add initial, rough, full-pivoting RRQR decomposition

lots of room for improvement!
and add Gael a (c) line in Householder.h

Eigen/QR

@@ -36,6 +36,7 @@ namespace Eigen {
   */
 
 #include "src/QR/QR.h"
+#include "src/QR/RRQR.h"
 #include "src/QR/Tridiagonalization.h"
 #include "src/QR/EigenSolver.h"
 #include "src/QR/SelfAdjointEigenSolver.h"

Eigen/src/Householder/Householder.h

@@ -2,6 +2,7 @@
 // for linear algebra.
 //
 // Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
+// Copyright (C) 2009 Gael Guennebaud <g.gael@free.fr>
 //
 // Eigen is free software; you can redistribute it and/or
 // modify it under the terms of the GNU Lesser General Public

Eigen/src/QR/QR.h

@@ -1,7 +1,7 @@
 // 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) 2008-2009 Gael Guennebaud <g.gael@free.fr>
 //
 // Eigen is free software; you can redistribute it and/or
 // modify it under the terms of the GNU Lesser General Public
@@ -39,7 +39,7 @@
   *
   * Note that no pivoting is performed. This is \b not a rank-revealing decomposition.
   *
-  * \sa MatrixBase::qr()
+  * \sa MatrixBase::householderQr()
   */
 template<typename MatrixType> class HouseholderQR
 {
@@ -54,6 +54,7 @@ template<typename MatrixType> class HouseholderQR
     typedef Block<MatrixType, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> MatrixRBlockType;
     typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> MatrixTypeR;
     typedef Matrix<Scalar, MinSizeAtCompileTime, 1> HCoeffsType;
+    typedef Matrix<Scalar, 1, MatrixType::ColsAtCompileTime> RowVectorType;
 
     /**
     * \brief Default Constructor.
@@ -125,12 +126,12 @@ HouseholderQR<MatrixType>& HouseholderQR<MatrixType>::compute(const MatrixType&
   m_qr = matrix;
   m_hCoeffs.resize(size);
 
-  Matrix<Scalar,1,MatrixType::ColsAtCompileTime> temp(cols);
+  RowVectorType temp(cols);
 
   for (int k = 0; k < size; ++k)
   {
     int remainingRows = rows - k;
-    int remainingCols = cols - k -1;
+    int remainingCols = cols - k - 1;
 
     RealScalar beta;
     m_qr.col(k).end(remainingRows).makeHouseholderInPlace(&m_hCoeffs.coeffRef(k), &beta);

Eigen/src/QR/RRQR.h

@@ -0,0 +1,286 @@
+// 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_RRQR_H
+#define EIGEN_RRQR_H
+
+/** \ingroup QR_Module
+  * \nonstableyet
+  *
+  * \class HouseholderRRQR
+  *
+  * \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::householderRrqr()
+  */
+template<typename MatrixType> class HouseholderRRQR
+{
+  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;
+
+    /**
+    * \brief Default Constructor.
+    *
+    * The default constructor is useful in cases in which the user intends to
+    * perform decompositions via HouseholderRRQR::compute(const MatrixType&).
+    */
+    HouseholderRRQR() : m_qr(), m_hCoeffs(), m_isInitialized(false) {}
+
+    HouseholderRRQR(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 HouseholderRRQR_solve.cpp
+      * Output: \verbinclude HouseholderRRQR_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; }
+
+    HouseholderRRQR& compute(const MatrixType& matrix);
+    
+    const IntRowVectorType& colsPermutation() const
+    {
+      ei_assert(m_isInitialized && "RRQR is not initialized.");
+      return m_cols_permutation;
+    }
+    
+    const IntColVectorType& rowsTranspositions() const
+    {
+      ei_assert(m_isInitialized && "RRQR is not initialized.");
+      return m_rows_transpositions;
+    }
+
+    inline int rank() const
+    {
+      ei_assert(m_isInitialized && "RRQR is not initialized.");
+      return m_rank;
+    }
+
+  protected:
+    MatrixType m_qr;
+    HCoeffsType m_hCoeffs;
+    IntColVectorType m_rows_transpositions;
+    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>
+HouseholderRRQR<MatrixType>& HouseholderRRQR<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);
+
+  // TODO: experiment to see the best formula
+  m_precision = epsilon<Scalar>() * size;
+
+  m_rows_transpositions.resize(matrix.rows());
+  IntRowVectorType cols_transpositions(matrix.cols());
+  m_cols_permutation.resize(matrix.cols());
+  int number_of_transpositions = 0;
+  
+  RealScalar biggest;
+  
+  for (int k = 0; k < size; ++k)
+  {
+    int row_of_biggest_in_corner, col_of_biggest_in_corner;
+    RealScalar biggest_in_corner;
+
+    biggest_in_corner = m_qr.corner(Eigen::BottomRight, rows-k, cols-k)
+                        .cwise().abs()
+                        .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
+    row_of_biggest_in_corner += k;
+    col_of_biggest_in_corner += k;
+    if(k==0) biggest = biggest_in_corner;
+    
+    // if the corner is negligible, then we have less than full rank, and we can finish early
+    if(ei_isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
+    {
+      m_rank = k;
+      for(int i = k; i < size; i++)
+      {
+        m_rows_transpositions.coeffRef(i) = i;
+        cols_transpositions.coeffRef(i) = i;
+        m_hCoeffs.coeffRef(i) = Scalar(0);
+      }
+      break;
+    }
+
+    m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
+    cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
+    if(k != row_of_biggest_in_corner) {
+      m_qr.row(k).end(cols-k).swap(m_qr.row(row_of_biggest_in_corner).end(cols-k));
+      ++number_of_transpositions;
+    }
+    if(k != col_of_biggest_in_corner) {
+      m_qr.col(k).swap(m_qr.col(col_of_biggest_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;
+
+    // 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));
+  }
+
+  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 HouseholderRRQR<MatrixType>::solve(
+  const MatrixBase<OtherDerived>& b,
+  ResultType *result
+) const
+{
+  ei_assert(m_isInitialized && "HouseholderRRQR 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.row(k).swap(c.row(m_rows_transpositions.coeff(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 HouseholderRRQR<MatrixType>::matrixQ() const
+{
+  ei_assert(m_isInitialized && "HouseholderRRQR 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));
+    res.row(k).swap(res.row(m_rows_transpositions.coeff(k)));
+  }
+  return res;
+}
+
+#endif // EIGEN_HIDE_HEAVY_CODE
+
+#if 0
+/** \return the Householder QR decomposition of \c *this.
+  *
+  * \sa class HouseholderRRQR
+  */
+template<typename Derived>
+const HouseholderRRQR<typename MatrixBase<Derived>::PlainMatrixType>
+MatrixBase<Derived>::householderQr() const
+{
+  return HouseholderRRQR<PlainMatrixType>(eval());
+}
+#endif
+
+
+#endif // EIGEN_QR_H

test/CMakeLists.txt

@@ -121,6 +121,7 @@ ei_add_test(lu ${EI_OFLAG})
 ei_add_test(determinant)
 ei_add_test(inverse ${EI_OFLAG})
 ei_add_test(qr)
+ei_add_test(rrqr)
 ei_add_test(eigensolver_selfadjoint " " "${GSL_LIBRARIES}")
 ei_add_test(eigensolver_generic " " "${GSL_LIBRARIES}")
 ei_add_test(svd)

test/rrqr.cpp

@@ -0,0 +1,116 @@
+// 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);
+  HouseholderRRQR<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()
+{
+  /* this test covers the following files: RRQR.h */
+  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();
+  }
+
+  HouseholderRRQR<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;
+
+  HouseholderRRQR<MatrixType> qr;
+  VERIFY_RAISES_ASSERT(qr.matrixR())
+  VERIFY_RAISES_ASSERT(qr.solve(tmp,&tmp))
+  VERIFY_RAISES_ASSERT(qr.matrixQ())
+}
+
+void test_rrqr()
+{
+ 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>());
+}