add ColPivotingHouseholderQR

rename RRQR to fullPivotingHouseholderQR

Eigen/QR

@@ -36,7 +36,8 @@ namespace Eigen {
   */
 
 #include "src/QR/QR.h"
-#include "src/QR/RRQR.h"
+#include "src/QR/FullPivotingHouseholderQR.h"
+#include "src/QR/ColPivotingHouseholderQR.h"
 #include "src/QR/Tridiagonalization.h"
 #include "src/QR/EigenSolver.h"
 #include "src/QR/SelfAdjointEigenSolver.h"

Eigen/src/Core/MatrixBase.h

@@ -745,6 +745,8 @@ template<typename Derived> class MatrixBase
 /////////// QR module ///////////
 
     const HouseholderQR<PlainMatrixType> householderQr() const;
+    const ColPivotingHouseholderQR<PlainMatrixType> colPivotingHouseholderQr() const;
+    const FullPivotingHouseholderQR<PlainMatrixType> fullPivotingHouseholderQr() const;
 
     EigenvaluesReturnType eigenvalues() const;
     RealScalar operatorNorm() const;

Eigen/src/Core/util/ForwardDeclarations.h

@@ -117,6 +117,8 @@ template<typename MatrixType, int Direction = BothDirections> class Reverse;
 template<typename MatrixType> class LU;
 template<typename MatrixType> class PartialLU;
 template<typename MatrixType> class HouseholderQR;
+template<typename MatrixType> class ColPivotingHouseholderQR;
+template<typename MatrixType> class FullPivotingHouseholderQR;
 template<typename MatrixType> class SVD;
 template<typename MatrixType, int UpLo = LowerTriangular> class LLT;
 template<typename MatrixType> class LDLT;

Eigen/src/QR/ColPivotingHouseholderQR.h

@@ -0,0 +1,266 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2008-2009 Gael Guennebaud <g.gael@free.fr>
+// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
+#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
+
+/** \ingroup QR_Module
+  * \nonstableyet
+  *
+  * \class ColPivotingHouseholderQR
+  *
+  * \brief Householder rank-revealing QR decomposition of a matrix
+  *
+  * \param MatrixType the type of the matrix of which we are computing the QR decomposition
+  *
+  * This class performs a rank-revealing QR decomposition using Householder transformations.
+  *
+  * This decomposition performs full-pivoting in order to be rank-revealing and achieve optimal
+  * numerical stability.
+  *
+  * \sa MatrixBase::colPivotingHouseholderQr()
+  */
+template<typename MatrixType> class ColPivotingHouseholderQR
+{
+  public:
+    
+    enum {
+      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+      Options = MatrixType::Options,
+      DiagSizeAtCompileTime = EIGEN_ENUM_MIN(ColsAtCompileTime,RowsAtCompileTime)
+    };
+    
+    typedef typename MatrixType::Scalar Scalar;
+    typedef typename MatrixType::RealScalar RealScalar;
+    typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime> MatrixQType;
+    typedef Matrix<Scalar, DiagSizeAtCompileTime, 1> HCoeffsType;
+    typedef Matrix<int, 1, ColsAtCompileTime> IntRowVectorType;
+    typedef Matrix<int, RowsAtCompileTime, 1> IntColVectorType;
+    typedef Matrix<Scalar, 1, ColsAtCompileTime> RowVectorType;
+    typedef Matrix<Scalar, RowsAtCompileTime, 1> ColVectorType;
+    typedef Matrix<RealScalar, 1, ColsAtCompileTime> RealRowVectorType;
+
+    /**
+    * \brief Default Constructor.
+    *
+    * The default constructor is useful in cases in which the user intends to
+    * perform decompositions via ColPivotingHouseholderQR::compute(const MatrixType&).
+    */
+    ColPivotingHouseholderQR() : m_qr(), m_hCoeffs(), m_isInitialized(false) {}
+
+    ColPivotingHouseholderQR(const MatrixType& matrix)
+      : m_qr(matrix.rows(), matrix.cols()),
+        m_hCoeffs(std::min(matrix.rows(),matrix.cols())),
+        m_isInitialized(false)
+    {
+      compute(matrix);
+    }
+
+    /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
+      * *this is the QR decomposition, if any exists.
+      *
+      * \param b the right-hand-side of the equation to solve.
+      *
+      * \param result a pointer to the vector/matrix in which to store the solution, if any exists.
+      *          Resized if necessary, so that result->rows()==A.cols() and result->cols()==b.cols().
+      *          If no solution exists, *result is left with undefined coefficients.
+      *
+      * \note The case where b is a matrix is not yet implemented. Also, this
+      *       code is space inefficient.
+      *
+      * Example: \include ColPivotingHouseholderQR_solve.cpp
+      * Output: \verbinclude ColPivotingHouseholderQR_solve.out
+      */
+    template<typename OtherDerived, typename ResultType>
+    void solve(const MatrixBase<OtherDerived>& b, ResultType *result) const;
+
+    MatrixType matrixQ(void) const;
+
+    /** \returns a reference to the matrix where the Householder QR decomposition is stored
+      */
+    const MatrixType& matrixQR() const { return m_qr; }
+
+    ColPivotingHouseholderQR& compute(const MatrixType& matrix);
+    
+    const IntRowVectorType& colsPermutation() const
+    {
+      ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized.");
+      return m_cols_permutation;
+    }
+    
+    inline int rank() const
+    {
+      ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized.");
+      return m_rank;
+    }
+
+  protected:
+    MatrixType m_qr;
+    HCoeffsType m_hCoeffs;
+    IntRowVectorType m_cols_permutation;
+    bool m_isInitialized;
+    RealScalar m_precision;
+    int m_rank;
+    int m_det_pq;
+};
+
+#ifndef EIGEN_HIDE_HEAVY_CODE
+
+template<typename MatrixType>
+ColPivotingHouseholderQR<MatrixType>& ColPivotingHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
+{
+  int rows = matrix.rows();
+  int cols = matrix.cols();
+  int size = std::min(rows,cols);
+  m_rank = size;
+  
+  m_qr = matrix;
+  m_hCoeffs.resize(size);
+
+  RowVectorType temp(cols);
+
+  m_precision = epsilon<Scalar>() * size;
+
+  IntRowVectorType cols_transpositions(matrix.cols());
+  m_cols_permutation.resize(matrix.cols());
+  int number_of_transpositions = 0;
+  
+  RealRowVectorType colSqNorms(cols);
+  for(int k = 0; k < cols; ++k)
+    colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();
+  RealScalar biggestColSqNorm = colSqNorms.maxCoeff();
+  
+  for (int k = 0; k < size; ++k)
+  {
+    int biggest_col_in_corner;
+    RealScalar biggestColSqNormInCorner = colSqNorms.end(cols-k).maxCoeff(&biggest_col_in_corner);
+    biggest_col_in_corner += k;
+    
+    // if the corner is negligible, then we have less than full rank, and we can finish early
+    if(ei_isMuchSmallerThan(biggestColSqNormInCorner, biggestColSqNorm, m_precision))
+    {
+      m_rank = k;
+      for(int i = k; i < size; i++)
+      {
+        cols_transpositions.coeffRef(i) = i;
+        m_hCoeffs.coeffRef(i) = Scalar(0);
+      }
+      break;
+    }
+    
+    cols_transpositions.coeffRef(k) = biggest_col_in_corner;
+    if(k != biggest_col_in_corner) {
+      m_qr.col(k).swap(m_qr.col(biggest_col_in_corner));
+      ++number_of_transpositions;
+    }
+
+    RealScalar beta;
+    m_qr.col(k).end(rows-k).makeHouseholderInPlace(&m_hCoeffs.coeffRef(k), &beta);
+    m_qr.coeffRef(k,k) = beta;
+
+    m_qr.corner(BottomRight, rows-k, cols-k-1)
+        .applyHouseholderOnTheLeft(m_qr.col(k).end(rows-k-1), m_hCoeffs.coeffRef(k), &temp.coeffRef(k+1));
+        
+    colSqNorms.end(cols-k-1) -= m_qr.row(k).end(cols-k-1).cwise().abs2();
+  }
+
+  for(int k = 0; k < matrix.cols(); ++k) m_cols_permutation.coeffRef(k) = k;
+  for(int k = 0; k < size; ++k)
+    std::swap(m_cols_permutation.coeffRef(k), m_cols_permutation.coeffRef(cols_transpositions.coeff(k)));
+
+  m_det_pq = (number_of_transpositions%2) ? -1 : 1;
+  m_isInitialized = true;
+  
+  return *this;
+}
+
+template<typename MatrixType>
+template<typename OtherDerived, typename ResultType>
+void ColPivotingHouseholderQR<MatrixType>::solve(
+  const MatrixBase<OtherDerived>& b,
+  ResultType *result
+) const
+{
+  ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized.");
+  const int rows = m_qr.rows();
+  const int cols = b.cols();
+  ei_assert(b.rows() == rows);
+  
+  typename OtherDerived::PlainMatrixType c(b);
+  
+  Matrix<Scalar,1,MatrixType::ColsAtCompileTime> temp(cols);
+  for (int k = 0; k < m_rank; ++k)
+  {
+    int remainingSize = rows-k;
+    c.corner(BottomRight, remainingSize, cols)
+     .applyHouseholderOnTheLeft(m_qr.col(k).end(remainingSize-1), m_hCoeffs.coeff(k), &temp.coeffRef(0));
+  }
+
+  m_qr.corner(TopLeft, m_rank, m_rank)
+      .template triangularView<UpperTriangular>()
+      .solveInPlace(c.corner(TopLeft, m_rank, c.cols()));
+
+  result->resize(m_qr.cols(), b.cols());
+  for(int i = 0; i < m_rank; ++i) result->row(m_cols_permutation.coeff(i)) = c.row(i);
+  for(int i = m_rank; i < m_qr.cols(); ++i) result->row(m_cols_permutation.coeff(i)).setZero();
+}
+
+/** \returns the matrix Q */
+template<typename MatrixType>
+MatrixType ColPivotingHouseholderQR<MatrixType>::matrixQ() const
+{
+  ei_assert(m_isInitialized && "ColPivotingHouseholderQR is not initialized.");
+  // compute the product H'_0 H'_1 ... H'_n-1,
+  // where H_k is the k-th Householder transformation I - h_k v_k v_k'
+  // and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
+  int rows = m_qr.rows();
+  int cols = m_qr.cols();
+  int size = std::min(rows,cols);
+  MatrixType res = MatrixType::Identity(rows, rows);
+  Matrix<Scalar,1,MatrixType::RowsAtCompileTime> temp(rows);
+  for (int k = size-1; k >= 0; k--)
+  {
+    res.block(k, k, rows-k, rows-k)
+       .applyHouseholderOnTheLeft(m_qr.col(k).end(rows-k-1), ei_conj(m_hCoeffs.coeff(k)), &temp.coeffRef(k));
+  }
+  return res;
+}
+
+#endif // EIGEN_HIDE_HEAVY_CODE
+
+/** \return the column-pivoting Householder QR decomposition of \c *this.
+  *
+  * \sa class ColPivotingHouseholderQR
+  */
+template<typename Derived>
+const ColPivotingHouseholderQR<typename MatrixBase<Derived>::PlainMatrixType>
+MatrixBase<Derived>::colPivotingHouseholderQr() const
+{
+  return ColPivotingHouseholderQR<PlainMatrixType>(eval());
+}
+
+
+#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H

Eigen/src/QR/FullPivotingHouseholderQR.h

@@ -23,13 +23,13 @@
 // License and a copy of the GNU General Public License along with
 // Eigen. If not, see <http://www.gnu.org/licenses/>.
 
-#ifndef EIGEN_RRQR_H
-#define EIGEN_RRQR_H
+#ifndef EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
+#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
 
 /** \ingroup QR_Module
   * \nonstableyet
   *
-  * \class HouseholderRRQR
+  * \class FullPivotingHouseholderQR
   *
   * \brief Householder rank-revealing QR decomposition of a matrix
   *
@@ -42,7 +42,7 @@
   *
   * \sa MatrixBase::householderRrqr()
   */
-template<typename MatrixType> class HouseholderRRQR
+template<typename MatrixType> class FullPivotingHouseholderQR
 {
   public:
     
@@ -66,11 +66,11 @@ template<typename MatrixType> class HouseholderRRQR
     * \brief Default Constructor.
     *
     * The default constructor is useful in cases in which the user intends to
-    * perform decompositions via HouseholderRRQR::compute(const MatrixType&).
+    * perform decompositions via FullPivotingHouseholderQR::compute(const MatrixType&).
     */
-    HouseholderRRQR() : m_qr(), m_hCoeffs(), m_isInitialized(false) {}
+    FullPivotingHouseholderQR() : m_qr(), m_hCoeffs(), m_isInitialized(false) {}
 
-    HouseholderRRQR(const MatrixType& matrix)
+    FullPivotingHouseholderQR(const MatrixType& matrix)
       : m_qr(matrix.rows(), matrix.cols()),
         m_hCoeffs(std::min(matrix.rows(),matrix.cols())),
         m_isInitialized(false)
@@ -90,8 +90,8 @@ template<typename MatrixType> class HouseholderRRQR
       * \note The case where b is a matrix is not yet implemented. Also, this
       *       code is space inefficient.
       *
-      * Example: \include HouseholderRRQR_solve.cpp
-      * Output: \verbinclude HouseholderRRQR_solve.out
+      * Example: \include FullPivotingHouseholderQR_solve.cpp
+      * Output: \verbinclude FullPivotingHouseholderQR_solve.out
       */
     template<typename OtherDerived, typename ResultType>
     void solve(const MatrixBase<OtherDerived>& b, ResultType *result) const;
@@ -102,23 +102,23 @@ template<typename MatrixType> class HouseholderRRQR
       */
     const MatrixType& matrixQR() const { return m_qr; }
 
-    HouseholderRRQR& compute(const MatrixType& matrix);
+    FullPivotingHouseholderQR& compute(const MatrixType& matrix);
     
     const IntRowVectorType& colsPermutation() const
     {
-      ei_assert(m_isInitialized && "RRQR is not initialized.");
+      ei_assert(m_isInitialized && "FULLPIVOTINGHOUSEHOLDERQR is not initialized.");
       return m_cols_permutation;
     }
     
     const IntColVectorType& rowsTranspositions() const
     {
-      ei_assert(m_isInitialized && "RRQR is not initialized.");
+      ei_assert(m_isInitialized && "FULLPIVOTINGHOUSEHOLDERQR is not initialized.");
       return m_rows_transpositions;
     }
 
     inline int rank() const
     {
-      ei_assert(m_isInitialized && "RRQR is not initialized.");
+      ei_assert(m_isInitialized && "FULLPIVOTINGHOUSEHOLDERQR is not initialized.");
       return m_rank;
     }
 
@@ -136,7 +136,7 @@ template<typename MatrixType> class HouseholderRRQR
 #ifndef EIGEN_HIDE_HEAVY_CODE
 
 template<typename MatrixType>
-HouseholderRRQR<MatrixType>& HouseholderRRQR<MatrixType>::compute(const MatrixType& matrix)
+FullPivotingHouseholderQR<MatrixType>& FullPivotingHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
 {
   int rows = matrix.rows();
   int cols = matrix.cols();
@@ -148,7 +148,6 @@ HouseholderRRQR<MatrixType>& HouseholderRRQR<MatrixType>::compute(const MatrixTy
 
   RowVectorType temp(cols);
 
-  // TODO: experiment to see the best formula
   m_precision = epsilon<Scalar>() * size;
 
   m_rows_transpositions.resize(matrix.rows());
@@ -198,7 +197,6 @@ HouseholderRRQR<MatrixType>& HouseholderRRQR<MatrixType>::compute(const MatrixTy
     m_qr.col(k).end(rows-k).makeHouseholderInPlace(&m_hCoeffs.coeffRef(k), &beta);
     m_qr.coeffRef(k,k) = beta;
 
-    // apply H to remaining part of m_qr from the left
     m_qr.corner(BottomRight, rows-k, cols-k-1)
         .applyHouseholderOnTheLeft(m_qr.col(k).end(rows-k-1), m_hCoeffs.coeffRef(k), &temp.coeffRef(k+1));
   }
@@ -215,12 +213,12 @@ HouseholderRRQR<MatrixType>& HouseholderRRQR<MatrixType>::compute(const MatrixTy
 
 template<typename MatrixType>
 template<typename OtherDerived, typename ResultType>
-void HouseholderRRQR<MatrixType>::solve(
+void FullPivotingHouseholderQR<MatrixType>::solve(
   const MatrixBase<OtherDerived>& b,
   ResultType *result
 ) const
 {
-  ei_assert(m_isInitialized && "HouseholderRRQR is not initialized.");
+  ei_assert(m_isInitialized && "FullPivotingHouseholderQR is not initialized.");
   const int rows = m_qr.rows();
   const int cols = b.cols();
   ei_assert(b.rows() == rows);
@@ -247,9 +245,9 @@ void HouseholderRRQR<MatrixType>::solve(
 
 /** \returns the matrix Q */
 template<typename MatrixType>
-MatrixType HouseholderRRQR<MatrixType>::matrixQ() const
+MatrixType FullPivotingHouseholderQR<MatrixType>::matrixQ() const
 {
-  ei_assert(m_isInitialized && "HouseholderRRQR is not initialized.");
+  ei_assert(m_isInitialized && "FullPivotingHouseholderQR is not initialized.");
   // compute the product H'_0 H'_1 ... H'_n-1,
   // where H_k is the k-th Householder transformation I - h_k v_k v_k'
   // and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
@@ -269,18 +267,15 @@ MatrixType HouseholderRRQR<MatrixType>::matrixQ() const
 
 #endif // EIGEN_HIDE_HEAVY_CODE
 
-#if 0
-/** \return the Householder QR decomposition of \c *this.
+/** \return the full-pivoting Householder QR decomposition of \c *this.
   *
-  * \sa class HouseholderRRQR
+  * \sa class FullPivotingHouseholderQR
   */
 template<typename Derived>
-const HouseholderRRQR<typename MatrixBase<Derived>::PlainMatrixType>
-MatrixBase<Derived>::householderQr() const
+const FullPivotingHouseholderQR<typename MatrixBase<Derived>::PlainMatrixType>
+MatrixBase<Derived>::fullPivotingHouseholderQr() const
 {
-  return HouseholderRRQR<PlainMatrixType>(eval());
+  return FullPivotingHouseholderQR<PlainMatrixType>(eval());
 }
-#endif
 
-
-#endif // EIGEN_QR_H
+#endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H

test/CMakeLists.txt

@@ -121,7 +121,8 @@ ei_add_test(lu ${EI_OFLAG})
 ei_add_test(determinant)
 ei_add_test(inverse ${EI_OFLAG})
 ei_add_test(qr)
-ei_add_test(rrqr)
+ei_add_test(qr_colpivoting)
+ei_add_test(qr_fullpivoting)
 ei_add_test(eigensolver_selfadjoint " " "${GSL_LIBRARIES}")
 ei_add_test(eigensolver_generic " " "${GSL_LIBRARIES}")
 ei_add_test(svd)

test/qr_colpivoting.cpp

@@ -37,7 +37,7 @@ template<typename MatrixType> void qr()
   typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
   MatrixType m1;
   createRandomMatrixOfRank(rank,rows,cols,m1);
-  HouseholderRRQR<MatrixType> qr(m1);
+  ColPivotingHouseholderQR<MatrixType> qr(m1);
   VERIFY_IS_APPROX(rank, qr.rank());
   
   MatrixType r = qr.matrixQR();
@@ -74,7 +74,7 @@ template<typename MatrixType> void qr_invertible()
     m1 += a * a.adjoint();
   }
 
-  HouseholderRRQR<MatrixType> qr(m1);
+  ColPivotingHouseholderQR<MatrixType> qr(m1);
   m3 = MatrixType::Random(size,size);
   qr.solve(m3, &m2);
   VERIFY_IS_APPROX(m3, m1*m2);
@@ -84,13 +84,13 @@ template<typename MatrixType> void qr_verify_assert()
 {
   MatrixType tmp;
 
-  HouseholderRRQR<MatrixType> qr;
+  ColPivotingHouseholderQR<MatrixType> qr;
   VERIFY_RAISES_ASSERT(qr.matrixR())
   VERIFY_RAISES_ASSERT(qr.solve(tmp,&tmp))
   VERIFY_RAISES_ASSERT(qr.matrixQ())
 }
 
-void test_rrqr()
+void test_qr_colpivoting()
 {
  for(int i = 0; i < 1; i++) {
     // FIXME : very weird bug here

test/qr_fullpivoting.cpp

@@ -0,0 +1,115 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
+// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#include "main.h"
+#include <Eigen/QR>
+
+template<typename MatrixType> void qr()
+{
+  /* this test covers the following files: QR.h */
+  int rows = ei_random<int>(20,200), cols = ei_random<int>(20,200), cols2 = ei_random<int>(20,200);
+  int rank = ei_random<int>(1, std::min(rows, cols)-1);
+
+  typedef typename MatrixType::Scalar Scalar;
+  typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> SquareMatrixType;
+  typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
+  MatrixType m1;
+  createRandomMatrixOfRank(rank,rows,cols,m1);
+  FullPivotingHouseholderQR<MatrixType> qr(m1);
+  VERIFY_IS_APPROX(rank, qr.rank());
+  
+  MatrixType r = qr.matrixQR();
+  // FIXME need better way to construct trapezoid
+  for(int i = 0; i < rows; i++) for(int j = 0; j < cols; j++) if(i>j) r(i,j) = Scalar(0);
+  
+  MatrixType b = qr.matrixQ() * r;
+
+  MatrixType c = MatrixType::Zero(rows,cols);
+  
+  for(int i = 0; i < cols; ++i) c.col(qr.colsPermutation().coeff(i)) = b.col(i);
+  VERIFY_IS_APPROX(m1, c);
+  
+  MatrixType m2 = MatrixType::Random(cols,cols2);
+  MatrixType m3 = m1*m2;
+  m2 = MatrixType::Random(cols,cols2);
+  qr.solve(m3, &m2);
+  VERIFY_IS_APPROX(m3, m1*m2);
+}
+
+template<typename MatrixType> void qr_invertible()
+{
+  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
+  int size = ei_random<int>(10,50);
+
+  MatrixType m1(size, size), m2(size, size), m3(size, size);
+  m1 = MatrixType::Random(size,size);
+
+  if (ei_is_same_type<RealScalar,float>::ret)
+  {
+    // let's build a matrix more stable to inverse
+    MatrixType a = MatrixType::Random(size,size*2);
+    m1 += a * a.adjoint();
+  }
+
+  FullPivotingHouseholderQR<MatrixType> qr(m1);
+  m3 = MatrixType::Random(size,size);
+  qr.solve(m3, &m2);
+  VERIFY_IS_APPROX(m3, m1*m2);
+}
+
+template<typename MatrixType> void qr_verify_assert()
+{
+  MatrixType tmp;
+
+  FullPivotingHouseholderQR<MatrixType> qr;
+  VERIFY_RAISES_ASSERT(qr.matrixR())
+  VERIFY_RAISES_ASSERT(qr.solve(tmp,&tmp))
+  VERIFY_RAISES_ASSERT(qr.matrixQ())
+}
+
+void test_qr_fullpivoting()
+{
+ for(int i = 0; i < 1; i++) {
+    // FIXME : very weird bug here
+//     CALL_SUBTEST( qr(Matrix2f()) );
+    CALL_SUBTEST( qr<MatrixXf>() );
+    CALL_SUBTEST( qr<MatrixXd>() );
+    CALL_SUBTEST( qr<MatrixXcd>() );
+  }
+
+  for(int i = 0; i < g_repeat; i++) {
+    CALL_SUBTEST( qr_invertible<MatrixXf>() );
+    CALL_SUBTEST( qr_invertible<MatrixXd>() );
+    CALL_SUBTEST( qr_invertible<MatrixXcf>() );
+    CALL_SUBTEST( qr_invertible<MatrixXcd>() );
+  }
+
+  CALL_SUBTEST(qr_verify_assert<Matrix3f>());
+  CALL_SUBTEST(qr_verify_assert<Matrix3d>());
+  CALL_SUBTEST(qr_verify_assert<MatrixXf>());
+  CALL_SUBTEST(qr_verify_assert<MatrixXd>());
+  CALL_SUBTEST(qr_verify_assert<MatrixXcf>());
+  CALL_SUBTEST(qr_verify_assert<MatrixXcd>());
+}