Added a SVD module:

- the decompostion code has been adfapted from JAMA
 - handles non square matrices of size MxN with M>=N
 - does not work for complex matrices
 - includes a solver where the parts corresponding to zero singular values are set to zero

Eigen/src/Core/MatrixBase.h

@@ -557,6 +557,10 @@ template<typename Derived> class MatrixBase
     EigenvaluesReturnType eigenvalues() const;
     RealScalar operatorNorm() const;
 
+/////////// SVD module ///////////
+
+    const SVD<EvalType> svd() const;
+
 /////////// Geometry module ///////////
 
     template<typename OtherDerived>

Eigen/src/Core/util/ForwardDeclarations.h

@@ -98,6 +98,7 @@ void ei_cache_friendly_product(
 
 template<typename MatrixType> class LU;
 template<typename MatrixType> class QR;
+template<typename MatrixType> class SVD;
 template<typename MatrixType> class Cholesky;
 template<typename MatrixType> class CholeskyWithoutSquareRoot;
 

Eigen/src/SVD/CMakeLists.txt

@@ -0,0 +1,6 @@
+FILE(GLOB Eigen_SVD_SRCS "*.h")
+
+INSTALL(FILES
+  ${Eigen_SVD_SRCS}
+  DESTINATION ${INCLUDE_INSTALL_DIR}/Eigen/src/SVD
+  )

Eigen/src/SVD/SVD.h

@@ -0,0 +1,506 @@
+// 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_SVD_H
+#define EIGEN_SVD_H
+
+/** \ingroup SVD_Module
+  *
+  * \class SVD
+  *
+  * \brief Standard SVD decomposition of a matrix and associated features
+  *
+  * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
+  *
+  * This class performs a standard SVD decomposition of a real matrix A of size \c M x \c N
+  * with \c M \>= \c N.
+  *
+  *
+  * \sa MatrixBase::SVD()
+  */
+template<typename MatrixType> class SVD
+{
+  private:
+    typedef typename MatrixType::Scalar Scalar;
+    typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
+
+    enum {
+      PacketSize = ei_packet_traits<Scalar>::size,
+      AlignmentMask = int(PacketSize)-1,
+      MinSize = EIGEN_ENUM_MIN(MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime)
+    };
+    
+    typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVector;
+    typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> RowVector;
+    
+    typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MinSize> MatrixUType;
+    typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> MatrixVType;
+    typedef Matrix<Scalar, MinSize, 1> SingularValuesType;
+
+  public:
+  
+    SVD(const MatrixType& matrix)
+      : m_matU(matrix.rows(), std::min(matrix.rows(), matrix.cols())),
+        m_matV(matrix.cols(),matrix.cols()),
+        m_sigma(std::min(matrix.rows(),matrix.cols()))
+    {
+      compute(matrix);
+    }
+
+    template<typename OtherDerived, typename ResultType>
+    void solve(const MatrixBase<OtherDerived> &b, ResultType* result) const;
+
+    const MatrixUType& matrixU() const { return m_matU; }
+    const SingularValuesType& singularValues() const { return m_sigma; }
+    const MatrixVType& matrixV() const { return m_matV; }
+
+    void compute(const MatrixType& matrix);
+
+  protected:
+    /** \internal */
+    MatrixUType m_matU;
+    /** \internal */
+    MatrixVType m_matV;
+    /** \internal */
+    SingularValuesType m_sigma;
+};
+
+/** Computes / recomputes the SVD decomposition A = U S V^* of \a matrix
+  *
+  * \note this code has been adapted from JAMA (public domain)
+  */
+template<typename MatrixType>
+void SVD<MatrixType>::compute(const MatrixType& matrix)
+{
+  const int m = matrix.rows();
+  const int n = matrix.cols();
+  const int nu = std::min(m,n);
+  
+  m_matU.resize(m, nu);
+  m_matU.setZero();
+  m_sigma.resize(std::min(m,n));
+  m_matV.resize(n,n);
+
+  RowVector e(n);
+  ColVector work(m);
+  MatrixType matA(matrix);
+  const bool wantu = true;
+  const bool wantv = true;
+  int i=0, j=0, k=0;
+
+  // Reduce A to bidiagonal form, storing the diagonal elements
+  // in s and the super-diagonal elements in e.
+  int nct = std::min(m-1,n);
+  int nrt = std::max(0,std::min(n-2,m));
+  for (k = 0; k < std::max(nct,nrt); k++)
+  {
+    if (k < nct)
+    {
+      // Compute the transformation for the k-th column and
+      // place the k-th diagonal in m_sigma[k].
+      m_sigma[k] = matA.col(k).end(m-k).norm();
+      if (m_sigma[k] != 0.0) // FIXME
+      {
+        if (matA(k,k) < 0.0)
+          m_sigma[k] = -m_sigma[k];
+        matA.col(k).end(m-k) /= m_sigma[k];
+        matA(k,k) += 1.0;
+      }
+      m_sigma[k] = -m_sigma[k];
+    }
+    
+    for (j = k+1; j < n; j++)
+    {
+      if ((k < nct) && (m_sigma[k] != 0.0))
+      {
+        // Apply the transformation.
+        Scalar t = matA.col(k).end(m-k).dot(matA.col(j).end(m-k)); // FIXME dot product or cwise prod + .sum() ??
+        t = -t/matA(k,k);
+        matA.col(j).end(m-k) += t * matA.col(k).end(m-k);
+      }
+
+      // Place the k-th row of A into e for the
+      // subsequent calculation of the row transformation.
+      e[j] = matA(k,j);
+    }
+
+    // Place the transformation in U for subsequent back multiplication.
+    if (wantu & (k < nct))
+      m_matU.col(k).end(m-k) = matA.col(k).end(m-k);
+
+    if (k < nrt)
+    {
+      // Compute the k-th row transformation and place the
+      // k-th super-diagonal in e[k].
+      e[k] = e.end(n-k-1).norm();
+      if (e[k] != 0.0)
+      {
+          if (e[k+1] < 0.0)
+            e[k] = -e[k];
+          e.end(n-k-1) /= e[k];
+          e[k+1] += 1.0;
+      }
+      e[k] = -e[k];
+      if ((k+1 < m) & (e[k] != 0.0))
+      {
+        // Apply the transformation.
+        work.end(m-k-1) = matA.corner(BottomRight,m-k-1,n-k-1) * e.end(n-k-1);
+        for (j = k+1; j < n; j++)
+          matA.col(j).end(m-k-1) += (-e[j]/e[k+1]) * work.end(m-k-1);
+      }
+
+      // Place the transformation in V for subsequent back multiplication.
+      if (wantv)
+        m_matV.col(k).end(n-k-1) = e.end(n-k-1);
+    }
+  }
+
+
+  // Set up the final bidiagonal matrix or order p.
+  int p = min(n,m+1);
+  if (nct < n)
+    m_sigma[nct] = matA(nct,nct);
+  if (m < p)
+    m_sigma[p-1] = 0.0;
+  if (nrt+1 < p)
+    e[nrt] = matA(nrt,p-1);
+  e[p-1] = 0.0;
+
+  // If required, generate U.
+  if (wantu)
+  {
+    for (j = nct; j < nu; j++)
+    {
+      m_matU.col(j).setZero();
+      m_matU(j,j) = 1.0;
+    }
+    for (k = nct-1; k >= 0; k--)
+    {
+      if (m_sigma[k] != 0.0)
+      {
+        for (j = k+1; j < nu; j++)
+        {
+          Scalar t = m_matU.col(k).end(m-k).dot(m_matU.col(j).end(m-k)); // FIXME is it really a dot product we want ?
+          t = -t/m_matU(k,k);
+          m_matU.col(j).end(m-k) += t * m_matU.col(k).end(m-k);
+        }
+        m_matU.col(k).end(m-k) = - m_matU.col(k).end(m-k);
+        m_matU(k,k) = 1.0 + m_matU(k,k);
+        if (k-1>0)
+          m_matU.col(k).start(k-1).setZero();
+      }
+      else
+      {
+        m_matU.col(k).setZero();
+        m_matU(k,k) = 1.0;
+      }
+    }
+  }
+
+  // If required, generate V.
+  if (wantv)
+  {
+    for (k = n-1; k >= 0; k--)
+    {
+      if ((k < nrt) & (e[k] != 0.0))
+      {
+        for (j = k+1; j < nu; j++)
+        {
+          Scalar t = m_matV.col(k).end(n-k-1).dot(m_matV.col(j).end(n-k-1)); // FIXME is it really a dot product we want ?
+          t = -t/m_matV(k+1,k);
+          m_matV.col(j).end(n-k-1) += t * m_matV.col(k).end(n-k-1);
+        }
+      }
+      m_matV.col(k).setZero();
+      m_matV(k,k) = 1.0;
+    }
+  }
+
+  // Main iteration loop for the singular values.
+  int pp = p-1;
+  int iter = 0;
+  Scalar eps(pow(2.0,-52.0));
+  while (p > 0)
+  {
+    int k=0;
+    int kase=0;
+
+    // Here is where a test for too many iterations would go.
+
+    // This section of the program inspects for
+    // negligible elements in the s and e arrays.  On
+    // completion the variables kase and k are set as follows.
+
+    // kase = 1     if s(p) and e[k-1] are negligible and k<p
+    // kase = 2     if s(k) is negligible and k<p
+    // kase = 3     if e[k-1] is negligible, k<p, and
+    //              s(k), ..., s(p) are not negligible (qr step).
+    // kase = 4     if e(p-1) is negligible (convergence).
+
+    for (k = p-2; k >= -1; k--)
+    {
+      if (k == -1)
+          break;
+      if (ei_abs(e[k]) <= eps*(ei_abs(m_sigma[k]) + ei_abs(m_sigma[k+1])))
+      {
+          e[k] = 0.0;
+          break;
+      }
+    }
+    if (k == p-2)
+    {
+      kase = 4;
+    }
+    else
+    {
+      int ks;
+      for (ks = p-1; ks >= k; ks--)
+      {
+        if (ks == k)
+          break;
+        Scalar t( (ks != p ? ei_abs(e[ks]) : 0.) + (ks != k+1 ? ei_abs(e[ks-1]) : 0.));
+        if (ei_abs(m_sigma[ks]) <= eps*t)
+        {
+          m_sigma[ks] = 0.0;
+          break;
+        }
+      }
+      if (ks == k)
+      {
+        kase = 3;
+      }
+      else if (ks == p-1)
+      {
+        kase = 1;
+      }
+      else
+      {
+        kase = 2;
+        k = ks;
+      }
+    }
+    k++;
+
+    // Perform the task indicated by kase.
+    switch (kase)
+    {
+
+      // Deflate negligible s(p).
+      case 1:
+      {
+        Scalar f(e[p-2]);
+        e[p-2] = 0.0;
+        for (j = p-2; j >= k; j--)
+        {
+          Scalar t(hypot(m_sigma[j],f));
+          Scalar cs(m_sigma[j]/t);
+          Scalar sn(f/t);
+          m_sigma[j] = t;
+          if (j != k)
+          {
+            f = -sn*e[j-1];
+            e[j-1] = cs*e[j-1];
+          }
+          if (wantv)
+          {
+            for (i = 0; i < n; i++)
+            {
+              t = cs*m_matV(i,j) + sn*m_matV(i,p-1);
+              m_matV(i,p-1) = -sn*m_matV(i,j) + cs*m_matV(i,p-1);
+              m_matV(i,j) = t;
+            }
+          }
+        }
+      }
+      break;
+
+      // Split at negligible s(k).
+      case 2:
+      {
+        Scalar f(e[k-1]);
+        e[k-1] = 0.0;
+        for (j = k; j < p; j++)
+        {
+          Scalar t(hypot(m_sigma[j],f));
+          Scalar cs( m_sigma[j]/t);
+          Scalar sn(f/t);
+          m_sigma[j] = t;
+          f = -sn*e[j];
+          e[j] = cs*e[j];
+          if (wantu)
+          {
+            for (i = 0; i < m; i++)
+            {
+              t = cs*m_matU(i,j) + sn*m_matU(i,k-1);
+              m_matU(i,k-1) = -sn*m_matU(i,j) + cs*m_matU(i,k-1);
+              m_matU(i,j) = t;
+            }
+          }
+        }
+      }
+      break;
+
+      // Perform one qr step.
+      case 3:
+      {
+        // Calculate the shift.
+        Scalar scale = std::max(std::max(std::max(std::max(
+                        ei_abs(m_sigma[p-1]),ei_abs(m_sigma[p-2])),ei_abs(e[p-2])),
+                        ei_abs(m_sigma[k])),ei_abs(e[k]));
+        Scalar sp = m_sigma[p-1]/scale;
+        Scalar spm1 = m_sigma[p-2]/scale;
+        Scalar epm1 = e[p-2]/scale;
+        Scalar sk = m_sigma[k]/scale;
+        Scalar ek = e[k]/scale;
+        Scalar b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
+        Scalar c = (sp*epm1)*(sp*epm1);
+        Scalar shift = 0.0;
+        if ((b != 0.0) || (c != 0.0))
+        {
+          shift = ei_sqrt(b*b + c);
+          if (b < 0.0)
+            shift = -shift;
+          shift = c/(b + shift);
+        }
+        Scalar f = (sk + sp)*(sk - sp) + shift;
+        Scalar g = sk*ek;
+
+        // Chase zeros.
+
+        for (j = k; j < p-1; j++)
+        {
+          Scalar t = hypot(f,g);
+          Scalar cs = f/t;
+          Scalar sn = g/t;
+          if (j != k)
+            e[j-1] = t;
+          f = cs*m_sigma[j] + sn*e[j];
+          e[j] = cs*e[j] - sn*m_sigma[j];
+          g = sn*m_sigma[j+1];
+          m_sigma[j+1] = cs*m_sigma[j+1];
+          if (wantv)
+          {
+            for (i = 0; i < n; i++)
+            {
+              t = cs*m_matV(i,j) + sn*m_matV(i,j+1);
+              m_matV(i,j+1) = -sn*m_matV(i,j) + cs*m_matV(i,j+1);
+              m_matV(i,j) = t;
+            }
+          }
+          t = hypot(f,g);
+          cs = f/t;
+          sn = g/t;
+          m_sigma[j] = t;
+          f = cs*e[j] + sn*m_sigma[j+1];
+          m_sigma[j+1] = -sn*e[j] + cs*m_sigma[j+1];
+          g = sn*e[j+1];
+          e[j+1] = cs*e[j+1];
+          if (wantu && (j < m-1))
+          {
+            for (i = 0; i < m; i++)
+            {
+              t = cs*m_matU(i,j) + sn*m_matU(i,j+1);
+              m_matU(i,j+1) = -sn*m_matU(i,j) + cs*m_matU(i,j+1);
+              m_matU(i,j) = t;
+            }
+          }
+        }
+        e[p-2] = f;
+        iter = iter + 1;
+      }
+      break;
+
+      // Convergence.
+      case 4:
+      {
+        // Make the singular values positive.
+        if (m_sigma[k] <= 0.0)
+        {
+          m_sigma[k] = (m_sigma[k] < 0.0 ? -m_sigma[k] : 0.0);
+          if (wantv)
+            m_matV.col(k).start(pp+1) = -m_matV.col(k).start(pp+1);
+        }
+
+        // Order the singular values.
+        while (k < pp)
+        {
+          if (m_sigma[k] >= m_sigma[k+1])
+            break;
+          Scalar t = m_sigma[k];
+          m_sigma[k] = m_sigma[k+1];
+          m_sigma[k+1] = t;
+          if (wantv && (k < n-1))
+            m_matV.col(k).swap(m_matV.col(k+1));
+          if (wantu && (k < m-1))
+            m_matU.col(k).swap(m_matU.col(k+1));
+          k++;
+        }
+        iter = 0;
+        p--;
+      }
+      break;
+    } // end big switch
+  } // end iterations
+}
+
+/** \returns the solution of \f$ A x = b \f$ using the current SVD decomposition of A.
+  * The parts of the solution corresponding to zero singular values are ignored.
+  *
+  * \sa MatrixBase::svd(), LU::solve(), Cholesky::solve()
+  */
+template<typename MatrixType>
+template<typename OtherDerived, typename ResultType>
+void SVD<MatrixType>::solve(const MatrixBase<OtherDerived> &b, ResultType* result) const
+{
+  const int rows = m_matU.rows();
+  ei_assert(b.rows() == rows);
+
+  for (int j=0; j<b.cols(); ++j)
+  {
+    Matrix<Scalar,MatrixUType::RowsAtCompileTime,1> aux = m_matU.transpose() * b.col(j);
+
+    for (int i = 0; i <m_matU.cols(); i++)
+    {
+      Scalar si = m_sigma.coeff(i);
+      if (si != 0)
+        aux.coeffRef(i) /= si;
+      else
+        aux.coeffRef(i) = 0;
+    }
+
+    result->col(j) = m_matV * aux;
+  }
+}
+
+/** \svd_module
+  * \returns the SVD decomposition of \c *this
+  */
+template<typename Derived>
+inline const SVD<typename MatrixBase<Derived>::EvalType>
+MatrixBase<Derived>::svd() const
+{
+  return SVD<typename ei_eval<Derived>::type>(derived());
+}
+
+#endif // EIGEN_SVD_H

doc/Doxyfile.in

@@ -201,6 +201,7 @@ ALIASES                = "only_for_vectors=This is only for vectors (either row-
                          "lu_module=This is defined in the %LU module. \code #include <Eigen/LU> \endcode" \
                          "cholesky_module=This is defined in the %Cholesky module. \code #include <Eigen/Cholesky> \endcode" \
                          "qr_module=This is defined in the %QR module. \code #include <Eigen/QR> \endcode" \
+                         "svd_module=This is defined in the %SVD module. \code #include <Eigen/SVD> \endcode" \
                          "geometry_module=This is defined in the %Geometry module. \code #include <Eigen/Geometry> \endcode" \
                          "regression_module=This is defined in the %Regression module. \code #include <Eigen/Regression> \endcode" \
                          "addexample=\anchor"  \

test/CMakeLists.txt

@@ -106,5 +106,6 @@ EI_ADD_TEST(qr)
 EI_ADD_TEST(eigensolver)
 EI_ADD_TEST(geometry)
 EI_ADD_TEST(regression)
+EI_ADD_TEST(svd)
 
 ENDIF(BUILD_TESTS)

test/svd.cpp

@@ -0,0 +1,68 @@
+// 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/>.
+
+#include "main.h"
+#include <Eigen/SVD>
+
+template<typename MatrixType> void svd(const MatrixType& m)
+{
+  /* this test covers the following files:
+     SVD.h
+  */
+  int rows = m.rows();
+  int cols = m.cols();
+
+  typedef typename MatrixType::Scalar Scalar;
+  MatrixType a = MatrixType::Random(rows,cols);
+  Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> b =
+    Matrix<Scalar, MatrixType::RowsAtCompileTime, 1>::Random(rows,1);
+  Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> x(cols,1), x2(cols,1);
+
+  SVD<MatrixType> svd(a);
+  MatrixType sigma = MatrixType::Zero(rows,cols);
+  MatrixType matU  = MatrixType::Zero(rows,rows);
+  sigma.block(0,0,cols,cols) = svd.singularValues().asDiagonal();
+  matU.block(0,0,rows,cols) = svd.matrixU();
+
+  VERIFY_IS_APPROX(a, matU * sigma * svd.matrixV().transpose());
+
+  if (rows==cols)
+  {
+    svd.solve(b, &x);
+    VERIFY_IS_APPROX(a * x, b);
+  }
+}
+
+void test_svd()
+{
+  for(int i = 0; i < g_repeat; i++) {
+    CALL_SUBTEST( svd(Matrix3f()) );
+    CALL_SUBTEST( svd(Matrix4d()) );
+    CALL_SUBTEST( svd(MatrixXf(7,7)) );
+    CALL_SUBTEST( svd(MatrixXf(14,7)) );
+    // complex are not implemented yet
+//     CALL_SUBTEST( svd(MatrixXcd(6,6)) );
+//     CALL_SUBTEST( svd(MatrixXcf(3,3)) );
+  }
+}