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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) 2008 Gael Guennebaud <g.gael@free.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_CHOLESKY_H
#define EIGEN_CHOLESKY_H
/** \ingroup Cholesky_Module
*
* \class Cholesky
*
* \brief Standard Cholesky decomposition of a matrix and associated features
*
* \param MatrixType the type of the matrix of which we are computing the Cholesky decomposition
*
* This class performs a standard Cholesky decomposition of a symmetric, positive definite
* matrix A such that A = LL^* = U^*U, where L is lower triangular.
*
* While the Cholesky decomposition is particularly useful to solve selfadjoint problems like D^*D x = b,
* for that purpose, we recommend the Cholesky decomposition without square root which is more stable
* and even faster. Nevertheless, this standard Cholesky decomposition remains useful in many other
* situations like generalised eigen problems with hermitian matrices.
*
* Note that during the decomposition, only the upper triangular part of A is considered. Therefore,
* the strict lower part does not have to store correct values.
*
* \sa MatrixBase::cholesky(), class CholeskyWithoutSquareRoot
*/
template<typename MatrixType> class Cholesky
{
private:
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
enum {
PacketSize = ei_packet_traits<Scalar>::size,
AlignmentMask = int(PacketSize)-1
};
public:
Cholesky(const MatrixType& matrix)
: m_matrix(matrix.rows(), matrix.cols())
{
compute(matrix);
}
inline Part<MatrixType, Lower> matrixL(void) const { return m_matrix; }
/** \returns true if the matrix is positive definite */
inline bool isPositiveDefinite(void) const { return m_isPositiveDefinite; }
template<typename Derived>
typename Derived::Eval solve(const MatrixBase<Derived> &b) const;
template<typename Derived>
bool solveInPlace(MatrixBase<Derived> &bAndX) const;
void compute(const MatrixType& matrix);
protected:
/** \internal
* Used to compute and store L
* The strict upper part is not used and even not initialized.
*/
MatrixType m_matrix;
bool m_isPositiveDefinite;
};
/** Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of \a matrix
*/
template<typename MatrixType>
void Cholesky<MatrixType>::compute(const MatrixType& a)
{
assert(a.rows()==a.cols());
const int size = a.rows();
m_matrix.resize(size, size);
const RealScalar eps = ei_sqrt(precision<Scalar>());
RealScalar x;
x = ei_real(a.coeff(0,0));
m_isPositiveDefinite = x > eps && ei_isMuchSmallerThan(ei_imag(a.coeff(0,0)), RealScalar(1));
m_matrix.coeffRef(0,0) = ei_sqrt(x);
m_matrix.col(0).end(size-1) = a.row(0).end(size-1).adjoint() / ei_real(m_matrix.coeff(0,0));
for (int j = 1; j < size; ++j)
{
Scalar tmp = ei_real(a.coeff(j,j)) - m_matrix.row(j).start(j).norm2();
x = ei_real(tmp);
if (x < eps || (!ei_isMuchSmallerThan(ei_imag(tmp), RealScalar(1))))
{
m_isPositiveDefinite = false;
return;
}
m_matrix.coeffRef(j,j) = x = ei_sqrt(x);
int endSize = size-j-1;
if (endSize>0) {
// Note that when all matrix columns have good alignment, then the following
// product is guaranteed to be optimal with respect to alignment.
m_matrix.col(j).end(endSize) =
(m_matrix.block(j+1, 0, endSize, j) * m_matrix.row(j).start(j).adjoint()).lazy();
// FIXME could use a.col instead of a.row
m_matrix.col(j).end(endSize) = (a.row(j).end(endSize).adjoint()
- m_matrix.col(j).end(endSize) ) / x;
}
}
}
/** \returns the solution of \f$ A x = b \f$ using the current decomposition of A.
* In other words, it returns \f$ A^{-1} b \f$ computing
* \f$ {L^{*}}^{-1} L^{-1} b \f$ from right to left.
* \param b the column vector \f$ b \f$, which can also be a matrix.
*
* Example: \include Cholesky_solve.cpp
* Output: \verbinclude Cholesky_solve.out
*
* \sa MatrixBase::cholesky(), CholeskyWithoutSquareRoot::solve()
*/
template<typename MatrixType>
template<typename Derived>
typename Derived::Eval Cholesky<MatrixType>::solve(const MatrixBase<Derived> &b) const
{
const int size = m_matrix.rows();
ei_assert(size==b.rows());
typename ei_eval_to_column_major<Derived>::type x(b);
solveInPlace(x);
return x;
//return m_matrix.adjoint().template part<Upper>().solveTriangular(matrixL().solveTriangular(b));
}
/** Computes the solution x of \f$ A x = b \f$ using the current decomposition of A.
* The result is stored in \a bAndx
*
* \returns true in case of success, false otherwise.
*
* In other words, it computes \f$ b = A^{-1} b \f$ with
* \f$ {L^{*}}^{-1} L^{-1} b \f$ from right to left.
* \param bAndX stores both the matrix \f$ b \f$ and the result \f$ x \f$
*
* Example: \include Cholesky_solve.cpp
* Output: \verbinclude Cholesky_solve.out
*
* \sa MatrixBase::cholesky(), Cholesky::solve()
*/
template<typename MatrixType>
template<typename Derived>
bool Cholesky<MatrixType>::solveInPlace(MatrixBase<Derived> &bAndX) const
{
const int size = m_matrix.rows();
ei_assert(size==bAndX.rows());
if (!m_isPositiveDefinite)
return false;
matrixL().solveTriangularInPlace(bAndX);
m_matrix.adjoint().template part<Upper>().solveTriangularInPlace(bAndX);
return true;
}
/** \cholesky_module
* \returns the Cholesky decomposition of \c *this
*/
template<typename Derived>
inline const Cholesky<typename MatrixBase<Derived>::EvalType>
MatrixBase<Derived>::cholesky() const
{
return Cholesky<typename ei_eval<Derived>::type>(derived());
}
#endif // EIGEN_CHOLESKY_H
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