1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2008-2011 Gael Guennebaud <gael.guennebaud@inria.fr>
5 // Copyright (C) 2009 Keir Mierle <mierle@gmail.com>
6 // Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
7 // Copyright (C) 2011 Timothy E. Holy <tim.holy@gmail.com >
8 //
9 // This Source Code Form is subject to the terms of the Mozilla
10 // Public License v. 2.0. If a copy of the MPL was not distributed
11 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
12 
13 #ifndef EIGEN_LDLT_H
14 #define EIGEN_LDLT_H
15 
16 namespace Eigen {
17 
18 namespace internal {
19   template<typename MatrixType, int UpLo> struct LDLT_Traits;
20 
21   // PositiveSemiDef means positive semi-definite and non-zero; same for NegativeSemiDef
22   enum SignMatrix { PositiveSemiDef, NegativeSemiDef, ZeroSign, Indefinite };
23 }
24 
25 /** \ingroup Cholesky_Module
26   *
27   * \class LDLT
28   *
29   * \brief Robust Cholesky decomposition of a matrix with pivoting
30   *
31   * \param MatrixType the type of the matrix of which to compute the LDL^T Cholesky decomposition
32   * \param UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
33   *             The other triangular part won't be read.
34   *
35   * Perform a robust Cholesky decomposition of a positive semidefinite or negative semidefinite
36   * matrix \f$ A \f$ such that \f$ A =  P^TLDL^*P \f$, where P is a permutation matrix, L
37   * is lower triangular with a unit diagonal and D is a diagonal matrix.
38   *
39   * The decomposition uses pivoting to ensure stability, so that L will have
40   * zeros in the bottom right rank(A) - n submatrix. Avoiding the square root
41   * on D also stabilizes the computation.
42   *
43   * Remember that Cholesky decompositions are not rank-revealing. Also, do not use a Cholesky
44   * decomposition to determine whether a system of equations has a solution.
45   *
46   * \sa MatrixBase::ldlt(), class LLT
47   */
48 template<typename _MatrixType, int _UpLo> class LDLT
49 {
50   public:
51     typedef _MatrixType MatrixType;
52     enum {
53       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
54       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
55       Options = MatrixType::Options & ~RowMajorBit, // these are the options for the TmpMatrixType, we need a ColMajor matrix here!
56       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
57       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
58       UpLo = _UpLo
59     };
60     typedef typename MatrixType::Scalar Scalar;
61     typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
62     typedef typename MatrixType::Index Index;
63     typedef Matrix<Scalar, RowsAtCompileTime, 1, Options, MaxRowsAtCompileTime, 1> TmpMatrixType;
64 
65     typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
66     typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
67 
68     typedef internal::LDLT_Traits<MatrixType,UpLo> Traits;
69 
70     /** \brief Default Constructor.
71       *
72       * The default constructor is useful in cases in which the user intends to
73       * perform decompositions via LDLT::compute(const MatrixType&).
74       */
LDLT()75     LDLT()
76       : m_matrix(),
77         m_transpositions(),
78         m_sign(internal::ZeroSign),
79         m_isInitialized(false)
80     {}
81 
82     /** \brief Default Constructor with memory preallocation
83       *
84       * Like the default constructor but with preallocation of the internal data
85       * according to the specified problem \a size.
86       * \sa LDLT()
87       */
LDLT(Index size)88     LDLT(Index size)
89       : m_matrix(size, size),
90         m_transpositions(size),
91         m_temporary(size),
92         m_sign(internal::ZeroSign),
93         m_isInitialized(false)
94     {}
95 
96     /** \brief Constructor with decomposition
97       *
98       * This calculates the decomposition for the input \a matrix.
99       * \sa LDLT(Index size)
100       */
LDLT(const MatrixType & matrix)101     LDLT(const MatrixType& matrix)
102       : m_matrix(matrix.rows(), matrix.cols()),
103         m_transpositions(matrix.rows()),
104         m_temporary(matrix.rows()),
105         m_sign(internal::ZeroSign),
106         m_isInitialized(false)
107     {
108       compute(matrix);
109     }
110 
111     /** Clear any existing decomposition
112      * \sa rankUpdate(w,sigma)
113      */
setZero()114     void setZero()
115     {
116       m_isInitialized = false;
117     }
118 
119     /** \returns a view of the upper triangular matrix U */
matrixU()120     inline typename Traits::MatrixU matrixU() const
121     {
122       eigen_assert(m_isInitialized && "LDLT is not initialized.");
123       return Traits::getU(m_matrix);
124     }
125 
126     /** \returns a view of the lower triangular matrix L */
matrixL()127     inline typename Traits::MatrixL matrixL() const
128     {
129       eigen_assert(m_isInitialized && "LDLT is not initialized.");
130       return Traits::getL(m_matrix);
131     }
132 
133     /** \returns the permutation matrix P as a transposition sequence.
134       */
transpositionsP()135     inline const TranspositionType& transpositionsP() const
136     {
137       eigen_assert(m_isInitialized && "LDLT is not initialized.");
138       return m_transpositions;
139     }
140 
141     /** \returns the coefficients of the diagonal matrix D */
vectorD()142     inline Diagonal<const MatrixType> vectorD() const
143     {
144       eigen_assert(m_isInitialized && "LDLT is not initialized.");
145       return m_matrix.diagonal();
146     }
147 
148     /** \returns true if the matrix is positive (semidefinite) */
isPositive()149     inline bool isPositive() const
150     {
151       eigen_assert(m_isInitialized && "LDLT is not initialized.");
152       return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign;
153     }
154 
155     #ifdef EIGEN2_SUPPORT
isPositiveDefinite()156     inline bool isPositiveDefinite() const
157     {
158       return isPositive();
159     }
160     #endif
161 
162     /** \returns true if the matrix is negative (semidefinite) */
isNegative(void)163     inline bool isNegative(void) const
164     {
165       eigen_assert(m_isInitialized && "LDLT is not initialized.");
166       return m_sign == internal::NegativeSemiDef || m_sign == internal::ZeroSign;
167     }
168 
169     /** \returns a solution x of \f$ A x = b \f$ using the current decomposition of A.
170       *
171       * This function also supports in-place solves using the syntax <tt>x = decompositionObject.solve(x)</tt> .
172       *
173       * \note_about_checking_solutions
174       *
175       * More precisely, this method solves \f$ A x = b \f$ using the decomposition \f$ A = P^T L D L^* P \f$
176       * by solving the systems \f$ P^T y_1 = b \f$, \f$ L y_2 = y_1 \f$, \f$ D y_3 = y_2 \f$,
177       * \f$ L^* y_4 = y_3 \f$ and \f$ P x = y_4 \f$ in succession. If the matrix \f$ A \f$ is singular, then
178       * \f$ D \f$ will also be singular (all the other matrices are invertible). In that case, the
179       * least-square solution of \f$ D y_3 = y_2 \f$ is computed. This does not mean that this function
180       * computes the least-square solution of \f$ A x = b \f$ is \f$ A \f$ is singular.
181       *
182       * \sa MatrixBase::ldlt()
183       */
184     template<typename Rhs>
185     inline const internal::solve_retval<LDLT, Rhs>
solve(const MatrixBase<Rhs> & b)186     solve(const MatrixBase<Rhs>& b) const
187     {
188       eigen_assert(m_isInitialized && "LDLT is not initialized.");
189       eigen_assert(m_matrix.rows()==b.rows()
190                 && "LDLT::solve(): invalid number of rows of the right hand side matrix b");
191       return internal::solve_retval<LDLT, Rhs>(*this, b.derived());
192     }
193 
194     #ifdef EIGEN2_SUPPORT
195     template<typename OtherDerived, typename ResultType>
solve(const MatrixBase<OtherDerived> & b,ResultType * result)196     bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const
197     {
198       *result = this->solve(b);
199       return true;
200     }
201     #endif
202 
203     template<typename Derived>
204     bool solveInPlace(MatrixBase<Derived> &bAndX) const;
205 
206     LDLT& compute(const MatrixType& matrix);
207 
208     template <typename Derived>
209     LDLT& rankUpdate(const MatrixBase<Derived>& w, const RealScalar& alpha=1);
210 
211     /** \returns the internal LDLT decomposition matrix
212       *
213       * TODO: document the storage layout
214       */
matrixLDLT()215     inline const MatrixType& matrixLDLT() const
216     {
217       eigen_assert(m_isInitialized && "LDLT is not initialized.");
218       return m_matrix;
219     }
220 
221     MatrixType reconstructedMatrix() const;
222 
rows()223     inline Index rows() const { return m_matrix.rows(); }
cols()224     inline Index cols() const { return m_matrix.cols(); }
225 
226     /** \brief Reports whether previous computation was successful.
227       *
228       * \returns \c Success if computation was succesful,
229       *          \c NumericalIssue if the matrix.appears to be negative.
230       */
info()231     ComputationInfo info() const
232     {
233       eigen_assert(m_isInitialized && "LDLT is not initialized.");
234       return Success;
235     }
236 
237   protected:
238 
check_template_parameters()239     static void check_template_parameters()
240     {
241       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
242     }
243 
244     /** \internal
245       * Used to compute and store the Cholesky decomposition A = L D L^* = U^* D U.
246       * The strict upper part is used during the decomposition, the strict lower
247       * part correspond to the coefficients of L (its diagonal is equal to 1 and
248       * is not stored), and the diagonal entries correspond to D.
249       */
250     MatrixType m_matrix;
251     TranspositionType m_transpositions;
252     TmpMatrixType m_temporary;
253     internal::SignMatrix m_sign;
254     bool m_isInitialized;
255 };
256 
257 namespace internal {
258 
259 template<int UpLo> struct ldlt_inplace;
260 
261 template<> struct ldlt_inplace<Lower>
262 {
263   template<typename MatrixType, typename TranspositionType, typename Workspace>
264   static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
265   {
266     using std::abs;
267     typedef typename MatrixType::Scalar Scalar;
268     typedef typename MatrixType::RealScalar RealScalar;
269     typedef typename MatrixType::Index Index;
270     eigen_assert(mat.rows()==mat.cols());
271     const Index size = mat.rows();
272 
273     if (size <= 1)
274     {
275       transpositions.setIdentity();
276       if (numext::real(mat.coeff(0,0)) > 0) sign = PositiveSemiDef;
277       else if (numext::real(mat.coeff(0,0)) < 0) sign = NegativeSemiDef;
278       else sign = ZeroSign;
279       return true;
280     }
281 
282     for (Index k = 0; k < size; ++k)
283     {
284       // Find largest diagonal element
285       Index index_of_biggest_in_corner;
286       mat.diagonal().tail(size-k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner);
287       index_of_biggest_in_corner += k;
288 
289       transpositions.coeffRef(k) = index_of_biggest_in_corner;
290       if(k != index_of_biggest_in_corner)
291       {
292         // apply the transposition while taking care to consider only
293         // the lower triangular part
294         Index s = size-index_of_biggest_in_corner-1; // trailing size after the biggest element
295         mat.row(k).head(k).swap(mat.row(index_of_biggest_in_corner).head(k));
296         mat.col(k).tail(s).swap(mat.col(index_of_biggest_in_corner).tail(s));
297         std::swap(mat.coeffRef(k,k),mat.coeffRef(index_of_biggest_in_corner,index_of_biggest_in_corner));
298         for(int i=k+1;i<index_of_biggest_in_corner;++i)
299         {
300           Scalar tmp = mat.coeffRef(i,k);
301           mat.coeffRef(i,k) = numext::conj(mat.coeffRef(index_of_biggest_in_corner,i));
302           mat.coeffRef(index_of_biggest_in_corner,i) = numext::conj(tmp);
303         }
304         if(NumTraits<Scalar>::IsComplex)
305           mat.coeffRef(index_of_biggest_in_corner,k) = numext::conj(mat.coeff(index_of_biggest_in_corner,k));
306       }
307 
308       // partition the matrix:
309       //       A00 |  -  |  -
310       // lu  = A10 | A11 |  -
311       //       A20 | A21 | A22
312       Index rs = size - k - 1;
313       Block<MatrixType,Dynamic,1> A21(mat,k+1,k,rs,1);
314       Block<MatrixType,1,Dynamic> A10(mat,k,0,1,k);
315       Block<MatrixType,Dynamic,Dynamic> A20(mat,k+1,0,rs,k);
316 
317       if(k>0)
318       {
319         temp.head(k) = mat.diagonal().real().head(k).asDiagonal() * A10.adjoint();
320         mat.coeffRef(k,k) -= (A10 * temp.head(k)).value();
321         if(rs>0)
322           A21.noalias() -= A20 * temp.head(k);
323       }
324 
325       // In some previous versions of Eigen (e.g., 3.2.1), the scaling was omitted if the pivot
326       // was smaller than the cutoff value. However, soince LDLT is not rank-revealing
327       // we should only make sure we do not introduce INF or NaN values.
328       // LAPACK also uses 0 as the cutoff value.
329       RealScalar realAkk = numext::real(mat.coeffRef(k,k));
330       if((rs>0) && (abs(realAkk) > RealScalar(0)))
331         A21 /= realAkk;
332 
333       if (sign == PositiveSemiDef) {
334         if (realAkk < 0) sign = Indefinite;
335       } else if (sign == NegativeSemiDef) {
336         if (realAkk > 0) sign = Indefinite;
337       } else if (sign == ZeroSign) {
338         if (realAkk > 0) sign = PositiveSemiDef;
339         else if (realAkk < 0) sign = NegativeSemiDef;
340       }
341     }
342 
343     return true;
344   }
345 
346   // Reference for the algorithm: Davis and Hager, "Multiple Rank
347   // Modifications of a Sparse Cholesky Factorization" (Algorithm 1)
348   // Trivial rearrangements of their computations (Timothy E. Holy)
349   // allow their algorithm to work for rank-1 updates even if the
350   // original matrix is not of full rank.
351   // Here only rank-1 updates are implemented, to reduce the
352   // requirement for intermediate storage and improve accuracy
353   template<typename MatrixType, typename WDerived>
354   static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w, const typename MatrixType::RealScalar& sigma=1)
355   {
356     using numext::isfinite;
357     typedef typename MatrixType::Scalar Scalar;
358     typedef typename MatrixType::RealScalar RealScalar;
359     typedef typename MatrixType::Index Index;
360 
361     const Index size = mat.rows();
362     eigen_assert(mat.cols() == size && w.size()==size);
363 
364     RealScalar alpha = 1;
365 
366     // Apply the update
367     for (Index j = 0; j < size; j++)
368     {
369       // Check for termination due to an original decomposition of low-rank
370       if (!(isfinite)(alpha))
371         break;
372 
373       // Update the diagonal terms
374       RealScalar dj = numext::real(mat.coeff(j,j));
375       Scalar wj = w.coeff(j);
376       RealScalar swj2 = sigma*numext::abs2(wj);
377       RealScalar gamma = dj*alpha + swj2;
378 
379       mat.coeffRef(j,j) += swj2/alpha;
380       alpha += swj2/dj;
381 
382 
383       // Update the terms of L
384       Index rs = size-j-1;
385       w.tail(rs) -= wj * mat.col(j).tail(rs);
386       if(gamma != 0)
387         mat.col(j).tail(rs) += (sigma*numext::conj(wj)/gamma)*w.tail(rs);
388     }
389     return true;
390   }
391 
392   template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
393   static bool update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w, const typename MatrixType::RealScalar& sigma=1)
394   {
395     // Apply the permutation to the input w
396     tmp = transpositions * w;
397 
398     return ldlt_inplace<Lower>::updateInPlace(mat,tmp,sigma);
399   }
400 };
401 
402 template<> struct ldlt_inplace<Upper>
403 {
404   template<typename MatrixType, typename TranspositionType, typename Workspace>
405   static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
406   {
407     Transpose<MatrixType> matt(mat);
408     return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign);
409   }
410 
411   template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
412   static EIGEN_STRONG_INLINE bool update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w, const typename MatrixType::RealScalar& sigma=1)
413   {
414     Transpose<MatrixType> matt(mat);
415     return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma);
416   }
417 };
418 
419 template<typename MatrixType> struct LDLT_Traits<MatrixType,Lower>
420 {
421   typedef const TriangularView<const MatrixType, UnitLower> MatrixL;
422   typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
423   static inline MatrixL getL(const MatrixType& m) { return m; }
424   static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
425 };
426 
427 template<typename MatrixType> struct LDLT_Traits<MatrixType,Upper>
428 {
429   typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
430   typedef const TriangularView<const MatrixType, UnitUpper> MatrixU;
431   static inline MatrixL getL(const MatrixType& m) { return m.adjoint(); }
432   static inline MatrixU getU(const MatrixType& m) { return m; }
433 };
434 
435 } // end namespace internal
436 
437 /** Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of \a matrix
438   */
439 template<typename MatrixType, int _UpLo>
440 LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::compute(const MatrixType& a)
441 {
442   check_template_parameters();
443 
444   eigen_assert(a.rows()==a.cols());
445   const Index size = a.rows();
446 
447   m_matrix = a;
448 
449   m_transpositions.resize(size);
450   m_isInitialized = false;
451   m_temporary.resize(size);
452   m_sign = internal::ZeroSign;
453 
454   internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign);
455 
456   m_isInitialized = true;
457   return *this;
458 }
459 
460 /** Update the LDLT decomposition:  given A = L D L^T, efficiently compute the decomposition of A + sigma w w^T.
461  * \param w a vector to be incorporated into the decomposition.
462  * \param sigma a scalar, +1 for updates and -1 for "downdates," which correspond to removing previously-added column vectors. Optional; default value is +1.
463  * \sa setZero()
464   */
465 template<typename MatrixType, int _UpLo>
466 template<typename Derived>
467 LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::rankUpdate(const MatrixBase<Derived>& w, const typename NumTraits<typename MatrixType::Scalar>::Real& sigma)
468 {
469   const Index size = w.rows();
470   if (m_isInitialized)
471   {
472     eigen_assert(m_matrix.rows()==size);
473   }
474   else
475   {
476     m_matrix.resize(size,size);
477     m_matrix.setZero();
478     m_transpositions.resize(size);
479     for (Index i = 0; i < size; i++)
480       m_transpositions.coeffRef(i) = i;
481     m_temporary.resize(size);
482     m_sign = sigma>=0 ? internal::PositiveSemiDef : internal::NegativeSemiDef;
483     m_isInitialized = true;
484   }
485 
486   internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);
487 
488   return *this;
489 }
490 
491 namespace internal {
492 template<typename _MatrixType, int _UpLo, typename Rhs>
493 struct solve_retval<LDLT<_MatrixType,_UpLo>, Rhs>
494   : solve_retval_base<LDLT<_MatrixType,_UpLo>, Rhs>
495 {
496   typedef LDLT<_MatrixType,_UpLo> LDLTType;
497   EIGEN_MAKE_SOLVE_HELPERS(LDLTType,Rhs)
498 
499   template<typename Dest> void evalTo(Dest& dst) const
500   {
501     eigen_assert(rhs().rows() == dec().matrixLDLT().rows());
502     // dst = P b
503     dst = dec().transpositionsP() * rhs();
504 
505     // dst = L^-1 (P b)
506     dec().matrixL().solveInPlace(dst);
507 
508     // dst = D^-1 (L^-1 P b)
509     // more precisely, use pseudo-inverse of D (see bug 241)
510     using std::abs;
511     using std::max;
512     typedef typename LDLTType::MatrixType MatrixType;
513     typedef typename LDLTType::RealScalar RealScalar;
514     const typename Diagonal<const MatrixType>::RealReturnType vectorD(dec().vectorD());
515     // In some previous versions, tolerance was set to the max of 1/highest and the maximal diagonal entry * epsilon
516     // as motivated by LAPACK's xGELSS:
517     // RealScalar tolerance = (max)(vectorD.array().abs().maxCoeff() *NumTraits<RealScalar>::epsilon(),RealScalar(1) / NumTraits<RealScalar>::highest());
518     // However, LDLT is not rank revealing, and so adjusting the tolerance wrt to the highest
519     // diagonal element is not well justified and to numerical issues in some cases.
520     // Moreover, Lapack's xSYTRS routines use 0 for the tolerance.
521     RealScalar tolerance = RealScalar(1) / NumTraits<RealScalar>::highest();
522 
523     for (Index i = 0; i < vectorD.size(); ++i) {
524       if(abs(vectorD(i)) > tolerance)
525         dst.row(i) /= vectorD(i);
526       else
527         dst.row(i).setZero();
528     }
529 
530     // dst = L^-T (D^-1 L^-1 P b)
531     dec().matrixU().solveInPlace(dst);
532 
533     // dst = P^-1 (L^-T D^-1 L^-1 P b) = A^-1 b
534     dst = dec().transpositionsP().transpose() * dst;
535   }
536 };
537 }
538 
539 /** \internal use x = ldlt_object.solve(x);
540   *
541   * This is the \em in-place version of solve().
542   *
543   * \param bAndX represents both the right-hand side matrix b and result x.
544   *
545   * \returns true always! If you need to check for existence of solutions, use another decomposition like LU, QR, or SVD.
546   *
547   * This version avoids a copy when the right hand side matrix b is not
548   * needed anymore.
549   *
550   * \sa LDLT::solve(), MatrixBase::ldlt()
551   */
552 template<typename MatrixType,int _UpLo>
553 template<typename Derived>
554 bool LDLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const
555 {
556   eigen_assert(m_isInitialized && "LDLT is not initialized.");
557   eigen_assert(m_matrix.rows() == bAndX.rows());
558 
559   bAndX = this->solve(bAndX);
560 
561   return true;
562 }
563 
564 /** \returns the matrix represented by the decomposition,
565  * i.e., it returns the product: P^T L D L^* P.
566  * This function is provided for debug purpose. */
567 template<typename MatrixType, int _UpLo>
568 MatrixType LDLT<MatrixType,_UpLo>::reconstructedMatrix() const
569 {
570   eigen_assert(m_isInitialized && "LDLT is not initialized.");
571   const Index size = m_matrix.rows();
572   MatrixType res(size,size);
573 
574   // P
575   res.setIdentity();
576   res = transpositionsP() * res;
577   // L^* P
578   res = matrixU() * res;
579   // D(L^*P)
580   res = vectorD().real().asDiagonal() * res;
581   // L(DL^*P)
582   res = matrixL() * res;
583   // P^T (LDL^*P)
584   res = transpositionsP().transpose() * res;
585 
586   return res;
587 }
588 
589 /** \cholesky_module
590   * \returns the Cholesky decomposition with full pivoting without square root of \c *this
591   */
592 template<typename MatrixType, unsigned int UpLo>
593 inline const LDLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo>
594 SelfAdjointView<MatrixType, UpLo>::ldlt() const
595 {
596   return LDLT<PlainObject,UpLo>(m_matrix);
597 }
598 
599 /** \cholesky_module
600   * \returns the Cholesky decomposition with full pivoting without square root of \c *this
601   */
602 template<typename Derived>
603 inline const LDLT<typename MatrixBase<Derived>::PlainObject>
604 MatrixBase<Derived>::ldlt() const
605 {
606   return LDLT<PlainObject>(derived());
607 }
608 
609 } // end namespace Eigen
610 
611 #endif // EIGEN_LDLT_H
612