1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
5//
6// This Source Code Form is subject to the terms of the Mozilla
7// Public License v. 2.0. If a copy of the MPL was not distributed
8// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10#ifndef EIGEN_LLT_H
11#define EIGEN_LLT_H
12
13namespace Eigen {
14
15namespace internal{
16template<typename MatrixType, int UpLo> struct LLT_Traits;
17}
18
19/** \ingroup Cholesky_Module
20  *
21  * \class LLT
22  *
23  * \brief Standard Cholesky decomposition (LL^T) of a matrix and associated features
24  *
25  * \param MatrixType the type of the matrix of which we are computing the LL^T Cholesky decomposition
26  * \param UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
27  *             The other triangular part won't be read.
28  *
29  * This class performs a LL^T Cholesky decomposition of a symmetric, positive definite
30  * matrix A such that A = LL^* = U^*U, where L is lower triangular.
31  *
32  * While the Cholesky decomposition is particularly useful to solve selfadjoint problems like  D^*D x = b,
33  * for that purpose, we recommend the Cholesky decomposition without square root which is more stable
34  * and even faster. Nevertheless, this standard Cholesky decomposition remains useful in many other
35  * situations like generalised eigen problems with hermitian matrices.
36  *
37  * Remember that Cholesky decompositions are not rank-revealing. This LLT decomposition is only stable on positive definite matrices,
38  * use LDLT instead for the semidefinite case. Also, do not use a Cholesky decomposition to determine whether a system of equations
39  * has a solution.
40  *
41  * Example: \include LLT_example.cpp
42  * Output: \verbinclude LLT_example.out
43  *
44  * \sa MatrixBase::llt(), class LDLT
45  */
46 /* HEY THIS DOX IS DISABLED BECAUSE THERE's A BUG EITHER HERE OR IN LDLT ABOUT THAT (OR BOTH)
47  * Note that during the decomposition, only the upper triangular part of A is considered. Therefore,
48  * the strict lower part does not have to store correct values.
49  */
50template<typename _MatrixType, int _UpLo> class LLT
51{
52  public:
53    typedef _MatrixType MatrixType;
54    enum {
55      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
56      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
57      Options = MatrixType::Options,
58      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
59    };
60    typedef typename MatrixType::Scalar Scalar;
61    typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
62    typedef typename MatrixType::Index Index;
63
64    enum {
65      PacketSize = internal::packet_traits<Scalar>::size,
66      AlignmentMask = int(PacketSize)-1,
67      UpLo = _UpLo
68    };
69
70    typedef internal::LLT_Traits<MatrixType,UpLo> Traits;
71
72    /**
73      * \brief Default Constructor.
74      *
75      * The default constructor is useful in cases in which the user intends to
76      * perform decompositions via LLT::compute(const MatrixType&).
77      */
78    LLT() : m_matrix(), m_isInitialized(false) {}
79
80    /** \brief Default Constructor with memory preallocation
81      *
82      * Like the default constructor but with preallocation of the internal data
83      * according to the specified problem \a size.
84      * \sa LLT()
85      */
86    LLT(Index size) : m_matrix(size, size),
87                    m_isInitialized(false) {}
88
89    LLT(const MatrixType& matrix)
90      : m_matrix(matrix.rows(), matrix.cols()),
91        m_isInitialized(false)
92    {
93      compute(matrix);
94    }
95
96    /** \returns a view of the upper triangular matrix U */
97    inline typename Traits::MatrixU matrixU() const
98    {
99      eigen_assert(m_isInitialized && "LLT is not initialized.");
100      return Traits::getU(m_matrix);
101    }
102
103    /** \returns a view of the lower triangular matrix L */
104    inline typename Traits::MatrixL matrixL() const
105    {
106      eigen_assert(m_isInitialized && "LLT is not initialized.");
107      return Traits::getL(m_matrix);
108    }
109
110    /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A.
111      *
112      * Since this LLT class assumes anyway that the matrix A is invertible, the solution
113      * theoretically exists and is unique regardless of b.
114      *
115      * Example: \include LLT_solve.cpp
116      * Output: \verbinclude LLT_solve.out
117      *
118      * \sa solveInPlace(), MatrixBase::llt()
119      */
120    template<typename Rhs>
121    inline const internal::solve_retval<LLT, Rhs>
122    solve(const MatrixBase<Rhs>& b) const
123    {
124      eigen_assert(m_isInitialized && "LLT is not initialized.");
125      eigen_assert(m_matrix.rows()==b.rows()
126                && "LLT::solve(): invalid number of rows of the right hand side matrix b");
127      return internal::solve_retval<LLT, Rhs>(*this, b.derived());
128    }
129
130    #ifdef EIGEN2_SUPPORT
131    template<typename OtherDerived, typename ResultType>
132    bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const
133    {
134      *result = this->solve(b);
135      return true;
136    }
137
138    bool isPositiveDefinite() const { return true; }
139    #endif
140
141    template<typename Derived>
142    void solveInPlace(MatrixBase<Derived> &bAndX) const;
143
144    LLT& compute(const MatrixType& matrix);
145
146    /** \returns the LLT decomposition matrix
147      *
148      * TODO: document the storage layout
149      */
150    inline const MatrixType& matrixLLT() const
151    {
152      eigen_assert(m_isInitialized && "LLT is not initialized.");
153      return m_matrix;
154    }
155
156    MatrixType reconstructedMatrix() const;
157
158
159    /** \brief Reports whether previous computation was successful.
160      *
161      * \returns \c Success if computation was succesful,
162      *          \c NumericalIssue if the matrix.appears to be negative.
163      */
164    ComputationInfo info() const
165    {
166      eigen_assert(m_isInitialized && "LLT is not initialized.");
167      return m_info;
168    }
169
170    inline Index rows() const { return m_matrix.rows(); }
171    inline Index cols() const { return m_matrix.cols(); }
172
173    template<typename VectorType>
174    LLT rankUpdate(const VectorType& vec, const RealScalar& sigma = 1);
175
176  protected:
177    /** \internal
178      * Used to compute and store L
179      * The strict upper part is not used and even not initialized.
180      */
181    MatrixType m_matrix;
182    bool m_isInitialized;
183    ComputationInfo m_info;
184};
185
186namespace internal {
187
188template<typename Scalar, int UpLo> struct llt_inplace;
189
190template<typename MatrixType, typename VectorType>
191static typename MatrixType::Index llt_rank_update_lower(MatrixType& mat, const VectorType& vec, const typename MatrixType::RealScalar& sigma)
192{
193  using std::sqrt;
194  typedef typename MatrixType::Scalar Scalar;
195  typedef typename MatrixType::RealScalar RealScalar;
196  typedef typename MatrixType::Index Index;
197  typedef typename MatrixType::ColXpr ColXpr;
198  typedef typename internal::remove_all<ColXpr>::type ColXprCleaned;
199  typedef typename ColXprCleaned::SegmentReturnType ColXprSegment;
200  typedef Matrix<Scalar,Dynamic,1> TempVectorType;
201  typedef typename TempVectorType::SegmentReturnType TempVecSegment;
202
203  Index n = mat.cols();
204  eigen_assert(mat.rows()==n && vec.size()==n);
205
206  TempVectorType temp;
207
208  if(sigma>0)
209  {
210    // This version is based on Givens rotations.
211    // It is faster than the other one below, but only works for updates,
212    // i.e., for sigma > 0
213    temp = sqrt(sigma) * vec;
214
215    for(Index i=0; i<n; ++i)
216    {
217      JacobiRotation<Scalar> g;
218      g.makeGivens(mat(i,i), -temp(i), &mat(i,i));
219
220      Index rs = n-i-1;
221      if(rs>0)
222      {
223        ColXprSegment x(mat.col(i).tail(rs));
224        TempVecSegment y(temp.tail(rs));
225        apply_rotation_in_the_plane(x, y, g);
226      }
227    }
228  }
229  else
230  {
231    temp = vec;
232    RealScalar beta = 1;
233    for(Index j=0; j<n; ++j)
234    {
235      RealScalar Ljj = numext::real(mat.coeff(j,j));
236      RealScalar dj = numext::abs2(Ljj);
237      Scalar wj = temp.coeff(j);
238      RealScalar swj2 = sigma*numext::abs2(wj);
239      RealScalar gamma = dj*beta + swj2;
240
241      RealScalar x = dj + swj2/beta;
242      if (x<=RealScalar(0))
243        return j;
244      RealScalar nLjj = sqrt(x);
245      mat.coeffRef(j,j) = nLjj;
246      beta += swj2/dj;
247
248      // Update the terms of L
249      Index rs = n-j-1;
250      if(rs)
251      {
252        temp.tail(rs) -= (wj/Ljj) * mat.col(j).tail(rs);
253        if(gamma != 0)
254          mat.col(j).tail(rs) = (nLjj/Ljj) * mat.col(j).tail(rs) + (nLjj * sigma*numext::conj(wj)/gamma)*temp.tail(rs);
255      }
256    }
257  }
258  return -1;
259}
260
261template<typename Scalar> struct llt_inplace<Scalar, Lower>
262{
263  typedef typename NumTraits<Scalar>::Real RealScalar;
264  template<typename MatrixType>
265  static typename MatrixType::Index unblocked(MatrixType& mat)
266  {
267    using std::sqrt;
268    typedef typename MatrixType::Index Index;
269
270    eigen_assert(mat.rows()==mat.cols());
271    const Index size = mat.rows();
272    for(Index k = 0; k < size; ++k)
273    {
274      Index rs = size-k-1; // remaining size
275
276      Block<MatrixType,Dynamic,1> A21(mat,k+1,k,rs,1);
277      Block<MatrixType,1,Dynamic> A10(mat,k,0,1,k);
278      Block<MatrixType,Dynamic,Dynamic> A20(mat,k+1,0,rs,k);
279
280      RealScalar x = numext::real(mat.coeff(k,k));
281      if (k>0) x -= A10.squaredNorm();
282      if (x<=RealScalar(0))
283        return k;
284      mat.coeffRef(k,k) = x = sqrt(x);
285      if (k>0 && rs>0) A21.noalias() -= A20 * A10.adjoint();
286      if (rs>0) A21 *= RealScalar(1)/x;
287    }
288    return -1;
289  }
290
291  template<typename MatrixType>
292  static typename MatrixType::Index blocked(MatrixType& m)
293  {
294    typedef typename MatrixType::Index Index;
295    eigen_assert(m.rows()==m.cols());
296    Index size = m.rows();
297    if(size<32)
298      return unblocked(m);
299
300    Index blockSize = size/8;
301    blockSize = (blockSize/16)*16;
302    blockSize = (std::min)((std::max)(blockSize,Index(8)), Index(128));
303
304    for (Index k=0; k<size; k+=blockSize)
305    {
306      // partition the matrix:
307      //       A00 |  -  |  -
308      // lu  = A10 | A11 |  -
309      //       A20 | A21 | A22
310      Index bs = (std::min)(blockSize, size-k);
311      Index rs = size - k - bs;
312      Block<MatrixType,Dynamic,Dynamic> A11(m,k,   k,   bs,bs);
313      Block<MatrixType,Dynamic,Dynamic> A21(m,k+bs,k,   rs,bs);
314      Block<MatrixType,Dynamic,Dynamic> A22(m,k+bs,k+bs,rs,rs);
315
316      Index ret;
317      if((ret=unblocked(A11))>=0) return k+ret;
318      if(rs>0) A11.adjoint().template triangularView<Upper>().template solveInPlace<OnTheRight>(A21);
319      if(rs>0) A22.template selfadjointView<Lower>().rankUpdate(A21,-1); // bottleneck
320    }
321    return -1;
322  }
323
324  template<typename MatrixType, typename VectorType>
325  static typename MatrixType::Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
326  {
327    return Eigen::internal::llt_rank_update_lower(mat, vec, sigma);
328  }
329};
330
331template<typename Scalar> struct llt_inplace<Scalar, Upper>
332{
333  typedef typename NumTraits<Scalar>::Real RealScalar;
334
335  template<typename MatrixType>
336  static EIGEN_STRONG_INLINE typename MatrixType::Index unblocked(MatrixType& mat)
337  {
338    Transpose<MatrixType> matt(mat);
339    return llt_inplace<Scalar, Lower>::unblocked(matt);
340  }
341  template<typename MatrixType>
342  static EIGEN_STRONG_INLINE typename MatrixType::Index blocked(MatrixType& mat)
343  {
344    Transpose<MatrixType> matt(mat);
345    return llt_inplace<Scalar, Lower>::blocked(matt);
346  }
347  template<typename MatrixType, typename VectorType>
348  static typename MatrixType::Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
349  {
350    Transpose<MatrixType> matt(mat);
351    return llt_inplace<Scalar, Lower>::rankUpdate(matt, vec.conjugate(), sigma);
352  }
353};
354
355template<typename MatrixType> struct LLT_Traits<MatrixType,Lower>
356{
357  typedef const TriangularView<const MatrixType, Lower> MatrixL;
358  typedef const TriangularView<const typename MatrixType::AdjointReturnType, Upper> MatrixU;
359  static inline MatrixL getL(const MatrixType& m) { return m; }
360  static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
361  static bool inplace_decomposition(MatrixType& m)
362  { return llt_inplace<typename MatrixType::Scalar, Lower>::blocked(m)==-1; }
363};
364
365template<typename MatrixType> struct LLT_Traits<MatrixType,Upper>
366{
367  typedef const TriangularView<const typename MatrixType::AdjointReturnType, Lower> MatrixL;
368  typedef const TriangularView<const MatrixType, Upper> MatrixU;
369  static inline MatrixL getL(const MatrixType& m) { return m.adjoint(); }
370  static inline MatrixU getU(const MatrixType& m) { return m; }
371  static bool inplace_decomposition(MatrixType& m)
372  { return llt_inplace<typename MatrixType::Scalar, Upper>::blocked(m)==-1; }
373};
374
375} // end namespace internal
376
377/** Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of \a matrix
378  *
379  * \returns a reference to *this
380  *
381  * Example: \include TutorialLinAlgComputeTwice.cpp
382  * Output: \verbinclude TutorialLinAlgComputeTwice.out
383  */
384template<typename MatrixType, int _UpLo>
385LLT<MatrixType,_UpLo>& LLT<MatrixType,_UpLo>::compute(const MatrixType& a)
386{
387  eigen_assert(a.rows()==a.cols());
388  const Index size = a.rows();
389  m_matrix.resize(size, size);
390  m_matrix = a;
391
392  m_isInitialized = true;
393  bool ok = Traits::inplace_decomposition(m_matrix);
394  m_info = ok ? Success : NumericalIssue;
395
396  return *this;
397}
398
399/** Performs a rank one update (or dowdate) of the current decomposition.
400  * If A = LL^* before the rank one update,
401  * then after it we have LL^* = A + sigma * v v^* where \a v must be a vector
402  * of same dimension.
403  */
404template<typename _MatrixType, int _UpLo>
405template<typename VectorType>
406LLT<_MatrixType,_UpLo> LLT<_MatrixType,_UpLo>::rankUpdate(const VectorType& v, const RealScalar& sigma)
407{
408  EIGEN_STATIC_ASSERT_VECTOR_ONLY(VectorType);
409  eigen_assert(v.size()==m_matrix.cols());
410  eigen_assert(m_isInitialized);
411  if(internal::llt_inplace<typename MatrixType::Scalar, UpLo>::rankUpdate(m_matrix,v,sigma)>=0)
412    m_info = NumericalIssue;
413  else
414    m_info = Success;
415
416  return *this;
417}
418
419namespace internal {
420template<typename _MatrixType, int UpLo, typename Rhs>
421struct solve_retval<LLT<_MatrixType, UpLo>, Rhs>
422  : solve_retval_base<LLT<_MatrixType, UpLo>, Rhs>
423{
424  typedef LLT<_MatrixType,UpLo> LLTType;
425  EIGEN_MAKE_SOLVE_HELPERS(LLTType,Rhs)
426
427  template<typename Dest> void evalTo(Dest& dst) const
428  {
429    dst = rhs();
430    dec().solveInPlace(dst);
431  }
432};
433}
434
435/** \internal use x = llt_object.solve(x);
436  *
437  * This is the \em in-place version of solve().
438  *
439  * \param bAndX represents both the right-hand side matrix b and result x.
440  *
441  * \returns true always! If you need to check for existence of solutions, use another decomposition like LU, QR, or SVD.
442  *
443  * This version avoids a copy when the right hand side matrix b is not
444  * needed anymore.
445  *
446  * \sa LLT::solve(), MatrixBase::llt()
447  */
448template<typename MatrixType, int _UpLo>
449template<typename Derived>
450void LLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const
451{
452  eigen_assert(m_isInitialized && "LLT is not initialized.");
453  eigen_assert(m_matrix.rows()==bAndX.rows());
454  matrixL().solveInPlace(bAndX);
455  matrixU().solveInPlace(bAndX);
456}
457
458/** \returns the matrix represented by the decomposition,
459 * i.e., it returns the product: L L^*.
460 * This function is provided for debug purpose. */
461template<typename MatrixType, int _UpLo>
462MatrixType LLT<MatrixType,_UpLo>::reconstructedMatrix() const
463{
464  eigen_assert(m_isInitialized && "LLT is not initialized.");
465  return matrixL() * matrixL().adjoint().toDenseMatrix();
466}
467
468/** \cholesky_module
469  * \returns the LLT decomposition of \c *this
470  */
471template<typename Derived>
472inline const LLT<typename MatrixBase<Derived>::PlainObject>
473MatrixBase<Derived>::llt() const
474{
475  return LLT<PlainObject>(derived());
476}
477
478/** \cholesky_module
479  * \returns the LLT decomposition of \c *this
480  */
481template<typename MatrixType, unsigned int UpLo>
482inline const LLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo>
483SelfAdjointView<MatrixType, UpLo>::llt() const
484{
485  return LLT<PlainObject,UpLo>(m_matrix);
486}
487
488} // end namespace Eigen
489
490#endif // EIGEN_LLT_H
491