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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra. Eigen itself is part of the KDE project.
//
// Copyright (C) 2006-2008 Benoit Jacob <jacob@math.jussieu.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_LU_H
#define EIGEN_LU_H
/** \ingroup LU_Module
*
* \class LU
*
* \brief LU decomposition of a matrix with complete 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 any matrix, with complete pivoting: the matrix A
* is decomposed as A = PLUQ where L is unit-lower-triangular, U is upper-triangular, and P and Q
* are permutation matrices. This is a rank-revealing LU decomposition. The eigenvalues of U are
* in non-increasing order.
*
* This decomposition provides the generic approach to solving systems of linear equations, computing
* the rank, invertibility, inverse, kernel, and determinant.
*
* The data of the LU decomposition can be directly accessed through the methods matrixLU(),
* permutationP(), permutationQ(). Convenience methods matrixL(), matrixU() are also provided.
*
* As an exemple, here is how the original matrix can be retrieved, in the square case:
* \include class_LU_1.cpp
* Output: \verbinclude class_LU_1.out
*
* When the matrix is not square, matrixL() is no longer very useful: if one needs it, one has
* to construct the L matrix by hand, as shown in this example:
* \include class_LU_2.cpp
* Output: \verbinclude class_LU_2.out
*
* \sa MatrixBase::lu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse()
*/
template<typename MatrixType> class LU
{
public:
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
typedef Matrix<int, 1, MatrixType::ColsAtCompileTime> IntRowVectorType;
typedef Matrix<int, MatrixType::RowsAtCompileTime, 1> IntColVectorType;
typedef Matrix<Scalar, 1, MatrixType::ColsAtCompileTime> RowVectorType;
typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVectorType;
enum { MaxSmallDimAtCompileTime = EIGEN_ENUM_MIN(
MatrixType::MaxColsAtCompileTime,
MatrixType::MaxRowsAtCompileTime)
};
typedef Matrix<typename MatrixType::Scalar, MatrixType::ColsAtCompileTime, Dynamic,
MatrixType::Flags&RowMajorBit,
MatrixType::MaxColsAtCompileTime, MaxSmallDimAtCompileTime> KernelReturnType;
/** Constructor.
*
* \param matrix the matrix of which to compute the LU decomposition.
*/
LU(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 LU).
*
* \sa matrixL(), matrixU()
*/
inline const MatrixType& matrixLU() const
{
return m_lu;
}
/** \returns an expression of the unit-lower-triangular part of the LU matrix. In the square case,
* this is the L matrix. In the non-square, actually obtaining the L matrix takes some
* more care, see the documentation of class LU.
*
* \sa matrixLU(), matrixU()
*/
inline const Part<MatrixType, UnitLower> matrixL() const
{
return m_lu;
}
/** \returns an expression of the U matrix, i.e. the upper-triangular part of the LU matrix.
*
* \note The eigenvalues of U are sorted in non-increasing order.
*
* \sa matrixLU(), matrixL()
*/
inline const Part<MatrixType, Upper> matrixU() const
{
return m_lu;
}
/** \returns a vector of integers, whose size is the number of rows of the matrix being decomposed,
* representing the P permutation i.e. the permutation of the rows. For its precise meaning,
* see the examples given in the documentation of class LU.
*
* \sa permutationQ()
*/
inline const IntColVectorType& permutationP() const
{
return m_p;
}
/** \returns a vector of integers, whose size is the number of columns of the matrix being
* decomposed, representing the Q permutation i.e. the permutation of the columns.
* For its precise meaning, see the examples given in the documentation of class LU.
*
* \sa permutationP()
*/
inline const IntRowVectorType& permutationQ() const
{
return m_q;
}
/** Computes the kernel of the matrix.
*
* \note: this method is only allowed on non-invertible matrices, as determined by
* isInvertible(). Calling it on an invertible matrice will make an assertion fail.
*
* \param result a pointer to the matrix in which to store the kernel. The columns of this
* matrix will be set to form a basis of the kernel (it will be resized
* if necessary).
*
* Example: \include LU_computeKernel.cpp
* Output: \verbinclude LU_computeKernel.out
*
* \sa kernel()
*/
void computeKernel(Matrix<typename MatrixType::Scalar,
MatrixType::ColsAtCompileTime, Dynamic,
MatrixType::Flags&RowMajorBit,
MatrixType::MaxColsAtCompileTime,
LU<MatrixType>::MaxSmallDimAtCompileTime
> *result) const;
/** \returns the kernel of the matrix. The columns of the returned matrix
* will form a basis of the kernel.
*
* \note: this method is only allowed on non-invertible matrices, as determined by
* isInvertible(). Calling it on an invertible matrice will make an assertion fail.
*
* \note: this method returns a matrix by value, which induces some inefficiency.
* If you prefer to avoid this overhead, use computeKernel() instead.
*
* Example: \include LU_kernel.cpp
* Output: \verbinclude LU_kernel.out
*
* \sa computeKernel()
*/
const KernelReturnType kernel() const;
/** This method finds a solution x to the equation Ax=b, where A is the matrix of which
* *this is the LU decomposition, if any exists.
*
* \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.
* \param result a pointer to the vector or matrix in which to store the solution, if any exists.
* Resized if necessary, so that result->rows()==A.cols() and result->cols()==b.cols().
* If no solution exists, *result is left with undefined coefficients.
*
* \returns true if any solution exists, false if no solution exists.
*
* \note If there exist more than one solution, this method will arbitrarily choose one.
* If you need a complete analysis of the space of solutions, take the one solution obtained * by this method and add to it elements of the kernel, as determined by kernel().
*
* Example: \include LU_solve.cpp
* Output: \verbinclude LU_solve.out
*
* \sa MatrixBase::solveTriangular(), kernel(), computeKernel(), inverse(), computeInverse()
*/
template<typename OtherDerived, typename ResultType>
bool solve(
const MatrixBase<OtherDerived>& b,
ResultType *result
) const;
/** \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 This is only for square matrices.
*
* \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 ei_traits<MatrixType>::Scalar determinant() const;
/** \returns the rank of the matrix of which *this is the LU decomposition.
*
* \note This is computed at the time of the construction of the LU decomposition. This
* method does not perform any further computation.
*/
inline int rank() const
{
return m_rank;
}
/** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition.
*
* \note Since the rank is computed at the time of the construction of the LU decomposition, this
* method almost does not perform any further computation.
*/
inline int dimensionOfKernel() const
{
return m_lu.cols() - m_rank;
}
/** \returns true if the matrix of which *this is the LU decomposition represents an injective
* linear map, i.e. has trivial kernel; false otherwise.
*
* \note Since the rank is computed at the time of the construction of the LU decomposition, this
* method almost does not perform any further computation.
*/
inline bool isInjective() const
{
return m_rank == m_lu.cols();
}
/** \returns true if the matrix of which *this is the LU decomposition represents a surjective
* linear map; false otherwise.
*
* \note Since the rank is computed at the time of the construction of the LU decomposition, this
* method almost does not perform any further computation.
*/
inline bool isSurjective() const
{
return m_rank == m_lu.rows();
}
/** \returns true if the matrix of which *this is the LU decomposition is invertible.
*
* \note Since the rank is computed at the time of the construction of the LU decomposition, this
* method almost does not perform any further computation.
*/
inline bool isInvertible() const
{
return isInjective() && isSurjective();
}
/** Computes the inverse of the matrix of which *this is the LU decomposition.
*
* \param result a pointer to the matrix into which to store the inverse. Resized if needed.
*
* \note If this matrix is not invertible, *result is left with undefined coefficients.
* Use isInvertible() to first determine whether this matrix is invertible.
*
* \sa MatrixBase::computeInverse(), inverse()
*/
inline void computeInverse(MatrixType *result) const
{
solve(MatrixType::Identity(m_lu.rows(), m_lu.cols()), result);
}
/** \returns the inverse of the matrix of which *this is the LU decomposition.
*
* \note If this matrix is not invertible, the returned matrix has undefined coefficients.
* Use isInvertible() to first determine whether this matrix is invertible.
*
* \sa computeInverse(), MatrixBase::inverse()
*/
inline MatrixType inverse() const
{
MatrixType result;
computeInverse(&result);
return result;
}
protected:
MatrixType m_lu;
IntColVectorType m_p;
IntRowVectorType m_q;
int m_det_pq;
int m_rank;
};
template<typename MatrixType>
LU<MatrixType>::LU(const MatrixType& matrix)
: m_lu(matrix),
m_p(matrix.rows()),
m_q(matrix.cols())
{
const int size = matrix.diagonal().size();
const int rows = matrix.rows();
const int cols = matrix.cols();
IntColVectorType rows_transpositions(matrix.rows());
IntRowVectorType cols_transpositions(matrix.cols());
int number_of_transpositions = 0;
RealScalar biggest = RealScalar(0);
for(int k = 0; k < size; k++)
{
int row_of_biggest_in_corner, col_of_biggest_in_corner;
RealScalar biggest_in_corner;
biggest_in_corner = m_lu.corner(Eigen::BottomRight, rows-k, cols-k)
.cwise().abs()
.maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
row_of_biggest_in_corner += k;
col_of_biggest_in_corner += k;
rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
if(k != row_of_biggest_in_corner) {
m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
number_of_transpositions++;
}
if(k != col_of_biggest_in_corner) {
m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
number_of_transpositions++;
}
if(k==0) biggest = biggest_in_corner;
const Scalar lu_k_k = m_lu.coeff(k,k);
if(ei_isMuchSmallerThan(lu_k_k, biggest)) continue;
if(k<rows-1)
m_lu.col(k).end(rows-k-1) /= lu_k_k;
if(k<size-1)
for( int col = k + 1; col < cols; col++ )
m_lu.col(col).end(rows-k-1) -= m_lu.col(k).end(rows-k-1) * m_lu.coeff(k,col);
}
for(int k = 0; k < matrix.rows(); k++) m_p.coeffRef(k) = k;
for(int k = size-1; k >= 0; k--)
std::swap(m_p.coeffRef(k), m_p.coeffRef(rows_transpositions.coeff(k)));
for(int k = 0; k < matrix.cols(); k++) m_q.coeffRef(k) = k;
for(int k = 0; k < size; k++)
std::swap(m_q.coeffRef(k), m_q.coeffRef(cols_transpositions.coeff(k)));
m_det_pq = (number_of_transpositions%2) ? -1 : 1;
for(m_rank = 0; m_rank < size; m_rank++)
if(ei_isMuchSmallerThan(m_lu.diagonal().coeff(m_rank), m_lu.diagonal().coeff(0)))
break;
}
template<typename MatrixType>
typename ei_traits<MatrixType>::Scalar LU<MatrixType>::determinant() const
{
return Scalar(m_det_pq) * m_lu.diagonal().redux(ei_scalar_product_op<Scalar>());
}
template<typename MatrixType>
void LU<MatrixType>::computeKernel(Matrix<typename MatrixType::Scalar,
MatrixType::ColsAtCompileTime, Dynamic,
MatrixType::Flags&RowMajorBit,
MatrixType::MaxColsAtCompileTime,
LU<MatrixType>::MaxSmallDimAtCompileTime
> *result) const
{
ei_assert(!isInvertible());
const int dimker = dimensionOfKernel(), cols = m_lu.cols();
result->resize(cols, dimker);
/* Let us use the following lemma:
*
* Lemma: If the matrix A has the LU decomposition PAQ = LU,
* then Ker A = Q( Ker U ).
*
* Proof: trivial: just keep in mind that P, Q, L are invertible.
*/
/* Thus, all we need to do is to compute Ker U, and then apply Q.
*
* U is upper triangular, with eigenvalues sorted in decreasing order of
* absolute value. Thus, the diagonal of U ends with exactly
* m_dimKer zero's. Let us use that to construct m_dimKer linearly
* independent vectors in Ker U.
*/
Matrix<Scalar, Dynamic, Dynamic, MatrixType::Flags&RowMajorBit,
MatrixType::MaxColsAtCompileTime, MaxSmallDimAtCompileTime>
y(-m_lu.corner(TopRight, m_rank, dimker));
m_lu.corner(TopLeft, m_rank, m_rank)
.template marked<Upper>()
.solveTriangularInPlace(y);
for(int i = 0; i < m_rank; i++)
result->row(m_q.coeff(i)) = y.row(i);
for(int i = m_rank; i < cols; i++) result->row(m_q.coeff(i)).setZero();
for(int k = 0; k < dimker; k++) result->coeffRef(m_q.coeff(m_rank+k), k) = Scalar(1);
}
template<typename MatrixType>
const typename LU<MatrixType>::KernelReturnType
LU<MatrixType>::kernel() const
{
Matrix<typename MatrixType::Scalar, MatrixType::ColsAtCompileTime, Dynamic,
MatrixType::Flags&RowMajorBit,
MatrixType::MaxColsAtCompileTime,
LU<MatrixType>::MaxSmallDimAtCompileTime> result(m_lu.cols(), dimensionOfKernel());
computeKernel(&result);
return result;
}
template<typename MatrixType>
template<typename OtherDerived, typename ResultType>
bool LU<MatrixType>::solve(
const MatrixBase<OtherDerived>& b,
ResultType *result
) const
{
/* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
* So we proceed as follows:
* Step 1: compute c = Pb.
* Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
* Step 3: compute d such that Ud = c. Check if such d really exists.
* Step 4: result = Qd;
*/
const int rows = m_lu.rows();
ei_assert(b.rows() == rows);
const int smalldim = std::min(rows, m_lu.cols());
typename OtherDerived::Eval c(b.rows(), b.cols());
// Step 1
for(int i = 0; i < rows; i++) c.row(m_p.coeff(i)) = b.row(i);
// Step 2
Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime,
MatrixType::Flags&RowMajorBit,
MatrixType::MaxRowsAtCompileTime,
MatrixType::MaxRowsAtCompileTime> l(rows, rows);
l.setZero();
l.corner(Eigen::TopLeft,rows,smalldim)
= m_lu.corner(Eigen::TopLeft,rows,smalldim);
l.template marked<UnitLower>().solveTriangularInPlace(c);
// Step 3
if(!isSurjective())
{
// is c is in the image of U ?
RealScalar biggest_in_c = c.corner(TopLeft, m_rank, c.cols()).cwise().abs().maxCoeff();
for(int col = 0; col < c.cols(); col++)
for(int row = m_rank; row < c.rows(); row++)
if(!ei_isMuchSmallerThan(c.coeff(row,col), biggest_in_c))
return false;
}
Matrix<Scalar, Dynamic, OtherDerived::ColsAtCompileTime,
MatrixType::Flags&RowMajorBit,
MatrixType::MaxRowsAtCompileTime, OtherDerived::MaxColsAtCompileTime>
d(c.corner(TopLeft, m_rank, c.cols()));
m_lu.corner(TopLeft, m_rank, m_rank)
.template marked<Upper>()
.solveTriangularInPlace(d);
// Step 4
result->resize(m_lu.cols(), b.cols());
for(int i = 0; i < m_rank; i++) result->row(m_q.coeff(i)) = d.row(i);
for(int i = m_rank; i < m_lu.cols(); i++) result->row(m_q.coeff(i)).setZero();
return true;
}
/** \lu_module
*
* \return the LU decomposition of \c *this.
*
* \sa class LU
*/
template<typename Derived>
inline const LU<typename MatrixBase<Derived>::EvalType>
MatrixBase<Derived>::lu() const
{
return eval();
}
#endif // EIGEN_LU_H
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