Added Triangular expression to extract upper or lower (strictly or not)

part of a matrix. Triangular also provide an optimised method for forward
and backward substitution. Further optimizations regarding assignments and
products might come later.

Updated determinant() to take into account triangular matrices.

Started the QR module with a QR decompostion algorithm.
Help needed to build a QR algorithm (eigen solver) based on it.

1 files changed, 153 insertions, 0 deletions

diff --git a/Eigen/src/QR/QR.h b/Eigen/src/QR/QR.h
new file mode 100644
index 000000000..d0121cc7a
--- /dev/null
+++ b/Eigen/src/QR/QR.h
@@ -0,0 +1,153 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra. Eigen itself is part of the KDE project.
+//
+// Copyright (C) 2008 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
+// 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_QR_H
+#define EIGEN_QR_H
+
+/** \class QR
+  *
+  * \brief QR decomposition of a matrix
+  *
+  * \param MatrixType the type of the matrix of which we are computing the QR decomposition
+  *
+  * This class performs a QR decomposition using Householder transformations. The result is
+  * stored in a compact way.
+  *
+  * \todo add convenient method to direclty use the result in a compact way. First need to determine
+  * typical use cases though.
+  *
+  * \todo what about complex matrices ?
+  *
+  * \sa MatrixBase::qr()
+  */
+template<typename MatrixType> class QR
+{
+  public:
+
+    typedef typename MatrixType::Scalar Scalar;
+    typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> RMatrixType;
+    typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
+
+    QR(const MatrixType& matrix)
+      : m_qr(matrix.rows(), matrix.cols()),
+        m_norms(matrix.cols())
+    {
+      _compute(matrix);
+    }
+
+    /** \returns whether or not the matrix is of full rank */
+    bool isFullRank() const { return ei_isMuchSmallerThan(m_norms.cwiseAbs().minCoeff(), Scalar(1)); }
+
+    RMatrixType matrixR(void) const;
+
+    MatrixType matrixQ(void) const;
+
+  private:
+
+    void _compute(const MatrixType& matrix);
+
+  protected:
+    MatrixType m_qr;
+    VectorType m_norms;
+};
+
+template<typename MatrixType>
+void QR<MatrixType>::_compute(const MatrixType& matrix)
+{
+  m_qr = matrix;
+  int rows = matrix.rows();
+  int cols = matrix.cols();
+
+  for (int k = 0; k < cols; k++)
+  {
+    int remainingSize = rows-k;
+
+    Scalar nrm = m_qr.col(k).end(remainingSize).norm();
+
+    if (nrm != Scalar(0))
+    {
+      // form k-th Householder vector
+      if (m_qr(k,k) < 0)
+        nrm = -nrm;
+
+      m_qr.col(k).end(rows-k) /= nrm;
+      m_qr(k,k) += 1.0;
+
+      // apply transformation to remaining columns
+      for (int j = k+1; j < cols; j++)
+      {
+        Scalar s = -(m_qr.col(k).end(remainingSize).transpose() * m_qr.col(j).end(remainingSize))(0,0) / m_qr(k,k);
+        m_qr.col(j).end(remainingSize) += s * m_qr.col(k).end(remainingSize);
+      }
+    }
+    m_norms[k] = -nrm;
+  }
+}
+
+/** \returns the matrix R */
+template<typename MatrixType>
+typename QR<MatrixType>::RMatrixType QR<MatrixType>::matrixR(void) const
+{
+  int cols = m_qr.cols();
+  RMatrixType res = m_qr.block(0,0,cols,cols).upperWithNullDiag();
+  res.diagonal() = m_norms;
+  return res;
+}
+
+/** \returns the matrix Q */
+template<typename MatrixType>
+MatrixType QR<MatrixType>::matrixQ(void) const
+{
+  int rows = m_qr.rows();
+  int cols = m_qr.cols();
+  MatrixType res = MatrixType::identity(rows, cols);
+  for (int k = cols-1; k >= 0; k--)
+  {
+    for (int j = k; j < cols; j++)
+    {
+      if (res(k,k) != Scalar(0))
+      {
+        int endLength = rows-k;
+        Scalar s = -(m_qr.col(k).end(endLength).transpose() * res.col(j).end(endLength))(0,0) / m_qr(k,k);
+
+        res.col(j).end(endLength) += s * m_qr.col(k).end(endLength);
+      }
+    }
+  }
+  return res;
+}
+
+/** \return the QR decomposition of \c *this.
+  *
+  * \sa class QR
+  */
+template<typename Derived>
+const QR<typename ei_eval<Derived>::type>
+MatrixBase<Derived>::qr() const
+{
+  return QR<typename ei_eval<Derived>::type>(derived());
+}
+
+
+#endif // EIGEN_QR_H