FullPivLU.h

00001 // This file is part of Eigen, a lightweight C++ template library
00002 // for linear algebra.
00003 //
00004 // Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
00005 //
00006 // This Source Code Form is subject to the terms of the Mozilla
00007 // Public License v. 2.0. If a copy of the MPL was not distributed
00008 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
00009 
00010 #ifndef EIGEN_LU_H
00011 #define EIGEN_LU_H
00012 
00013 namespace Eigen { 
00014 
00015 /** \ingroup LU_Module
00016   *
00017   * \class FullPivLU
00018   *
00019   * \brief LU decomposition of a matrix with complete pivoting, and related features
00020   *
00021   * \param MatrixType the type of the matrix of which we are computing the LU decomposition
00022   *
00023   * This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A is
00024   * decomposed as \f$ A = P^{-1} L U Q^{-1} \f$ where L is unit-lower-triangular, U is
00025   * upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU
00026   * decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any
00027   * zeros are at the end.
00028   *
00029   * This decomposition provides the generic approach to solving systems of linear equations, computing
00030   * the rank, invertibility, inverse, kernel, and determinant.
00031   *
00032   * This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD
00033   * decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix,
00034   * working with the SVD allows to select the smallest singular values of the matrix, something that
00035   * the LU decomposition doesn't see.
00036   *
00037   * The data of the LU decomposition can be directly accessed through the methods matrixLU(),
00038   * permutationP(), permutationQ().
00039   *
00040   * As an exemple, here is how the original matrix can be retrieved:
00041   * \include class_FullPivLU.cpp
00042   * Output: \verbinclude class_FullPivLU.out
00043   *
00044   * \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse()
00045   */
00046 template<typename _MatrixType> class FullPivLU
00047 {
00048   public:
00049     typedef _MatrixType MatrixType;
00050     enum {
00051       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00052       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00053       Options = MatrixType::Options,
00054       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00055       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
00056     };
00057     typedef typename MatrixType::Scalar Scalar;
00058     typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
00059     typedef typename internal::traits<MatrixType>::StorageKind StorageKind;
00060     typedef typename MatrixType::Index Index;
00061     typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
00062     typedef typename internal::plain_col_type<MatrixType, Index>::type IntColVectorType;
00063     typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime>  PermutationQType ;
00064     typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime>  PermutationPType ;
00065 
00066     /**
00067       * \brief Default Constructor.
00068       *
00069       * The default constructor is useful in cases in which the user intends to
00070       * perform decompositions via LU::compute(const MatrixType&).
00071       */
00072     FullPivLU();
00073 
00074     /** \brief Default Constructor with memory preallocation
00075       *
00076       * Like the default constructor but with preallocation of the internal data
00077       * according to the specified problem \a size.
00078       * \sa FullPivLU()
00079       */
00080     FullPivLU(Index rows, Index cols);
00081 
00082     /** Constructor.
00083       *
00084       * \param matrix the matrix of which to compute the LU decomposition.
00085       *               It is required to be nonzero.
00086       */
00087     FullPivLU(const MatrixType& matrix);
00088 
00089     /** Computes the LU decomposition of the given matrix.
00090       *
00091       * \param matrix the matrix of which to compute the LU decomposition.
00092       *               It is required to be nonzero.
00093       *
00094       * \returns a reference to *this
00095       */
00096     FullPivLU& compute(const MatrixType& matrix);
00097 
00098     /** \returns the LU decomposition matrix: the upper-triangular part is U, the
00099       * unit-lower-triangular part is L (at least for square matrices; in the non-square
00100       * case, special care is needed, see the documentation of class FullPivLU).
00101       *
00102       * \sa matrixL(), matrixU()
00103       */
00104     inline const MatrixType& matrixLU () const
00105     {
00106       eigen_assert(m_isInitialized && "LU is not initialized.");
00107       return m_lu;
00108     }
00109 
00110     /** \returns the number of nonzero pivots in the LU decomposition.
00111       * Here nonzero is meant in the exact sense, not in a fuzzy sense.
00112       * So that notion isn't really intrinsically interesting, but it is
00113       * still useful when implementing algorithms.
00114       *
00115       * \sa rank()
00116       */
00117     inline Index nonzeroPivots () const
00118     {
00119       eigen_assert(m_isInitialized && "LU is not initialized.");
00120       return m_nonzero_pivots;
00121     }
00122 
00123     /** \returns the absolute value of the biggest pivot, i.e. the biggest
00124       *          diagonal coefficient of U.
00125       */
00126     RealScalar maxPivot () const { return m_maxpivot; }
00127 
00128     /** \returns the permutation matrix P
00129       *
00130       * \sa permutationQ()
00131       */
00132     inline const PermutationPType & permutationP () const
00133     {
00134       eigen_assert(m_isInitialized && "LU is not initialized.");
00135       return m_p;
00136     }
00137 
00138     /** \returns the permutation matrix Q
00139       *
00140       * \sa permutationP()
00141       */
00142     inline const PermutationQType & permutationQ () const
00143     {
00144       eigen_assert(m_isInitialized && "LU is not initialized.");
00145       return m_q;
00146     }
00147 
00148     /** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix
00149       * will form a basis of the kernel.
00150       *
00151       * \note If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.
00152       *
00153       * \note This method has to determine which pivots should be considered nonzero.
00154       *       For that, it uses the threshold value that you can control by calling
00155       *       setThreshold(const RealScalar&).
00156       *
00157       * Example: \include FullPivLU_kernel.cpp
00158       * Output: \verbinclude FullPivLU_kernel.out
00159       *
00160       * \sa image()
00161       */
00162     inline const internal::kernel_retval<FullPivLU> kernel () const
00163     {
00164       eigen_assert(m_isInitialized && "LU is not initialized.");
00165       return internal::kernel_retval<FullPivLU>(*this);
00166     }
00167 
00168     /** \returns the image of the matrix, also called its column-space. The columns of the returned matrix
00169       * will form a basis of the kernel.
00170       *
00171       * \param originalMatrix the original matrix, of which *this is the LU decomposition.
00172       *                       The reason why it is needed to pass it here, is that this allows
00173       *                       a large optimization, as otherwise this method would need to reconstruct it
00174       *                       from the LU decomposition.
00175       *
00176       * \note If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.
00177       *
00178       * \note This method has to determine which pivots should be considered nonzero.
00179       *       For that, it uses the threshold value that you can control by calling
00180       *       setThreshold(const RealScalar&).
00181       *
00182       * Example: \include FullPivLU_image.cpp
00183       * Output: \verbinclude FullPivLU_image.out
00184       *
00185       * \sa kernel()
00186       */
00187     inline const internal::image_retval<FullPivLU>
00188       image (const MatrixType& originalMatrix) const
00189     {
00190       eigen_assert(m_isInitialized && "LU is not initialized.");
00191       return internal::image_retval<FullPivLU>(*this, originalMatrix);
00192     }
00193 
00194     /** \return a solution x to the equation Ax=b, where A is the matrix of which
00195       * *this is the LU decomposition.
00196       *
00197       * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
00198       *          the only requirement in order for the equation to make sense is that
00199       *          b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
00200       *
00201       * \returns a solution.
00202       *
00203       * \note_about_checking_solutions
00204       *
00205       * \note_about_arbitrary_choice_of_solution
00206       * \note_about_using_kernel_to_study_multiple_solutions
00207       *
00208       * Example: \include FullPivLU_solve.cpp
00209       * Output: \verbinclude FullPivLU_solve.out
00210       *
00211       * \sa TriangularView::solve(), kernel(), inverse()
00212       */
00213     template<typename Rhs>
00214     inline const internal::solve_retval<FullPivLU, Rhs>
00215     solve (const MatrixBase<Rhs>& b) const
00216     {
00217       eigen_assert(m_isInitialized && "LU is not initialized.");
00218       return internal::solve_retval<FullPivLU, Rhs>(*this, b.derived());
00219     }
00220 
00221     /** \returns the determinant of the matrix of which
00222       * *this is the LU decomposition. It has only linear complexity
00223       * (that is, O(n) where n is the dimension of the square matrix)
00224       * as the LU decomposition has already been computed.
00225       *
00226       * \note This is only for square matrices.
00227       *
00228       * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
00229       *       optimized paths.
00230       *
00231       * \warning a determinant can be very big or small, so for matrices
00232       * of large enough dimension, there is a risk of overflow/underflow.
00233       *
00234       * \sa MatrixBase::determinant()
00235       */
00236     typename internal::traits<MatrixType>::Scalar determinant () const;
00237 
00238     /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
00239       * who need to determine when pivots are to be considered nonzero. This is not used for the
00240       * LU decomposition itself.
00241       *
00242       * When it needs to get the threshold value, Eigen calls threshold(). By default, this
00243       * uses a formula to automatically determine a reasonable threshold.
00244       * Once you have called the present method setThreshold(const RealScalar&),
00245       * your value is used instead.
00246       *
00247       * \param threshold The new value to use as the threshold.
00248       *
00249       * A pivot will be considered nonzero if its absolute value is strictly greater than
00250       *  \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
00251       * where maxpivot is the biggest pivot.
00252       *
00253       * If you want to come back to the default behavior, call setThreshold(Default_t)
00254       */
00255     FullPivLU& setThreshold(const RealScalar& threshold)
00256     {
00257       m_usePrescribedThreshold = true;
00258       m_prescribedThreshold = threshold;
00259       return *this;
00260     }
00261 
00262     /** Allows to come back to the default behavior, letting Eigen use its default formula for
00263       * determining the threshold.
00264       *
00265       * You should pass the special object Eigen::Default as parameter here.
00266       * \code lu.setThreshold(Eigen::Default); \endcode
00267       *
00268       * See the documentation of setThreshold(const RealScalar&).
00269       */
00270     FullPivLU& setThreshold(Default_t)
00271     {
00272       m_usePrescribedThreshold = false;
00273       return *this;
00274     }
00275 
00276     /** Returns the threshold that will be used by certain methods such as rank().
00277       *
00278       * See the documentation of setThreshold(const RealScalar&).
00279       */
00280     RealScalar threshold() const
00281     {
00282       eigen_assert(m_isInitialized || m_usePrescribedThreshold);
00283       return m_usePrescribedThreshold ? m_prescribedThreshold
00284       // this formula comes from experimenting (see "LU precision tuning" thread on the list)
00285       // and turns out to be identical to Higham's formula used already in LDLt.
00286                                       : NumTraits<Scalar>::epsilon() * m_lu.diagonalSize();
00287     }
00288 
00289     /** \returns the rank of the matrix of which *this is the LU decomposition.
00290       *
00291       * \note This method has to determine which pivots should be considered nonzero.
00292       *       For that, it uses the threshold value that you can control by calling
00293       *       setThreshold(const RealScalar&).
00294       */
00295     inline Index rank () const
00296     {
00297       using std::abs;
00298       eigen_assert(m_isInitialized && "LU is not initialized.");
00299       RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
00300       Index result = 0;
00301       for(Index i = 0; i < m_nonzero_pivots; ++i)
00302         result += (abs(m_lu.coeff(i,i)) > premultiplied_threshold);
00303       return result;
00304     }
00305 
00306     /** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition.
00307       *
00308       * \note This method has to determine which pivots should be considered nonzero.
00309       *       For that, it uses the threshold value that you can control by calling
00310       *       setThreshold(const RealScalar&).
00311       */
00312     inline Index dimensionOfKernel () const
00313     {
00314       eigen_assert(m_isInitialized && "LU is not initialized.");
00315       return cols() - rank ();
00316     }
00317 
00318     /** \returns true if the matrix of which *this is the LU decomposition represents an injective
00319       *          linear map, i.e. has trivial kernel; false otherwise.
00320       *
00321       * \note This method has to determine which pivots should be considered nonzero.
00322       *       For that, it uses the threshold value that you can control by calling
00323       *       setThreshold(const RealScalar&).
00324       */
00325     inline bool isInjective () const
00326     {
00327       eigen_assert(m_isInitialized && "LU is not initialized.");
00328       return rank () == cols();
00329     }
00330 
00331     /** \returns true if the matrix of which *this is the LU decomposition represents a surjective
00332       *          linear map; false otherwise.
00333       *
00334       * \note This method has to determine which pivots should be considered nonzero.
00335       *       For that, it uses the threshold value that you can control by calling
00336       *       setThreshold(const RealScalar&).
00337       */
00338     inline bool isSurjective () const
00339     {
00340       eigen_assert(m_isInitialized && "LU is not initialized.");
00341       return rank () == rows();
00342     }
00343 
00344     /** \returns true if the matrix of which *this is the LU decomposition is invertible.
00345       *
00346       * \note This method has to determine which pivots should be considered nonzero.
00347       *       For that, it uses the threshold value that you can control by calling
00348       *       setThreshold(const RealScalar&).
00349       */
00350     inline bool isInvertible () const
00351     {
00352       eigen_assert(m_isInitialized && "LU is not initialized.");
00353       return isInjective () && (m_lu.rows() == m_lu.cols());
00354     }
00355 
00356     /** \returns the inverse of the matrix of which *this is the LU decomposition.
00357       *
00358       * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
00359       *       Use isInvertible() to first determine whether this matrix is invertible.
00360       *
00361       * \sa MatrixBase::inverse()
00362       */
00363     inline const internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> inverse () const
00364     {
00365       eigen_assert(m_isInitialized && "LU is not initialized.");
00366       eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!");
00367       return internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType>
00368                (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
00369     }
00370 
00371     MatrixType reconstructedMatrix () const;
00372 
00373     inline Index rows() const { return m_lu.rows(); }
00374     inline Index cols() const { return m_lu.cols(); }
00375 
00376   protected:
00377     
00378     static void check_template_parameters()
00379     {
00380       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
00381     }
00382     
00383     MatrixType m_lu;
00384     PermutationPType m_p;
00385     PermutationQType m_q;
00386     IntColVectorType m_rowsTranspositions;
00387     IntRowVectorType m_colsTranspositions;
00388     Index m_det_pq, m_nonzero_pivots;
00389     RealScalar m_maxpivot, m_prescribedThreshold;
00390     bool m_isInitialized, m_usePrescribedThreshold;
00391 };
00392 
00393 template<typename MatrixType>
00394 FullPivLU<MatrixType>::FullPivLU()
00395   : m_isInitialized(false), m_usePrescribedThreshold(false)
00396 {
00397 }
00398 
00399 template<typename MatrixType>
00400 FullPivLU<MatrixType>::FullPivLU(Index rows, Index cols)
00401   : m_lu(rows, cols),
00402     m_p(rows),
00403     m_q(cols),
00404     m_rowsTranspositions(rows),
00405     m_colsTranspositions(cols),
00406     m_isInitialized(false),
00407     m_usePrescribedThreshold(false)
00408 {
00409 }
00410 
00411 template<typename MatrixType>
00412 FullPivLU<MatrixType>::FullPivLU(const MatrixType& matrix)
00413   : m_lu(matrix.rows(), matrix.cols()),
00414     m_p(matrix.rows()),
00415     m_q(matrix.cols()),
00416     m_rowsTranspositions(matrix.rows()),
00417     m_colsTranspositions(matrix.cols()),
00418     m_isInitialized(false),
00419     m_usePrescribedThreshold(false)
00420 {
00421   compute(matrix);
00422 }
00423 
00424 template<typename MatrixType>
00425 FullPivLU<MatrixType>& FullPivLU<MatrixType>::compute(const MatrixType& matrix)
00426 {
00427   check_template_parameters();
00428   
00429   // the permutations are stored as int indices, so just to be sure:
00430   eigen_assert(matrix.rows()<=NumTraits<int>::highest() && matrix.cols()<=NumTraits<int>::highest());
00431   
00432   m_isInitialized = true;
00433   m_lu = matrix;
00434 
00435   const Index size = matrix.diagonalSize();
00436   const Index rows = matrix.rows();
00437   const Index cols = matrix.cols();
00438 
00439   // will store the transpositions, before we accumulate them at the end.
00440   // can't accumulate on-the-fly because that will be done in reverse order for the rows.
00441   m_rowsTranspositions.resize(matrix.rows());
00442   m_colsTranspositions.resize(matrix.cols());
00443   Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i
00444 
00445   m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
00446   m_maxpivot = RealScalar(0);
00447 
00448   for(Index k = 0; k < size; ++k)
00449   {
00450     // First, we need to find the pivot.
00451 
00452     // biggest coefficient in the remaining bottom-right corner (starting at row k, col k)
00453     Index row_of_biggest_in_corner, col_of_biggest_in_corner;
00454     RealScalar biggest_in_corner;
00455     biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k)
00456                         .cwiseAbs()
00457                         .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
00458     row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner,
00459     col_of_biggest_in_corner += k; // need to add k to them.
00460 
00461     if(biggest_in_corner==RealScalar(0))
00462     {
00463       // before exiting, make sure to initialize the still uninitialized transpositions
00464       // in a sane state without destroying what we already have.
00465       m_nonzero_pivots = k;
00466       for(Index i = k; i < size; ++i)
00467       {
00468         m_rowsTranspositions.coeffRef(i) = i;
00469         m_colsTranspositions.coeffRef(i) = i;
00470       }
00471       break;
00472     }
00473 
00474     if(biggest_in_corner > m_maxpivot) m_maxpivot = biggest_in_corner;
00475 
00476     // Now that we've found the pivot, we need to apply the row/col swaps to
00477     // bring it to the location (k,k).
00478 
00479     m_rowsTranspositions.coeffRef(k) = row_of_biggest_in_corner;
00480     m_colsTranspositions.coeffRef(k) = col_of_biggest_in_corner;
00481     if(k != row_of_biggest_in_corner) {
00482       m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
00483       ++number_of_transpositions;
00484     }
00485     if(k != col_of_biggest_in_corner) {
00486       m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
00487       ++number_of_transpositions;
00488     }
00489 
00490     // Now that the pivot is at the right location, we update the remaining
00491     // bottom-right corner by Gaussian elimination.
00492 
00493     if(k<rows-1)
00494       m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k);
00495     if(k<size-1)
00496       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);
00497   }
00498 
00499   // the main loop is over, we still have to accumulate the transpositions to find the
00500   // permutations P and Q
00501 
00502   m_p.setIdentity(rows);
00503   for(Index k = size-1; k >= 0; --k)
00504     m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));
00505 
00506   m_q.setIdentity(cols);
00507   for(Index k = 0; k < size; ++k)
00508     m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));
00509 
00510   m_det_pq = (number_of_transpositions%2) ? -1 : 1;
00511   return *this;
00512 }
00513 
00514 template<typename MatrixType>
00515 typename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant () const
00516 {
00517   eigen_assert(m_isInitialized && "LU is not initialized.");
00518   eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!");
00519   return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
00520 }
00521 
00522 /** \returns the matrix represented by the decomposition,
00523  * i.e., it returns the product: \f$ P^{-1} L U Q^{-1} \f$.
00524  * This function is provided for debug purposes. */
00525 template<typename MatrixType>
00526 MatrixType FullPivLU<MatrixType>::reconstructedMatrix () const
00527 {
00528   eigen_assert(m_isInitialized && "LU is not initialized.");
00529   const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols());
00530   // LU
00531   MatrixType res(m_lu.rows(),m_lu.cols());
00532   // FIXME the .toDenseMatrix() should not be needed...
00533   res = m_lu.leftCols(smalldim)
00534             .template triangularView<UnitLower>().toDenseMatrix()
00535       * m_lu.topRows(smalldim)
00536             .template triangularView<Upper>().toDenseMatrix();
00537 
00538   // P^{-1}(LU)
00539   res = m_p.inverse() * res;
00540 
00541   // (P^{-1}LU)Q^{-1}
00542   res = res * m_q.inverse();
00543 
00544   return res;
00545 }
00546 
00547 /********* Implementation of kernel() **************************************************/
00548 
00549 namespace internal {
00550 template<typename _MatrixType>
00551 struct kernel_retval<FullPivLU<_MatrixType> >
00552   : kernel_retval_base<FullPivLU<_MatrixType> >
00553 {
00554   EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>)
00555 
00556   enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
00557             MatrixType::MaxColsAtCompileTime,
00558             MatrixType::MaxRowsAtCompileTime)
00559   };
00560 
00561   template<typename Dest> void evalTo(Dest& dst) const
00562   {
00563     using std::abs;
00564     const Index cols = dec().matrixLU ().cols(), dimker = cols - rank();
00565     if(dimker == 0)
00566     {
00567       // The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's
00568       // avoid crashing/asserting as that depends on floating point calculations. Let's
00569       // just return a single column vector filled with zeros.
00570       dst.setZero();
00571       return;
00572     }
00573 
00574     /* Let us use the following lemma:
00575       *
00576       * Lemma: If the matrix A has the LU decomposition PAQ = LU,
00577       * then Ker A = Q(Ker U).
00578       *
00579       * Proof: trivial: just keep in mind that P, Q, L are invertible.
00580       */
00581 
00582     /* Thus, all we need to do is to compute Ker U, and then apply Q.
00583       *
00584       * U is upper triangular, with eigenvalues sorted so that any zeros appear at the end.
00585       * Thus, the diagonal of U ends with exactly
00586       * dimKer zero's. Let us use that to construct dimKer linearly
00587       * independent vectors in Ker U.
00588       */
00589 
00590     Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
00591     RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
00592     Index p = 0;
00593     for(Index i = 0; i < dec().nonzeroPivots(); ++i)
00594       if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
00595         pivots.coeffRef(p++) = i;
00596     eigen_internal_assert(p == rank());
00597 
00598     // we construct a temporaty trapezoid matrix m, by taking the U matrix and
00599     // permuting the rows and cols to bring the nonnegligible pivots to the top of
00600     // the main diagonal. We need that to be able to apply our triangular solvers.
00601     // FIXME when we get triangularView-for-rectangular-matrices, this can be simplified
00602     Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options,
00603            MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime>
00604       m(dec().matrixLU().block(0, 0, rank(), cols));
00605     for(Index i = 0; i < rank(); ++i)
00606     {
00607       if(i) m.row(i).head(i).setZero();
00608       m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i);
00609     }
00610     m.block(0, 0, rank(), rank());
00611     m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero();
00612     for(Index i = 0; i < rank(); ++i)
00613       m.col(i).swap(m.col(pivots.coeff(i)));
00614 
00615     // ok, we have our trapezoid matrix, we can apply the triangular solver.
00616     // notice that the math behind this suggests that we should apply this to the
00617     // negative of the RHS, but for performance we just put the negative sign elsewhere, see below.
00618     m.topLeftCorner(rank(), rank())
00619      .template triangularView<Upper>().solveInPlace(
00620         m.topRightCorner(rank(), dimker)
00621       );
00622 
00623     // now we must undo the column permutation that we had applied!
00624     for(Index i = rank()-1; i >= 0; --i)
00625       m.col(i).swap(m.col(pivots.coeff(i)));
00626 
00627     // see the negative sign in the next line, that's what we were talking about above.
00628     for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker);
00629     for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero();
00630     for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1);
00631   }
00632 };
00633 
00634 /***** Implementation of image() *****************************************************/
00635 
00636 template<typename _MatrixType>
00637 struct image_retval<FullPivLU<_MatrixType> >
00638   : image_retval_base<FullPivLU<_MatrixType> >
00639 {
00640   EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>)
00641 
00642   enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
00643             MatrixType::MaxColsAtCompileTime,
00644             MatrixType::MaxRowsAtCompileTime)
00645   };
00646 
00647   template<typename Dest> void evalTo(Dest& dst) const
00648   {
00649     using std::abs;
00650     if(rank() == 0)
00651     {
00652       // The Image is just {0}, so it doesn't have a basis properly speaking, but let's
00653       // avoid crashing/asserting as that depends on floating point calculations. Let's
00654       // just return a single column vector filled with zeros.
00655       dst.setZero();
00656       return;
00657     }
00658 
00659     Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
00660     RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
00661     Index p = 0;
00662     for(Index i = 0; i < dec().nonzeroPivots(); ++i)
00663       if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
00664         pivots.coeffRef(p++) = i;
00665     eigen_internal_assert(p == rank());
00666 
00667     for(Index i = 0; i < rank(); ++i)
00668       dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i)));
00669   }
00670 };
00671 
00672 /***** Implementation of solve() *****************************************************/
00673 
00674 template<typename _MatrixType, typename Rhs>
00675 struct solve_retval<FullPivLU<_MatrixType>, Rhs>
00676   : solve_retval_base<FullPivLU<_MatrixType>, Rhs>
00677 {
00678   EIGEN_MAKE_SOLVE_HELPERS(FullPivLU<_MatrixType>,Rhs)
00679 
00680   template<typename Dest> void evalTo(Dest& dst) const
00681   {
00682     /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
00683      * So we proceed as follows:
00684      * Step 1: compute c = P * rhs.
00685      * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
00686      * Step 3: replace c by the solution x to Ux = c. May or may not exist.
00687      * Step 4: result = Q * c;
00688      */
00689 
00690     const Index rows = dec().rows(), cols = dec().cols(),
00691               nonzero_pivots = dec().rank();
00692     eigen_assert(rhs().rows() == rows);
00693     const Index smalldim = (std::min)(rows, cols);
00694 
00695     if(nonzero_pivots == 0)
00696     {
00697       dst.setZero();
00698       return;
00699     }
00700 
00701     typename Rhs::PlainObject c(rhs().rows(), rhs().cols());
00702 
00703     // Step 1
00704     c = dec().permutationP() * rhs();
00705 
00706     // Step 2
00707     dec().matrixLU()
00708         .topLeftCorner(smalldim,smalldim)
00709         .template triangularView<UnitLower>()
00710         .solveInPlace(c.topRows(smalldim));
00711     if(rows>cols)
00712     {
00713       c.bottomRows(rows-cols)
00714         -= dec().matrixLU().bottomRows(rows-cols)
00715          * c.topRows(cols);
00716     }
00717 
00718     // Step 3
00719     dec().matrixLU()
00720         .topLeftCorner(nonzero_pivots, nonzero_pivots)
00721         .template triangularView<Upper>()
00722         .solveInPlace(c.topRows(nonzero_pivots));
00723 
00724     // Step 4
00725     for(Index i = 0; i < nonzero_pivots; ++i)
00726       dst.row(dec().permutationQ().indices().coeff(i)) = c.row(i);
00727     for(Index i = nonzero_pivots; i < dec().matrixLU().cols(); ++i)
00728       dst.row(dec().permutationQ().indices().coeff(i)).setZero();
00729   }
00730 };
00731 
00732 } // end namespace internal
00733 
00734 /******* MatrixBase methods *****************************************************************/
00735 
00736 /** \lu_module
00737   *
00738   * \return the full-pivoting LU decomposition of \c *this.
00739   *
00740   * \sa class FullPivLU
00741   */
00742 template<typename Derived>
00743 inline const FullPivLU<typename MatrixBase<Derived>::PlainObject>
00744 MatrixBase<Derived>::fullPivLu () const
00745 {
00746   return FullPivLU<PlainObject>(eval());
00747 }
00748 
00749 } // end namespace Eigen
00750 
00751 #endif // EIGEN_LU_H