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LDLT.h
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 
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;
64 
67 
68  typedef internal::LDLT_Traits<MatrixType,UpLo> Traits;
69 
75  LDLT()
76  : m_matrix(),
77  m_transpositions(),
78  m_sign(internal::ZeroSign),
79  m_isInitialized(false)
80  {}
81 
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 
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 
114  void setZero()
115  {
116  m_isInitialized = false;
117  }
118 
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 
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 
135  inline const TranspositionType& transpositionsP() const
136  {
137  eigen_assert(m_isInitialized && "LDLT is not initialized.");
138  return m_transpositions;
139  }
140 
143  {
144  eigen_assert(m_isInitialized && "LDLT is not initialized.");
145  return m_matrix.diagonal();
146  }
147 
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
156  inline bool isPositiveDefinite() const
157  {
158  return isPositive();
159  }
160  #endif
161 
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 
184  template<typename Rhs>
185  inline const internal::solve_retval<LDLT, Rhs>
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>
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 
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 
223  inline Index rows() const { return m_matrix.rows(); }
224  inline Index cols() const { return m_matrix.cols(); }
225 
232  {
233  eigen_assert(m_isInitialized && "LDLT is not initialized.");
234  return Success;
235  }
236 
237  protected:
238 
245  MatrixType m_matrix;
246  TranspositionType m_transpositions;
247  TmpMatrixType m_temporary;
248  internal::SignMatrix m_sign;
249  bool m_isInitialized;
250 };
251 
252 namespace internal {
253 
254 template<int UpLo> struct ldlt_inplace;
255 
256 template<> struct ldlt_inplace<Lower>
257 {
258  template<typename MatrixType, typename TranspositionType, typename Workspace>
259  static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
260  {
261  using std::abs;
262  typedef typename MatrixType::Scalar Scalar;
263  typedef typename MatrixType::RealScalar RealScalar;
264  typedef typename MatrixType::Index Index;
265  eigen_assert(mat.rows()==mat.cols());
266  const Index size = mat.rows();
267 
268  if (size <= 1)
269  {
270  transpositions.setIdentity();
271  if (numext::real(mat.coeff(0,0)) > 0) sign = PositiveSemiDef;
272  else if (numext::real(mat.coeff(0,0)) < 0) sign = NegativeSemiDef;
273  else sign = ZeroSign;
274  return true;
275  }
276 
277  for (Index k = 0; k < size; ++k)
278  {
279  // Find largest diagonal element
280  Index index_of_biggest_in_corner;
281  mat.diagonal().tail(size-k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner);
282  index_of_biggest_in_corner += k;
283 
284  transpositions.coeffRef(k) = index_of_biggest_in_corner;
285  if(k != index_of_biggest_in_corner)
286  {
287  // apply the transposition while taking care to consider only
288  // the lower triangular part
289  Index s = size-index_of_biggest_in_corner-1; // trailing size after the biggest element
290  mat.row(k).head(k).swap(mat.row(index_of_biggest_in_corner).head(k));
291  mat.col(k).tail(s).swap(mat.col(index_of_biggest_in_corner).tail(s));
292  std::swap(mat.coeffRef(k,k),mat.coeffRef(index_of_biggest_in_corner,index_of_biggest_in_corner));
293  for(int i=k+1;i<index_of_biggest_in_corner;++i)
294  {
295  Scalar tmp = mat.coeffRef(i,k);
296  mat.coeffRef(i,k) = numext::conj(mat.coeffRef(index_of_biggest_in_corner,i));
297  mat.coeffRef(index_of_biggest_in_corner,i) = numext::conj(tmp);
298  }
299  if(NumTraits<Scalar>::IsComplex)
300  mat.coeffRef(index_of_biggest_in_corner,k) = numext::conj(mat.coeff(index_of_biggest_in_corner,k));
301  }
302 
303  // partition the matrix:
304  // A00 | - | -
305  // lu = A10 | A11 | -
306  // A20 | A21 | A22
307  Index rs = size - k - 1;
308  Block<MatrixType,Dynamic,1> A21(mat,k+1,k,rs,1);
309  Block<MatrixType,1,Dynamic> A10(mat,k,0,1,k);
310  Block<MatrixType,Dynamic,Dynamic> A20(mat,k+1,0,rs,k);
311 
312  if(k>0)
313  {
314  temp.head(k) = mat.diagonal().real().head(k).asDiagonal() * A10.adjoint();
315  mat.coeffRef(k,k) -= (A10 * temp.head(k)).value();
316  if(rs>0)
317  A21.noalias() -= A20 * temp.head(k);
318  }
319 
320  // In some previous versions of Eigen (e.g., 3.2.1), the scaling was omitted if the pivot
321  // was smaller than the cutoff value. However, soince LDLT is not rank-revealing
322  // we should only make sure we do not introduce INF or NaN values.
323  // LAPACK also uses 0 as the cutoff value.
324  RealScalar realAkk = numext::real(mat.coeffRef(k,k));
325  if((rs>0) && (abs(realAkk) > RealScalar(0)))
326  A21 /= realAkk;
327 
328  if (sign == PositiveSemiDef) {
329  if (realAkk < 0) sign = Indefinite;
330  } else if (sign == NegativeSemiDef) {
331  if (realAkk > 0) sign = Indefinite;
332  } else if (sign == ZeroSign) {
333  if (realAkk > 0) sign = PositiveSemiDef;
334  else if (realAkk < 0) sign = NegativeSemiDef;
335  }
336  }
337 
338  return true;
339  }
340 
341  // Reference for the algorithm: Davis and Hager, "Multiple Rank
342  // Modifications of a Sparse Cholesky Factorization" (Algorithm 1)
343  // Trivial rearrangements of their computations (Timothy E. Holy)
344  // allow their algorithm to work for rank-1 updates even if the
345  // original matrix is not of full rank.
346  // Here only rank-1 updates are implemented, to reduce the
347  // requirement for intermediate storage and improve accuracy
348  template<typename MatrixType, typename WDerived>
349  static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w, const typename MatrixType::RealScalar& sigma=1)
350  {
351  using numext::isfinite;
352  typedef typename MatrixType::Scalar Scalar;
353  typedef typename MatrixType::RealScalar RealScalar;
354  typedef typename MatrixType::Index Index;
355 
356  const Index size = mat.rows();
357  eigen_assert(mat.cols() == size && w.size()==size);
358 
359  RealScalar alpha = 1;
360 
361  // Apply the update
362  for (Index j = 0; j < size; j++)
363  {
364  // Check for termination due to an original decomposition of low-rank
365  if (!(isfinite)(alpha))
366  break;
367 
368  // Update the diagonal terms
369  RealScalar dj = numext::real(mat.coeff(j,j));
370  Scalar wj = w.coeff(j);
371  RealScalar swj2 = sigma*numext::abs2(wj);
372  RealScalar gamma = dj*alpha + swj2;
373 
374  mat.coeffRef(j,j) += swj2/alpha;
375  alpha += swj2/dj;
376 
377 
378  // Update the terms of L
379  Index rs = size-j-1;
380  w.tail(rs) -= wj * mat.col(j).tail(rs);
381  if(gamma != 0)
382  mat.col(j).tail(rs) += (sigma*numext::conj(wj)/gamma)*w.tail(rs);
383  }
384  return true;
385  }
386 
387  template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
388  static bool update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w, const typename MatrixType::RealScalar& sigma=1)
389  {
390  // Apply the permutation to the input w
391  tmp = transpositions * w;
392 
393  return ldlt_inplace<Lower>::updateInPlace(mat,tmp,sigma);
394  }
395 };
396 
397 template<> struct ldlt_inplace<Upper>
398 {
399  template<typename MatrixType, typename TranspositionType, typename Workspace>
400  static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
401  {
402  Transpose<MatrixType> matt(mat);
403  return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign);
404  }
405 
406  template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
407  static EIGEN_STRONG_INLINE bool update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w, const typename MatrixType::RealScalar& sigma=1)
408  {
409  Transpose<MatrixType> matt(mat);
410  return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma);
411  }
412 };
413 
414 template<typename MatrixType> struct LDLT_Traits<MatrixType,Lower>
415 {
416  typedef const TriangularView<const MatrixType, UnitLower> MatrixL;
417  typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
418  static inline MatrixL getL(const MatrixType& m) { return m; }
419  static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
420 };
421 
422 template<typename MatrixType> struct LDLT_Traits<MatrixType,Upper>
423 {
424  typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
425  typedef const TriangularView<const MatrixType, UnitUpper> MatrixU;
426  static inline MatrixL getL(const MatrixType& m) { return m.adjoint(); }
427  static inline MatrixU getU(const MatrixType& m) { return m; }
428 };
429 
430 } // end namespace internal
431 
434 template<typename MatrixType, int _UpLo>
436 {
437  eigen_assert(a.rows()==a.cols());
438  const Index size = a.rows();
439 
440  m_matrix = a;
441 
442  m_transpositions.resize(size);
443  m_isInitialized = false;
444  m_temporary.resize(size);
445 
446  internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign);
447 
448  m_isInitialized = true;
449  return *this;
450 }
451 
457 template<typename MatrixType, int _UpLo>
458 template<typename Derived>
460 {
461  const Index size = w.rows();
462  if (m_isInitialized)
463  {
464  eigen_assert(m_matrix.rows()==size);
465  }
466  else
467  {
468  m_matrix.resize(size,size);
469  m_matrix.setZero();
470  m_transpositions.resize(size);
471  for (Index i = 0; i < size; i++)
472  m_transpositions.coeffRef(i) = i;
473  m_temporary.resize(size);
474  m_sign = sigma>=0 ? internal::PositiveSemiDef : internal::NegativeSemiDef;
475  m_isInitialized = true;
476  }
477 
478  internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);
479 
480  return *this;
481 }
482 
483 namespace internal {
484 template<typename _MatrixType, int _UpLo, typename Rhs>
485 struct solve_retval<LDLT<_MatrixType,_UpLo>, Rhs>
486  : solve_retval_base<LDLT<_MatrixType,_UpLo>, Rhs>
487 {
488  typedef LDLT<_MatrixType,_UpLo> LDLTType;
489  EIGEN_MAKE_SOLVE_HELPERS(LDLTType,Rhs)
490 
491  template<typename Dest> void evalTo(Dest& dst) const
492  {
493  eigen_assert(rhs().rows() == dec().matrixLDLT().rows());
494  // dst = P b
495  dst = dec().transpositionsP() * rhs();
496 
497  // dst = L^-1 (P b)
498  dec().matrixL().solveInPlace(dst);
499 
500  // dst = D^-1 (L^-1 P b)
501  // more precisely, use pseudo-inverse of D (see bug 241)
502  using std::abs;
503  using std::max;
504  typedef typename LDLTType::MatrixType MatrixType;
505  typedef typename LDLTType::RealScalar RealScalar;
506  const typename Diagonal<const MatrixType>::RealReturnType vectorD(dec().vectorD());
507  // In some previous versions, tolerance was set to the max of 1/highest and the maximal diagonal entry * epsilon
508  // as motivated by LAPACK's xGELSS:
509  // RealScalar tolerance = (max)(vectorD.array().abs().maxCoeff() *NumTraits<RealScalar>::epsilon(),RealScalar(1) / NumTraits<RealScalar>::highest());
510  // However, LDLT is not rank revealing, and so adjusting the tolerance wrt to the highest
511  // diagonal element is not well justified and to numerical issues in some cases.
512  // Moreover, Lapack's xSYTRS routines use 0 for the tolerance.
513  RealScalar tolerance = RealScalar(1) / NumTraits<RealScalar>::highest();
514 
515  for (Index i = 0; i < vectorD.size(); ++i) {
516  if(abs(vectorD(i)) > tolerance)
517  dst.row(i) /= vectorD(i);
518  else
519  dst.row(i).setZero();
520  }
521 
522  // dst = L^-T (D^-1 L^-1 P b)
523  dec().matrixU().solveInPlace(dst);
524 
525  // dst = P^-1 (L^-T D^-1 L^-1 P b) = A^-1 b
526  dst = dec().transpositionsP().transpose() * dst;
527  }
528 };
529 }
530 
544 template<typename MatrixType,int _UpLo>
545 template<typename Derived>
546 bool LDLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const
547 {
548  eigen_assert(m_isInitialized && "LDLT is not initialized.");
549  eigen_assert(m_matrix.rows() == bAndX.rows());
550 
551  bAndX = this->solve(bAndX);
552 
553  return true;
554 }
555 
559 template<typename MatrixType, int _UpLo>
561 {
562  eigen_assert(m_isInitialized && "LDLT is not initialized.");
563  const Index size = m_matrix.rows();
564  MatrixType res(size,size);
565 
566  // P
567  res.setIdentity();
568  res = transpositionsP() * res;
569  // L^* P
570  res = matrixU() * res;
571  // D(L^*P)
572  res = vectorD().real().asDiagonal() * res;
573  // L(DL^*P)
574  res = matrixL() * res;
575  // P^T (LDL^*P)
576  res = transpositionsP().transpose() * res;
577 
578  return res;
579 }
580 
584 template<typename MatrixType, unsigned int UpLo>
587 {
588  return LDLT<PlainObject,UpLo>(m_matrix);
589 }
590 
594 template<typename Derived>
597 {
598  return LDLT<PlainObject>(derived());
599 }
600 
601 } // end namespace Eigen
602 
603 #endif // EIGEN_LDLT_H
Robust Cholesky decomposition of a matrix with pivoting.
Definition: LDLT.h:48
ComputationInfo info() const
Reports whether previous computation was successful.
Definition: LDLT.h:231
LDLT(const MatrixType &matrix)
Constructor with decomposition.
Definition: LDLT.h:101
MatrixType reconstructedMatrix() const
Definition: LDLT.h:560
const TranspositionType & transpositionsP() const
Definition: LDLT.h:135
Holds information about the various numeric (i.e. scalar) types allowed by Eigen. ...
Definition: NumTraits.h:88
Traits::MatrixL matrixL() const
Definition: LDLT.h:127
const internal::solve_retval< LDLT, Rhs > solve(const MatrixBase< Rhs > &b) const
Definition: LDLT.h:186
bool isPositive() const
Definition: LDLT.h:149
const LDLT< PlainObject, UpLo > ldlt() const
Definition: LDLT.h:586
Definition: Constants.h:169
Definition: Constants.h:167
LDLT & compute(const MatrixType &matrix)
Definition: LDLT.h:435
LDLT(Index size)
Default Constructor with memory preallocation.
Definition: LDLT.h:88
void setZero()
Definition: LDLT.h:114
Diagonal< const MatrixType > vectorD() const
Definition: LDLT.h:142
LDLT()
Default Constructor.
Definition: LDLT.h:75
Definition: Constants.h:376
bool isNegative(void) const
Definition: LDLT.h:163
const unsigned int RowMajorBit
Definition: Constants.h:53
Expression of a diagonal/subdiagonal/superdiagonal in a matrix.
Definition: Diagonal.h:64
ComputationInfo
Definition: Constants.h:374
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:48
const LDLT< PlainObject > ldlt() const
Definition: LDLT.h:596
Traits::MatrixU matrixU() const
Definition: LDLT.h:120
const MatrixType & matrixLDLT() const
Definition: LDLT.h:215