LLT, LLT with hacks and LDLT algorithms
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cholesky.h
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cholesky.h
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/*
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This library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with this library; if not, write to the Free Software
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Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
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Copyright (C) 2010 Thales group
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*/
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/*
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Authors:
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Johann Dréo <johann.dreo@thalesgroup.com>
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Caner Candan <caner.candan@thalesgroup.com>
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*/
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namespace cholesky {
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/** Cholesky decomposition, given a matrix V, return a matrix L
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* such as V = L L^T (L^T being the transposed of L).
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*
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* Need a symmetric and positive definite matrix as an input, which
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* should be the case of a non-ill-conditionned covariance matrix.
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* Thus, expect a (lower) triangular matrix.
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*/
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template< typename T >
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class CholeskyBase
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{
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public:
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//! The covariance-matrix is symetric
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typedef ublas::symmetric_matrix< T, ublas::lower > CovarMat;
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//! The factorization matrix is triangular
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// FIXME check if triangular types behaviour is like having 0
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typedef ublas::matrix< T > FactorMat;
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/** Instanciate without computing anything, you are responsible of
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* calling the algorithm and getting the result with operator()
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* */
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Cholesky( size_t s1 = 1, size_t s2 = 1 ) :
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_L(ublas::zero_matrix<T>(s1,s2))
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{}
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/** Computation is made at instanciation and then cached in a member variable,
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* use decomposition() to get the result.
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*/
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Cholesky(const CovarMat& V) :
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_L(ublas::zero_matrix<T>(V.size1(),V.size2()))
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{
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(*this)( V );
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}
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/** Compute the factorization and cache the result */
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virtual void operator()( const CovarMat& V ) = 0;
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/** Compute the factorization and return the result */
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virtual const FactorMat& operator()( const CovarMat& V )
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{
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(*this)( V );
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return decomposition();
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}
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//! The decomposition of the covariance matrix
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const FactorMat & decomposition() const
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{
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return _L;
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}
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protected:
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/** Assert that the covariance matrix have the required properties and returns its dimension.
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*
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* Note: if compiled with NDEBUG, will not assert anything and just return the dimension.
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*/
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unsigned assert_properties( const CovarMat& V )
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{
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unsigned int Vl = V.size1(); // number of lines
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// the result goes in _L
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_L = ublas::zero_matrix<T>(Vl,Vl);
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#ifndef NDEBUG
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assert(Vl > 0);
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unsigned int Vc = V.size2(); // number of columns
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assert(Vc > 0);
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assert( Vl == Vc );
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// partial assert that V is semi-positive definite
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// assert that all diagonal elements are positives
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for( unsigned int i=0; i < Vl; ++i ) {
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assert( V(i,i) > 0 );
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}
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/* FIXME what is the more efficient way to check semi-positive definite? Candidates are:
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* perform the cholesky factorization
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* check if all eigenvalues are positives
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* check if all of the leading principal minors are positive
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*/
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#endif
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return Vl;
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}
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//! The decomposition is a (lower) symetric matrix, just like the covariance matrix
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FactorMat _L;
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};
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/** This standard algorithm makes use of square root and is thus subject
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* to round-off errors if the covariance matrix is very ill-conditioned.
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*
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* Compute L such that V = L L^T
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*
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* When compiled in debug mode and called on ill-conditionned matrix,
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* will raise an assert before calling the square root on a negative number.
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*/
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template< typename T >
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class CholeskyLLT : public CholeskyBase<T>
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{
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public:
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virtual void operator()( const CovarMat& V )
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{
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unsigned int N = assert_properties( V );
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unsigned int i=0, j=0, k;
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_L(0, 0) = sqrt( V(0, 0) );
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// end of the column
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for ( j = 1; j < N; ++j ) {
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_L(j, 0) = V(0, j) / _L(0, 0);
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}
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// end of the matrix
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for ( i = 1; i < N; ++i ) { // each column
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// diagonal
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double sum = 0.0;
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for ( k = 0; k < i; ++k) {
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sum += _L(i, k) * _L(i, k);
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}
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_L(i,i) = L_i_i( V, i, sum );
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for ( j = i + 1; j < N; ++j ) { // rows
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// one element
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sum = 0.0;
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for ( k = 0; k < i; ++k ) {
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sum += _L(j, k) * _L(i, k);
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}
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_L(j, i) = (V(j, i) - sum) / _L(i, i);
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} // for j in ]i,N[
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} // for i in [1,N[
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}
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/** The step of the standard LLT algorithm where round off errors may appear */
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inline virtual L_i_i( const CovarMat& V, const unsigned int& i, const double& sum ) const
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{
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// round-off errors may appear here
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assert( V(i,i) - sum >= 0 );
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return sqrt( V(i,i) - sum );
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}
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};
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/** This standard algorithm makes use of square root but do not fail
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* if the covariance matrix is very ill-conditioned.
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* Here, we propagate the error by using the absolute value before
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* computing the square root.
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*
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* Be aware that this increase round-off errors, this is just a ugly
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* hack to avoid crash.
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*/
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template< typename T >
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class CholeskyLLTabs : public CholeskyLLT<T>
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{
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public:
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inline virtual L_i_i( const CovarMat& V, const unsigned int& i, const double& sum ) const
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{
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/***** ugly hack *****/
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return sqrt( fabs( V(i,i) - sum) );
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}
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};
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/** This standard algorithm makes use of square root but do not fail
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* if the covariance matrix is very ill-conditioned.
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* Here, if the diagonal difference ir negative, we set it to zero.
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*
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* Be aware that this increase round-off errors, this is just a ugly
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* hack to avoid crash.
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*/
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template< typename T >
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class CholeskyLLTzero : public CholeskyLLT<T>
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{
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public:
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inline virtual L_i_i( const CovarMat& V, const unsigned int& i, const double& sum ) const
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{
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T Lii;
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if( V(i,i) - sum >= 0 ) {
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Lii = sqrt( V(i,i) - sum);
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} else {
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/***** ugly hack *****/
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Lii = 0;
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}
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return Lii;
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}
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};
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/** This alternative algorithm do not use square root in an inner loop,
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* but only for some diagonal elements of the matrix D.
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*
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* Computes L and D such as V = L D L^T.
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* The factorized matrix is (L D^1/2), because V = (L D^1/2) (L D^1/2)^T
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*/
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template< typename T >
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class CholeskyLDLT : public CholeskyBase<T>
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{
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public:
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virtual void operator()( const CovarMat& V )
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{
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// use "int" everywhere, because of the "j-1" operation
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int N = assert_properties( V );
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// example of an invertible matrix whose decomposition is undefined
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assert( V(0,0) != 0 );
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FactorMat L = ublas::zero_matrix<T>(N,N);
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FactorMat D = ublas::zero_matrix<T>(N,N);
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D(0,0) = V(0,0);
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for( int j=0; j<N; ++j ) { // each columns
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L(j, j) = 1;
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D(j,j) = V(j,j);
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for( int k=0; k<=j-1; ++k) { // sum
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D(j,j) -= L(j,k) * L(j,k) * D(k,k);
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}
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for( int i=j+1; i<N; ++i ) { // remaining rows
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L(i,j) = V(i,j);
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for( int k=0; k<=j-1; ++k) { // sum
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L(i,j) -= L(i,k)*L(j,k) * D(k,k);
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}
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L(i,j) /= D(j,j);
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} // for i in rows
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} // for j in columns
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_L = root( L, D );
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}
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inline FactorMat root( FactorMat& L, FactorMat& D )
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{
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// now compute the final symetric matrix: _L = L D^1/2
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// remember that V = ( L D^1/2) ( L D^1/2)^T
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// fortunately, the square root of a diagonal matrix is the square
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// root of all its elements
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FactorMat sqrt_D = D;
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for(int i=0; i<N; ++i) {
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sqrt_D(i,i) = sqrt(D(i,i));
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}
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// the factorization is thus _L*D^1/2
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return ublas::prod( L, sqrt_D );
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}
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};
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} // namespace cholesky
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