Eigen  3.2.6
IncompleteLUT.h
1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@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_INCOMPLETE_LUT_H
11 #define EIGEN_INCOMPLETE_LUT_H
12 
13 
14 namespace Eigen {
15 
16 namespace internal {
17 
27 template <typename VectorV, typename VectorI, typename Index>
28 Index QuickSplit(VectorV &row, VectorI &ind, Index ncut)
29 {
30  typedef typename VectorV::RealScalar RealScalar;
31  using std::swap;
32  using std::abs;
33  Index mid;
34  Index n = row.size(); /* length of the vector */
35  Index first, last ;
36 
37  ncut--; /* to fit the zero-based indices */
38  first = 0;
39  last = n-1;
40  if (ncut < first || ncut > last ) return 0;
41 
42  do {
43  mid = first;
44  RealScalar abskey = abs(row(mid));
45  for (Index j = first + 1; j <= last; j++) {
46  if ( abs(row(j)) > abskey) {
47  ++mid;
48  swap(row(mid), row(j));
49  swap(ind(mid), ind(j));
50  }
51  }
52  /* Interchange for the pivot element */
53  swap(row(mid), row(first));
54  swap(ind(mid), ind(first));
55 
56  if (mid > ncut) last = mid - 1;
57  else if (mid < ncut ) first = mid + 1;
58  } while (mid != ncut );
59 
60  return 0; /* mid is equal to ncut */
61 }
62 
63 }// end namespace internal
64 
95 template <typename _Scalar>
96 class IncompleteLUT : internal::noncopyable
97 {
98  typedef _Scalar Scalar;
99  typedef typename NumTraits<Scalar>::Real RealScalar;
103  typedef typename FactorType::Index Index;
104 
105  public:
107 
108  IncompleteLUT()
109  : m_droptol(NumTraits<Scalar>::dummy_precision()), m_fillfactor(10),
110  m_analysisIsOk(false), m_factorizationIsOk(false), m_isInitialized(false)
111  {}
112 
113  template<typename MatrixType>
114  IncompleteLUT(const MatrixType& mat, const RealScalar& droptol=NumTraits<Scalar>::dummy_precision(), int fillfactor = 10)
115  : m_droptol(droptol),m_fillfactor(fillfactor),
116  m_analysisIsOk(false),m_factorizationIsOk(false),m_isInitialized(false)
117  {
118  eigen_assert(fillfactor != 0);
119  compute(mat);
120  }
121 
122  Index rows() const { return m_lu.rows(); }
123 
124  Index cols() const { return m_lu.cols(); }
125 
132  {
133  eigen_assert(m_isInitialized && "IncompleteLUT is not initialized.");
134  return m_info;
135  }
136 
137  template<typename MatrixType>
138  void analyzePattern(const MatrixType& amat);
139 
140  template<typename MatrixType>
141  void factorize(const MatrixType& amat);
142 
148  template<typename MatrixType>
149  IncompleteLUT<Scalar>& compute(const MatrixType& amat)
150  {
151  analyzePattern(amat);
152  factorize(amat);
153  return *this;
154  }
155 
156  void setDroptol(const RealScalar& droptol);
157  void setFillfactor(int fillfactor);
158 
159  template<typename Rhs, typename Dest>
160  void _solve(const Rhs& b, Dest& x) const
161  {
162  x = m_Pinv * b;
163  x = m_lu.template triangularView<UnitLower>().solve(x);
164  x = m_lu.template triangularView<Upper>().solve(x);
165  x = m_P * x;
166  }
167 
168  template<typename Rhs> inline const internal::solve_retval<IncompleteLUT, Rhs>
169  solve(const MatrixBase<Rhs>& b) const
170  {
171  eigen_assert(m_isInitialized && "IncompleteLUT is not initialized.");
172  eigen_assert(cols()==b.rows()
173  && "IncompleteLUT::solve(): invalid number of rows of the right hand side matrix b");
174  return internal::solve_retval<IncompleteLUT, Rhs>(*this, b.derived());
175  }
176 
177 protected:
178 
180  struct keep_diag {
181  inline bool operator() (const Index& row, const Index& col, const Scalar&) const
182  {
183  return row!=col;
184  }
185  };
186 
187 protected:
188 
189  FactorType m_lu;
190  RealScalar m_droptol;
191  int m_fillfactor;
192  bool m_analysisIsOk;
193  bool m_factorizationIsOk;
194  bool m_isInitialized;
195  ComputationInfo m_info;
196  PermutationMatrix<Dynamic,Dynamic,Index> m_P; // Fill-reducing permutation
197  PermutationMatrix<Dynamic,Dynamic,Index> m_Pinv; // Inverse permutation
198 };
199 
204 template<typename Scalar>
205 void IncompleteLUT<Scalar>::setDroptol(const RealScalar& droptol)
206 {
207  this->m_droptol = droptol;
208 }
209 
214 template<typename Scalar>
216 {
217  this->m_fillfactor = fillfactor;
218 }
219 
220 template <typename Scalar>
221 template<typename _MatrixType>
222 void IncompleteLUT<Scalar>::analyzePattern(const _MatrixType& amat)
223 {
224  // Compute the Fill-reducing permutation
226  SparseMatrix<Scalar,ColMajor, Index> mat2 = amat.transpose();
227  // Symmetrize the pattern
228  // FIXME for a matrix with nearly symmetric pattern, mat2+mat1 is the appropriate choice.
229  // on the other hand for a really non-symmetric pattern, mat2*mat1 should be prefered...
230  SparseMatrix<Scalar,ColMajor, Index> AtA = mat2 + mat1;
231  AtA.prune(keep_diag());
232  internal::minimum_degree_ordering<Scalar, Index>(AtA, m_P); // Then compute the AMD ordering...
233 
234  m_Pinv = m_P.inverse(); // ... and the inverse permutation
235 
236  m_analysisIsOk = true;
237  m_factorizationIsOk = false;
238  m_isInitialized = false;
239 }
240 
241 template <typename Scalar>
242 template<typename _MatrixType>
243 void IncompleteLUT<Scalar>::factorize(const _MatrixType& amat)
244 {
245  using std::sqrt;
246  using std::swap;
247  using std::abs;
248 
249  eigen_assert((amat.rows() == amat.cols()) && "The factorization should be done on a square matrix");
250  Index n = amat.cols(); // Size of the matrix
251  m_lu.resize(n,n);
252  // Declare Working vectors and variables
253  Vector u(n) ; // real values of the row -- maximum size is n --
254  VectorXi ju(n); // column position of the values in u -- maximum size is n
255  VectorXi jr(n); // Indicate the position of the nonzero elements in the vector u -- A zero location is indicated by -1
256 
257  // Apply the fill-reducing permutation
258  eigen_assert(m_analysisIsOk && "You must first call analyzePattern()");
259  SparseMatrix<Scalar,RowMajor, Index> mat;
260  mat = amat.twistedBy(m_Pinv);
261 
262  // Initialization
263  jr.fill(-1);
264  ju.fill(0);
265  u.fill(0);
266 
267  // number of largest elements to keep in each row:
268  Index fill_in = static_cast<Index> (amat.nonZeros()*m_fillfactor)/n+1;
269  if (fill_in > n) fill_in = n;
270 
271  // number of largest nonzero elements to keep in the L and the U part of the current row:
272  Index nnzL = fill_in/2;
273  Index nnzU = nnzL;
274  m_lu.reserve(n * (nnzL + nnzU + 1));
275 
276  // global loop over the rows of the sparse matrix
277  for (Index ii = 0; ii < n; ii++)
278  {
279  // 1 - copy the lower and the upper part of the row i of mat in the working vector u
280 
281  Index sizeu = 1; // number of nonzero elements in the upper part of the current row
282  Index sizel = 0; // number of nonzero elements in the lower part of the current row
283  ju(ii) = ii;
284  u(ii) = 0;
285  jr(ii) = ii;
286  RealScalar rownorm = 0;
287 
288  typename FactorType::InnerIterator j_it(mat, ii); // Iterate through the current row ii
289  for (; j_it; ++j_it)
290  {
291  Index k = j_it.index();
292  if (k < ii)
293  {
294  // copy the lower part
295  ju(sizel) = k;
296  u(sizel) = j_it.value();
297  jr(k) = sizel;
298  ++sizel;
299  }
300  else if (k == ii)
301  {
302  u(ii) = j_it.value();
303  }
304  else
305  {
306  // copy the upper part
307  Index jpos = ii + sizeu;
308  ju(jpos) = k;
309  u(jpos) = j_it.value();
310  jr(k) = jpos;
311  ++sizeu;
312  }
313  rownorm += numext::abs2(j_it.value());
314  }
315 
316  // 2 - detect possible zero row
317  if(rownorm==0)
318  {
319  m_info = NumericalIssue;
320  return;
321  }
322  // Take the 2-norm of the current row as a relative tolerance
323  rownorm = sqrt(rownorm);
324 
325  // 3 - eliminate the previous nonzero rows
326  Index jj = 0;
327  Index len = 0;
328  while (jj < sizel)
329  {
330  // In order to eliminate in the correct order,
331  // we must select first the smallest column index among ju(jj:sizel)
332  Index k;
333  Index minrow = ju.segment(jj,sizel-jj).minCoeff(&k); // k is relative to the segment
334  k += jj;
335  if (minrow != ju(jj))
336  {
337  // swap the two locations
338  Index j = ju(jj);
339  swap(ju(jj), ju(k));
340  jr(minrow) = jj; jr(j) = k;
341  swap(u(jj), u(k));
342  }
343  // Reset this location
344  jr(minrow) = -1;
345 
346  // Start elimination
347  typename FactorType::InnerIterator ki_it(m_lu, minrow);
348  while (ki_it && ki_it.index() < minrow) ++ki_it;
349  eigen_internal_assert(ki_it && ki_it.col()==minrow);
350  Scalar fact = u(jj) / ki_it.value();
351 
352  // drop too small elements
353  if(abs(fact) <= m_droptol)
354  {
355  jj++;
356  continue;
357  }
358 
359  // linear combination of the current row ii and the row minrow
360  ++ki_it;
361  for (; ki_it; ++ki_it)
362  {
363  Scalar prod = fact * ki_it.value();
364  Index j = ki_it.index();
365  Index jpos = jr(j);
366  if (jpos == -1) // fill-in element
367  {
368  Index newpos;
369  if (j >= ii) // dealing with the upper part
370  {
371  newpos = ii + sizeu;
372  sizeu++;
373  eigen_internal_assert(sizeu<=n);
374  }
375  else // dealing with the lower part
376  {
377  newpos = sizel;
378  sizel++;
379  eigen_internal_assert(sizel<=ii);
380  }
381  ju(newpos) = j;
382  u(newpos) = -prod;
383  jr(j) = newpos;
384  }
385  else
386  u(jpos) -= prod;
387  }
388  // store the pivot element
389  u(len) = fact;
390  ju(len) = minrow;
391  ++len;
392 
393  jj++;
394  } // end of the elimination on the row ii
395 
396  // reset the upper part of the pointer jr to zero
397  for(Index k = 0; k <sizeu; k++) jr(ju(ii+k)) = -1;
398 
399  // 4 - partially sort and insert the elements in the m_lu matrix
400 
401  // sort the L-part of the row
402  sizel = len;
403  len = (std::min)(sizel, nnzL);
404  typename Vector::SegmentReturnType ul(u.segment(0, sizel));
405  typename VectorXi::SegmentReturnType jul(ju.segment(0, sizel));
406  internal::QuickSplit(ul, jul, len);
407 
408  // store the largest m_fill elements of the L part
409  m_lu.startVec(ii);
410  for(Index k = 0; k < len; k++)
411  m_lu.insertBackByOuterInnerUnordered(ii,ju(k)) = u(k);
412 
413  // store the diagonal element
414  // apply a shifting rule to avoid zero pivots (we are doing an incomplete factorization)
415  if (u(ii) == Scalar(0))
416  u(ii) = sqrt(m_droptol) * rownorm;
417  m_lu.insertBackByOuterInnerUnordered(ii, ii) = u(ii);
418 
419  // sort the U-part of the row
420  // apply the dropping rule first
421  len = 0;
422  for(Index k = 1; k < sizeu; k++)
423  {
424  if(abs(u(ii+k)) > m_droptol * rownorm )
425  {
426  ++len;
427  u(ii + len) = u(ii + k);
428  ju(ii + len) = ju(ii + k);
429  }
430  }
431  sizeu = len + 1; // +1 to take into account the diagonal element
432  len = (std::min)(sizeu, nnzU);
433  typename Vector::SegmentReturnType uu(u.segment(ii+1, sizeu-1));
434  typename VectorXi::SegmentReturnType juu(ju.segment(ii+1, sizeu-1));
435  internal::QuickSplit(uu, juu, len);
436 
437  // store the largest elements of the U part
438  for(Index k = ii + 1; k < ii + len; k++)
439  m_lu.insertBackByOuterInnerUnordered(ii,ju(k)) = u(k);
440  }
441 
442  m_lu.finalize();
443  m_lu.makeCompressed();
444 
445  m_factorizationIsOk = true;
446  m_isInitialized = m_factorizationIsOk;
447  m_info = Success;
448 }
449 
450 namespace internal {
451 
452 template<typename _MatrixType, typename Rhs>
453 struct solve_retval<IncompleteLUT<_MatrixType>, Rhs>
454  : solve_retval_base<IncompleteLUT<_MatrixType>, Rhs>
455 {
456  typedef IncompleteLUT<_MatrixType> Dec;
457  EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
458 
459  template<typename Dest> void evalTo(Dest& dst) const
460  {
461  dec()._solve(rhs(),dst);
462  }
463 };
464 
465 } // end namespace internal
466 
467 } // end namespace Eigen
468 
469 #endif // EIGEN_INCOMPLETE_LUT_H
Index rows() const
Definition: SparseMatrix.h:119
Index cols() const
Definition: SparseMatrix.h:121
ComputationInfo info() const
Reports whether previous computation was successful.
Definition: IncompleteLUT.h:131
void fill(const Scalar &value)
Definition: CwiseNullaryOp.h:322
Definition: Constants.h:378
Definition: LDLT.h:16
Holds information about the various numeric (i.e. scalar) types allowed by Eigen. ...
Definition: NumTraits.h:88
Definition: IncompleteLUT.h:180
void setFillfactor(int fillfactor)
Definition: IncompleteLUT.h:215
Incomplete LU factorization with dual-threshold strategy.
Definition: IncompleteLUT.h:96
IncompleteLUT< Scalar > & compute(const MatrixType &amat)
Definition: IncompleteLUT.h:149
Definition: Eigen_Colamd.h:54
void prune(const Scalar &reference, const RealScalar &epsilon=NumTraits< RealScalar >::dummy_precision())
Definition: SparseMatrix.h:490
Definition: Constants.h:376
void setDroptol(const RealScalar &droptol)
Definition: IncompleteLUT.h:205
Matrix< int, Dynamic, 1 > VectorXi
Definition: Matrix.h:393
ComputationInfo
Definition: Constants.h:374