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lammps/lib/smd/Eigen/src/LU/PartialPivLU.h
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PartialPivLU.h
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@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_PARTIALLU_H
#define EIGEN_PARTIALLU_H
namespace Eigen {
/** \ingroup LU_Module
*
* \class PartialPivLU
*
* \brief LU decomposition of a matrix with partial pivoting, and related features
*
* \param MatrixType the type of the matrix of which we are computing the LU decomposition
*
* This class represents a LU decomposition of a \b square \b invertible matrix, with partial pivoting: the matrix A
* is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
* is a permutation matrix.
*
* Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible
* matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class
* does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the
* matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.
*
* The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided
* by class FullPivLU.
*
* This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
* such as rank computation. If you need these features, use class FullPivLU.
*
* This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses
* in the general case.
* On the other hand, it is \b not suitable to determine whether a given matrix is invertible.
*
* The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
*
* \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU
*/
template<typename _MatrixType> class PartialPivLU
{
public:
typedef _MatrixType MatrixType;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
Options = MatrixType::Options,
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
typedef typename internal::traits<MatrixType>::StorageKind StorageKind;
typedef typename MatrixType::Index Index;
typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
/**
* \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via PartialPivLU::compute(const MatrixType&).
*/
PartialPivLU();
/** \brief Default Constructor with memory preallocation
*
* Like the default constructor but with preallocation of the internal data
* according to the specified problem \a size.
* \sa PartialPivLU()
*/
PartialPivLU(Index size);
/** Constructor.
*
* \param matrix the matrix of which to compute the LU decomposition.
*
* \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
* If you need to deal with non-full rank, use class FullPivLU instead.
*/
PartialPivLU(const MatrixType& matrix);
PartialPivLU& compute(const MatrixType& matrix);
/** \returns the LU decomposition matrix: the upper-triangular part is U, the
* unit-lower-triangular part is L (at least for square matrices; in the non-square
* case, special care is needed, see the documentation of class FullPivLU).
*
* \sa matrixL(), matrixU()
*/
inline const MatrixType& matrixLU() const
{
eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
return m_lu;
}
/** \returns the permutation matrix P.
*/
inline const PermutationType& permutationP() const
{
eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
return m_p;
}
/** This method returns the solution x to the equation Ax=b, where A is the matrix of which
* *this is the LU decomposition.
*
* \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
* the only requirement in order for the equation to make sense is that
* b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
*
* \returns the solution.
*
* Example: \include PartialPivLU_solve.cpp
* Output: \verbinclude PartialPivLU_solve.out
*
* Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution
* theoretically exists and is unique regardless of b.
*
* \sa TriangularView::solve(), inverse(), computeInverse()
*/
template<typename Rhs>
inline const internal::solve_retval<PartialPivLU, Rhs>
solve(const MatrixBase<Rhs>& b) const
{
eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
return internal::solve_retval<PartialPivLU, Rhs>(*this, b.derived());
}
/** \returns the inverse of the matrix of which *this is the LU decomposition.
*
* \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
* invertibility, use class FullPivLU instead.
*
* \sa MatrixBase::inverse(), LU::inverse()
*/
inline const internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType> inverse() const
{
eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
return internal::solve_retval<PartialPivLU,typename MatrixType::IdentityReturnType>
(*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
}
/** \returns the determinant of the matrix of which
* *this is the LU decomposition. It has only linear complexity
* (that is, O(n) where n is the dimension of the square matrix)
* as the LU decomposition has already been computed.
*
* \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
* optimized paths.
*
* \warning a determinant can be very big or small, so for matrices
* of large enough dimension, there is a risk of overflow/underflow.
*
* \sa MatrixBase::determinant()
*/
typename internal::traits<MatrixType>::Scalar determinant() const;
MatrixType reconstructedMatrix() const;
inline Index rows() const { return m_lu.rows(); }
inline Index cols() const { return m_lu.cols(); }
protected:
static void check_template_parameters()
{
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
}
MatrixType m_lu;
PermutationType m_p;
TranspositionType m_rowsTranspositions;
Index m_det_p;
bool m_isInitialized;
};
template<typename MatrixType>
PartialPivLU<MatrixType>::PartialPivLU()
: m_lu(),
m_p(),
m_rowsTranspositions(),
m_det_p(0),
m_isInitialized(false)
{
}
template<typename MatrixType>
PartialPivLU<MatrixType>::PartialPivLU(Index size)
: m_lu(size, size),
m_p(size),
m_rowsTranspositions(size),
m_det_p(0),
m_isInitialized(false)
{
}
template<typename MatrixType>
PartialPivLU<MatrixType>::PartialPivLU(const MatrixType& matrix)
: m_lu(matrix.rows(), matrix.rows()),
m_p(matrix.rows()),
m_rowsTranspositions(matrix.rows()),
m_det_p(0),
m_isInitialized(false)
{
compute(matrix);
}
namespace internal {
/** \internal This is the blocked version of fullpivlu_unblocked() */
template<typename Scalar, int StorageOrder, typename PivIndex>
struct partial_lu_impl
{
// FIXME add a stride to Map, so that the following mapping becomes easier,
// another option would be to create an expression being able to automatically
// warp any Map, Matrix, and Block expressions as a unique type, but since that's exactly
// a Map + stride, why not adding a stride to Map, and convenient ctors from a Matrix,
// and Block.
typedef Map<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > MapLU;
typedef Block<MapLU, Dynamic, Dynamic> MatrixType;
typedef Block<MatrixType,Dynamic,Dynamic> BlockType;
typedef typename MatrixType::RealScalar RealScalar;
typedef typename MatrixType::Index Index;
/** \internal performs the LU decomposition in-place of the matrix \a lu
* using an unblocked algorithm.
*
* In addition, this function returns the row transpositions in the
* vector \a row_transpositions which must have a size equal to the number
* of columns of the matrix \a lu, and an integer \a nb_transpositions
* which returns the actual number of transpositions.
*
* \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
*/
static Index unblocked_lu(MatrixType& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions)
{
const Index rows = lu.rows();
const Index cols = lu.cols();
const Index size = (std::min)(rows,cols);
nb_transpositions = 0;
Index first_zero_pivot = -1;
for(Index k = 0; k < size; ++k)
{
Index rrows = rows-k-1;
Index rcols = cols-k-1;
Index row_of_biggest_in_col;
RealScalar biggest_in_corner
= lu.col(k).tail(rows-k).cwiseAbs().maxCoeff(&row_of_biggest_in_col);
row_of_biggest_in_col += k;
row_transpositions[k] = PivIndex(row_of_biggest_in_col);
if(biggest_in_corner != RealScalar(0))
{
if(k != row_of_biggest_in_col)
{
lu.row(k).swap(lu.row(row_of_biggest_in_col));
++nb_transpositions;
}
// FIXME shall we introduce a safe quotient expression in cas 1/lu.coeff(k,k)
// overflow but not the actual quotient?
lu.col(k).tail(rrows) /= lu.coeff(k,k);
}
else if(first_zero_pivot==-1)
{
// the pivot is exactly zero, we record the index of the first pivot which is exactly 0,
// and continue the factorization such we still have A = PLU
first_zero_pivot = k;
}
if(k<rows-1)
lu.bottomRightCorner(rrows,rcols).noalias() -= lu.col(k).tail(rrows) * lu.row(k).tail(rcols);
}
return first_zero_pivot;
}
/** \internal performs the LU decomposition in-place of the matrix represented
* by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
* recursive, blocked algorithm.
*
* In addition, this function returns the row transpositions in the
* vector \a row_transpositions which must have a size equal to the number
* of columns of the matrix \a lu, and an integer \a nb_transpositions
* which returns the actual number of transpositions.
*
* \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
*
* \note This very low level interface using pointers, etc. is to:
* 1 - reduce the number of instanciations to the strict minimum
* 2 - avoid infinite recursion of the instanciations with Block<Block<Block<...> > >
*/
static Index blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions, PivIndex& nb_transpositions, Index maxBlockSize=256)
{
MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols);
MatrixType lu(lu1,0,0,rows,cols);
const Index size = (std::min)(rows,cols);
// if the matrix is too small, no blocking:
if(size<=16)
{
return unblocked_lu(lu, row_transpositions, nb_transpositions);
}
// automatically adjust the number of subdivisions to the size
// of the matrix so that there is enough sub blocks:
Index blockSize;
{
blockSize = size/8;
blockSize = (blockSize/16)*16;
blockSize = (std::min)((std::max)(blockSize,Index(8)), maxBlockSize);
}
nb_transpositions = 0;
Index first_zero_pivot = -1;
for(Index k = 0; k < size; k+=blockSize)
{
Index bs = (std::min)(size-k,blockSize); // actual size of the block
Index trows = rows - k - bs; // trailing rows
Index tsize = size - k - bs; // trailing size
// partition the matrix:
// A00 | A01 | A02
// lu = A_0 | A_1 | A_2 = A10 | A11 | A12
// A20 | A21 | A22
BlockType A_0(lu,0,0,rows,k);
BlockType A_2(lu,0,k+bs,rows,tsize);
BlockType A11(lu,k,k,bs,bs);
BlockType A12(lu,k,k+bs,bs,tsize);
BlockType A21(lu,k+bs,k,trows,bs);
BlockType A22(lu,k+bs,k+bs,trows,tsize);
PivIndex nb_transpositions_in_panel;
// recursively call the blocked LU algorithm on [A11^T A21^T]^T
// with a very small blocking size:
Index ret = blocked_lu(trows+bs, bs, &lu.coeffRef(k,k), luStride,
row_transpositions+k, nb_transpositions_in_panel, 16);
if(ret>=0 && first_zero_pivot==-1)
first_zero_pivot = k+ret;
nb_transpositions += nb_transpositions_in_panel;
// update permutations and apply them to A_0
for(Index i=k; i<k+bs; ++i)
{
Index piv = (row_transpositions[i] += k);
A_0.row(i).swap(A_0.row(piv));
}
if(trows)
{
// apply permutations to A_2
for(Index i=k;i<k+bs; ++i)
A_2.row(i).swap(A_2.row(row_transpositions[i]));
// A12 = A11^-1 A12
A11.template triangularView<UnitLower>().solveInPlace(A12);
A22.noalias() -= A21 * A12;
}
}
return first_zero_pivot;
}
};
/** \internal performs the LU decomposition with partial pivoting in-place.
*/
template<typename MatrixType, typename TranspositionType>
void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions, typename TranspositionType::Index& nb_transpositions)
{
eigen_assert(lu.cols() == row_transpositions.size());
eigen_assert((&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);
partial_lu_impl
<typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor, typename TranspositionType::Index>
::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.outerStride(), &row_transpositions.coeffRef(0), nb_transpositions);
}
} // end namespace internal
template<typename MatrixType>
PartialPivLU<MatrixType>& PartialPivLU<MatrixType>::compute(const MatrixType& matrix)
{
check_template_parameters();
// the row permutation is stored as int indices, so just to be sure:
eigen_assert(matrix.rows()<NumTraits<int>::highest());
m_lu = matrix;
eigen_assert(matrix.rows() == matrix.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
const Index size = matrix.rows();
m_rowsTranspositions.resize(size);
typename TranspositionType::Index nb_transpositions;
internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
m_det_p = (nb_transpositions%2) ? -1 : 1;
m_p = m_rowsTranspositions;
m_isInitialized = true;
return *this;
}
template<typename MatrixType>
typename internal::traits<MatrixType>::Scalar PartialPivLU<MatrixType>::determinant() const
{
eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
return Scalar(m_det_p) * m_lu.diagonal().prod();
}
/** \returns the matrix represented by the decomposition,
* i.e., it returns the product: P^{-1} L U.
* This function is provided for debug purpose. */
template<typename MatrixType>
MatrixType PartialPivLU<MatrixType>::reconstructedMatrix() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
// LU
MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix()
* m_lu.template triangularView<Upper>();
// P^{-1}(LU)
res = m_p.inverse() * res;
return res;
}
/***** Implementation of solve() *****************************************************/
namespace internal {
template<typename _MatrixType, typename Rhs>
struct solve_retval<PartialPivLU<_MatrixType>, Rhs>
: solve_retval_base<PartialPivLU<_MatrixType>, Rhs>
{
EIGEN_MAKE_SOLVE_HELPERS(PartialPivLU<_MatrixType>,Rhs)
template<typename Dest> void evalTo(Dest& dst) const
{
/* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
* So we proceed as follows:
* Step 1: compute c = Pb.
* Step 2: replace c by the solution x to Lx = c.
* Step 3: replace c by the solution x to Ux = c.
*/
eigen_assert(rhs().rows() == dec().matrixLU().rows());
// Step 1
dst = dec().permutationP() * rhs();
// Step 2
dec().matrixLU().template triangularView<UnitLower>().solveInPlace(dst);
// Step 3
dec().matrixLU().template triangularView<Upper>().solveInPlace(dst);
}
};
} // end namespace internal
/******** MatrixBase methods *******/
/** \lu_module
*
* \return the partial-pivoting LU decomposition of \c *this.
*
* \sa class PartialPivLU
*/
template<typename Derived>
inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::partialPivLu() const
{
return PartialPivLU<PlainObject>(eval());
}
#if EIGEN2_SUPPORT_STAGE > STAGE20_RESOLVE_API_CONFLICTS
/** \lu_module
*
* Synonym of partialPivLu().
*
* \return the partial-pivoting LU decomposition of \c *this.
*
* \sa class PartialPivLU
*/
template<typename Derived>
inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::lu() const
{
return PartialPivLU<PlainObject>(eval());
}
#endif
} // end namespace Eigen
#endif // EIGEN_PARTIALLU_H
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