] matrix Matrix.
 * @return Matrix.
 *
 * Assigns the content of another matrix to the actual matrix instance.
 ***************************************************************************/
GMatrixSymmetric& GMatrixSymmetric::operator=(const GMatrixSymmetric& matrix)
{
    // Execute only if object is not identical
    if (this != &matrix) {
 
        // Assign base class members. Note that this method will also perform
        // the allocation of the matrix memory and the copy of the matrix
        // attributes.
        this->GMatrixBase::operator=(matrix);
 
        // Free members
        free_members();
 
        // Initialise members
        init_members();
 
        // Copy members
        copy_members(matrix);
 
    } // endif: object was not identical
 
    // Return this object
    return *this;
}
 
 
/***********************************************************************//**
 * @brief Value assignment operator
 *
 * @param[in] value Value.
 * @return Matrix.
 *
 * Assigns the specified @p value to all elements of the matrix.
 ***************************************************************************/
GMatrixSymmetric& GMatrixSymmetric::operator=(const double& value)
{
    // Assign value
    double* ptr = m_data;
    for (int i = 0; i < m_elements; ++i) {
        *ptr++ = value;
    }
 
    // Return this object
    return *this;
}
 
 
/***********************************************************************//**
 * @brief Return reference to matrix element
 *
 * @param[in] row Matrix row [0,...,rows()-1].
 * @param[in] column Matrix column [0,...,columns()-1].
 * @return Reference to matrix element.
 ***************************************************************************/
double& GMatrixSymmetric::operator()(const int& row, const int& column)
{
    // Get element index
    int inx = (row >= column) ? m_colstart[column]+(row-column)
                              : m_colstart[row]+(column-row);
 
    // Return element
    return m_data[inx];
}
 
 
/***********************************************************************//**
 * @brief Return reference to matrix element (const version)
 *
 * @param[in] row Matrix row [0,...,rows()-1].
 * @param[in] column Matrix column [0,...,columns()-1].
 * @return Reference to matrix element.
 ***************************************************************************/
const double& GMatrixSymmetric::operator()(const int& row,
                                           const int& column) const
{
    // Get element index
    int inx = (row >= column) ? m_colstart[column]+(row-column)
                              : m_colstart[row]+(column-row);
 
    // Return element
    return m_data[inx];
}
 
 
/***********************************************************************//**
 * @brief Vector multiplication
 *
 * @param[in] vector Vector.
 *
 * @exception GException::invalid_argument
 *            Vector size differs from number of columns in matrix.
 *
 * This method performs a vector multiplication of a matrix. The vector
 * multiplication will produce a vector. The multiplication can only be
 * performed when the number of matrix columns is identical to the size
 * of the vector. If the size of the vector differs an exception will be
 * thrown.
 ***************************************************************************/
GVector GMatrixSymmetric::operator*(const GVector& vector) const
{
    // Raise an exception if the matrix and vector dimensions are not compatible
    if (m_cols != vector.size()) {
        std::string msg = "Vector size "+gammalib::str(vector.size())+" does "
                          "not match "+gammalib::str(m_cols)+" matrix columns. "
                          "Please specify a vector of size "+
                          gammalib::str(m_cols)+".";
        throw GException::invalid_argument(G_OP_MUL_VEC, msg);
    }
 
    // Perform vector multiplication
    GVector result(m_rows);
    for (int row = 0; row < m_rows; ++row) {
        double sum = 0.0;
        for (int col = 0; col < m_cols; ++col) {
            sum += (*this)(row,col) * vector[col];
        }
        result[row] = sum;
    }
 
    // Return result
    return result;
}
 
 
/***********************************************************************//**
 * @brief Negate matrix elements
 *
 * @return Matrix with negated elements.
 *
 * Returns a matrix where each element has been replaced by its negative
 * element.
 ***************************************************************************/
GMatrixSymmetric GMatrixSymmetric::operator-(void) const
{
    // Copy matrix
    GMatrixSymmetric matrix = *this;
 
    // Take the absolute value of all matrix elements
    double* ptr = matrix.m_data;
    for (int i = 0; i < matrix.m_elements; ++i, ++ptr) {
        *ptr = -(*ptr);
    }
 
    // Return matrix
    return matrix;
}
 
 
/***********************************************************************//**
 * @brief Unary matrix addition operator
 *
 * @param[in] matrix Matrix.
 *
 * @exception GException::invalid_argument
 *            Incompatible number of matrix rows or columns.
 *
 * This method performs a matrix addition. The operation can only succeed
 * when the dimensions of both matrices are identical.
 ***************************************************************************/
GMatrixSymmetric& GMatrixSymmetric::operator+=(const GMatrixSymmetric& matrix)
{
    // Throw an exception if the matrix dimensions are not compatible
    if (m_rows != matrix.m_rows) {
        std::string msg = "Number of matrix rows "+gammalib::str(matrix.m_rows)+
                          " differs from the expected number of "+
                          gammalib::str(m_rows)+" rows. Please specify a "
                          "matrix with "+gammalib::str(m_rows)+" rows.";
        throw GException::invalid_argument(G_OP_ADD, msg);
    }
    if (m_cols != matrix.m_cols) {
        std::string msg = "Number of matrix columns "+gammalib::str(matrix.m_cols)+
                          " differs from the expected number of "+
                          gammalib::str(m_cols)+" columns. Please specify a "
                          "matrix with "+gammalib::str(m_cols)+" columns.";
        throw GException::invalid_argument(G_OP_ADD, msg);
    }
 
    // Add all matrix elements
    const double* src = matrix.m_data;
    double*       dst = m_data;
    for (int i = 0; i < m_elements; ++i) {
        *dst++ += *src++;
    }
 
    // Return result
    return *this;
}
 
 
/***********************************************************************//**
 * @brief Unary matrix scalar addition operator
 *
 * @param[in] scalar Scalar.
 *
 * Adds a @p scalar to each matrix element.
 ***************************************************************************/
GMatrixSymmetric& GMatrixSymmetric::operator+=(const double& scalar)
{
    // Add all matrix elements
    double* dst = m_data;
    for (int i = 0; i < m_elements; ++i) {
        *dst++ += scalar;
    }
 
    // Return result
    return *this;
}
 
 
/***********************************************************************//**
 * @brief Unary matrix subtraction operator
 *
 * @param[in] matrix Matrix.
 *
 * @exception GException::invalid_argument
 *            Incompatible number of matrix rows or columns.
 *
 * This method performs a matrix addition. The operation can only succeed
 * when the dimensions of both matrices are identical.
 ***************************************************************************/
GMatrixSymmetric& GMatrixSymmetric::operator-=(const GMatrixSymmetric& matrix)
{
    // Throw an exception if the matrix dimensions are not compatible
    if (m_rows != matrix.m_rows) {
        std::string msg = "Number of matrix rows "+gammalib::str(matrix.m_rows)+
                          " differs from the expected number of "+
                          gammalib::str(m_rows)+" rows. Please specify a "
                          "matrix with "+gammalib::str(m_rows)+" rows.";
        throw GException::invalid_argument(G_OP_SUB, msg);
    }
    if (m_cols != matrix.m_cols) {
        std::string msg = "Number of matrix columns "+gammalib::str(matrix.m_cols)+
                          " differs from the expected number of "+
                          gammalib::str(m_cols)+" columns. Please specify a "
                          "matrix with "+gammalib::str(m_cols)+" columns.";
        throw GException::invalid_argument(G_OP_SUB, msg);
    }
 
    // Subtract all matrix elements
    const double* src = matrix.m_data;
    double*       dst = m_data;
    for (int i = 0; i < m_elements; ++i) {
        *dst++ -= *src++;
    }
 
    // Return result
    return *this;
}
 
 
/***********************************************************************//**
 * @brief Unary matrix scalar subtraction operator
 *
 * @param[in] scalar Scalar.
 *
 * Subtracts a @p scalar from each matrix element.
 ***************************************************************************/
GMatrixSymmetric& GMatrixSymmetric::operator-=(const double& scalar)
{
    // Add all matrix elements
    double* dst = m_data;
    for (int i = 0; i < m_elements; ++i) {
        *dst++ -= scalar;
    }
 
    // Return result
    return *this;
}
 
 
/*==========================================================================
 =                                                                         =
 =                              Public methods                             =
 =                                                                         =
 ==========================================================================*/
 
/***********************************************************************//**
 * @brief Clear matrix
 ***************************************************************************/
void GMatrixSymmetric::clear(void)
{
    // Free members
    free_members();
 
    // Initialise private members
    init_members();
    
    // Return
    return; 
}
 
 
/***********************************************************************//**
 * @brief Clone matrix
 *
 * @return Pointer to deep copy of matrix.
 ***************************************************************************/
GMatrixSymmetric* GMatrixSymmetric::clone(void) const
{
    // Clone matrix
    return new GMatrixSymmetric(*this);
}
 
 
/***********************************************************************//**
 * @brief Return reference to matrix element
 *
 * @param[in] row Matrix row [0,...,rows()[.
 * @param[in] column Matrix column [0,...,columns()[.
 * @return Reference to matrix element.
 *
 * @exception GException::out_of_range
 *            Matrix row or column out of range.
 ***************************************************************************/
double& GMatrixSymmetric::at(const int& row, const int& column)
{
    // Throw exception if row or column is out of range
    if (row < 0 || row >= m_rows) {
        throw GException::out_of_range(G_AT, "Matrix row", row, m_rows);
    }
    if (column < 0 || column >= m_cols) {
        throw GException::out_of_range(G_AT, "Matrix column", column, m_cols);
    }
 
    // Get element index
    int inx = (row >= column) ? m_colstart[column]+(row-column)
                              : m_colstart[row]+(column-row);
 
    // Return element
    return m_data[inx];
}
 
 
/***********************************************************************//**
 * @brief Return reference to matrix element (const version)
 *
 * @param[in] row Matrix row [0,...,rows()[.
 * @param[in] column Matrix column [0,...,columns()[.
 * @return Reference to matrix element.
 *
 * @exception GException::out_of_range
 *            Matrix row or column out of range.
 ***************************************************************************/
const double& GMatrixSymmetric::at(const int& row, const int& column) const
{
    // Throw exception if row or column is out of range
    if (row < 0 || row >= m_rows) {
        throw GException::out_of_range(G_AT, "Matrix row", row, m_rows);
    }
    if (column < 0 || column >= m_cols) {
        throw GException::out_of_range(G_AT, "Matrix column", column, m_cols);
    }
 
    // Get element index
    int inx = (row >= column) ? m_colstart[column]+(row-column)
                              : m_colstart[row]+(column-row);
 
    // Return element
    return m_data[inx];
}
 
 
/***********************************************************************//**
 * @brief Extract row as vector from matrix
 *
 * @param[in] row Matrix row [0,...,rows()[.
 * @return Vector of matrix row elements (columns() elements).
 *
 * @exception GException::out_of_range
 *            Invalid matrix row specified.
 *
 * This method extracts a matrix row into a vector.
 ***************************************************************************/
GVector GMatrixSymmetric::row(const int& row) const
{
    // Throw exception if the row is invalid
    #if defined(G_RANGE_CHECK)
    if (row < 0 || row >= m_rows) {
        throw GException::out_of_range(G_EXTRACT_ROW, "Matrix row", row, m_rows);
    }
    #endif
 
    // Create result vector
    GVector result(m_cols);
 
    // Extract row into vector
    for (int col = 0; col < m_cols; ++col) {
        result[col] = (*this)(row,col);
    }
 
    // Return vector
    return result;
}
 
 
/***********************************************************************//**
 * @brief Set row in matrix
 *
 * @param[in] row Matrix row [0,...,rows()[.
 * @param[in] vector Vector of matrix row elements (columns() elements).
 *
 * @todo To be implemented.
 ***************************************************************************/
void GMatrixSymmetric::row(const int& row, const GVector& vector)
{
    // Throw an exception if the row is invalid
    #if defined(G_RANGE_CHECK)
    if (row < 0 || row >= m_rows) {
        throw GException::out_of_range(G_SET_ROW, "Matrix row", row, m_rows);
    }
    #endif
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Extract column as vector from matrix
 *
 * @param[in] column Matrix column [0,...,columns()[.
 * @return Vector of matrix column elements (rows() elements).
 *
 * @exception GException::out_of_range
 *            Invalid matrix column specified.
 *
 * This method extracts a matrix column into a vector.
 ***************************************************************************/
GVector GMatrixSymmetric::column(const int& column) const
{
    // Throw exception if the column is invalid
    #if defined(G_RANGE_CHECK)
    if (column < 0 || column >= m_cols) {
        throw GException::out_of_range(G_EXTRACT_COLUMN, "Matrix column",
                                       column, m_cols);
    }
    #endif
 
    // Create result vector
    GVector result(m_rows);
 
    // Extract column into vector
    for (int row = 0; row < m_rows; ++row) {
        result[row] = (*this)(row, column);
    }
 
    // Return vector
    return result;
}
 
 
/***********************************************************************//**
 * @brief Set matrix column from vector
 *
 * @param[in] column Matrix column [0,...,columns()[.
 * @param[in] vector Vector.
 *
 * @exception GException::out_of_range
 *            Invalid matrix column specified.
 * @exception GException::invalid_argument
 *            Vector size does not match number of matrix rows.
 *
 * Inserts the content of a vector into a matrix column. Any previous
 * content in the matrix column will be overwritted.
 ***************************************************************************/
void GMatrixSymmetric::column(const int& column, const GVector& vector)
{
    // Throw exception if the column is invalid
    #if defined(G_RANGE_CHECK)
    if (column < 0 || column >= m_cols) {
        throw GException::out_of_range(G_SET_COLUMN, "Matrix column",
                                       column, m_cols);
    }
    #endif
 
    // Throw exception if the vector size does not match the number of rows
    // in the matrix
    if (m_rows != vector.size()) {
        std::string msg = "Vector size "+gammalib::str(vector.size())+
                          " does not match "+gammalib::str(m_rows)+" matrix "
                          "rows. Please specify a vector of size "+
                          gammalib::str(m_rows)+".";
        throw GException::invalid_argument(G_SET_COLUMN, msg);
    }
 
    // Insert column into vector
    for (int row = 0; row < m_rows; ++row) {
        (*this)(row, column) = vector[row];
    }
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Add row to matrix elements
 *
 * @param[in] row Matrix row [0,...,rows()[.
 * @param[in] vector Vector of matrix row elements (columns() elements).
 *
 * @exception GException::out_of_range
 *            Invalid matrix row specified.
 * @exception GException::matrix_vector_mismatch
 *            Vector does not match the matrix dimensions.
 *
 * @todo To be implemented.
 ***************************************************************************/
void GMatrixSymmetric::add_to_row(const int& row, const GVector& vector)
{
    // Throw exception if the row is invalid
    #if defined(G_RANGE_CHECK)
    if (row < 0 || row >= m_rows) {
        throw GException::out_of_range(G_ADD_TO_ROW, "Matrix row", row, m_rows);
 
    }
    #endif
 
    // Throw exception if the matrix and vector dimensions are not
    // compatible
    if (m_cols != vector.size()) {
        std::string msg = "Vector size "+gammalib::str(vector.size())+
                          " does not match "+gammalib::str(m_cols)+" matrix "
                          "columns. Please specify a vector of size "+
                          gammalib::str(m_cols)+".";
        throw GException::invalid_argument(G_ADD_TO_ROW, msg);
    }
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Add vector column into matrix
 *
 * @param[in] column Matrix column [0,...,columns()[.
 * @param[in] vector Vector.
 *
 * @exception GException::out_of_range
 *            Invalid matrix column specified.
 * @exception GException::matrix_vector_mismatch
 *            Vector size does not match number of matrix rows.
 *
 * Adds the content of a vector to a matrix column.
 ***************************************************************************/
void GMatrixSymmetric::add_to_column(const int& column, const GVector& vector)
{
    // Raise an exception if the column is invalid
    #if defined(G_RANGE_CHECK)
    if (column < 0 || column >= m_cols) {
        throw GException::out_of_range(G_ADD_TO_COLUMN, "Matrix column",
                                       column, m_cols);
 
    }
    #endif
 
    // Raise an exception if the matrix and vector dimensions are not
    // compatible
    if (m_rows != vector.size()) {
        std::string msg = "Vector size "+gammalib::str(vector.size())+
                          " does not match "+gammalib::str(m_rows)+" matrix "
                          "rows. Please specify a vector of size "+
                          gammalib::str(m_rows)+".";
        throw GException::invalid_argument(G_ADD_TO_COLUMN, msg);
    }
 
    // Insert column into vector
    for (int row = 0; row < m_rows; ++row) {
        (*this)(row, column) += vector[row];
    }
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Return inverted matrix
 *
 * @return Inverted matrix.
 *
 * Returns inverse of matrix. Inversion is done for the moment using Cholesky
 * decomposition. This does not work on any kind of matrix.
 *
 * @todo Specify in documentation for which kind of matrix the method works.
 ***************************************************************************/
GMatrixSymmetric GMatrixSymmetric::invert(void) const
{
    // Allocate result matrix
    GMatrixSymmetric matrix(m_cols, m_rows);
 
    // Invert matrix
    matrix.cholesky_invert(true);
    
    // Return matrix
    return matrix;
}
 
 
/***********************************************************************//**
 * @brief Solves linear matrix equation
 *
 * @param[in] vector Solution vector.
 * 
 * Solves the linear equation
 *
 * \f[M \times {\tt solution} = {\tt vector} \f]
 *
 * where \f$M\f$ is the matrix, \f${\tt vector}\f$ is the result, and
 * \f${\tt solution}\f$ is the solution. Solving is done using Cholesky
 * decomposition. This does not work on any kind of matrix.
 *
 * @todo Specify in documentation for which kind of matrix the method works.
 ***************************************************************************/
GVector GMatrixSymmetric::solve(const GVector& vector) const
{
    // Get Cholesky decomposition of matrix
    GMatrixSymmetric decomposition = cholesky_decompose(true);
 
    // Solve linear equation
    GVector result = decomposition.cholesky_solver(vector);
 
    // Return result
    return result;
}
 
 
/***********************************************************************//**
 * @brief Return absolute of matrix
 *
 * @return Absolute of matrix
 *
 * Returns matrix where all elements of the matrix have been replaced by
 * their absolute values.
 ***************************************************************************/
GMatrixSymmetric GMatrixSymmetric::abs(void) const
{
    // Allocate result matrix
    GMatrixSymmetric matrix(m_rows, m_cols);
 
    // Take the absolute value of all matrix elements
    double* src = m_data;
    double* dst = matrix.m_data;
    for (int i = 0; i < m_elements; ++i) {
        *dst++ = std::abs(*src++);
    }
 
    // Return matrix
    return matrix;
}
 
 
/***********************************************************************//**
 * @brief Determine fill of matrix
 *
 * @return Matrix fill (between 0 and 1).
 *
 * The fill of a matrix is defined as the number of non-zero elements
 * devided by the total number of matrix elements.
 ***************************************************************************/
double GMatrixSymmetric::fill(void) const
{
    // Determine the number of zero elements
    int zero = 0;
    for (int col = 0, i = 0; col < m_cols; ++col) {
        if (m_data[i++] == 0.0) {                    // Count diag. once
            zero++;
        }
        for (int row = col+1; row < m_rows; ++row) {
            if (m_data[i++] == 0.0) {                // Count off-diag. twice
                zero +=2;
            }
        }
    }
 
    // Return the fill
    return (1.0-double(zero)/double(m_elements));
}
 
 
/***********************************************************************//**
 * @brief Sum matrix elements
 *
 * @return Sum of all matrix elements.
 ***************************************************************************/
double GMatrixSymmetric::sum(void) const
{
    // Initialise matrix sums (diagonal and off-diagonal)
    double diag     = 0.0;
    double off_diag = 0.0;
 
    // Calulate sum over diagonal elements
    for (int row = 0; row < m_rows; ++row) {
        diag += m_data[m_colstart[row]];
    }
 
    // Calulate sum over off-diagonal elements
    for (int row = 0; row < m_rows; ++row) {
        for (int col = row+1; col < m_cols; ++col) {
            off_diag += m_data[m_colstart[row]+(col-row)];
        }
    }
 
    // Calculate total
    double result = diag + 2.0 * off_diag;
 
    // Return result
    return result;
}
 
 
/***********************************************************************//**
 * @brief Extract lower triangle of matrix
 *
 * This method extracts the lower triangle of a matrix into another matrix.
 * (including the diagonal elements).
 * All remaining matrix elements will be zero.
 ***************************************************************************/
GMatrix GMatrixSymmetric::extract_lower_triangle(void) const
{
    // Define result matrix
    GMatrix result(m_rows, m_cols);
 
    // Extract all elements
    for (int row = 0; row < m_rows; ++row) {
        for (int col = 0; col <= row; ++col) {
            result(row,col) = m_data[m_colstart[col]+(row-col)];
        }
    }
 
    // Return result
    return result;
}
 
 
/***********************************************************************//**
 * @brief Extract upper triangle of matrix
 *
 * This method extracts the upper triangle of a matrix into another matrix.
 * (including the diagonal elements).
 * All remaining matrix elements will be zero.
 ***************************************************************************/
GMatrix GMatrixSymmetric::extract_upper_triangle(void) const
{
    // Define result matrix
    GMatrix result(m_rows, m_cols);
 
    // Extract all elements
    for (int row = 0; row < m_rows; ++row) {
        for (int col = row; col < m_cols; ++col) {
            result(row,col) = m_data[m_colstart[row]+(col-row)];
        }
    }
 
    // Return result
    return result;
}
 
 
/***********************************************************************//**
 * @brief Return Cholesky decomposition
 *
 * @param[in] compress Use zero-row/column compression (defaults to true).
 * @return Cholesky decomposition of matrix
 *
 * @exception GException::runtime_error
 *            Matrix is not positive definite.
 *
 * Returns the Cholesky decomposition of a sparse matrix. The decomposition
 * is stored within a GMatrixSymmetric object.
 *
 * The method is inspired by the algorithm found in Numerical Recipes.
 * The decomposition, which is a matrix occupying only the lower triange,
 * is stored in the elements of the symmetric matrix. To visualise the
 * matrix one has to use 'lower_triangle()' to extract the relevant part.
 * Case A operates on a full matrix, Case B operates on a (logically)
 * compressed matrix where zero rows/columns have been removed.
 ***************************************************************************/
GMatrixSymmetric GMatrixSymmetric::cholesky_decompose(const bool& compress) const
{
    // Create copy of matrix
    GMatrixSymmetric matrix = *this;
 
    // Set-up incides of non zero rows if matrix compression is requested
    if (compress) {
        matrix.set_inx();
    }
 
    // Check if zero-row/col compression is needed  
    int no_zeros = ((compress && (matrix.m_num_inx == matrix.m_rows)) || !compress);
 
    // Case A: no zero-row/col compression needed
    if (no_zeros) {
 
        // Loop over upper triangle (col >= row)
        double diag = 0.0;
        for (int row = 0; row < matrix.m_rows; ++row) {
            double* ptr = matrix.m_data + matrix.m_colstart[row];
            for (int col = row; col < matrix.m_cols; ++col, ++ptr) {
                // sum = M(row,col)
                double sum = *ptr;
                for (int k = 0; k < row; ++k) {
                    int offset = matrix.m_colstart[k] - k; // is always positive
                    sum -= matrix.m_data[offset+row] * matrix.m_data[offset+col]; // sum -= M(row,k)*M(col,k)
                }
                if (row == col) {
                    if (sum <= 0.0) {
                        std::string msg = "Matrix is not positive definite, "
                                          "the sum "+gammalib::str(sum)+
                                          " occured in row and column "+
                                          gammalib::str(row)+".";
                        throw GException::runtime_error(G_CHOL_DECOMP, msg);
                    }
                    *ptr = std::sqrt(sum);  // M(row,row) = sqrt(sum)
                    diag = 1.0/(*ptr);
                }
                else {
                    *ptr = sum*diag; // M(row,col) = sum/M(row,row)
                }
            }
        }
    } // endif: there were no zero rows/cols in matrix
 
    // Case B: zero-row/col compression needed
    else if (matrix.m_num_inx > 0) {
 
        // Allocate loop variables and pointers
        int  row;
        int  col;
        int  k;
        int* row_ptr;
        int* col_ptr;
        int* k_ptr;
 
        // Loop over upper triangle (col >= row)
        double diag = 0.0;
        for (row = 0, row_ptr = matrix.m_inx; row < matrix.m_num_inx; ++row, ++row_ptr) {
            double* ptr_0 = matrix.m_data + matrix.m_colstart[*row_ptr] - *row_ptr;
            for (col = row, col_ptr = matrix.m_inx + row; col < matrix.m_num_inx; ++col, ++col_ptr) {
                double* ptr = ptr_0 + *col_ptr;
                double  sum = *ptr;                                  // sum = M(row,col)
                for (k = 0, k_ptr = matrix.m_inx; k < row; ++k, ++k_ptr) {
                    int offset = matrix.m_colstart[*k_ptr] - *k_ptr;        // is always positive
                    sum       -= matrix.m_data[offset+*row_ptr] *
                                 matrix.m_data[offset+*col_ptr];
                                                                     // sum -= M(row,k)*M(col,k)
                }
                if (*row_ptr == *col_ptr) {
                    if (sum <= 0.0) {
                        std::string msg = "Matrix is not positive definite, "
                                          "the sum "+gammalib::str(sum)+
                                          " occured in row and column "+
                                          gammalib::str(*row_ptr)+".";
                        throw GException::runtime_error(G_CHOL_DECOMP, msg);
                    }
                    *ptr = std::sqrt(sum); // M(row,row) = sqrt(sum)
                    diag = 1.0/(*ptr);
                }
                else {
                    *ptr = sum*diag; // M(row,col) = sum/M(row,row)
                }
            }
        }
    } // endelse: zero-row/col compression needed
 
    // Case C: all matrix elements are zero
    else {
        std::string msg = "Matrix is not positive definite, all matrix "
                          "elements are zero.";
        throw GException::runtime_error(G_CHOL_DECOMP, msg);
    }
 
    // Return matrix
    return matrix;
}
 
 
/***********************************************************************//**
 * @brief Cholesky solver
 *
 * @param[in] vector Vector for which should be solved.
 * @param[in] compress Use zero-row/column compression (default: true).
 *
 * @exception GException::invalid_argument
 *            Matrix and vector do not match.
 * @exception GException::runtime_error
 *            All matrix elements are zero.
 *
 * Solves the linear equation A*x=b using a Cholesky decomposition of A.
 * This function is to be applied on a decomposition GMatrixSymmetric matrix
 * that is produced by 'cholesky_decompose'. Case A operates on a full
 * matrix, Case B on a zero rows/columns (logically) compressed matrix.
 ***************************************************************************/
GVector GMatrixSymmetric::cholesky_solver(const GVector& vector,
                                          const bool& compress) const
{
    // Raise an exception if the matrix and vector dimensions are not compatible
    if (m_rows != vector.size()) {
        std::string msg = "Vector size "+gammalib::str(vector.size())+
                          " does not match "+gammalib::str(m_rows)+" matrix "
                          "rows. Please specify a vector of size "+
                          gammalib::str(m_rows)+".";
        throw GException::invalid_argument(G_CHOL_SOLVE, msg);
    }
 
    // Allocate result vector
    GVector x(m_rows);
 
    // Check if zero-row/col compression is needed  
    int no_zeros = ((compress && (m_num_inx == m_rows)) || !compress);
 
    // Case A: no zero-row/col compression needed
    if (no_zeros) {
 
        // Solve L*y=b, storing y in x (row>k)
        for (int row = 0; row < m_rows; ++row) {
            double sum = vector[row];
            for (int k = 0; k < row; ++k) {
                sum -= m_data[m_colstart[k]+(row-k)] * x[k]; // sum -= M(row,k) * x(k)
            }
            x[row] = sum/m_data[m_colstart[row]];            // x(row) = sum/M(row,row)
        }
 
        // Solve trans(L)*x=y (k>row)
        for (int row = m_rows-1; row >= 0; --row) {
            double  sum = x[row];
            double* ptr = m_data + m_colstart[row] + 1;
            for (int k = row+1; k < m_rows; ++k) {
                sum -= *ptr++ * x[k];               // sum -= M(k,row) * x(k)
            }
            x[row] = sum/m_data[m_colstart[row]];   // x(row) = sum/M(row,row)
        }
    } // endif: no zero-row/col compression needed
 
    // Case B: zero-row/col compression needed
    else if (m_num_inx > 0) {
 
        // Allocate loop variables and pointers
        int  row;
        int  k;
        int* row_ptr;
        int* k_ptr;
 
        // Solve L*y=b, storing y in x (row>k)
        for (row = 0, row_ptr = m_inx; row < m_num_inx; ++row, ++row_ptr) {
            double  sum = vector[*row_ptr];
            double* ptr = m_data + *row_ptr;
            for (k = 0, k_ptr = m_inx; k < row; ++k, ++k_ptr) {
                sum -= *(ptr + m_colstart[*k_ptr] - *k_ptr) * x[*k_ptr]; // sum -= M(row,k) * x(k)
            }
            x[*row_ptr] = sum/m_data[m_colstart[*row_ptr]];              // x(row) = sum/M(row,row)
        }
 
        // Solve trans(L)*x=y (k>row)
        for (row = m_num_inx-1, row_ptr = m_inx+m_num_inx-1; row >= 0; --row, --row_ptr) {
            double  sum      = x[*row_ptr];
            double* ptr_diag = m_data + m_colstart[*row_ptr];
            double* ptr      = ptr_diag - *row_ptr;
            for (k = row+1, k_ptr = m_inx+row+1; k < m_num_inx; ++k, ++k_ptr) {
                sum -= *(ptr + *k_ptr) * x[*k_ptr];              // sum -= M(k,row) * x(k)
            }
            x[*row_ptr] = sum/(*ptr_diag);                       // x(row) = sum/M(row,row)
        }
    } // endelse: zero-row/col compression needed
 
    // Case C: all matrix elements are zero
    else {
        std::string msg = "All matrix elements are zero.";
        throw GException::runtime_error(G_CHOL_SOLVE, msg);
    }
 
  // Return result vector
  return x;
}
 
 
/***********************************************************************//**
 * @brief Invert matrix using a Cholesky decomposition
 *
 * @param[in] compress Use zero-row/column compression (defaults to true).
 * @return Inverted matrix.
 *
 * @exception GException::runtime_error
 *            All matrix elements are zero.
 *
 * Inverts the matrix using a Cholesky decomposition.
 *
 * The method distinguish two cases. Case A operates on a full matrix while
 * Case B operates on a (logically) compressed matrix where all zero
 * rows/columns are skipped.
 ***************************************************************************/
GMatrixSymmetric GMatrixSymmetric::cholesky_invert(const bool& compress) const
{
    // Generate Cholesky decomposition of matrix
    GMatrixSymmetric matrix = cholesky_decompose(compress);
 
    // Check if zero-row/col compression is needed
    int no_zeros = ((compress && (matrix.m_num_inx == matrix.m_rows)) || !compress);
 
    // Case A: no zero-row/col compression needed
    if (no_zeros) {
 
        // Generate inverse of Cholesky decomposition (col>row)
        for (int row = 0; row < matrix.m_rows; ++row) {
 
            // M(row,row) = 1/M(row,row)
            double* ptr = matrix.m_data + matrix.m_colstart[row];
            *ptr        = 1.0/(*ptr);
 
            for (int col = row+1; col < matrix.m_cols; ++col) {
 
                // sum -= M(col,k)*M(k,row)
                double   sum = 0.0;
                double* ptr1 = matrix.m_data + col - row;
                double* ptr2 = ptr;
                for (int k = row; k < col; ++k) {
                    sum -= *(ptr1-- + matrix.m_colstart[k]) * *ptr2++;
                }
 
                // M(col,row) = sum/M(col,col)
                *(ptr+col-row) = sum/matrix.m_data[matrix.m_colstart[col]];
            }
        }
 
        // Matrix multiplication (col>=row)
        for (int row = 0; row < matrix.m_rows; ++row) {
            double* ptr = matrix.m_data + matrix.m_colstart[row];
            for (int col = row; col < matrix.m_cols; ++col) {
                // sum += M(row,k)*M(k,col)
                double   sum = 0.0;
                double* ptr1 = ptr + col - row;
                double* ptr2 = matrix.m_data + matrix.m_colstart[col];
                for (int k = col; k < matrix.m_cols; ++k) {
                    sum += *ptr1++ * *ptr2++;
                }
                // M(row,col) = sum
                *(ptr+col-row) = sum;
            }
        }
    } // endif: no zero-row/col compression needed
 
    // Case B: zero-row/col compression needed
    else if (matrix.m_num_inx > 0) {
 
        // Allocate loop variables and pointers
        int  row;
        int  col;
        int  k;
        int* row_ptr;
        int* col_ptr;
        int* k_ptr;
 
        // Generate inverse of Cholesky decomposition (col>row)
        for (row = 0, row_ptr = matrix.m_inx;
             row < matrix.m_num_inx; ++row, ++row_ptr) {
 
            // M(row,row) = 1/M(row,row)
            double* ptr_diag = matrix.m_data + matrix.m_colstart[*row_ptr];
            double* ptr_2    = ptr_diag - *row_ptr;
            *ptr_diag        = 1.0/(*ptr_diag);
                                     
            for (col = row+1, col_ptr = matrix.m_inx+row+1;
                 col < matrix.m_num_inx; ++col, ++col_ptr) {
 
                // sum -= M(col,k)*M(k,row)
                double  sum   = 0.0;
                double* ptr_1 = matrix.m_data + *col_ptr;
                for (k = row, k_ptr = matrix.m_inx+row;
                     k < col; ++k, ++k_ptr) {
                    sum -= *(ptr_1 + matrix.m_colstart[*k_ptr] - *k_ptr) *
                           *(ptr_2 + *k_ptr);
                }
 
                // M(col,row) = sum/M(col,col)
                *(ptr_2 + *col_ptr) =
                    sum/matrix.m_data[matrix.m_colstart[*col_ptr]];
            }
        }
 
        // Matrix multiplication (col>=row)
        for (row = 0, row_ptr = matrix.m_inx;
             row < matrix.m_num_inx; ++row, ++row_ptr) {
            double* ptr_diag = matrix.m_data + matrix.m_colstart[*row_ptr];
            double* ptr_1    = ptr_diag - *row_ptr;
 
            for (col = row, col_ptr = matrix.m_inx+row;
                 col < matrix.m_num_inx; ++col, ++col_ptr) {
 
                // sum += M(row,k)*M(k,col)
                double  sum   = 0.0;
                double* ptr_2 = matrix.m_data + matrix.m_colstart[*col_ptr] - *col_ptr;
                for (k = col, k_ptr = matrix.m_inx+col;
                     k < matrix.m_num_inx; ++k, ++k_ptr) {
                    sum += *(ptr_1 + *k_ptr) * *(ptr_2 + *k_ptr);
                }
 
                // M(row,col) = sum
                *(ptr_1 + *col_ptr) = sum;
            }
        }
    } // endelse: zero-row/col compression needed
 
    // Case C: all matrix elements are zero
    else {
        std::string msg = "All matrix elements are zero.";
        throw GException::runtime_error(G_CHOL_INVERT, msg);
    }
 
    // Return matrix
    return matrix;
}
 
 
/***********************************************************************//**
 * @brief Print matrix
 *
 * @param[in] chatter Chattiness.
 * @return String containing matrix information
 ***************************************************************************/
std::string GMatrixSymmetric::print(const GChatter& chatter) const
{
    // Initialise result string
    std::string result;
 
    // Append header
    result.append("=== GMatrixSymmetric ===");
 
    // Continue only if chatter is not silent
    if (chatter != SILENT) {
 
        // Append information
        result.append("\n"+gammalib::parformat("Number of rows"));
        result.append(gammalib::str(m_rows));
        if (m_rowsel != NULL) {
            result.append(" (compressed "+gammalib::str(m_num_rowsel)+")");
        }
        result.append("\n"+gammalib::parformat("Number of columns"));
        result.append(gammalib::str(m_cols));
        if (m_colsel != NULL) {
            result.append(" (compressed "+gammalib::str(m_num_colsel)+")");
        }
        result.append("\n"+gammalib::parformat("Number of elements"));
        result.append(gammalib::str(m_elements));
        result.append("\n"+gammalib::parformat("Number of allocated cells"));
        result.append(gammalib::str(m_alloc));
 
        // Append elements and compression schemes
        result.append(print_elements(chatter));
        result.append(print_row_compression(chatter));
        result.append(print_col_compression(chatter));
 
    } // endif: chatter was not silent
 
    // Return result
    return result;
}
 
 
/*==========================================================================
 =                                                                         =
 =                             Private methods                             =
 =                                                                         =
 ==========================================================================*/
 
/***********************************************************************//**
 * @brief Initialise class mambers
 ***************************************************************************/
void GMatrixSymmetric::init_members(void)
{
    // Initialise members
    m_num_inx = 0;
    m_inx     = NULL;
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Copy class members
 *
 * @param[in] matrix Matrix.
 ***************************************************************************/
void GMatrixSymmetric::copy_members(const GMatrixSymmetric& matrix)
{
    // Copy attributes
    m_num_inx = matrix.m_num_inx;
 
    // Copy index selection
    if (m_cols > 0) {
        m_inx     = new int[m_cols];
        for (int i = 0; i < m_cols; ++i) {
            m_inx[i] = matrix.m_inx[i];
        }
    }
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Delete class members
 ***************************************************************************/
void GMatrixSymmetric::free_members(void)
{
    // De-allocate only if memory has indeed been allocated by derived class
    if (m_inx != NULL) delete [] m_inx;
 
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Allocates matrix memory
 *
 * @param[in] rows Number of rows.
 * @param[in] columns Number of columns.
 *
 * @exception GException::invalid_argument
 *            Matrix is not symmetric.
 *
 * This method is the main constructor code that allocates and initialises
 * memory for matrix elements. The method assumes that no memory has been
 * allocated for the matrix elements, the column start index array and the
 * index array.
 * The method allocates the memory for matrix elements, the column start
 * indices and the index array, sets all matrix elements to 0.0, and sets
 * the column start indices. The content of the index array is undefined.
 *
 * @todo Verify if the index array m_inx should be initialized.
 ***************************************************************************/
void GMatrixSymmetric::alloc_members(const int& rows, const int& columns)
{
    // Determine number of physical elements in matrix
    int elements = rows*(rows+1)/2;
 
    // Throw exception if number of rows and columns is not identical
    if (rows != columns) {
        std::string msg = "Number of rows "+gammalib::str(rows)+" differs from "
                          "number of columns "+gammalib::str(columns)+". Please "
                          "specify a symmetric matrix.";
        throw GException::invalid_argument(G_ALLOC_MEMBERS, msg);
    }
 
    // Continue only if number of elements is positive
    if (elements > 0) {
 
        // Free any existing memory
        if (m_data     != NULL) delete [] m_data;
        if (m_colstart != NULL) delete [] m_colstart;
        if (m_inx      != NULL) delete [] m_inx;
 
        // Allocate matrix array and column start index array.
        m_data     = new double[elements];
        m_colstart = new int[columns+1];
        m_inx      = new int[columns];
 
        // Store matrix size (logical and physical)
        m_rows     = rows;
        m_cols     = columns;
        m_elements = elements;
        m_alloc    = elements;
 
        // Set-up column start indices
        m_colstart[0]   = 0;
        int offset = rows;
        for (int col = 1; col <= m_cols; ++col) {
            m_colstart[col] = m_colstart[col-1] + offset--;
        }
 
        // Initialise matrix elements to 0.0
        for (int i = 0; i < m_elements; ++i) {
            m_data[i] = 0.0;
        }
 
    } // endif: number of elements was positive
    
    // Return
    return;
}
 
 
/***********************************************************************//**
 * @brief Set index selection
 *
 * Determines the non-zero rows/columns in matrix and set up index array
 * that points to these rows/columns. This array is used for compressed
 * matrix calculations (Case B in the above routines).
 ***************************************************************************/
void GMatrixSymmetric::set_inx(void)
{
    // Allocate loop variables and pointers
    int row;
    int col;
 
    // Find all nonzero rows/cols
    m_num_inx = 0;
    for (row = 0; row < m_rows; ++row) {
 
        // Check first for zero diagonal. If we have one then check the rest of
        // the row
        if (m_data[m_colstart[row]] == 0.0) {
            for (col = 0; col < row; ++col) {
                if (m_data[m_colstart[col]+(row-col)] != 0.0) {
                    break;
                }
            }
            // Found a non-zero element
            if (col < row) {
                m_inx[m_num_inx++] = row;
            }
            else {
                for (col = row+1; col < m_cols; ++col) {
                    if (m_data[m_colstart[row]+(col-row)] != 0.0) {
                        break;
                    }
                }
                // Found a non-zero element
                if (col < m_cols) {
                    m_inx[m_num_inx++] = row;
                }
            }
        }
        else {
            m_inx[m_num_inx++] = row;
        }
    }
 
    // Return
    return;
}
 
 
/*==========================================================================
 =                                                                         =
 =                           Friend functions                              =
 =                                                                         =
 ==========================================================================*/
 
/***********************************************************************//**
 * @brief Binary matrix multiplication operator
 *
 * @param[in] matrix Matrix to be multiplied.
 *
 * @exception GException::invalid_argument
 *            Number of rows in @p matrix is different from number of
 *            matrix columns.
 *
 * This method performs a matrix multiplication. Since the product of two
 * symmetric matrices is not necessarily symmetric, this method returns a
 * GMatrix object.
 *
 * The operation can only succeed when the dimensions of both matrices are
 * compatible.
 ***************************************************************************/
GMatrix GMatrixSymmetric::operator*(const GMatrixSymmetric& matrix) const
{
    // Raise an exception if the matrix dimensions are not compatible
    if (m_cols != matrix.m_rows) {
        std::string msg = "Number of "+gammalib::str(m_cols)+" columns in "
                          "the first matrix differs from number of "+
                          gammalib::str(matrix.m_rows)+" rows in the second "
                          "matrix. Please specify a second matrix with "+
                          gammalib::str(m_cols)+" rows.";
        throw GException::invalid_argument(G_OP_MAT_MUL, msg);
    }
 
    // Allocate result matrix
    GMatrix result(m_rows, matrix.m_cols);
 
    // Loop over all elements of result matrix
    for (int row = 0; row < m_rows; ++row) {
        for (int col = 0; col < matrix.m_cols; ++col) {
            double sum = 0.0;
            for (int i = 0; i < m_cols; ++i) {
                sum += (*this)(row,i) * matrix(i,col);
            }
            result(row,col) = sum;
        }
    }
    
    // Return result
    return result;
}