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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_WITHOUT_SQUARE_ROOT_H
#define EIGEN_CHOLESKY_WITHOUT_SQUARE_ROOT_H
/** \ingroup Cholesky_Module
*
* \class CholeskyWithoutSquareRoot
*
* \brief Robust 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 Cholesky decomposition without square root of a symmetric, positive definite
* matrix A such that A = L D L^* = U^* D U, where L is lower triangular with a unit diagonal
* and D is a diagonal matrix.
*
* Compared to a standard Cholesky decomposition, avoiding the square roots allows for faster and more
* stable computation.
*
* 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::choleskyNoSqrt(), class Cholesky
*/
template<typename MatrixType> class CholeskyWithoutSquareRoot
{
public:
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
CholeskyWithoutSquareRoot(const MatrixType& matrix)
: m_matrix(matrix.rows(), matrix.cols())
{
compute(matrix);
}
/** \returns the lower triangular matrix L */
inline Part<MatrixType, UnitLower> matrixL(void) const { return m_matrix; }
/** \returns the coefficients of the diagonal matrix D */
inline DiagonalCoeffs<MatrixType> vectorD(void) const { return m_matrix.diagonal(); }
/** \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;
void compute(const MatrixType& matrix);
protected:
/** \internal
* Used to compute and store the cholesky decomposition A = L D L^* = U^* D U.
* The strict upper part is used during the decomposition, the strict lower
* part correspond to the coefficients of L (its diagonal is equal to 1 and
* is not stored), and the diagonal entries correspond to D.
*/
MatrixType m_matrix;
bool m_isPositiveDefinite;
};
/** Compute / recompute the Cholesky decomposition A = L D L^* = U^* D U of \a matrix
*/
template<typename MatrixType>
void CholeskyWithoutSquareRoot<MatrixType>::compute(const MatrixType& a)
{
assert(a.rows()==a.cols());
const int size = a.rows();
m_matrix.resize(size, size);
m_isPositiveDefinite = true;
// Let's preallocate a temporay vector to evaluate the matrix-vector product into it.
// Unlike the standard Cholesky decomposition, here we cannot evaluate it to the destination
// matrix because it a sub-row which is not compatible suitable for efficient packet evaluation.
// (at least if we assume the matrix is col-major)
Matrix<Scalar,MatrixType::RowsAtCompileTime,1> _temporary(size);
// Note that, in this algorithm the rows of the strict upper part of m_matrix is used to store
// column vector, thus the strange .conjugate() and .transpose()...
m_matrix.row(0) = a.row(0).conjugate();
m_matrix.col(0).end(size-1) = m_matrix.row(0).end(size-1) / m_matrix.coeff(0,0);
for (int j = 1; j < size; ++j)
{
RealScalar tmp = ei_real(a.coeff(j,j) - (m_matrix.row(j).start(j) * m_matrix.col(j).start(j).conjugate()).coeff(0,0));
m_matrix.coeffRef(j,j) = tmp;
if (ei_isMuchSmallerThan(tmp,RealScalar(1)))
{
m_isPositiveDefinite = false;
return;
}
int endSize = size-j-1;
if (endSize>0)
{
_temporary.end(endSize) = ( m_matrix.block(j+1,0, endSize, j)
* m_matrix.col(j).start(j).conjugate() ).lazy();
m_matrix.row(j).end(endSize) = a.row(j).end(endSize).conjugate()
- _temporary.end(endSize).transpose();
m_matrix.col(j).end(endSize) = m_matrix.row(j).end(endSize) / tmp;
}
}
}
/** \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} D^{-1} L^{-1} b \f$ from right to left.
* \param b the column vector \f$ b \f$, which can also be a matrix.
*
* See Cholesky::solve() for a example.
*
* \sa MatrixBase::choleskyNoSqrt()
*/
template<typename MatrixType>
template<typename Derived>
typename Derived::Eval CholeskyWithoutSquareRoot<MatrixType>::solve(const MatrixBase<Derived> &b) const
{
const int size = m_matrix.rows();
ei_assert(size==b.rows());
return m_matrix.adjoint().template part<UnitUpper>()
.inverseProduct(
(matrixL()
.inverseProduct(b))
.cwise()/m_matrix.diagonal()
);
}
/** \cholesky_module
* \returns the Cholesky decomposition without square root of \c *this
*/
template<typename Derived>
inline const CholeskyWithoutSquareRoot<typename ei_eval<Derived>::type>
MatrixBase<Derived>::choleskyNoSqrt() const
{
return derived();
}
#endif // EIGEN_CHOLESKY_WITHOUT_SQUARE_ROOT_H
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