SparseQR.h
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1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2012-2013 Desire Nuentsa <desire.nuentsa_wakam@inria.fr>
5 // Copyright (C) 2012-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
6 //
7 // This Source Code Form is subject to the terms of the Mozilla
8 // Public License v. 2.0. If a copy of the MPL was not distributed
9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10 
11 #ifndef EIGEN_SPARSE_QR_H
12 #define EIGEN_SPARSE_QR_H
13 
14 namespace Eigen {
15 
16 template<typename MatrixType, typename OrderingType> class SparseQR;
17 template<typename SparseQRType> struct SparseQRMatrixQReturnType;
18 template<typename SparseQRType> struct SparseQRMatrixQTransposeReturnType;
19 template<typename SparseQRType, typename Derived> struct SparseQR_QProduct;
20 namespace internal {
21  template <typename SparseQRType> struct traits<SparseQRMatrixQReturnType<SparseQRType> >
22  {
24  typedef typename ReturnType::StorageIndex StorageIndex;
25  typedef typename ReturnType::StorageKind StorageKind;
26  enum {
27  RowsAtCompileTime = Dynamic,
28  ColsAtCompileTime = Dynamic
29  };
30  };
31  template <typename SparseQRType> struct traits<SparseQRMatrixQTransposeReturnType<SparseQRType> >
32  {
34  };
35  template <typename SparseQRType, typename Derived> struct traits<SparseQR_QProduct<SparseQRType, Derived> >
36  {
37  typedef typename Derived::PlainObject ReturnType;
38  };
39 } // End namespace internal
40 
83 template<typename _MatrixType, typename _OrderingType>
84 class SparseQR : public SparseSolverBase<SparseQR<_MatrixType,_OrderingType> >
85 {
86  protected:
89  public:
90  using Base::_solve_impl;
91  typedef _MatrixType MatrixType;
92  typedef _OrderingType OrderingType;
93  typedef typename MatrixType::Scalar Scalar;
95  typedef typename MatrixType::StorageIndex StorageIndex;
100 
101  enum {
102  ColsAtCompileTime = MatrixType::ColsAtCompileTime,
103  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
104  };
105 
106  public:
108  { }
109 
116  explicit SparseQR(const MatrixType& mat) : m_analysisIsok(false), m_lastError(""), m_useDefaultThreshold(true),m_isQSorted(false),m_isEtreeOk(false)
117  {
118  compute(mat);
119  }
120 
127  void compute(const MatrixType& mat)
128  {
130  factorize(mat);
131  }
132  void analyzePattern(const MatrixType& mat);
133  void factorize(const MatrixType& mat);
134 
137  inline Index rows() const { return m_pmat.rows(); }
138 
141  inline Index cols() const { return m_pmat.cols();}
142 
156  const QRMatrixType& matrixR() const { return m_R; }
157 
162  Index rank() const
163  {
164  eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
165  return m_nonzeropivots;
166  }
167 
187  { return SparseQRMatrixQReturnType<SparseQR>(*this); }
188 
193  {
194  eigen_assert(m_isInitialized && "Decomposition is not initialized.");
195  return m_outputPerm_c;
196  }
197 
201  std::string lastErrorMessage() const { return m_lastError; }
202 
204  template<typename Rhs, typename Dest>
205  bool _solve_impl(const MatrixBase<Rhs> &B, MatrixBase<Dest> &dest) const
206  {
207  eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
208  eigen_assert(this->rows() == B.rows() && "SparseQR::solve() : invalid number of rows in the right hand side matrix");
209 
210  Index rank = this->rank();
211 
212  // Compute Q^* * b;
213  typename Dest::PlainObject y, b;
214  y = this->matrixQ().adjoint() * B;
215  b = y;
216 
217  // Solve with the triangular matrix R
218  y.resize((std::max<Index>)(cols(),y.rows()),y.cols());
219  y.topRows(rank) = this->matrixR().topLeftCorner(rank, rank).template triangularView<Upper>().solve(b.topRows(rank));
220  y.bottomRows(y.rows()-rank).setZero();
221 
222  // Apply the column permutation
223  if (m_perm_c.size()) dest = colsPermutation() * y.topRows(cols());
224  else dest = y.topRows(cols());
225 
226  m_info = Success;
227  return true;
228  }
229 
235  void setPivotThreshold(const RealScalar& threshold)
236  {
237  m_useDefaultThreshold = false;
238  m_threshold = threshold;
239  }
240 
245  template<typename Rhs>
246  inline const Solve<SparseQR, Rhs> solve(const MatrixBase<Rhs>& B) const
247  {
248  eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
249  eigen_assert(this->rows() == B.rows() && "SparseQR::solve() : invalid number of rows in the right hand side matrix");
250  return Solve<SparseQR, Rhs>(*this, B.derived());
251  }
252  template<typename Rhs>
254  {
255  eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
256  eigen_assert(this->rows() == B.rows() && "SparseQR::solve() : invalid number of rows in the right hand side matrix");
257  return Solve<SparseQR, Rhs>(*this, B.derived());
258  }
259 
269  {
270  eigen_assert(m_isInitialized && "Decomposition is not initialized.");
271  return m_info;
272  }
273 
274 
276  inline void _sort_matrix_Q()
277  {
278  if(this->m_isQSorted) return;
279  // The matrix Q is sorted during the transposition
281  this->m_Q = mQrm;
282  this->m_isQSorted = true;
283  }
284 
285 
286  protected:
290  std::string m_lastError;
291  QRMatrixType m_pmat; // Temporary matrix
292  QRMatrixType m_R; // The triangular factor matrix
293  QRMatrixType m_Q; // The orthogonal reflectors
294  ScalarVector m_hcoeffs; // The Householder coefficients
295  PermutationType m_perm_c; // Fill-reducing Column permutation
296  PermutationType m_pivotperm; // The permutation for rank revealing
297  PermutationType m_outputPerm_c; // The final column permutation
298  RealScalar m_threshold; // Threshold to determine null Householder reflections
299  bool m_useDefaultThreshold; // Use default threshold
300  Index m_nonzeropivots; // Number of non zero pivots found
301  IndexVector m_etree; // Column elimination tree
302  IndexVector m_firstRowElt; // First element in each row
303  bool m_isQSorted; // whether Q is sorted or not
304  bool m_isEtreeOk; // whether the elimination tree match the initial input matrix
305 
306  template <typename, typename > friend struct SparseQR_QProduct;
307 
308 };
309 
319 template <typename MatrixType, typename OrderingType>
321 {
322  eigen_assert(mat.isCompressed() && "SparseQR requires a sparse matrix in compressed mode. Call .makeCompressed() before passing it to SparseQR");
323  // Copy to a column major matrix if the input is rowmajor
325  // Compute the column fill reducing ordering
326  OrderingType ord;
327  ord(matCpy, m_perm_c);
328  Index n = mat.cols();
329  Index m = mat.rows();
330  Index diagSize = (std::min)(m,n);
331 
332  if (!m_perm_c.size())
333  {
334  m_perm_c.resize(n);
335  m_perm_c.indices().setLinSpaced(n, 0,StorageIndex(n-1));
336  }
337 
338  // Compute the column elimination tree of the permuted matrix
339  m_outputPerm_c = m_perm_c.inverse();
340  internal::coletree(matCpy, m_etree, m_firstRowElt, m_outputPerm_c.indices().data());
341  m_isEtreeOk = true;
342 
343  m_R.resize(m, n);
344  m_Q.resize(m, diagSize);
345 
346  // Allocate space for nonzero elements: rough estimation
347  m_R.reserve(2*mat.nonZeros()); //FIXME Get a more accurate estimation through symbolic factorization with the etree
348  m_Q.reserve(2*mat.nonZeros());
349  m_hcoeffs.resize(diagSize);
350  m_analysisIsok = true;
351 }
352 
360 template <typename MatrixType, typename OrderingType>
362 {
363  using std::abs;
364 
365  eigen_assert(m_analysisIsok && "analyzePattern() should be called before this step");
366  StorageIndex m = StorageIndex(mat.rows());
367  StorageIndex n = StorageIndex(mat.cols());
368  StorageIndex diagSize = (std::min)(m,n);
369  IndexVector mark((std::max)(m,n)); mark.setConstant(-1); // Record the visited nodes
370  IndexVector Ridx(n), Qidx(m); // Store temporarily the row indexes for the current column of R and Q
371  Index nzcolR, nzcolQ; // Number of nonzero for the current column of R and Q
372  ScalarVector tval(m); // The dense vector used to compute the current column
373  RealScalar pivotThreshold = m_threshold;
374 
375  m_R.setZero();
376  m_Q.setZero();
377  m_pmat = mat;
378  if(!m_isEtreeOk)
379  {
380  m_outputPerm_c = m_perm_c.inverse();
381  internal::coletree(m_pmat, m_etree, m_firstRowElt, m_outputPerm_c.indices().data());
382  m_isEtreeOk = true;
383  }
384 
385  m_pmat.uncompress(); // To have the innerNonZeroPtr allocated
386 
387  // Apply the fill-in reducing permutation lazily:
388  {
389  // If the input is row major, copy the original column indices,
390  // otherwise directly use the input matrix
391  //
392  IndexVector originalOuterIndicesCpy;
393  const StorageIndex *originalOuterIndices = mat.outerIndexPtr();
394  if(MatrixType::IsRowMajor)
395  {
396  originalOuterIndicesCpy = IndexVector::Map(m_pmat.outerIndexPtr(),n+1);
397  originalOuterIndices = originalOuterIndicesCpy.data();
398  }
399 
400  for (int i = 0; i < n; i++)
401  {
402  Index p = m_perm_c.size() ? m_perm_c.indices()(i) : i;
403  m_pmat.outerIndexPtr()[p] = originalOuterIndices[i];
404  m_pmat.innerNonZeroPtr()[p] = originalOuterIndices[i+1] - originalOuterIndices[i];
405  }
406  }
407 
408  /* Compute the default threshold as in MatLab, see:
409  * Tim Davis, "Algorithm 915, SuiteSparseQR: Multifrontal Multithreaded Rank-Revealing
410  * Sparse QR Factorization, ACM Trans. on Math. Soft. 38(1), 2011, Page 8:3
411  */
412  if(m_useDefaultThreshold)
413  {
414  RealScalar max2Norm = 0.0;
415  for (int j = 0; j < n; j++) max2Norm = numext::maxi(max2Norm, m_pmat.col(j).norm());
416  if(max2Norm==RealScalar(0))
417  max2Norm = RealScalar(1);
418  pivotThreshold = 20 * (m + n) * max2Norm * NumTraits<RealScalar>::epsilon();
419  }
420 
421  // Initialize the numerical permutation
422  m_pivotperm.setIdentity(n);
423 
424  StorageIndex nonzeroCol = 0; // Record the number of valid pivots
425  m_Q.startVec(0);
426 
427  // Left looking rank-revealing QR factorization: compute a column of R and Q at a time
428  for (StorageIndex col = 0; col < n; ++col)
429  {
430  mark.setConstant(-1);
431  m_R.startVec(col);
432  mark(nonzeroCol) = col;
433  Qidx(0) = nonzeroCol;
434  nzcolR = 0; nzcolQ = 1;
435  bool found_diag = nonzeroCol>=m;
436  tval.setZero();
437 
438  // Symbolic factorization: find the nonzero locations of the column k of the factors R and Q, i.e.,
439  // all the nodes (with indexes lower than rank) reachable through the column elimination tree (etree) rooted at node k.
440  // Note: if the diagonal entry does not exist, then its contribution must be explicitly added,
441  // thus the trick with found_diag that permits to do one more iteration on the diagonal element if this one has not been found.
442  for (typename QRMatrixType::InnerIterator itp(m_pmat, col); itp || !found_diag; ++itp)
443  {
444  StorageIndex curIdx = nonzeroCol;
445  if(itp) curIdx = StorageIndex(itp.row());
446  if(curIdx == nonzeroCol) found_diag = true;
447 
448  // Get the nonzeros indexes of the current column of R
449  StorageIndex st = m_firstRowElt(curIdx); // The traversal of the etree starts here
450  if (st < 0 )
451  {
452  m_lastError = "Empty row found during numerical factorization";
453  m_info = InvalidInput;
454  return;
455  }
456 
457  // Traverse the etree
458  Index bi = nzcolR;
459  for (; mark(st) != col; st = m_etree(st))
460  {
461  Ridx(nzcolR) = st; // Add this row to the list,
462  mark(st) = col; // and mark this row as visited
463  nzcolR++;
464  }
465 
466  // Reverse the list to get the topological ordering
467  Index nt = nzcolR-bi;
468  for(Index i = 0; i < nt/2; i++) std::swap(Ridx(bi+i), Ridx(nzcolR-i-1));
469 
470  // Copy the current (curIdx,pcol) value of the input matrix
471  if(itp) tval(curIdx) = itp.value();
472  else tval(curIdx) = Scalar(0);
473 
474  // Compute the pattern of Q(:,k)
475  if(curIdx > nonzeroCol && mark(curIdx) != col )
476  {
477  Qidx(nzcolQ) = curIdx; // Add this row to the pattern of Q,
478  mark(curIdx) = col; // and mark it as visited
479  nzcolQ++;
480  }
481  }
482 
483  // Browse all the indexes of R(:,col) in reverse order
484  for (Index i = nzcolR-1; i >= 0; i--)
485  {
486  Index curIdx = Ridx(i);
487 
488  // Apply the curIdx-th householder vector to the current column (temporarily stored into tval)
489  Scalar tdot(0);
490 
491  // First compute q' * tval
492  tdot = m_Q.col(curIdx).dot(tval);
493 
494  tdot *= m_hcoeffs(curIdx);
495 
496  // Then update tval = tval - q * tau
497  // FIXME: tval -= tdot * m_Q.col(curIdx) should amount to the same (need to check/add support for efficient "dense ?= sparse")
498  for (typename QRMatrixType::InnerIterator itq(m_Q, curIdx); itq; ++itq)
499  tval(itq.row()) -= itq.value() * tdot;
500 
501  // Detect fill-in for the current column of Q
502  if(m_etree(Ridx(i)) == nonzeroCol)
503  {
504  for (typename QRMatrixType::InnerIterator itq(m_Q, curIdx); itq; ++itq)
505  {
506  StorageIndex iQ = StorageIndex(itq.row());
507  if (mark(iQ) != col)
508  {
509  Qidx(nzcolQ++) = iQ; // Add this row to the pattern of Q,
510  mark(iQ) = col; // and mark it as visited
511  }
512  }
513  }
514  } // End update current column
515 
516  Scalar tau = RealScalar(0);
517  RealScalar beta = 0;
518 
519  if(nonzeroCol < diagSize)
520  {
521  // Compute the Householder reflection that eliminate the current column
522  // FIXME this step should call the Householder module.
523  Scalar c0 = nzcolQ ? tval(Qidx(0)) : Scalar(0);
524 
525  // First, the squared norm of Q((col+1):m, col)
526  RealScalar sqrNorm = 0.;
527  for (Index itq = 1; itq < nzcolQ; ++itq) sqrNorm += numext::abs2(tval(Qidx(itq)));
528  if(sqrNorm == RealScalar(0) && numext::imag(c0) == RealScalar(0))
529  {
530  beta = numext::real(c0);
531  tval(Qidx(0)) = 1;
532  }
533  else
534  {
535  using std::sqrt;
536  beta = sqrt(numext::abs2(c0) + sqrNorm);
537  if(numext::real(c0) >= RealScalar(0))
538  beta = -beta;
539  tval(Qidx(0)) = 1;
540  for (Index itq = 1; itq < nzcolQ; ++itq)
541  tval(Qidx(itq)) /= (c0 - beta);
542  tau = numext::conj((beta-c0) / beta);
543 
544  }
545  }
546 
547  // Insert values in R
548  for (Index i = nzcolR-1; i >= 0; i--)
549  {
550  Index curIdx = Ridx(i);
551  if(curIdx < nonzeroCol)
552  {
553  m_R.insertBackByOuterInnerUnordered(col, curIdx) = tval(curIdx);
554  tval(curIdx) = Scalar(0.);
555  }
556  }
557 
558  if(nonzeroCol < diagSize && abs(beta) >= pivotThreshold)
559  {
560  m_R.insertBackByOuterInner(col, nonzeroCol) = beta;
561  // The householder coefficient
562  m_hcoeffs(nonzeroCol) = tau;
563  // Record the householder reflections
564  for (Index itq = 0; itq < nzcolQ; ++itq)
565  {
566  Index iQ = Qidx(itq);
567  m_Q.insertBackByOuterInnerUnordered(nonzeroCol,iQ) = tval(iQ);
568  tval(iQ) = Scalar(0.);
569  }
570  nonzeroCol++;
571  if(nonzeroCol<diagSize)
572  m_Q.startVec(nonzeroCol);
573  }
574  else
575  {
576  // Zero pivot found: move implicitly this column to the end
577  for (Index j = nonzeroCol; j < n-1; j++)
578  std::swap(m_pivotperm.indices()(j), m_pivotperm.indices()[j+1]);
579 
580  // Recompute the column elimination tree
581  internal::coletree(m_pmat, m_etree, m_firstRowElt, m_pivotperm.indices().data());
582  m_isEtreeOk = false;
583  }
584  }
585 
586  m_hcoeffs.tail(diagSize-nonzeroCol).setZero();
587 
588  // Finalize the column pointers of the sparse matrices R and Q
589  m_Q.finalize();
590  m_Q.makeCompressed();
591  m_R.finalize();
592  m_R.makeCompressed();
593  m_isQSorted = false;
594 
595  m_nonzeropivots = nonzeroCol;
596 
597  if(nonzeroCol<n)
598  {
599  // Permute the triangular factor to put the 'dead' columns to the end
600  QRMatrixType tempR(m_R);
601  m_R = tempR * m_pivotperm;
602 
603  // Update the column permutation
604  m_outputPerm_c = m_outputPerm_c * m_pivotperm;
605  }
606 
607  m_isInitialized = true;
608  m_factorizationIsok = true;
609  m_info = Success;
610 }
611 
612 template <typename SparseQRType, typename Derived>
613 struct SparseQR_QProduct : ReturnByValue<SparseQR_QProduct<SparseQRType, Derived> >
614 {
615  typedef typename SparseQRType::QRMatrixType MatrixType;
616  typedef typename SparseQRType::Scalar Scalar;
617  // Get the references
618  SparseQR_QProduct(const SparseQRType& qr, const Derived& other, bool transpose) :
619  m_qr(qr),m_other(other),m_transpose(transpose) {}
620  inline Index rows() const { return m_qr.matrixQ().rows(); }
621  inline Index cols() const { return m_other.cols(); }
622 
623  // Assign to a vector
624  template<typename DesType>
625  void evalTo(DesType& res) const
626  {
627  Index m = m_qr.rows();
628  Index n = m_qr.cols();
629  Index diagSize = (std::min)(m,n);
630  res = m_other;
631  if (m_transpose)
632  {
633  eigen_assert(m_qr.m_Q.rows() == m_other.rows() && "Non conforming object sizes");
634  //Compute res = Q' * other column by column
635  for(Index j = 0; j < res.cols(); j++){
636  for (Index k = 0; k < diagSize; k++)
637  {
638  Scalar tau = Scalar(0);
639  tau = m_qr.m_Q.col(k).dot(res.col(j));
640  if(tau==Scalar(0)) continue;
641  tau = tau * m_qr.m_hcoeffs(k);
642  res.col(j) -= tau * m_qr.m_Q.col(k);
643  }
644  }
645  }
646  else
647  {
648  eigen_assert(m_qr.matrixQ().cols() == m_other.rows() && "Non conforming object sizes");
649 
650  res.conservativeResize(rows(), cols());
651 
652  // Compute res = Q * other column by column
653  for(Index j = 0; j < res.cols(); j++)
654  {
655  Index start_k = internal::is_identity<Derived>::value ? numext::mini(j,diagSize-1) : diagSize-1;
656  for (Index k = start_k; k >=0; k--)
657  {
658  Scalar tau = Scalar(0);
659  tau = m_qr.m_Q.col(k).dot(res.col(j));
660  if(tau==Scalar(0)) continue;
661  tau = tau * numext::conj(m_qr.m_hcoeffs(k));
662  res.col(j) -= tau * m_qr.m_Q.col(k);
663  }
664  }
665  }
666  }
667 
668  const SparseQRType& m_qr;
669  const Derived& m_other;
670  bool m_transpose; // TODO this actually means adjoint
671 };
672 
673 template<typename SparseQRType>
674 struct SparseQRMatrixQReturnType : public EigenBase<SparseQRMatrixQReturnType<SparseQRType> >
675 {
676  typedef typename SparseQRType::Scalar Scalar;
678  enum {
681  };
682  explicit SparseQRMatrixQReturnType(const SparseQRType& qr) : m_qr(qr) {}
683  template<typename Derived>
685  {
686  return SparseQR_QProduct<SparseQRType,Derived>(m_qr,other.derived(),false);
687  }
688  // To use for operations with the adjoint of Q
690  {
692  }
693  inline Index rows() const { return m_qr.rows(); }
694  inline Index cols() const { return m_qr.rows(); }
695  // To use for operations with the transpose of Q FIXME this is the same as adjoint at the moment
697  {
699  }
700  const SparseQRType& m_qr;
701 };
702 
703 // TODO this actually represents the adjoint of Q
704 template<typename SparseQRType>
706 {
707  explicit SparseQRMatrixQTransposeReturnType(const SparseQRType& qr) : m_qr(qr) {}
708  template<typename Derived>
710  {
711  return SparseQR_QProduct<SparseQRType,Derived>(m_qr,other.derived(), true);
712  }
713  const SparseQRType& m_qr;
714 };
715 
716 namespace internal {
717 
718 template<typename SparseQRType>
720 {
724 };
725 
726 template< typename DstXprType, typename SparseQRType>
727 struct Assignment<DstXprType, SparseQRMatrixQReturnType<SparseQRType>, internal::assign_op<typename DstXprType::Scalar,typename DstXprType::Scalar>, Sparse2Sparse>
728 {
730  typedef typename DstXprType::Scalar Scalar;
731  typedef typename DstXprType::StorageIndex StorageIndex;
732  static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<Scalar,Scalar> &/*func*/)
733  {
734  typename DstXprType::PlainObject idMat(src.rows(), src.cols());
735  idMat.setIdentity();
736  // Sort the sparse householder reflectors if needed
737  const_cast<SparseQRType *>(&src.m_qr)->_sort_matrix_Q();
738  dst = SparseQR_QProduct<SparseQRType, DstXprType>(src.m_qr, idMat, false);
739  }
740 };
741 
742 template< typename DstXprType, typename SparseQRType>
743 struct Assignment<DstXprType, SparseQRMatrixQReturnType<SparseQRType>, internal::assign_op<typename DstXprType::Scalar,typename DstXprType::Scalar>, Sparse2Dense>
744 {
746  typedef typename DstXprType::Scalar Scalar;
747  typedef typename DstXprType::StorageIndex StorageIndex;
748  static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<Scalar,Scalar> &/*func*/)
749  {
750  dst = src.m_qr.matrixQ() * DstXprType::Identity(src.m_qr.rows(), src.m_qr.rows());
751  }
752 };
753 
754 } // end namespace internal
755 
756 } // end namespace Eigen
757 
758 #endif
Eigen::SparseMatrix::cols
Index cols() const
Definition: SparseMatrix.h:140
Eigen::SparseQR::setPivotThreshold
void setPivotThreshold(const RealScalar &threshold)
Definition: SparseQR.h:235
Eigen::internal::evaluator_traits< SparseQRMatrixQReturnType< SparseQRType > >::MatrixType
SparseQRType::MatrixType MatrixType
Definition: SparseQR.h:721
Eigen::SparseQR::m_analysisIsok
bool m_analysisIsok
Definition: SparseQR.h:287
Eigen::SparseQR::ScalarVector
Matrix< Scalar, Dynamic, 1 > ScalarVector
Definition: SparseQR.h:98
Eigen
Namespace containing all symbols from the Eigen library.
Definition: jet.h:637
Eigen::SparseMatrix< Scalar, ColMajor, StorageIndex >
Eigen::SparseQRMatrixQReturnType::adjoint
SparseQRMatrixQTransposeReturnType< SparseQRType > adjoint() const
Definition: SparseQR.h:689
Eigen::ReturnByValue
Definition: ReturnByValue.h:50
Eigen::SparseQR::m_factorizationIsok
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