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1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
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_LU_H
11 #define EIGEN_LU_H
12 
13 namespace Eigen {
14 
15 /** \ingroup LU_Module
16   *
17   * \class FullPivLU
18   *
19   * \brief LU decomposition of a matrix with complete pivoting, and related features
20   *
21   * \param MatrixType the type of the matrix of which we are computing the LU decomposition
22   *
23   * This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A
24   * is decomposed as A = PLUQ where L is unit-lower-triangular, U is upper-triangular, and P and Q
25   * are permutation matrices. This is a rank-revealing LU decomposition. The eigenvalues (diagonal
26   * coefficients) of U are sorted in such a way that any zeros are at the end.
27   *
28   * This decomposition provides the generic approach to solving systems of linear equations, computing
29   * the rank, invertibility, inverse, kernel, and determinant.
30   *
31   * This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD
32   * decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix,
33   * working with the SVD allows to select the smallest singular values of the matrix, something that
34   * the LU decomposition doesn't see.
35   *
36   * The data of the LU decomposition can be directly accessed through the methods matrixLU(),
37   * permutationP(), permutationQ().
38   *
39   * As an exemple, here is how the original matrix can be retrieved:
40   * \include class_FullPivLU.cpp
41   * Output: \verbinclude class_FullPivLU.out
42   *
43   * \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse()
44   */
45 template<typename _MatrixType> class FullPivLU
46 {
47   public:
48     typedef _MatrixType MatrixType;
49     enum {
50       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
51       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
52       Options = MatrixType::Options,
53       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
54       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
55     };
56     typedef typename MatrixType::Scalar Scalar;
57     typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
58     typedef typename internal::traits<MatrixType>::StorageKind StorageKind;
59     typedef typename MatrixType::Index Index;
60     typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
61     typedef typename internal::plain_col_type<MatrixType, Index>::type IntColVectorType;
62     typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType;
63     typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType;
64 
65     /**
66       * \brief Default Constructor.
67       *
68       * The default constructor is useful in cases in which the user intends to
69       * perform decompositions via LU::compute(const MatrixType&).
70       */
71     FullPivLU();
72 
73     /** \brief Default Constructor with memory preallocation
74       *
75       * Like the default constructor but with preallocation of the internal data
76       * according to the specified problem \a size.
77       * \sa FullPivLU()
78       */
79     FullPivLU(Index rows, Index cols);
80 
81     /** Constructor.
82       *
83       * \param matrix the matrix of which to compute the LU decomposition.
84       *               It is required to be nonzero.
85       */
86     FullPivLU(const MatrixType& matrix);
87 
88     /** Computes the LU decomposition of the given matrix.
89       *
90       * \param matrix the matrix of which to compute the LU decomposition.
91       *               It is required to be nonzero.
92       *
93       * \returns a reference to *this
94       */
95     FullPivLU& compute(const MatrixType& matrix);
96 
97     /** \returns the LU decomposition matrix: the upper-triangular part is U, the
98       * unit-lower-triangular part is L (at least for square matrices; in the non-square
99       * case, special care is needed, see the documentation of class FullPivLU).
100       *
101       * \sa matrixL(), matrixU()
102       */
matrixLU()103     inline const MatrixType& matrixLU() const
104     {
105       eigen_assert(m_isInitialized && "LU is not initialized.");
106       return m_lu;
107     }
108 
109     /** \returns the number of nonzero pivots in the LU decomposition.
110       * Here nonzero is meant in the exact sense, not in a fuzzy sense.
111       * So that notion isn't really intrinsically interesting, but it is
112       * still useful when implementing algorithms.
113       *
114       * \sa rank()
115       */
nonzeroPivots()116     inline Index nonzeroPivots() const
117     {
118       eigen_assert(m_isInitialized && "LU is not initialized.");
119       return m_nonzero_pivots;
120     }
121 
122     /** \returns the absolute value of the biggest pivot, i.e. the biggest
123       *          diagonal coefficient of U.
124       */
maxPivot()125     RealScalar maxPivot() const { return m_maxpivot; }
126 
127     /** \returns the permutation matrix P
128       *
129       * \sa permutationQ()
130       */
permutationP()131     inline const PermutationPType& permutationP() const
132     {
133       eigen_assert(m_isInitialized && "LU is not initialized.");
134       return m_p;
135     }
136 
137     /** \returns the permutation matrix Q
138       *
139       * \sa permutationP()
140       */
permutationQ()141     inline const PermutationQType& permutationQ() const
142     {
143       eigen_assert(m_isInitialized && "LU is not initialized.");
144       return m_q;
145     }
146 
147     /** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix
148       * will form a basis of the kernel.
149       *
150       * \note If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.
151       *
152       * \note This method has to determine which pivots should be considered nonzero.
153       *       For that, it uses the threshold value that you can control by calling
154       *       setThreshold(const RealScalar&).
155       *
156       * Example: \include FullPivLU_kernel.cpp
157       * Output: \verbinclude FullPivLU_kernel.out
158       *
159       * \sa image()
160       */
kernel()161     inline const internal::kernel_retval<FullPivLU> kernel() const
162     {
163       eigen_assert(m_isInitialized && "LU is not initialized.");
164       return internal::kernel_retval<FullPivLU>(*this);
165     }
166 
167     /** \returns the image of the matrix, also called its column-space. The columns of the returned matrix
168       * will form a basis of the kernel.
169       *
170       * \param originalMatrix the original matrix, of which *this is the LU decomposition.
171       *                       The reason why it is needed to pass it here, is that this allows
172       *                       a large optimization, as otherwise this method would need to reconstruct it
173       *                       from the LU decomposition.
174       *
175       * \note If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.
176       *
177       * \note This method has to determine which pivots should be considered nonzero.
178       *       For that, it uses the threshold value that you can control by calling
179       *       setThreshold(const RealScalar&).
180       *
181       * Example: \include FullPivLU_image.cpp
182       * Output: \verbinclude FullPivLU_image.out
183       *
184       * \sa kernel()
185       */
186     inline const internal::image_retval<FullPivLU>
image(const MatrixType & originalMatrix)187       image(const MatrixType& originalMatrix) const
188     {
189       eigen_assert(m_isInitialized && "LU is not initialized.");
190       return internal::image_retval<FullPivLU>(*this, originalMatrix);
191     }
192 
193     /** \return a solution x to the equation Ax=b, where A is the matrix of which
194       * *this is the LU decomposition.
195       *
196       * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
197       *          the only requirement in order for the equation to make sense is that
198       *          b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
199       *
200       * \returns a solution.
201       *
202       * \note_about_checking_solutions
203       *
204       * \note_about_arbitrary_choice_of_solution
205       * \note_about_using_kernel_to_study_multiple_solutions
206       *
207       * Example: \include FullPivLU_solve.cpp
208       * Output: \verbinclude FullPivLU_solve.out
209       *
210       * \sa TriangularView::solve(), kernel(), inverse()
211       */
212     template<typename Rhs>
213     inline const internal::solve_retval<FullPivLU, Rhs>
solve(const MatrixBase<Rhs> & b)214     solve(const MatrixBase<Rhs>& b) const
215     {
216       eigen_assert(m_isInitialized && "LU is not initialized.");
217       return internal::solve_retval<FullPivLU, Rhs>(*this, b.derived());
218     }
219 
220     /** \returns the determinant of the matrix of which
221       * *this is the LU decomposition. It has only linear complexity
222       * (that is, O(n) where n is the dimension of the square matrix)
223       * as the LU decomposition has already been computed.
224       *
225       * \note This is only for square matrices.
226       *
227       * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
228       *       optimized paths.
229       *
230       * \warning a determinant can be very big or small, so for matrices
231       * of large enough dimension, there is a risk of overflow/underflow.
232       *
233       * \sa MatrixBase::determinant()
234       */
235     typename internal::traits<MatrixType>::Scalar determinant() const;
236 
237     /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
238       * who need to determine when pivots are to be considered nonzero. This is not used for the
239       * LU decomposition itself.
240       *
241       * When it needs to get the threshold value, Eigen calls threshold(). By default, this
242       * uses a formula to automatically determine a reasonable threshold.
243       * Once you have called the present method setThreshold(const RealScalar&),
244       * your value is used instead.
245       *
246       * \param threshold The new value to use as the threshold.
247       *
248       * A pivot will be considered nonzero if its absolute value is strictly greater than
249       *  \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
250       * where maxpivot is the biggest pivot.
251       *
252       * If you want to come back to the default behavior, call setThreshold(Default_t)
253       */
setThreshold(const RealScalar & threshold)254     FullPivLU& setThreshold(const RealScalar& threshold)
255     {
256       m_usePrescribedThreshold = true;
257       m_prescribedThreshold = threshold;
258       return *this;
259     }
260 
261     /** Allows to come back to the default behavior, letting Eigen use its default formula for
262       * determining the threshold.
263       *
264       * You should pass the special object Eigen::Default as parameter here.
265       * \code lu.setThreshold(Eigen::Default); \endcode
266       *
267       * See the documentation of setThreshold(const RealScalar&).
268       */
setThreshold(Default_t)269     FullPivLU& setThreshold(Default_t)
270     {
271       m_usePrescribedThreshold = false;
272       return *this;
273     }
274 
275     /** Returns the threshold that will be used by certain methods such as rank().
276       *
277       * See the documentation of setThreshold(const RealScalar&).
278       */
threshold()279     RealScalar threshold() const
280     {
281       eigen_assert(m_isInitialized || m_usePrescribedThreshold);
282       return m_usePrescribedThreshold ? m_prescribedThreshold
283       // this formula comes from experimenting (see "LU precision tuning" thread on the list)
284       // and turns out to be identical to Higham's formula used already in LDLt.
285                                       : NumTraits<Scalar>::epsilon() * m_lu.diagonalSize();
286     }
287 
288     /** \returns the rank of the matrix of which *this is the LU decomposition.
289       *
290       * \note This method has to determine which pivots should be considered nonzero.
291       *       For that, it uses the threshold value that you can control by calling
292       *       setThreshold(const RealScalar&).
293       */
rank()294     inline Index rank() const
295     {
296       eigen_assert(m_isInitialized && "LU is not initialized.");
297       RealScalar premultiplied_threshold = internal::abs(m_maxpivot) * threshold();
298       Index result = 0;
299       for(Index i = 0; i < m_nonzero_pivots; ++i)
300         result += (internal::abs(m_lu.coeff(i,i)) > premultiplied_threshold);
301       return result;
302     }
303 
304     /** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition.
305       *
306       * \note This method has to determine which pivots should be considered nonzero.
307       *       For that, it uses the threshold value that you can control by calling
308       *       setThreshold(const RealScalar&).
309       */
dimensionOfKernel()310     inline Index dimensionOfKernel() const
311     {
312       eigen_assert(m_isInitialized && "LU is not initialized.");
313       return cols() - rank();
314     }
315 
316     /** \returns true if the matrix of which *this is the LU decomposition represents an injective
317       *          linear map, i.e. has trivial kernel; false otherwise.
318       *
319       * \note This method has to determine which pivots should be considered nonzero.
320       *       For that, it uses the threshold value that you can control by calling
321       *       setThreshold(const RealScalar&).
322       */
isInjective()323     inline bool isInjective() const
324     {
325       eigen_assert(m_isInitialized && "LU is not initialized.");
326       return rank() == cols();
327     }
328 
329     /** \returns true if the matrix of which *this is the LU decomposition represents a surjective
330       *          linear map; false otherwise.
331       *
332       * \note This method has to determine which pivots should be considered nonzero.
333       *       For that, it uses the threshold value that you can control by calling
334       *       setThreshold(const RealScalar&).
335       */
isSurjective()336     inline bool isSurjective() const
337     {
338       eigen_assert(m_isInitialized && "LU is not initialized.");
339       return rank() == rows();
340     }
341 
342     /** \returns true if the matrix of which *this is the LU decomposition is invertible.
343       *
344       * \note This method has to determine which pivots should be considered nonzero.
345       *       For that, it uses the threshold value that you can control by calling
346       *       setThreshold(const RealScalar&).
347       */
isInvertible()348     inline bool isInvertible() const
349     {
350       eigen_assert(m_isInitialized && "LU is not initialized.");
351       return isInjective() && (m_lu.rows() == m_lu.cols());
352     }
353 
354     /** \returns the inverse of the matrix of which *this is the LU decomposition.
355       *
356       * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
357       *       Use isInvertible() to first determine whether this matrix is invertible.
358       *
359       * \sa MatrixBase::inverse()
360       */
inverse()361     inline const internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> inverse() const
362     {
363       eigen_assert(m_isInitialized && "LU is not initialized.");
364       eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!");
365       return internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType>
366                (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
367     }
368 
369     MatrixType reconstructedMatrix() const;
370 
rows()371     inline Index rows() const { return m_lu.rows(); }
cols()372     inline Index cols() const { return m_lu.cols(); }
373 
374   protected:
375     MatrixType m_lu;
376     PermutationPType m_p;
377     PermutationQType m_q;
378     IntColVectorType m_rowsTranspositions;
379     IntRowVectorType m_colsTranspositions;
380     Index m_det_pq, m_nonzero_pivots;
381     RealScalar m_maxpivot, m_prescribedThreshold;
382     bool m_isInitialized, m_usePrescribedThreshold;
383 };
384 
385 template<typename MatrixType>
FullPivLU()386 FullPivLU<MatrixType>::FullPivLU()
387   : m_isInitialized(false), m_usePrescribedThreshold(false)
388 {
389 }
390 
391 template<typename MatrixType>
FullPivLU(Index rows,Index cols)392 FullPivLU<MatrixType>::FullPivLU(Index rows, Index cols)
393   : m_lu(rows, cols),
394     m_p(rows),
395     m_q(cols),
396     m_rowsTranspositions(rows),
397     m_colsTranspositions(cols),
398     m_isInitialized(false),
399     m_usePrescribedThreshold(false)
400 {
401 }
402 
403 template<typename MatrixType>
FullPivLU(const MatrixType & matrix)404 FullPivLU<MatrixType>::FullPivLU(const MatrixType& matrix)
405   : m_lu(matrix.rows(), matrix.cols()),
406     m_p(matrix.rows()),
407     m_q(matrix.cols()),
408     m_rowsTranspositions(matrix.rows()),
409     m_colsTranspositions(matrix.cols()),
410     m_isInitialized(false),
411     m_usePrescribedThreshold(false)
412 {
413   compute(matrix);
414 }
415 
416 template<typename MatrixType>
compute(const MatrixType & matrix)417 FullPivLU<MatrixType>& FullPivLU<MatrixType>::compute(const MatrixType& matrix)
418 {
419   m_isInitialized = true;
420   m_lu = matrix;
421 
422   const Index size = matrix.diagonalSize();
423   const Index rows = matrix.rows();
424   const Index cols = matrix.cols();
425 
426   // will store the transpositions, before we accumulate them at the end.
427   // can't accumulate on-the-fly because that will be done in reverse order for the rows.
428   m_rowsTranspositions.resize(matrix.rows());
429   m_colsTranspositions.resize(matrix.cols());
430   Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i
431 
432   m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
433   m_maxpivot = RealScalar(0);
434 
435   for(Index k = 0; k < size; ++k)
436   {
437     // First, we need to find the pivot.
438 
439     // biggest coefficient in the remaining bottom-right corner (starting at row k, col k)
440     Index row_of_biggest_in_corner, col_of_biggest_in_corner;
441     RealScalar biggest_in_corner;
442     biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k)
443                         .cwiseAbs()
444                         .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
445     row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner,
446     col_of_biggest_in_corner += k; // need to add k to them.
447 
448     if(biggest_in_corner==RealScalar(0))
449     {
450       // before exiting, make sure to initialize the still uninitialized transpositions
451       // in a sane state without destroying what we already have.
452       m_nonzero_pivots = k;
453       for(Index i = k; i < size; ++i)
454       {
455         m_rowsTranspositions.coeffRef(i) = i;
456         m_colsTranspositions.coeffRef(i) = i;
457       }
458       break;
459     }
460 
461     if(biggest_in_corner > m_maxpivot) m_maxpivot = biggest_in_corner;
462 
463     // Now that we've found the pivot, we need to apply the row/col swaps to
464     // bring it to the location (k,k).
465 
466     m_rowsTranspositions.coeffRef(k) = row_of_biggest_in_corner;
467     m_colsTranspositions.coeffRef(k) = col_of_biggest_in_corner;
468     if(k != row_of_biggest_in_corner) {
469       m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
470       ++number_of_transpositions;
471     }
472     if(k != col_of_biggest_in_corner) {
473       m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
474       ++number_of_transpositions;
475     }
476 
477     // Now that the pivot is at the right location, we update the remaining
478     // bottom-right corner by Gaussian elimination.
479 
480     if(k<rows-1)
481       m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k);
482     if(k<size-1)
483       m_lu.block(k+1,k+1,rows-k-1,cols-k-1).noalias() -= m_lu.col(k).tail(rows-k-1) * m_lu.row(k).tail(cols-k-1);
484   }
485 
486   // the main loop is over, we still have to accumulate the transpositions to find the
487   // permutations P and Q
488 
489   m_p.setIdentity(rows);
490   for(Index k = size-1; k >= 0; --k)
491     m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));
492 
493   m_q.setIdentity(cols);
494   for(Index k = 0; k < size; ++k)
495     m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));
496 
497   m_det_pq = (number_of_transpositions%2) ? -1 : 1;
498   return *this;
499 }
500 
501 template<typename MatrixType>
determinant()502 typename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant() const
503 {
504   eigen_assert(m_isInitialized && "LU is not initialized.");
505   eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!");
506   return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
507 }
508 
509 /** \returns the matrix represented by the decomposition,
510  * i.e., it returns the product: P^{-1} L U Q^{-1}.
511  * This function is provided for debug purpose. */
512 template<typename MatrixType>
reconstructedMatrix()513 MatrixType FullPivLU<MatrixType>::reconstructedMatrix() const
514 {
515   eigen_assert(m_isInitialized && "LU is not initialized.");
516   const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols());
517   // LU
518   MatrixType res(m_lu.rows(),m_lu.cols());
519   // FIXME the .toDenseMatrix() should not be needed...
520   res = m_lu.leftCols(smalldim)
521             .template triangularView<UnitLower>().toDenseMatrix()
522       * m_lu.topRows(smalldim)
523             .template triangularView<Upper>().toDenseMatrix();
524 
525   // P^{-1}(LU)
526   res = m_p.inverse() * res;
527 
528   // (P^{-1}LU)Q^{-1}
529   res = res * m_q.inverse();
530 
531   return res;
532 }
533 
534 /********* Implementation of kernel() **************************************************/
535 
536 namespace internal {
537 template<typename _MatrixType>
538 struct kernel_retval<FullPivLU<_MatrixType> >
539   : kernel_retval_base<FullPivLU<_MatrixType> >
540 {
541   EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>)
542 
543   enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
544             MatrixType::MaxColsAtCompileTime,
545             MatrixType::MaxRowsAtCompileTime)
546   };
547 
548   template<typename Dest> void evalTo(Dest& dst) const
549   {
550     const Index cols = dec().matrixLU().cols(), dimker = cols - rank();
551     if(dimker == 0)
552     {
553       // The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's
554       // avoid crashing/asserting as that depends on floating point calculations. Let's
555       // just return a single column vector filled with zeros.
556       dst.setZero();
557       return;
558     }
559 
560     /* Let us use the following lemma:
561       *
562       * Lemma: If the matrix A has the LU decomposition PAQ = LU,
563       * then Ker A = Q(Ker U).
564       *
565       * Proof: trivial: just keep in mind that P, Q, L are invertible.
566       */
567 
568     /* Thus, all we need to do is to compute Ker U, and then apply Q.
569       *
570       * U is upper triangular, with eigenvalues sorted so that any zeros appear at the end.
571       * Thus, the diagonal of U ends with exactly
572       * dimKer zero's. Let us use that to construct dimKer linearly
573       * independent vectors in Ker U.
574       */
575 
576     Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
577     RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
578     Index p = 0;
579     for(Index i = 0; i < dec().nonzeroPivots(); ++i)
580       if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
581         pivots.coeffRef(p++) = i;
582     eigen_internal_assert(p == rank());
583 
584     // we construct a temporaty trapezoid matrix m, by taking the U matrix and
585     // permuting the rows and cols to bring the nonnegligible pivots to the top of
586     // the main diagonal. We need that to be able to apply our triangular solvers.
587     // FIXME when we get triangularView-for-rectangular-matrices, this can be simplified
588     Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options,
589            MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime>
590       m(dec().matrixLU().block(0, 0, rank(), cols));
591     for(Index i = 0; i < rank(); ++i)
592     {
593       if(i) m.row(i).head(i).setZero();
594       m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i);
595     }
596     m.block(0, 0, rank(), rank());
597     m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero();
598     for(Index i = 0; i < rank(); ++i)
599       m.col(i).swap(m.col(pivots.coeff(i)));
600 
601     // ok, we have our trapezoid matrix, we can apply the triangular solver.
602     // notice that the math behind this suggests that we should apply this to the
603     // negative of the RHS, but for performance we just put the negative sign elsewhere, see below.
604     m.topLeftCorner(rank(), rank())
605      .template triangularView<Upper>().solveInPlace(
606         m.topRightCorner(rank(), dimker)
607       );
608 
609     // now we must undo the column permutation that we had applied!
610     for(Index i = rank()-1; i >= 0; --i)
611       m.col(i).swap(m.col(pivots.coeff(i)));
612 
613     // see the negative sign in the next line, that's what we were talking about above.
614     for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker);
615     for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero();
616     for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1);
617   }
618 };
619 
620 /***** Implementation of image() *****************************************************/
621 
622 template<typename _MatrixType>
623 struct image_retval<FullPivLU<_MatrixType> >
624   : image_retval_base<FullPivLU<_MatrixType> >
625 {
626   EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>)
627 
628   enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
629             MatrixType::MaxColsAtCompileTime,
630             MatrixType::MaxRowsAtCompileTime)
631   };
632 
633   template<typename Dest> void evalTo(Dest& dst) const
634   {
635     if(rank() == 0)
636     {
637       // The Image is just {0}, so it doesn't have a basis properly speaking, but let's
638       // avoid crashing/asserting as that depends on floating point calculations. Let's
639       // just return a single column vector filled with zeros.
640       dst.setZero();
641       return;
642     }
643 
644     Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
645     RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
646     Index p = 0;
647     for(Index i = 0; i < dec().nonzeroPivots(); ++i)
648       if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
649         pivots.coeffRef(p++) = i;
650     eigen_internal_assert(p == rank());
651 
652     for(Index i = 0; i < rank(); ++i)
653       dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i)));
654   }
655 };
656 
657 /***** Implementation of solve() *****************************************************/
658 
659 template<typename _MatrixType, typename Rhs>
660 struct solve_retval<FullPivLU<_MatrixType>, Rhs>
661   : solve_retval_base<FullPivLU<_MatrixType>, Rhs>
662 {
663   EIGEN_MAKE_SOLVE_HELPERS(FullPivLU<_MatrixType>,Rhs)
664 
665   template<typename Dest> void evalTo(Dest& dst) const
666   {
667     /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
668      * So we proceed as follows:
669      * Step 1: compute c = P * rhs.
670      * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
671      * Step 3: replace c by the solution x to Ux = c. May or may not exist.
672      * Step 4: result = Q * c;
673      */
674 
675     const Index rows = dec().rows(), cols = dec().cols(),
676               nonzero_pivots = dec().nonzeroPivots();
677     eigen_assert(rhs().rows() == rows);
678     const Index smalldim = (std::min)(rows, cols);
679 
680     if(nonzero_pivots == 0)
681     {
682       dst.setZero();
683       return;
684     }
685 
686     typename Rhs::PlainObject c(rhs().rows(), rhs().cols());
687 
688     // Step 1
689     c = dec().permutationP() * rhs();
690 
691     // Step 2
692     dec().matrixLU()
693         .topLeftCorner(smalldim,smalldim)
694         .template triangularView<UnitLower>()
695         .solveInPlace(c.topRows(smalldim));
696     if(rows>cols)
697     {
698       c.bottomRows(rows-cols)
699         -= dec().matrixLU().bottomRows(rows-cols)
700          * c.topRows(cols);
701     }
702 
703     // Step 3
704     dec().matrixLU()
705         .topLeftCorner(nonzero_pivots, nonzero_pivots)
706         .template triangularView<Upper>()
707         .solveInPlace(c.topRows(nonzero_pivots));
708 
709     // Step 4
710     for(Index i = 0; i < nonzero_pivots; ++i)
711       dst.row(dec().permutationQ().indices().coeff(i)) = c.row(i);
712     for(Index i = nonzero_pivots; i < dec().matrixLU().cols(); ++i)
713       dst.row(dec().permutationQ().indices().coeff(i)).setZero();
714   }
715 };
716 
717 } // end namespace internal
718 
719 /******* MatrixBase methods *****************************************************************/
720 
721 /** \lu_module
722   *
723   * \return the full-pivoting LU decomposition of \c *this.
724   *
725   * \sa class FullPivLU
726   */
727 template<typename Derived>
728 inline const FullPivLU<typename MatrixBase<Derived>::PlainObject>
729 MatrixBase<Derived>::fullPivLu() const
730 {
731   return FullPivLU<PlainObject>(eval());
732 }
733 
734 } // end namespace Eigen
735 
736 #endif // EIGEN_LU_H
737