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Algorithm to equilibrate rows and columns of a square matrix

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Desire NUENTSA 2012-03-22 16:18:34 +01:00
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unsupported/Eigen/src/IterativeSolvers/Scaling.h Normal file
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012 Desire NUENTSA WAKAM <desire.nuentsa_wakam@inria.fr
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, 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 2 of
// the License, or (at your option) any later version.
//
// Eigen 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 Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#ifndef EIGEN_SCALING_H
#define EIGEN_SCALING_H
/**
* \ingroup IterativeSolvers_Module
* \brief iterative scaling algorithm to equilibrate rows and column norms in matrices
*
* This class can be used as a preprocessing tool to accelerate the convergence of iterative methods
*
* This feature is useful to limit the pivoting amount during LU/ILU factorization
* The scaling strategy as presented here preserves the symmetry of the problem
* NOTE It is assumed that the matrix does not have empty row or column,
*
* Example with key steps
* \code
* VectorXd x(n), b(n);
* SparseMatrix<double> A;
* // fill A and b;
* Scaling<SparseMatrix<double> > scal;
* // Compute the left and right scaling vectors. The matrix is equilibrated at output
* scal.computeRef(A);
* // Scale the right hand side
* b = scal.LeftScaling().cwiseProduct(b);
* // Now, solve the equilibrated linear system with any available solver
*
* // Scale back the computed solution
* x = scal.RightScaling().cwiseProduct(x);
* \endcode
*
* \tparam _MatrixType the type of the matrix. It should be a real square sparsematrix
*
* References : D. Ruiz and B. Ucar, A Symmetry Preserving Algorithm for Matrix Scaling, INRIA Research report RR-7552
*
* \sa \ref IncompleteLUT
*/
using std::abs;
using namespace Eigen;
template<typename _MatrixType>
class Scaling
{
public:
typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::Index Index;
public:
Scaling() { init(); }
Scaling(const MatrixType& matrix)
{
init();
compute(matrix);
}
~Scaling() { }
/**
* Compute the left and right diagonal matrices to scale the input matrix @p mat
*
* FIXME This algorithm will be modified such that the diagonal elements are permuted on the diagonal.
*
* \sa LeftScaling() RightScaling()
*/
void compute (const MatrixType& mat)
{
int m = mat.rows();
int n = mat.cols();
assert((m>0 && m == n) && "Please give a non - empty matrix");
m_left.resize(m);
m_right.resize(n);
m_left.setOnes();
m_right.setOnes();
m_matrix = mat;
VectorXd Dr, Dc, DrRes, DcRes; // Temporary Left and right scaling vectors
Dr.resize(m); Dc.resize(n);
DrRes.resize(m); DcRes.resize(n);
double EpsRow = 1.0, EpsCol = 1.0;
int its = 0;
do
{ // Iterate until the infinite norm of each row and column is approximately 1
// Get the maximum value in each row and column
Dr.setZero(); Dc.setZero();
for (int k=0; k<m_matrix.outerSize(); ++k)
{
for (typename MatrixType::InnerIterator it(m_matrix, k); it; ++it)
{
if ( Dr(it.row()) < abs(it.value()) )
Dr(it.row()) = abs(it.value());
if ( Dc(it.col()) < abs(it.value()) )
Dc(it.col()) = abs(it.value());
}
}
for (int i = 0; i < m; ++i)
{
Dr(i) = std::sqrt(Dr(i));
Dc(i) = std::sqrt(Dc(i));
}
// Save the scaling factors
for (int i = 0; i < m; ++i)
{
m_left(i) /= Dr(i);
m_right(i) /= Dc(i);
}
// Scale the rows and the columns of the matrix
DrRes.setZero(); DcRes.setZero();
for (int k=0; k<m_matrix.outerSize(); ++k)
{
for (typename MatrixType::InnerIterator it(m_matrix, k); it; ++it)
{
it.valueRef() = it.value()/( Dr(it.row()) * Dc(it.col()) );
// Accumulate the norms of the row and column vectors
if ( DrRes(it.row()) < abs(it.value()) )
DrRes(it.row()) = abs(it.value());
if ( DcRes(it.col()) < abs(it.value()) )
DcRes(it.col()) = abs(it.value());
}
}
DrRes.array() = (1-DrRes.array()).abs();
EpsRow = DrRes.maxCoeff();
DcRes.array() = (1-DcRes.array()).abs();
EpsCol = DcRes.maxCoeff();
its++;
}while ( (EpsRow >m_tol || EpsCol > m_tol) && (its < m_maxits) );
m_isInitialized = true;
}
/** Compute the left and right vectors to scale the vectors
* the input matrix is scaled with the computed vectors at output
*
* \sa compute()
*/
void computeRef (MatrixType& mat)
{
compute (mat);
mat = m_matrix;
}
/** Get the vector to scale the rows of the matrix
*/
VectorXd& LeftScaling()
{
return m_left;
}
/** Get the vector to scale the columns of the matrix
*/
VectorXd& RightScaling()
{
return m_right;
}
/** Set the tolerance for the convergence of the iterative scaling algorithm
*/
void setTolerance(double tol)
{
m_tol = tol;
}
protected:
void init()
{
m_tol = 1e-10;
m_maxits = 5;
m_isInitialized = false;
}
MatrixType m_matrix;
mutable ComputationInfo m_info;
bool m_isInitialized;
VectorXd m_left; // Left scaling vector
VectorXd m_right; // m_right scaling vector
double m_tol;
int m_maxits; // Maximum number of iterations allowed
};
#endif