Diff: src/SVD/JacobiSVD.h

Revision:

0:13a5d365ba16

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/SVD/JacobiSVD.h	Thu Oct 13 04:07:23 2016 +0000
@@ -0,0 +1,976 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_JACOBISVD_H
+#define EIGEN_JACOBISVD_H
+
+namespace Eigen { 
+
+namespace internal {
+// forward declaration (needed by ICC)
+// the empty body is required by MSVC
+template<typename MatrixType, int QRPreconditioner,
+         bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
+struct svd_precondition_2x2_block_to_be_real {};
+
+/*** QR preconditioners (R-SVD)
+ ***
+ *** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
+ *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
+ *** JacobiSVD which by itself is only able to work on square matrices.
+ ***/
+
+enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };
+
+template<typename MatrixType, int QRPreconditioner, int Case>
+struct qr_preconditioner_should_do_anything
+{
+  enum { a = MatrixType::RowsAtCompileTime != Dynamic &&
+             MatrixType::ColsAtCompileTime != Dynamic &&
+             MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
+         b = MatrixType::RowsAtCompileTime != Dynamic &&
+             MatrixType::ColsAtCompileTime != Dynamic &&
+             MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
+         ret = !( (QRPreconditioner == NoQRPreconditioner) ||
+                  (Case == PreconditionIfMoreColsThanRows && bool(a)) ||
+                  (Case == PreconditionIfMoreRowsThanCols && bool(b)) )
+  };
+};
+
+template<typename MatrixType, int QRPreconditioner, int Case,
+         bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
+> struct qr_preconditioner_impl {};
+
+template<typename MatrixType, int QRPreconditioner, int Case>
+class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
+{
+public:
+  typedef typename MatrixType::Index Index;
+  void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {}
+  bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&)
+  {
+    return false;
+  }
+};
+
+/*** preconditioner using FullPivHouseholderQR ***/
+
+template<typename MatrixType>
+class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
+{
+public:
+  typedef typename MatrixType::Index Index;
+  typedef typename MatrixType::Scalar Scalar;
+  enum
+  {
+    RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
+  };
+  typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType;
+
+  void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
+  {
+    if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
+    {
+      m_qr.~QRType();
+      ::new (&m_qr) QRType(svd.rows(), svd.cols());
+    }
+    if (svd.m_computeFullU) m_workspace.resize(svd.rows());
+  }
+
+  bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
+  {
+    if(matrix.rows() > matrix.cols())
+    {
+      m_qr.compute(matrix);
+      svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
+      if(svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace);
+      if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
+      return true;
+    }
+    return false;
+  }
+private:
+  typedef FullPivHouseholderQR<MatrixType> QRType;
+  QRType m_qr;
+  WorkspaceType m_workspace;
+};
+
+template<typename MatrixType>
+class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
+{
+public:
+  typedef typename MatrixType::Index Index;
+  typedef typename MatrixType::Scalar Scalar;
+  enum
+  {
+    RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+    ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
+    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
+    Options = MatrixType::Options
+  };
+  typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
+          TransposeTypeWithSameStorageOrder;
+
+  void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
+  {
+    if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
+    {
+      m_qr.~QRType();
+      ::new (&m_qr) QRType(svd.cols(), svd.rows());
+    }
+    m_adjoint.resize(svd.cols(), svd.rows());
+    if (svd.m_computeFullV) m_workspace.resize(svd.cols());
+  }
+
+  bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
+  {
+    if(matrix.cols() > matrix.rows())
+    {
+      m_adjoint = matrix.adjoint();
+      m_qr.compute(m_adjoint);
+      svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
+      if(svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace);
+      if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
+      return true;
+    }
+    else return false;
+  }
+private:
+  typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
+  QRType m_qr;
+  TransposeTypeWithSameStorageOrder m_adjoint;
+  typename internal::plain_row_type<MatrixType>::type m_workspace;
+};
+
+/*** preconditioner using ColPivHouseholderQR ***/
+
+template<typename MatrixType>
+class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
+{
+public:
+  typedef typename MatrixType::Index Index;
+
+  void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
+  {
+    if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
+    {
+      m_qr.~QRType();
+      ::new (&m_qr) QRType(svd.rows(), svd.cols());
+    }
+    if (svd.m_computeFullU) m_workspace.resize(svd.rows());
+    else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
+  }
+
+  bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
+  {
+    if(matrix.rows() > matrix.cols())
+    {
+      m_qr.compute(matrix);
+      svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
+      if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
+      else if(svd.m_computeThinU)
+      {
+        svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
+        m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
+      }
+      if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
+      return true;
+    }
+    return false;
+  }
+
+private:
+  typedef ColPivHouseholderQR<MatrixType> QRType;
+  QRType m_qr;
+  typename internal::plain_col_type<MatrixType>::type m_workspace;
+};
+
+template<typename MatrixType>
+class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
+{
+public:
+  typedef typename MatrixType::Index Index;
+  typedef typename MatrixType::Scalar Scalar;
+  enum
+  {
+    RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+    ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
+    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
+    Options = MatrixType::Options
+  };
+
+  typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
+          TransposeTypeWithSameStorageOrder;
+
+  void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
+  {
+    if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
+    {
+      m_qr.~QRType();
+      ::new (&m_qr) QRType(svd.cols(), svd.rows());
+    }
+    if (svd.m_computeFullV) m_workspace.resize(svd.cols());
+    else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
+    m_adjoint.resize(svd.cols(), svd.rows());
+  }
+
+  bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
+  {
+    if(matrix.cols() > matrix.rows())
+    {
+      m_adjoint = matrix.adjoint();
+      m_qr.compute(m_adjoint);
+
+      svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
+      if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
+      else if(svd.m_computeThinV)
+      {
+        svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
+        m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
+      }
+      if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
+      return true;
+    }
+    else return false;
+  }
+
+private:
+  typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
+  QRType m_qr;
+  TransposeTypeWithSameStorageOrder m_adjoint;
+  typename internal::plain_row_type<MatrixType>::type m_workspace;
+};
+
+/*** preconditioner using HouseholderQR ***/
+
+template<typename MatrixType>
+class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
+{
+public:
+  typedef typename MatrixType::Index Index;
+
+  void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
+  {
+    if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
+    {
+      m_qr.~QRType();
+      ::new (&m_qr) QRType(svd.rows(), svd.cols());
+    }
+    if (svd.m_computeFullU) m_workspace.resize(svd.rows());
+    else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
+  }
+
+  bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
+  {
+    if(matrix.rows() > matrix.cols())
+    {
+      m_qr.compute(matrix);
+      svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
+      if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
+      else if(svd.m_computeThinU)
+      {
+        svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
+        m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
+      }
+      if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
+      return true;
+    }
+    return false;
+  }
+private:
+  typedef HouseholderQR<MatrixType> QRType;
+  QRType m_qr;
+  typename internal::plain_col_type<MatrixType>::type m_workspace;
+};
+
+template<typename MatrixType>
+class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
+{
+public:
+  typedef typename MatrixType::Index Index;
+  typedef typename MatrixType::Scalar Scalar;
+  enum
+  {
+    RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+    ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
+    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
+    Options = MatrixType::Options
+  };
+
+  typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
+          TransposeTypeWithSameStorageOrder;
+
+  void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
+  {
+    if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
+    {
+      m_qr.~QRType();
+      ::new (&m_qr) QRType(svd.cols(), svd.rows());
+    }
+    if (svd.m_computeFullV) m_workspace.resize(svd.cols());
+    else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
+    m_adjoint.resize(svd.cols(), svd.rows());
+  }
+
+  bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
+  {
+    if(matrix.cols() > matrix.rows())
+    {
+      m_adjoint = matrix.adjoint();
+      m_qr.compute(m_adjoint);
+
+      svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
+      if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
+      else if(svd.m_computeThinV)
+      {
+        svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
+        m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
+      }
+      if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
+      return true;
+    }
+    else return false;
+  }
+
+private:
+  typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
+  QRType m_qr;
+  TransposeTypeWithSameStorageOrder m_adjoint;
+  typename internal::plain_row_type<MatrixType>::type m_workspace;
+};
+
+/*** 2x2 SVD implementation
+ ***
+ *** JacobiSVD consists in performing a series of 2x2 SVD subproblems
+ ***/
+
+template<typename MatrixType, int QRPreconditioner>
+struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false>
+{
+  typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
+  typedef typename SVD::Index Index;
+  static void run(typename SVD::WorkMatrixType&, SVD&, Index, Index) {}
+};
+
+template<typename MatrixType, int QRPreconditioner>
+struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true>
+{
+  typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar RealScalar;
+  typedef typename SVD::Index Index;
+  static void run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q)
+  {
+    using std::sqrt;
+    Scalar z;
+    JacobiRotation<Scalar> rot;
+    RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p,p)) + numext::abs2(work_matrix.coeff(q,p)));
+    
+    if(n==0)
+    {
+      z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
+      work_matrix.row(p) *= z;
+      if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
+      if(work_matrix.coeff(q,q)!=Scalar(0))
+      {
+        z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
+        work_matrix.row(q) *= z;
+        if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
+      }
+      // otherwise the second row is already zero, so we have nothing to do.
+    }
+    else
+    {
+      rot.c() = conj(work_matrix.coeff(p,p)) / n;
+      rot.s() = work_matrix.coeff(q,p) / n;
+      work_matrix.applyOnTheLeft(p,q,rot);
+      if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint());
+      if(work_matrix.coeff(p,q) != Scalar(0))
+      {
+        Scalar z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
+        work_matrix.col(q) *= z;
+        if(svd.computeV()) svd.m_matrixV.col(q) *= z;
+      }
+      if(work_matrix.coeff(q,q) != Scalar(0))
+      {
+        z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
+        work_matrix.row(q) *= z;
+        if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
+      }
+    }
+  }
+};
+
+template<typename MatrixType, typename RealScalar, typename Index>
+void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
+                            JacobiRotation<RealScalar> *j_left,
+                            JacobiRotation<RealScalar> *j_right)
+{
+  using std::sqrt;
+  using std::abs;
+  Matrix<RealScalar,2,2> m;
+  m << numext::real(matrix.coeff(p,p)), numext::real(matrix.coeff(p,q)),
+       numext::real(matrix.coeff(q,p)), numext::real(matrix.coeff(q,q));
+  JacobiRotation<RealScalar> rot1;
+  RealScalar t = m.coeff(0,0) + m.coeff(1,1);
+  RealScalar d = m.coeff(1,0) - m.coeff(0,1);
+  if(t == RealScalar(0))
+  {
+    rot1.c() = RealScalar(0);
+    rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
+  }
+  else
+  {
+    RealScalar t2d2 = numext::hypot(t,d);
+    rot1.c() = abs(t)/t2d2;
+    rot1.s() = d/t2d2;
+    if(t<RealScalar(0))
+      rot1.s() = -rot1.s();
+  }
+  m.applyOnTheLeft(0,1,rot1);
+  j_right->makeJacobi(m,0,1);
+  *j_left  = rot1 * j_right->transpose();
+}
+
+} // end namespace internal
+
+/** \ingroup SVD_Module
+  *
+  *
+  * \class JacobiSVD
+  *
+  * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix
+  *
+  * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
+  * \param QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally
+  *                        for the R-SVD step for non-square matrices. See discussion of possible values below.
+  *
+  * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
+  *   \f[ A = U S V^* \f]
+  * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero outside of its main diagonal;
+  * the diagonal entries of S are known as the \em singular \em values of \a A and the columns of \a U and \a V are known as the left
+  * and right \em singular \em vectors of \a A respectively.
+  *
+  * Singular values are always sorted in decreasing order.
+  *
+  * This JacobiSVD decomposition computes only the singular values by default. If you want \a U or \a V, you need to ask for them explicitly.
+  *
+  * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting \a m be the
+  * smaller value among \a n and \a p, there are only \a m singular vectors; the remaining columns of \a U and \a V do not correspond to actual
+  * singular vectors. Asking for \em thin \a U or \a V means asking for only their \a m first columns to be formed. So \a U is then a n-by-m matrix,
+  * and \a V is then a p-by-m matrix. Notice that thin \a U and \a V are all you need for (least squares) solving.
+  *
+  * Here's an example demonstrating basic usage:
+  * \include JacobiSVD_basic.cpp
+  * Output: \verbinclude JacobiSVD_basic.out
+  *
+  * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than
+  * bidiagonalizing SVD algorithms for large square matrices; however its complexity is still \f$ O(n^2p) \f$ where \a n is the smaller dimension and
+  * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms.
+  * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
+  *
+  * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
+  * terminate in finite (and reasonable) time.
+  *
+  * The possible values for QRPreconditioner are:
+  * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
+  * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
+  *     Contrary to other QRs, it doesn't allow computing thin unitaries.
+  * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR.
+  *     This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization
+  *     is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive
+  *     process is more reliable than the optimized bidiagonal SVD iterations.
+  * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing
+  *     JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in
+  *     faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking
+  *     if QR preconditioning is needed before applying it anyway.
+  *
+  * \sa MatrixBase::jacobiSvd()
+  */
+template<typename _MatrixType, int QRPreconditioner> class JacobiSVD
+{
+  public:
+
+    typedef _MatrixType MatrixType;
+    typedef typename MatrixType::Scalar Scalar;
+    typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
+    typedef typename MatrixType::Index Index;
+    enum {
+      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+      DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime),
+      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
+      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
+      MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime),
+      MatrixOptions = MatrixType::Options
+    };
+
+    typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime,
+                   MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
+            MatrixUType;
+    typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime,
+                   MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
+            MatrixVType;
+    typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
+    typedef typename internal::plain_row_type<MatrixType>::type RowType;
+    typedef typename internal::plain_col_type<MatrixType>::type ColType;
+    typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime,
+                   MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
+            WorkMatrixType;
+
+    /** \brief Default Constructor.
+      *
+      * The default constructor is useful in cases in which the user intends to
+      * perform decompositions via JacobiSVD::compute(const MatrixType&).
+      */
+    JacobiSVD()
+      : m_isInitialized(false),
+        m_isAllocated(false),
+        m_usePrescribedThreshold(false),
+        m_computationOptions(0),
+        m_rows(-1), m_cols(-1), m_diagSize(0)
+    {}
+
+
+    /** \brief Default Constructor with memory preallocation
+      *
+      * Like the default constructor but with preallocation of the internal data
+      * according to the specified problem size.
+      * \sa JacobiSVD()
+      */
+    JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0)
+      : m_isInitialized(false),
+        m_isAllocated(false),
+        m_usePrescribedThreshold(false),
+        m_computationOptions(0),
+        m_rows(-1), m_cols(-1)
+    {
+      allocate(rows, cols, computationOptions);
+    }
+
+    /** \brief Constructor performing the decomposition of given matrix.
+     *
+     * \param matrix the matrix to decompose
+     * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
+     *                           By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
+     *                           #ComputeFullV, #ComputeThinV.
+     *
+     * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
+     * available with the (non-default) FullPivHouseholderQR preconditioner.
+     */
+    JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
+      : m_isInitialized(false),
+        m_isAllocated(false),
+        m_usePrescribedThreshold(false),
+        m_computationOptions(0),
+        m_rows(-1), m_cols(-1)
+    {
+      compute(matrix, computationOptions);
+    }
+
+    /** \brief Method performing the decomposition of given matrix using custom options.
+     *
+     * \param matrix the matrix to decompose
+     * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
+     *                           By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
+     *                           #ComputeFullV, #ComputeThinV.
+     *
+     * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
+     * available with the (non-default) FullPivHouseholderQR preconditioner.
+     */
+    JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
+
+    /** \brief Method performing the decomposition of given matrix using current options.
+     *
+     * \param matrix the matrix to decompose
+     *
+     * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
+     */
+    JacobiSVD& compute(const MatrixType& matrix)
+    {
+      return compute(matrix, m_computationOptions);
+    }
+
+    /** \returns the \a U matrix.
+     *
+     * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
+     * the U matrix is n-by-n if you asked for #ComputeFullU, and is n-by-m if you asked for #ComputeThinU.
+     *
+     * The \a m first columns of \a U are the left singular vectors of the matrix being decomposed.
+     *
+     * This method asserts that you asked for \a U to be computed.
+     */
+    const MatrixUType& matrixU() const
+    {
+      eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
+      eigen_assert(computeU() && "This JacobiSVD decomposition didn't compute U. Did you ask for it?");
+      return m_matrixU;
+    }
+
+    /** \returns the \a V matrix.
+     *
+     * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
+     * the V matrix is p-by-p if you asked for #ComputeFullV, and is p-by-m if you asked for ComputeThinV.
+     *
+     * The \a m first columns of \a V are the right singular vectors of the matrix being decomposed.
+     *
+     * This method asserts that you asked for \a V to be computed.
+     */
+    const MatrixVType& matrixV() const
+    {
+      eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
+      eigen_assert(computeV() && "This JacobiSVD decomposition didn't compute V. Did you ask for it?");
+      return m_matrixV;
+    }
+
+    /** \returns the vector of singular values.
+     *
+     * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, the
+     * returned vector has size \a m.  Singular values are always sorted in decreasing order.
+     */
+    const SingularValuesType& singularValues() const
+    {
+      eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
+      return m_singularValues;
+    }
+
+    /** \returns true if \a U (full or thin) is asked for in this SVD decomposition */
+    inline bool computeU() const { return m_computeFullU || m_computeThinU; }
+    /** \returns true if \a V (full or thin) is asked for in this SVD decomposition */
+    inline bool computeV() const { return m_computeFullV || m_computeThinV; }
+
+    /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
+      *
+      * \param b the right-hand-side of the equation to solve.
+      *
+      * \note Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V.
+      *
+      * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving.
+      * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
+      */
+    template<typename Rhs>
+    inline const internal::solve_retval<JacobiSVD, Rhs>
+    solve(const MatrixBase<Rhs>& b) const
+    {
+      eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
+      eigen_assert(computeU() && computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
+      return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived());
+    }
+
+    /** \returns the number of singular values that are not exactly 0 */
+    Index nonzeroSingularValues() const
+    {
+      eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
+      return m_nonzeroSingularValues;
+    }
+    
+    /** \returns the rank of the matrix of which \c *this is the SVD.
+      *
+      * \note This method has to determine which singular values should be considered nonzero.
+      *       For that, it uses the threshold value that you can control by calling
+      *       setThreshold(const RealScalar&).
+      */
+    inline Index rank() const
+    {
+      using std::abs;
+      eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
+      if(m_singularValues.size()==0) return 0;
+      RealScalar premultiplied_threshold = m_singularValues.coeff(0) * threshold();
+      Index i = m_nonzeroSingularValues-1;
+      while(i>=0 && m_singularValues.coeff(i) < premultiplied_threshold) --i;
+      return i+1;
+    }
+    
+    /** Allows to prescribe a threshold to be used by certain methods, such as rank() and solve(),
+      * which need to determine when singular values are to be considered nonzero.
+      * This is not used for the SVD decomposition itself.
+      *
+      * When it needs to get the threshold value, Eigen calls threshold().
+      * The default is \c NumTraits<Scalar>::epsilon()
+      *
+      * \param threshold The new value to use as the threshold.
+      *
+      * A singular value will be considered nonzero if its value is strictly greater than
+      *  \f$ \vert singular value \vert \leqslant threshold \times \vert max singular value \vert \f$.
+      *
+      * If you want to come back to the default behavior, call setThreshold(Default_t)
+      */
+    JacobiSVD& setThreshold(const RealScalar& threshold)
+    {
+      m_usePrescribedThreshold = true;
+      m_prescribedThreshold = threshold;
+      return *this;
+    }
+
+    /** Allows to come back to the default behavior, letting Eigen use its default formula for
+      * determining the threshold.
+      *
+      * You should pass the special object Eigen::Default as parameter here.
+      * \code svd.setThreshold(Eigen::Default); \endcode
+      *
+      * See the documentation of setThreshold(const RealScalar&).
+      */
+    JacobiSVD& setThreshold(Default_t)
+    {
+      m_usePrescribedThreshold = false;
+      return *this;
+    }
+
+    /** Returns the threshold that will be used by certain methods such as rank().
+      *
+      * See the documentation of setThreshold(const RealScalar&).
+      */
+    RealScalar threshold() const
+    {
+      eigen_assert(m_isInitialized || m_usePrescribedThreshold);
+      return m_usePrescribedThreshold ? m_prescribedThreshold
+                                      : (std::max<Index>)(1,m_diagSize)*NumTraits<Scalar>::epsilon();
+    }
+
+    inline Index rows() const { return m_rows; }
+    inline Index cols() const { return m_cols; }
+
+  private:
+    void allocate(Index rows, Index cols, unsigned int computationOptions);
+    
+    static void check_template_parameters()
+    {
+      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
+    }
+
+  protected:
+    MatrixUType m_matrixU;
+    MatrixVType m_matrixV;
+    SingularValuesType m_singularValues;
+    WorkMatrixType m_workMatrix;
+    bool m_isInitialized, m_isAllocated, m_usePrescribedThreshold;
+    bool m_computeFullU, m_computeThinU;
+    bool m_computeFullV, m_computeThinV;
+    unsigned int m_computationOptions;
+    Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize;
+    RealScalar m_prescribedThreshold;
+
+    template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
+    friend struct internal::svd_precondition_2x2_block_to_be_real;
+    template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
+    friend struct internal::qr_preconditioner_impl;
+
+    internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols;
+    internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows;
+    MatrixType m_scaledMatrix;
+};
+
+template<typename MatrixType, int QRPreconditioner>
+void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
+{
+  eigen_assert(rows >= 0 && cols >= 0);
+
+  if (m_isAllocated &&
+      rows == m_rows &&
+      cols == m_cols &&
+      computationOptions == m_computationOptions)
+  {
+    return;
+  }
+
+  m_rows = rows;
+  m_cols = cols;
+  m_isInitialized = false;
+  m_isAllocated = true;
+  m_computationOptions = computationOptions;
+  m_computeFullU = (computationOptions & ComputeFullU) != 0;
+  m_computeThinU = (computationOptions & ComputeThinU) != 0;
+  m_computeFullV = (computationOptions & ComputeFullV) != 0;
+  m_computeThinV = (computationOptions & ComputeThinV) != 0;
+  eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U");
+  eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V");
+  eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) &&
+              "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns.");
+  if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
+  {
+      eigen_assert(!(m_computeThinU || m_computeThinV) &&
+              "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
+              "Use the ColPivHouseholderQR preconditioner instead.");
+  }
+  m_diagSize = (std::min)(m_rows, m_cols);
+  m_singularValues.resize(m_diagSize);
+  if(RowsAtCompileTime==Dynamic)
+    m_matrixU.resize(m_rows, m_computeFullU ? m_rows
+                            : m_computeThinU ? m_diagSize
+                            : 0);
+  if(ColsAtCompileTime==Dynamic)
+    m_matrixV.resize(m_cols, m_computeFullV ? m_cols
+                            : m_computeThinV ? m_diagSize
+                            : 0);
+  m_workMatrix.resize(m_diagSize, m_diagSize);
+  
+  if(m_cols>m_rows)   m_qr_precond_morecols.allocate(*this);
+  if(m_rows>m_cols)   m_qr_precond_morerows.allocate(*this);
+  if(m_rows!=m_cols)  m_scaledMatrix.resize(rows,cols);
+}
+
+template<typename MatrixType, int QRPreconditioner>
+JacobiSVD<MatrixType, QRPreconditioner>&
+JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
+{
+  check_template_parameters();
+  
+  using std::abs;
+  allocate(matrix.rows(), matrix.cols(), computationOptions);
+
+  // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
+  // only worsening the precision of U and V as we accumulate more rotations
+  const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();
+
+  // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
+  const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min();
+
+  // Scaling factor to reduce over/under-flows
+  RealScalar scale = matrix.cwiseAbs().maxCoeff();
+  if(scale==RealScalar(0)) scale = RealScalar(1);
+  
+  /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
+
+  if(m_rows!=m_cols)
+  {
+    m_scaledMatrix = matrix / scale;
+    m_qr_precond_morecols.run(*this, m_scaledMatrix);
+    m_qr_precond_morerows.run(*this, m_scaledMatrix);
+  }
+  else
+  {
+    m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize) / scale;
+    if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows);
+    if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize);
+    if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols);
+    if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize);
+  }
+
+  /*** step 2. The main Jacobi SVD iteration. ***/
+
+  bool finished = false;
+  while(!finished)
+  {
+    finished = true;
+
+    // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix
+
+    for(Index p = 1; p < m_diagSize; ++p)
+    {
+      for(Index q = 0; q < p; ++q)
+      {
+        // if this 2x2 sub-matrix is not diagonal already...
+        // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
+        // keep us iterating forever. Similarly, small denormal numbers are considered zero.
+        using std::max;
+        RealScalar threshold = (max)(considerAsZero, precision * (max)(abs(m_workMatrix.coeff(p,p)),
+                                                                       abs(m_workMatrix.coeff(q,q))));
+        // We compare both values to threshold instead of calling max to be robust to NaN (See bug 791)
+        if(abs(m_workMatrix.coeff(p,q))>threshold || abs(m_workMatrix.coeff(q,p)) > threshold)
+        {
+          finished = false;
+
+          // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
+          internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
+          JacobiRotation<RealScalar> j_left, j_right;
+          internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);
+
+          // accumulate resulting Jacobi rotations
+          m_workMatrix.applyOnTheLeft(p,q,j_left);
+          if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose());
+
+          m_workMatrix.applyOnTheRight(p,q,j_right);
+          if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right);
+        }
+      }
+    }
+  }
+
+  /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/
+
+  for(Index i = 0; i < m_diagSize; ++i)
+  {
+    RealScalar a = abs(m_workMatrix.coeff(i,i));
+    m_singularValues.coeffRef(i) = a;
+    if(computeU() && (a!=RealScalar(0))) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a;
+  }
+
+  /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/
+
+  m_nonzeroSingularValues = m_diagSize;
+  for(Index i = 0; i < m_diagSize; i++)
+  {
+    Index pos;
+    RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos);
+    if(maxRemainingSingularValue == RealScalar(0))
+    {
+      m_nonzeroSingularValues = i;
+      break;
+    }
+    if(pos)
+    {
+      pos += i;
+      std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos));
+      if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i));
+      if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i));
+    }
+  }
+  
+  m_singularValues *= scale;
+
+  m_isInitialized = true;
+  return *this;
+}
+
+namespace internal {
+template<typename _MatrixType, int QRPreconditioner, typename Rhs>
+struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
+  : solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
+{
+  typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType;
+  EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs)
+
+  template<typename Dest> void evalTo(Dest& dst) const
+  {
+    eigen_assert(rhs().rows() == dec().rows());
+
+    // A = U S V^*
+    // So A^{-1} = V S^{-1} U^*
+
+    Matrix<Scalar, Dynamic, Rhs::ColsAtCompileTime, 0, _MatrixType::MaxRowsAtCompileTime, Rhs::MaxColsAtCompileTime> tmp;
+    Index rank = dec().rank();
+    
+    tmp.noalias() = dec().matrixU().leftCols(rank).adjoint() * rhs();
+    tmp = dec().singularValues().head(rank).asDiagonal().inverse() * tmp;
+    dst = dec().matrixV().leftCols(rank) * tmp;
+  }
+};
+} // end namespace internal
+
+/** \svd_module
+  *
+  * \return the singular value decomposition of \c *this computed by two-sided
+  * Jacobi transformations.
+  *
+  * \sa class JacobiSVD
+  */
+template<typename Derived>
+JacobiSVD<typename MatrixBase<Derived>::PlainObject>
+MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
+{
+  return JacobiSVD<PlainObject>(*this, computationOptions);
+}
+
+} // end namespace Eigen
+
+#endif // EIGEN_JACOBISVD_H
\ No newline at end of file