SparseQR.h

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012-2013 Desire Nuentsa <desire.nuentsa_wakam@inria.fr>
// Copyright (C) 2012-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// 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_SPARSE_QR_H
#define EIGEN_SPARSE_QR_H
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
namespace Eigen {
 
template <typename MatrixType, typename OrderingType>
class SparseQR;
template <typename SparseQRType>
struct SparseQRMatrixQReturnType;
template <typename SparseQRType>
struct SparseQRMatrixQTransposeReturnType;
template <typename SparseQRType, typename Derived>
struct SparseQR_QProduct;
namespace internal {
template <typename SparseQRType>
struct traits<SparseQRMatrixQReturnType<SparseQRType> > {
  typedef typename SparseQRType::MatrixType ReturnType;
  typedef typename ReturnType::StorageIndex StorageIndex;
  typedef typename ReturnType::StorageKind StorageKind;
  enum { RowsAtCompileTime = Dynamic, ColsAtCompileTime = Dynamic };
};
template <typename SparseQRType>
struct traits<SparseQRMatrixQTransposeReturnType<SparseQRType> > {
  typedef typename SparseQRType::MatrixType ReturnType;
};
template <typename SparseQRType, typename Derived>
struct traits<SparseQR_QProduct<SparseQRType, Derived> > {
  typedef typename Derived::PlainObject ReturnType;
};
}  // End namespace internal
 
template <typename MatrixType_, typename OrderingType_>
class SparseQR : public SparseSolverBase<SparseQR<MatrixType_, OrderingType_> > {
 protected:
  typedef SparseSolverBase<SparseQR<MatrixType_, OrderingType_> > Base;
  using Base::m_isInitialized;
 
 public:
  using Base::_solve_impl;
  typedef MatrixType_ MatrixType;
  typedef OrderingType_ OrderingType;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef typename MatrixType::StorageIndex StorageIndex;
  typedef SparseMatrix<Scalar, ColMajor, StorageIndex> QRMatrixType;
  typedef Matrix<StorageIndex, Dynamic, 1> IndexVector;
  typedef Matrix<Scalar, Dynamic, 1> ScalarVector;
  typedef PermutationMatrix<Dynamic, Dynamic, StorageIndex> PermutationType;
 
  enum { ColsAtCompileTime = MatrixType::ColsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime };
 
 public:
  SparseQR()
      : m_analysisIsok(false), m_lastError(""), m_useDefaultThreshold(true), m_isQSorted(false), m_isEtreeOk(false) {}
 
  explicit SparseQR(const MatrixType& mat)
      : m_analysisIsok(false), m_lastError(""), m_useDefaultThreshold(true), m_isQSorted(false), m_isEtreeOk(false) {
    compute(mat);
  }
 
  void compute(const MatrixType& mat) {
    analyzePattern(mat);
    factorize(mat);
  }
  void analyzePattern(const MatrixType& mat);
  void factorize(const MatrixType& mat);
 
  inline Index rows() const { return m_pmat.rows(); }
 
  inline Index cols() const { return m_pmat.cols(); }
 
  const QRMatrixType& matrixR() const { return m_R; }
 
  Index rank() const {
    eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
    return m_nonzeropivots;
  }
 
  SparseQRMatrixQReturnType<SparseQR> matrixQ() const { return SparseQRMatrixQReturnType<SparseQR>(*this); }
 
  const PermutationType& colsPermutation() const {
    eigen_assert(m_isInitialized && "Decomposition is not initialized.");
    return m_outputPerm_c;
  }
 
  std::string lastErrorMessage() const { return m_lastError; }
 
  template <typename Rhs, typename Dest>
  bool _solve_impl(const MatrixBase<Rhs>& B, MatrixBase<Dest>& dest) const {
    eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
    eigen_assert(this->rows() == B.rows() &&
                 "SparseQR::solve() : invalid number of rows in the right hand side matrix");
 
    Index rank = this->rank();
 
    // Compute Q^* * b;
    typename Dest::PlainObject y, b;
    y = this->matrixQ().adjoint() * B;
    b = y;
 
    // Solve with the triangular matrix R
    y.resize((std::max<Index>)(cols(), y.rows()), y.cols());
    y.topRows(rank) = this->matrixR().topLeftCorner(rank, rank).template triangularView<Upper>().solve(b.topRows(rank));
    y.bottomRows(y.rows() - rank).setZero();
 
    // Apply the column permutation
    if (m_perm_c.size())
      dest = colsPermutation() * y.topRows(cols());
    else
      dest = y.topRows(cols());
 
    m_info = Success;
    return true;
  }
 
  void setPivotThreshold(const RealScalar& threshold) {
    m_useDefaultThreshold = false;
    m_threshold = threshold;
  }
 
  template <typename Rhs>
  inline const Solve<SparseQR, Rhs> solve(const MatrixBase<Rhs>& B) const {
    eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
    eigen_assert(this->rows() == B.rows() &&
                 "SparseQR::solve() : invalid number of rows in the right hand side matrix");
    return Solve<SparseQR, Rhs>(*this, B.derived());
  }
  template <typename Rhs>
  inline const Solve<SparseQR, Rhs> solve(const SparseMatrixBase<Rhs>& B) const {
    eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
    eigen_assert(this->rows() == B.rows() &&
                 "SparseQR::solve() : invalid number of rows in the right hand side matrix");
    return Solve<SparseQR, Rhs>(*this, B.derived());
  }
 
  ComputationInfo info() const {
    eigen_assert(m_isInitialized && "Decomposition is not initialized.");
    return m_info;
  }
 
  inline void _sort_matrix_Q() {
    if (this->m_isQSorted) return;
    // The matrix Q is sorted during the transposition
    SparseMatrix<Scalar, RowMajor, Index> mQrm(this->m_Q);
    this->m_Q = mQrm;
    this->m_isQSorted = true;
  }
 
 protected:
  bool m_analysisIsok;
  bool m_factorizationIsok;
  mutable ComputationInfo m_info;
  std::string m_lastError;
  QRMatrixType m_pmat;             // Temporary matrix
  QRMatrixType m_R;                // The triangular factor matrix
  QRMatrixType m_Q;                // The orthogonal reflectors
  ScalarVector m_hcoeffs;          // The Householder coefficients
  PermutationType m_perm_c;        // Fill-reducing  Column  permutation
  PermutationType m_pivotperm;     // The permutation for rank revealing
  PermutationType m_outputPerm_c;  // The final column permutation
  RealScalar m_threshold;          // Threshold to determine null Householder reflections
  bool m_useDefaultThreshold;      // Use default threshold
  Index m_nonzeropivots;           // Number of non zero pivots found
  IndexVector m_etree;             // Column elimination tree
  IndexVector m_firstRowElt;       // First element in each row
  bool m_isQSorted;                // whether Q is sorted or not
  bool m_isEtreeOk;                // whether the elimination tree match the initial input matrix
 
  template <typename, typename>
  friend struct SparseQR_QProduct;
};
 
template <typename MatrixType, typename OrderingType>
void SparseQR<MatrixType, OrderingType>::analyzePattern(const MatrixType& mat) {
  eigen_assert(
      mat.isCompressed() &&
      "SparseQR requires a sparse matrix in compressed mode. Call .makeCompressed() before passing it to SparseQR");
  // Copy to a column major matrix if the input is rowmajor
  std::conditional_t<MatrixType::IsRowMajor, QRMatrixType, const MatrixType&> matCpy(mat);
  // Compute the column fill reducing ordering
  OrderingType ord;
  ord(matCpy, m_perm_c);
  Index n = mat.cols();
  Index m = mat.rows();
  Index diagSize = (std::min)(m, n);
 
  if (!m_perm_c.size()) {
    m_perm_c.resize(n);
    m_perm_c.indices().setLinSpaced(n, 0, StorageIndex(n - 1));
  }
 
  // Compute the column elimination tree of the permuted matrix
  m_outputPerm_c = m_perm_c.inverse();
  internal::coletree(matCpy, m_etree, m_firstRowElt, m_outputPerm_c.indices().data());
  m_isEtreeOk = true;
 
  m_R.resize(m, n);
  m_Q.resize(m, diagSize);
 
  // Allocate space for nonzero elements: rough estimation
  m_R.reserve(2 * mat.nonZeros());  // FIXME Get a more accurate estimation through symbolic factorization with the
                                    // etree
  m_Q.reserve(2 * mat.nonZeros());
  m_hcoeffs.resize(diagSize);
  m_analysisIsok = true;
}
 
template <typename MatrixType, typename OrderingType>
void SparseQR<MatrixType, OrderingType>::factorize(const MatrixType& mat) {
  using std::abs;
 
  eigen_assert(m_analysisIsok && "analyzePattern() should be called before this step");
  StorageIndex m = StorageIndex(mat.rows());
  StorageIndex n = StorageIndex(mat.cols());
  StorageIndex diagSize = (std::min)(m, n);
  IndexVector mark((std::max)(m, n));
  mark.setConstant(-1);          // Record the visited nodes
  IndexVector Ridx(n), Qidx(m);  // Store temporarily the row indexes for the current column of R and Q
  Index nzcolR, nzcolQ;          // Number of nonzero for the current column of R and Q
  ScalarVector tval(m);          // The dense vector used to compute the current column
 
  m_R.setZero();
  m_Q.setZero();
  m_pmat = mat;
  if (!m_isEtreeOk) {
    m_outputPerm_c = m_perm_c.inverse();
    internal::coletree(m_pmat, m_etree, m_firstRowElt, m_outputPerm_c.indices().data());
    m_isEtreeOk = true;
  }
 
  m_pmat.uncompress();  // To have the innerNonZeroPtr allocated
 
  // Apply the fill-in reducing permutation lazily:
  {
    // If the input is row major, copy the original column indices,
    // otherwise directly use the input matrix
    //
    IndexVector originalOuterIndicesCpy;
    const StorageIndex* originalOuterIndices = mat.outerIndexPtr();
    if (MatrixType::IsRowMajor) {
      originalOuterIndicesCpy = IndexVector::Map(m_pmat.outerIndexPtr(), n + 1);
      originalOuterIndices = originalOuterIndicesCpy.data();
    }
 
    for (int i = 0; i < n; i++) {
      Index p = m_perm_c.size() ? m_perm_c.indices()(i) : i;
      m_pmat.outerIndexPtr()[p] = originalOuterIndices[i];
      m_pmat.innerNonZeroPtr()[p] = originalOuterIndices[i + 1] - originalOuterIndices[i];
    }
  }
 
  /* Compute the default threshold as in MatLab, see:
   * Tim Davis, "Algorithm 915, SuiteSparseQR: Multifrontal Multithreaded Rank-Revealing
   * Sparse QR Factorization, ACM Trans. on Math. Soft. 38(1), 2011, Page 8:3
   */
  RealScalar pivotThreshold;
  if (m_useDefaultThreshold) {
    RealScalar max2Norm = 0.0;
    for (int j = 0; j < n; j++) max2Norm = numext::maxi(max2Norm, m_pmat.col(j).norm());
    if (max2Norm == RealScalar(0)) max2Norm = RealScalar(1);
    pivotThreshold = 20 * (m + n) * max2Norm * NumTraits<RealScalar>::epsilon();
  } else {
    pivotThreshold = m_threshold;
  }
 
  // Initialize the numerical permutation
  m_pivotperm.setIdentity(n);
 
  StorageIndex nonzeroCol = 0;  // Record the number of valid pivots
  m_Q.startVec(0);
 
  // Left looking rank-revealing QR factorization: compute a column of R and Q at a time
  for (StorageIndex col = 0; col < n; ++col) {
    mark.setConstant(-1);
    m_R.startVec(col);
    mark(nonzeroCol) = col;
    Qidx(0) = nonzeroCol;
    nzcolR = 0;
    nzcolQ = 1;
    bool found_diag = nonzeroCol >= m;
    tval.setZero();
 
    // Symbolic factorization: find the nonzero locations of the column k of the factors R and Q, i.e.,
    // all the nodes (with indexes lower than rank) reachable through the column elimination tree (etree) rooted at node
    // k. Note: if the diagonal entry does not exist, then its contribution must be explicitly added, thus the trick
    // with found_diag that permits to do one more iteration on the diagonal element if this one has not been found.
    for (typename QRMatrixType::InnerIterator itp(m_pmat, col); itp || !found_diag; ++itp) {
      StorageIndex curIdx = nonzeroCol;
      if (itp) curIdx = StorageIndex(itp.row());
      if (curIdx == nonzeroCol) found_diag = true;
 
      // Get the nonzeros indexes of the current column of R
      StorageIndex st = m_firstRowElt(curIdx);  // The traversal of the etree starts here
      if (st < 0) {
        m_lastError = "Empty row found during numerical factorization";
        m_info = InvalidInput;
        return;
      }
 
      // Traverse the etree
      Index bi = nzcolR;
      for (; mark(st) != col; st = m_etree(st)) {
        Ridx(nzcolR) = st;  // Add this row to the list,
        mark(st) = col;     // and mark this row as visited
        nzcolR++;
      }
 
      // Reverse the list to get the topological ordering
      Index nt = nzcolR - bi;
      for (Index i = 0; i < nt / 2; i++) std::swap(Ridx(bi + i), Ridx(nzcolR - i - 1));
 
      // Copy the current (curIdx,pcol) value of the input matrix
      if (itp)
        tval(curIdx) = itp.value();
      else
        tval(curIdx) = Scalar(0);
 
      // Compute the pattern of Q(:,k)
      if (curIdx > nonzeroCol && mark(curIdx) != col) {
        Qidx(nzcolQ) = curIdx;  // Add this row to the pattern of Q,
        mark(curIdx) = col;     // and mark it as visited
        nzcolQ++;
      }
    }
 
    // Browse all the indexes of R(:,col) in reverse order
    for (Index i = nzcolR - 1; i >= 0; i--) {
      Index curIdx = Ridx(i);
 
      // Apply the curIdx-th householder vector to the current column (temporarily stored into tval)
      Scalar tdot(0);
 
      // First compute q' * tval
      tdot = m_Q.col(curIdx).dot(tval);
 
      tdot *= m_hcoeffs(curIdx);
 
      // Then update tval = tval - q * tau
      tval -= tdot * m_Q.col(curIdx);
 
      // Detect fill-in for the current column of Q
      if (m_etree(Ridx(i)) == nonzeroCol) {
        for (typename QRMatrixType::InnerIterator itq(m_Q, curIdx); itq; ++itq) {
          StorageIndex iQ = StorageIndex(itq.row());
          if (mark(iQ) != col) {
            Qidx(nzcolQ++) = iQ;  // Add this row to the pattern of Q,
            mark(iQ) = col;       // and mark it as visited
          }
        }
      }
    }  // End update current column
 
    Scalar tau = RealScalar(0);
    RealScalar beta = 0;
 
    if (nonzeroCol < diagSize) {
      // Compute the Householder reflection that eliminate the current column
      // FIXME this step should call the Householder module.
      Scalar c0 = nzcolQ ? tval(Qidx(0)) : Scalar(0);
 
      // First, the squared norm of Q((col+1):m, col)
      RealScalar sqrNorm = 0.;
      for (Index itq = 1; itq < nzcolQ; ++itq) sqrNorm += numext::abs2(tval(Qidx(itq)));
      if (sqrNorm == RealScalar(0) && numext::imag(c0) == RealScalar(0)) {
        beta = numext::real(c0);
        tval(Qidx(0)) = 1;
      } else {
        using std::sqrt;
        beta = sqrt(numext::abs2(c0) + sqrNorm);
        if (numext::real(c0) >= RealScalar(0)) beta = -beta;
        tval(Qidx(0)) = 1;
        for (Index itq = 1; itq < nzcolQ; ++itq) tval(Qidx(itq)) /= (c0 - beta);
        tau = numext::conj((beta - c0) / beta);
      }
    }
 
    // Insert values in R
    for (Index i = nzcolR - 1; i >= 0; i--) {
      Index curIdx = Ridx(i);
      if (curIdx < nonzeroCol) {
        m_R.insertBackByOuterInnerUnordered(col, curIdx) = tval(curIdx);
        tval(curIdx) = Scalar(0.);
      }
    }
 
    if (nonzeroCol < diagSize && abs(beta) >= pivotThreshold) {
      m_R.insertBackByOuterInner(col, nonzeroCol) = beta;
      // The householder coefficient
      m_hcoeffs(nonzeroCol) = tau;
      // Record the householder reflections
      for (Index itq = 0; itq < nzcolQ; ++itq) {
        Index iQ = Qidx(itq);
        m_Q.insertBackByOuterInnerUnordered(nonzeroCol, iQ) = tval(iQ);
        tval(iQ) = Scalar(0.);
      }
      nonzeroCol++;
      if (nonzeroCol < diagSize) m_Q.startVec(nonzeroCol);
    } else {
      // Zero pivot found: move implicitly this column to the end
      for (Index j = nonzeroCol; j < n - 1; j++) std::swap(m_pivotperm.indices()(j), m_pivotperm.indices()[j + 1]);
 
      // Recompute the column elimination tree
      internal::coletree(m_pmat, m_etree, m_firstRowElt, m_pivotperm.indices().data());
      m_isEtreeOk = false;
    }
  }
 
  m_hcoeffs.tail(diagSize - nonzeroCol).setZero();
 
  // Finalize the column pointers of the sparse matrices R and Q
  m_Q.finalize();
  m_Q.makeCompressed();
  m_R.finalize();
  m_R.makeCompressed();
  m_isQSorted = false;
 
  m_nonzeropivots = nonzeroCol;
 
  if (nonzeroCol < n) {
    // Permute the triangular factor to put the 'dead' columns to the end
    QRMatrixType tempR(m_R);
    m_R = tempR * m_pivotperm;
 
    // Update the column permutation
    m_outputPerm_c = m_outputPerm_c * m_pivotperm;
  }
 
  m_isInitialized = true;
  m_factorizationIsok = true;
  m_info = Success;
}
 
template <typename SparseQRType, typename Derived>
struct SparseQR_QProduct : ReturnByValue<SparseQR_QProduct<SparseQRType, Derived> > {
  typedef typename SparseQRType::QRMatrixType MatrixType;
  typedef typename SparseQRType::Scalar Scalar;
  // Get the references
  SparseQR_QProduct(const SparseQRType& qr, const Derived& other, bool transpose)
      : m_qr(qr), m_other(other), m_transpose(transpose) {}
  inline Index rows() const { return m_qr.matrixQ().rows(); }
  inline Index cols() const { return m_other.cols(); }
 
  // Assign to a vector
  template <typename DesType>
  void evalTo(DesType& res) const {
    Index m = m_qr.rows();
    Index n = m_qr.cols();
    Index diagSize = (std::min)(m, n);
    res = m_other;
    if (m_transpose) {
      eigen_assert(m_qr.m_Q.rows() == m_other.rows() && "Non conforming object sizes");
      // Compute res = Q' * other column by column
      for (Index j = 0; j < res.cols(); j++) {
        for (Index k = 0; k < diagSize; k++) {
          Scalar tau = Scalar(0);
          tau = m_qr.m_Q.col(k).dot(res.col(j));
          if (tau == Scalar(0)) continue;
          tau = tau * m_qr.m_hcoeffs(k);
          res.col(j) -= tau * m_qr.m_Q.col(k);
        }
      }
    } else {
      eigen_assert(m_qr.matrixQ().cols() == m_other.rows() && "Non conforming object sizes");
 
      res.conservativeResize(rows(), cols());
 
      // Compute res = Q * other column by column
      for (Index j = 0; j < res.cols(); j++) {
        Index start_k = internal::is_identity<Derived>::value ? numext::mini(j, diagSize - 1) : diagSize - 1;
        for (Index k = start_k; k >= 0; k--) {
          Scalar tau = Scalar(0);
          tau = m_qr.m_Q.col(k).dot(res.col(j));
          if (tau == Scalar(0)) continue;
          tau = tau * numext::conj(m_qr.m_hcoeffs(k));
          res.col(j) -= tau * m_qr.m_Q.col(k);
        }
      }
    }
  }
 
  const SparseQRType& m_qr;
  const Derived& m_other;
  bool m_transpose;  // TODO this actually means adjoint
};
 
template <typename SparseQRType>
struct SparseQRMatrixQReturnType : public EigenBase<SparseQRMatrixQReturnType<SparseQRType> > {
  typedef typename SparseQRType::Scalar Scalar;
  typedef Matrix<Scalar, Dynamic, Dynamic> DenseMatrix;
  enum { RowsAtCompileTime = Dynamic, ColsAtCompileTime = Dynamic };
  explicit SparseQRMatrixQReturnType(const SparseQRType& qr) : m_qr(qr) {}
  template <typename Derived>
  SparseQR_QProduct<SparseQRType, Derived> operator*(const MatrixBase<Derived>& other) {
    return SparseQR_QProduct<SparseQRType, Derived>(m_qr, other.derived(), false);
  }
  // To use for operations with the adjoint of Q
  SparseQRMatrixQTransposeReturnType<SparseQRType> adjoint() const {
    return SparseQRMatrixQTransposeReturnType<SparseQRType>(m_qr);
  }
  inline Index rows() const { return m_qr.rows(); }
  inline Index cols() const { return m_qr.rows(); }
  // To use for operations with the transpose of Q FIXME this is the same as adjoint at the moment
  SparseQRMatrixQTransposeReturnType<SparseQRType> transpose() const {
    return SparseQRMatrixQTransposeReturnType<SparseQRType>(m_qr);
  }
  const SparseQRType& m_qr;
};
 
// TODO this actually represents the adjoint of Q
template <typename SparseQRType>
struct SparseQRMatrixQTransposeReturnType {
  explicit SparseQRMatrixQTransposeReturnType(const SparseQRType& qr) : m_qr(qr) {}
  template <typename Derived>
  SparseQR_QProduct<SparseQRType, Derived> operator*(const MatrixBase<Derived>& other) {
    return SparseQR_QProduct<SparseQRType, Derived>(m_qr, other.derived(), true);
  }
  const SparseQRType& m_qr;
};
 
namespace internal {
 
template <typename SparseQRType>
struct evaluator_traits<SparseQRMatrixQReturnType<SparseQRType> > {
  typedef typename SparseQRType::MatrixType MatrixType;
  typedef typename storage_kind_to_evaluator_kind<typename MatrixType::StorageKind>::Kind Kind;
  typedef SparseShape Shape;
};
 
template <typename DstXprType, typename SparseQRType>
struct Assignment<DstXprType, SparseQRMatrixQReturnType<SparseQRType>,
                  internal::assign_op<typename DstXprType::Scalar, typename DstXprType::Scalar>, Sparse2Sparse> {
  typedef SparseQRMatrixQReturnType<SparseQRType> SrcXprType;
  typedef typename DstXprType::Scalar Scalar;
  typedef typename DstXprType::StorageIndex StorageIndex;
  static void run(DstXprType& dst, const SrcXprType& src, const internal::assign_op<Scalar, Scalar>& /*func*/) {
    typename DstXprType::PlainObject idMat(src.rows(), src.cols());
    idMat.setIdentity();
    // Sort the sparse householder reflectors if needed
    const_cast<SparseQRType*>(&src.m_qr)->_sort_matrix_Q();
    dst = SparseQR_QProduct<SparseQRType, DstXprType>(src.m_qr, idMat, false);
  }
};
 
template <typename DstXprType, typename SparseQRType>
struct Assignment<DstXprType, SparseQRMatrixQReturnType<SparseQRType>,
                  internal::assign_op<typename DstXprType::Scalar, typename DstXprType::Scalar>, Sparse2Dense> {
  typedef SparseQRMatrixQReturnType<SparseQRType> SrcXprType;
  typedef typename DstXprType::Scalar Scalar;
  typedef typename DstXprType::StorageIndex StorageIndex;
  static void run(DstXprType& dst, const SrcXprType& src, const internal::assign_op<Scalar, Scalar>& /*func*/) {
    dst = src.m_qr.matrixQ() * DstXprType::Identity(src.m_qr.rows(), src.m_qr.rows());
  }
};
 
}  // end namespace internal
 
}  // end namespace Eigen
 
#endif