// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_INCOMPLETE_LUT_H
#define EIGEN_INCOMPLETE_LUT_H
namespace Eigen {
namespace internal {
/** \internal
* Compute a quick-sort split of a vector
* On output, the vector row is permuted such that its elements satisfy
* abs(row(i)) >= abs(row(ncut)) if i<ncut
* abs(row(i)) <= abs(row(ncut)) if i>ncut
* \param row The vector of values
* \param ind The array of index for the elements in @p row
* \param ncut The number of largest elements to keep
**/
template <typename VectorV, typename VectorI, typename Index>
Index QuickSplit(VectorV &row, VectorI &ind, Index ncut)
{
typedef typename VectorV::RealScalar RealScalar;
using std::swap;
using std::abs;
Index mid;
Index n = row.size(); /* length of the vector */
Index first, last ;
ncut--; /* to fit the zero-based indices */
first = 0;
last = n-1;
if (ncut < first || ncut > last ) return 0;
do {
mid = first;
RealScalar abskey = abs(row(mid));
for (Index j = first + 1; j <= last; j++) {
if ( abs(row(j)) > abskey) {
++mid;
swap(row(mid), row(j));
swap(ind(mid), ind(j));
}
}
/* Interchange for the pivot element */
swap(row(mid), row(first));
swap(ind(mid), ind(first));
if (mid > ncut) last = mid - 1;
else if (mid < ncut ) first = mid + 1;
} while (mid != ncut );
return 0; /* mid is equal to ncut */
}
}// end namespace internal
/** \ingroup IterativeLinearSolvers_Module
* \class IncompleteLUT
* \brief Incomplete LU factorization with dual-threshold strategy
*
* During the numerical factorization, two dropping rules are used :
* 1) any element whose magnitude is less than some tolerance is dropped.
* This tolerance is obtained by multiplying the input tolerance @p droptol
* by the average magnitude of all the original elements in the current row.
* 2) After the elimination of the row, only the @p fill largest elements in
* the L part and the @p fill largest elements in the U part are kept
* (in addition to the diagonal element ). Note that @p fill is computed from
* the input parameter @p fillfactor which is used the ratio to control the fill_in
* relatively to the initial number of nonzero elements.
*
* The two extreme cases are when @p droptol=0 (to keep all the @p fill*2 largest elements)
* and when @p fill=n/2 with @p droptol being different to zero.
*
* References : Yousef Saad, ILUT: A dual threshold incomplete LU factorization,
* Numerical Linear Algebra with Applications, 1(4), pp 387-402, 1994.
*
* NOTE : The following implementation is derived from the ILUT implementation
* in the SPARSKIT package, Copyright (C) 2005, the Regents of the University of Minnesota
* released under the terms of the GNU LGPL:
* http://www-users.cs.umn.edu/~saad/software/SPARSKIT/README
* However, Yousef Saad gave us permission to relicense his ILUT code to MPL2.
* See the Eigen mailing list archive, thread: ILUT, date: July 8, 2012:
* http://listengine.tuxfamily.org/lists.tuxfamily.org/eigen/2012/07/msg00064.html
* alternatively, on GMANE:
* http://comments.gmane.org/gmane.comp.lib.eigen/3302
*/
template <typename _Scalar>
class IncompleteLUT : internal::noncopyable
{
typedef _Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef Matrix<Scalar,Dynamic,1> Vector;
typedef SparseMatrix<Scalar,RowMajor> FactorType;
typedef SparseMatrix<Scalar,ColMajor> PermutType;
typedef typename FactorType::Index Index;
public:
typedef Matrix<Scalar,Dynamic,Dynamic> MatrixType;
IncompleteLUT()
: m_droptol(NumTraits<Scalar>::dummy_precision()), m_fillfactor(10),
m_analysisIsOk(false), m_factorizationIsOk(false), m_isInitialized(false)
{}
template<typename MatrixType>
IncompleteLUT(const MatrixType& mat, const RealScalar& droptol=NumTraits<Scalar>::dummy_precision(), int fillfactor = 10)
: m_droptol(droptol),m_fillfactor(fillfactor),
m_analysisIsOk(false),m_factorizationIsOk(false),m_isInitialized(false)
{
eigen_assert(fillfactor != 0);
compute(mat);
}
Index rows() const { return m_lu.rows(); }
Index cols() const { return m_lu.cols(); }
/** \brief Reports whether previous computation was successful.
*
* \returns \c Success if computation was succesful,
* \c NumericalIssue if the matrix.appears to be negative.
*/
ComputationInfo info() const
{
eigen_assert(m_isInitialized && "IncompleteLUT is not initialized.");
return m_info;
}
template<typename MatrixType>
void analyzePattern(const MatrixType& amat);
template<typename MatrixType>
void factorize(const MatrixType& amat);
/**
* Compute an incomplete LU factorization with dual threshold on the matrix mat
* No pivoting is done in this version
*
**/
template<typename MatrixType>
IncompleteLUT<Scalar>& compute(const MatrixType& amat)
{
analyzePattern(amat);
factorize(amat);
return *this;
}
void setDroptol(const RealScalar& droptol);
void setFillfactor(int fillfactor);
template<typename Rhs, typename Dest>
void _solve(const Rhs& b, Dest& x) const
{
x = m_Pinv * b;
x = m_lu.template triangularView<UnitLower>().solve(x);
x = m_lu.template triangularView<Upper>().solve(x);
x = m_P * x;
}
template<typename Rhs> inline const internal::solve_retval<IncompleteLUT, Rhs>
solve(const MatrixBase<Rhs>& b) const
{
eigen_assert(m_isInitialized && "IncompleteLUT is not initialized.");
eigen_assert(cols()==b.rows()
&& "IncompleteLUT::solve(): invalid number of rows of the right hand side matrix b");
return internal::solve_retval<IncompleteLUT, Rhs>(*this, b.derived());
}
protected:
/** keeps off-diagonal entries; drops diagonal entries */
struct keep_diag {
inline bool operator() (const Index& row, const Index& col, const Scalar&) const
{
return row!=col;
}
};
protected:
FactorType m_lu;
RealScalar m_droptol;
int m_fillfactor;
bool m_analysisIsOk;
bool m_factorizationIsOk;
bool m_isInitialized;
ComputationInfo m_info;
PermutationMatrix<Dynamic,Dynamic,Index> m_P; // Fill-reducing permutation
PermutationMatrix<Dynamic,Dynamic,Index> m_Pinv; // Inverse permutation
};
/**
* Set control parameter droptol
* \param droptol Drop any element whose magnitude is less than this tolerance
**/
template<typename Scalar>
void IncompleteLUT<Scalar>::setDroptol(const RealScalar& droptol)
{
this->m_droptol = droptol;
}
/**
* Set control parameter fillfactor
* \param fillfactor This is used to compute the number @p fill_in of largest elements to keep on each row.
**/
template<typename Scalar>
void IncompleteLUT<Scalar>::setFillfactor(int fillfactor)
{
this->m_fillfactor = fillfactor;
}
template <typename Scalar>
template<typename _MatrixType>
void IncompleteLUT<Scalar>::analyzePattern(const _MatrixType& amat)
{
// Compute the Fill-reducing permutation
SparseMatrix<Scalar,ColMajor, Index> mat1 = amat;
SparseMatrix<Scalar,ColMajor, Index> mat2 = amat.transpose();
// Symmetrize the pattern
// FIXME for a matrix with nearly symmetric pattern, mat2+mat1 is the appropriate choice.
// on the other hand for a really non-symmetric pattern, mat2*mat1 should be prefered...
SparseMatrix<Scalar,ColMajor, Index> AtA = mat2 + mat1;
AtA.prune(keep_diag());
internal::minimum_degree_ordering<Scalar, Index>(AtA, m_P); // Then compute the AMD ordering...
m_Pinv = m_P.inverse(); // ... and the inverse permutation
m_analysisIsOk = true;
m_factorizationIsOk = false;
m_isInitialized = false;
}
template <typename Scalar>
template<typename _MatrixType>
void IncompleteLUT<Scalar>::factorize(const _MatrixType& amat)
{
using std::sqrt;
using std::swap;
using std::abs;
eigen_assert((amat.rows() == amat.cols()) && "The factorization should be done on a square matrix");
Index n = amat.cols(); // Size of the matrix
m_lu.resize(n,n);
// Declare Working vectors and variables
Vector u(n) ; // real values of the row -- maximum size is n --
VectorXi ju(n); // column position of the values in u -- maximum size is n
VectorXi jr(n); // Indicate the position of the nonzero elements in the vector u -- A zero location is indicated by -1
// Apply the fill-reducing permutation
eigen_assert(m_analysisIsOk && "You must first call analyzePattern()");
SparseMatrix<Scalar,RowMajor, Index> mat;
mat = amat.twistedBy(m_Pinv);
// Initialization
jr.fill(-1);
ju.fill(0);
u.fill(0);
// number of largest elements to keep in each row:
Index fill_in = static_cast<Index> (amat.nonZeros()*m_fillfactor)/n+1;
if (fill_in > n) fill_in = n;
// number of largest nonzero elements to keep in the L and the U part of the current row:
Index nnzL = fill_in/2;
Index nnzU = nnzL;
m_lu.reserve(n * (nnzL + nnzU + 1));
// global loop over the rows of the sparse matrix
for (Index ii = 0; ii < n; ii++)
{
// 1 - copy the lower and the upper part of the row i of mat in the working vector u
Index sizeu = 1; // number of nonzero elements in the upper part of the current row
Index sizel = 0; // number of nonzero elements in the lower part of the current row
ju(ii) = ii;
u(ii) = 0;
jr(ii) = ii;
RealScalar rownorm = 0;
typename FactorType::InnerIterator j_it(mat, ii); // Iterate through the current row ii
for (; j_it; ++j_it)
{
Index k = j_it.index();
if (k < ii)
{
// copy the lower part
ju(sizel) = k;
u(sizel) = j_it.value();
jr(k) = sizel;
++sizel;
}
else if (k == ii)
{
u(ii) = j_it.value();
}
else
{
// copy the upper part
Index jpos = ii + sizeu;
ju(jpos) = k;
u(jpos) = j_it.value();
jr(k) = jpos;
++sizeu;
}
rownorm += numext::abs2(j_it.value());
}
// 2 - detect possible zero row
if(rownorm==0)
{
m_info = NumericalIssue;
return;
}
// Take the 2-norm of the current row as a relative tolerance
rownorm = sqrt(rownorm);
// 3 - eliminate the previous nonzero rows
Index jj = 0;
Index len = 0;
while (jj < sizel)
{
// In order to eliminate in the correct order,
// we must select first the smallest column index among ju(jj:sizel)
Index k;
Index minrow = ju.segment(jj,sizel-jj).minCoeff(&k); // k is relative to the segment
k += jj;
if (minrow != ju(jj))
{
// swap the two locations
Index j = ju(jj);
swap(ju(jj), ju(k));
jr(minrow) = jj; jr(j) = k;
swap(u(jj), u(k));
}
// Reset this location
jr(minrow) = -1;
// Start elimination
typename FactorType::InnerIterator ki_it(m_lu, minrow);
while (ki_it && ki_it.index() < minrow) ++ki_it;
eigen_internal_assert(ki_it && ki_it.col()==minrow);
Scalar fact = u(jj) / ki_it.value();
// drop too small elements
if(abs(fact) <= m_droptol)
{
jj++;
continue;
}
// linear combination of the current row ii and the row minrow
++ki_it;
for (; ki_it; ++ki_it)
{
Scalar prod = fact * ki_it.value();
Index j = ki_it.index();
Index jpos = jr(j);
if (jpos == -1) // fill-in element
{
Index newpos;
if (j >= ii) // dealing with the upper part
{
newpos = ii + sizeu;
sizeu++;
eigen_internal_assert(sizeu<=n);
}
else // dealing with the lower part
{
newpos = sizel;
sizel++;
eigen_internal_assert(sizel<=ii);
}
ju(newpos) = j;
u(newpos) = -prod;
jr(j) = newpos;
}
else
u(jpos) -= prod;
}
// store the pivot element
u(len) = fact;
ju(len) = minrow;
++len;
jj++;
} // end of the elimination on the row ii
// reset the upper part of the pointer jr to zero
for(Index k = 0; k <sizeu; k++) jr(ju(ii+k)) = -1;
// 4 - partially sort and insert the elements in the m_lu matrix
// sort the L-part of the row
sizel = len;
len = (std::min)(sizel, nnzL);
typename Vector::SegmentReturnType ul(u.segment(0, sizel));
typename VectorXi::SegmentReturnType jul(ju.segment(0, sizel));
internal::QuickSplit(ul, jul, len);
// store the largest m_fill elements of the L part
m_lu.startVec(ii);
for(Index k = 0; k < len; k++)
m_lu.insertBackByOuterInnerUnordered(ii,ju(k)) = u(k);
// store the diagonal element
// apply a shifting rule to avoid zero pivots (we are doing an incomplete factorization)
if (u(ii) == Scalar(0))
u(ii) = sqrt(m_droptol) * rownorm;
m_lu.insertBackByOuterInnerUnordered(ii, ii) = u(ii);
// sort the U-part of the row
// apply the dropping rule first
len = 0;
for(Index k = 1; k < sizeu; k++)
{
if(abs(u(ii+k)) > m_droptol * rownorm )
{
++len;
u(ii + len) = u(ii + k);
ju(ii + len) = ju(ii + k);
}
}
sizeu = len + 1; // +1 to take into account the diagonal element
len = (std::min)(sizeu, nnzU);
typename Vector::SegmentReturnType uu(u.segment(ii+1, sizeu-1));
typename VectorXi::SegmentReturnType juu(ju.segment(ii+1, sizeu-1));
internal::QuickSplit(uu, juu, len);
// store the largest elements of the U part
for(Index k = ii + 1; k < ii + len; k++)
m_lu.insertBackByOuterInnerUnordered(ii,ju(k)) = u(k);
}
m_lu.finalize();
m_lu.makeCompressed();
m_factorizationIsOk = true;
m_isInitialized = m_factorizationIsOk;
m_info = Success;
}
namespace internal {
template<typename _MatrixType, typename Rhs>
struct solve_retval<IncompleteLUT<_MatrixType>, Rhs>
: solve_retval_base<IncompleteLUT<_MatrixType>, Rhs>
{
typedef IncompleteLUT<_MatrixType> Dec;
EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
template<typename Dest> void evalTo(Dest& dst) const
{
dec()._solve(rhs(),dst);
}
};
} // end namespace internal
} // end namespace Eigen
#endif // EIGEN_INCOMPLETE_LUT_H