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 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2010 Benoit Jacob <jacob.benoit.1@gmail.com>
 // Copyright (C) 2013-2014 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_BIDIAGONALIZATION_H
 #define EIGEN_BIDIAGONALIZATION_H
 
 namespace Eigen { 
 
 namespace internal {
 // UpperBidiagonalization will probably be replaced by a Bidiagonalization class, don't want to make it stable API.
 // At the same time, it's useful to keep for now as it's about the only thing that is testing the BandMatrix class.
 
 template<typename _MatrixType> class UpperBidiagonalization
 {
   public:
 
     typedef _MatrixType MatrixType;
     enum {
       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
       ColsAtCompileTimeMinusOne = internal::decrement_size<ColsAtCompileTime>::ret
     };
     typedef typename MatrixType::Scalar Scalar;
     typedef typename MatrixType::RealScalar RealScalar;
     typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
     typedef Matrix<Scalar, 1, ColsAtCompileTime> RowVectorType;
     typedef Matrix<Scalar, RowsAtCompileTime, 1> ColVectorType;
     typedef BandMatrix<RealScalar, ColsAtCompileTime, ColsAtCompileTime, 1, 0, RowMajor> BidiagonalType;
     typedef Matrix<Scalar, ColsAtCompileTime, 1> DiagVectorType;
     typedef Matrix<Scalar, ColsAtCompileTimeMinusOne, 1> SuperDiagVectorType;
     typedef HouseholderSequence<
               const MatrixType,
               const typename internal::remove_all<typename Diagonal<const MatrixType,0>::ConjugateReturnType>::type
             > HouseholderUSequenceType;
     typedef HouseholderSequence<
               const typename internal::remove_all<typename MatrixType::ConjugateReturnType>::type,
               Diagonal<const MatrixType,1>,
               OnTheRight
             > HouseholderVSequenceType;
     
     /**
     * \brief Default Constructor.
     *
     * The default constructor is useful in cases in which the user intends to
     * perform decompositions via Bidiagonalization::compute(const MatrixType&).
     */
     UpperBidiagonalization() : m_householder(), m_bidiagonal(), m_isInitialized(false) {}
 
     explicit UpperBidiagonalization(const MatrixType& matrix)
       : m_householder(matrix.rows(), matrix.cols()),
         m_bidiagonal(matrix.cols(), matrix.cols()),
         m_isInitialized(false)
     {
       compute(matrix);
     }
     
     UpperBidiagonalization& compute(const MatrixType& matrix);
     UpperBidiagonalization& computeUnblocked(const MatrixType& matrix);
     
     const MatrixType& householder() const { return m_householder; }
     const BidiagonalType& bidiagonal() const { return m_bidiagonal; }
     
     const HouseholderUSequenceType householderU() const
     {
       eigen_assert(m_isInitialized && "UpperBidiagonalization is not initialized.");
       return HouseholderUSequenceType(m_householder, m_householder.diagonal().conjugate());
     }
 
     const HouseholderVSequenceType householderV() // const here gives nasty errors and i'm lazy
     {
       eigen_assert(m_isInitialized && "UpperBidiagonalization is not initialized.");
       return HouseholderVSequenceType(m_householder.conjugate(), m_householder.const_derived().template diagonal<1>())
              .setLength(m_householder.cols()-1)
              .setShift(1);
     }
     
   protected:
     MatrixType m_householder;
     BidiagonalType m_bidiagonal;
     bool m_isInitialized;
 };
 
 // Standard upper bidiagonalization without fancy optimizations
 // This version should be faster for small matrix size
 template<typename MatrixType>
 void upperbidiagonalization_inplace_unblocked(MatrixType& mat,
                                               typename MatrixType::RealScalar *diagonal,
                                               typename MatrixType::RealScalar *upper_diagonal,
                                               typename MatrixType::Scalar* tempData = 0)
 {
   typedef typename MatrixType::Scalar Scalar;
 
   Index rows = mat.rows();
   Index cols = mat.cols();
 
   typedef Matrix<Scalar,Dynamic,1,ColMajor,MatrixType::MaxRowsAtCompileTime,1> TempType;
   TempType tempVector;
   if(tempData==0)
   {
     tempVector.resize(rows);
     tempData = tempVector.data();
   }
 
   for (Index k = 0; /* breaks at k==cols-1 below */ ; ++k)
   {
     Index remainingRows = rows - k;
     Index remainingCols = cols - k - 1;
 
     // construct left householder transform in-place in A
     mat.col(k).tail(remainingRows)
        .makeHouseholderInPlace(mat.coeffRef(k,k), diagonal[k]);
     // apply householder transform to remaining part of A on the left
     mat.bottomRightCorner(remainingRows, remainingCols)
        .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows-1), mat.coeff(k,k), tempData);
 
     if(k == cols-1) break;
 
     // construct right householder transform in-place in mat
     mat.row(k).tail(remainingCols)
        .makeHouseholderInPlace(mat.coeffRef(k,k+1), upper_diagonal[k]);
     // apply householder transform to remaining part of mat on the left
     mat.bottomRightCorner(remainingRows-1, remainingCols)
        .applyHouseholderOnTheRight(mat.row(k).tail(remainingCols-1).adjoint(), mat.coeff(k,k+1), tempData);
   }
 }
 
 /** \internal
   * Helper routine for the block reduction to upper bidiagonal form.
   *
   * Let's partition the matrix A:
   * 
   *      | A00 A01 |
   *  A = |         |
   *      | A10 A11 |
   *
   * This function reduces to bidiagonal form the left \c rows x \a blockSize vertical panel [A00/A10]
   * and the \a blockSize x \c cols horizontal panel [A00 A01] of the matrix \a A. The bottom-right block A11
   * is updated using matrix-matrix products:
   *   A22 -= V * Y^T - X * U^T
   * where V and U contains the left and right Householder vectors. U and V are stored in A10, and A01
   * respectively, and the update matrices X and Y are computed during the reduction.
   * 
   */
 template<typename MatrixType>
 void upperbidiagonalization_blocked_helper(MatrixType& A,
                                            typename MatrixType::RealScalar *diagonal,
                                            typename MatrixType::RealScalar *upper_diagonal,
                                            Index bs,
                                            Ref<Matrix<typename MatrixType::Scalar, Dynamic, Dynamic,
                                                       traits<MatrixType>::Flags & RowMajorBit> > X,
                                            Ref<Matrix<typename MatrixType::Scalar, Dynamic, Dynamic,
                                                       traits<MatrixType>::Flags & RowMajorBit> > Y)
 {
   typedef typename MatrixType::Scalar Scalar;
   typedef typename MatrixType::RealScalar RealScalar;
   typedef typename NumTraits<RealScalar>::Literal Literal;
   enum { StorageOrder = traits<MatrixType>::Flags & RowMajorBit };
   typedef InnerStride<int(StorageOrder) == int(ColMajor) ? 1 : Dynamic> ColInnerStride;
   typedef InnerStride<int(StorageOrder) == int(ColMajor) ? Dynamic : 1> RowInnerStride;
   typedef Ref<Matrix<Scalar, Dynamic, 1>, 0, ColInnerStride>    SubColumnType;
   typedef Ref<Matrix<Scalar, 1, Dynamic>, 0, RowInnerStride>    SubRowType;
   typedef Ref<Matrix<Scalar, Dynamic, Dynamic, StorageOrder > > SubMatType;
   
   Index brows = A.rows();
   Index bcols = A.cols();
 
   Scalar tau_u, tau_u_prev(0), tau_v;
 
   for(Index k = 0; k < bs; ++k)
   {
     Index remainingRows = brows - k;
     Index remainingCols = bcols - k - 1;
 
     SubMatType X_k1( X.block(k,0, remainingRows,k) );
     SubMatType V_k1( A.block(k,0, remainingRows,k) );
 
     // 1 - update the k-th column of A
     SubColumnType v_k = A.col(k).tail(remainingRows);
           v_k -= V_k1 * Y.row(k).head(k).adjoint();
     if(k) v_k -= X_k1 * A.col(k).head(k);
     
     // 2 - construct left Householder transform in-place
     v_k.makeHouseholderInPlace(tau_v, diagonal[k]);
        
     if(k+1<bcols)
     {
       SubMatType Y_k  ( Y.block(k+1,0, remainingCols, k+1) );
       SubMatType U_k1 ( A.block(0,k+1, k,remainingCols) );
       
       // this eases the application of Householder transforAions
       // A(k,k) will store tau_v later
       A(k,k) = Scalar(1);
 
       // 3 - Compute y_k^T = tau_v * ( A^T*v_k - Y_k-1*V_k-1^T*v_k - U_k-1*X_k-1^T*v_k )
       {
         SubColumnType y_k( Y.col(k).tail(remainingCols) );
         
         // let's use the beginning of column k of Y as a temporary vector
         SubColumnType tmp( Y.col(k).head(k) );
         y_k.noalias()  = A.block(k,k+1, remainingRows,remainingCols).adjoint() * v_k; // bottleneck
         tmp.noalias()  = V_k1.adjoint()  * v_k;
         y_k.noalias() -= Y_k.leftCols(k) * tmp;
         tmp.noalias()  = X_k1.adjoint()  * v_k;
         y_k.noalias() -= U_k1.adjoint()  * tmp;
         y_k *= numext::conj(tau_v);
       }
 
       // 4 - update k-th row of A (it will become u_k)
       SubRowType u_k( A.row(k).tail(remainingCols) );
       u_k = u_k.conjugate();
       {
         u_k -= Y_k * A.row(k).head(k+1).adjoint();
         if(k) u_k -= U_k1.adjoint() * X.row(k).head(k).adjoint();
       }
 
       // 5 - construct right Householder transform in-place
       u_k.makeHouseholderInPlace(tau_u, upper_diagonal[k]);
 
       // this eases the application of Householder transformations
       // A(k,k+1) will store tau_u later
       A(k,k+1) = Scalar(1);
 
       // 6 - Compute x_k = tau_u * ( A*u_k - X_k-1*U_k-1^T*u_k - V_k*Y_k^T*u_k )
       {
         SubColumnType x_k ( X.col(k).tail(remainingRows-1) );
         
         // let's use the beginning of column k of X as a temporary vectors
         // note that tmp0 and tmp1 overlaps
         SubColumnType tmp0 ( X.col(k).head(k) ),
                       tmp1 ( X.col(k).head(k+1) );
                     
         x_k.noalias()   = A.block(k+1,k+1, remainingRows-1,remainingCols) * u_k.transpose(); // bottleneck
         tmp0.noalias()  = U_k1 * u_k.transpose();
         x_k.noalias()  -= X_k1.bottomRows(remainingRows-1) * tmp0;
         tmp1.noalias()  = Y_k.adjoint() * u_k.transpose();
         x_k.noalias()  -= A.block(k+1,0, remainingRows-1,k+1) * tmp1;
         x_k *= numext::conj(tau_u);
         tau_u = numext::conj(tau_u);
         u_k = u_k.conjugate();
       }
 
       if(k>0) A.coeffRef(k-1,k) = tau_u_prev;
       tau_u_prev = tau_u;
     }
     else
       A.coeffRef(k-1,k) = tau_u_prev;
 
     A.coeffRef(k,k) = tau_v;
   }
   
   if(bs<bcols)
     A.coeffRef(bs-1,bs) = tau_u_prev;
 
   // update A22
   if(bcols>bs && brows>bs)
   {
     SubMatType A11( A.bottomRightCorner(brows-bs,bcols-bs) );
     SubMatType A10( A.block(bs,0, brows-bs,bs) );
     SubMatType A01( A.block(0,bs, bs,bcols-bs) );
     Scalar tmp = A01(bs-1,0);
     A01(bs-1,0) = Literal(1);
     A11.noalias() -= A10 * Y.topLeftCorner(bcols,bs).bottomRows(bcols-bs).adjoint();
     A11.noalias() -= X.topLeftCorner(brows,bs).bottomRows(brows-bs) * A01;
     A01(bs-1,0) = tmp;
   }
 }
 
 /** \internal
   *
   * Implementation of a block-bidiagonal reduction.
   * It is based on the following paper:
   *   The Design of a Parallel Dense Linear Algebra Software Library: Reduction to Hessenberg, Tridiagonal, and Bidiagonal Form.
   *   by Jaeyoung Choi, Jack J. Dongarra, David W. Walker. (1995)
   *   section 3.3
   */
 template<typename MatrixType, typename BidiagType>
 void upperbidiagonalization_inplace_blocked(MatrixType& A, BidiagType& bidiagonal,
                                             Index maxBlockSize=32,
                                             typename MatrixType::Scalar* /*tempData*/ = 0)
 {
   typedef typename MatrixType::Scalar Scalar;
   typedef Block<MatrixType,Dynamic,Dynamic> BlockType;
 
   Index rows = A.rows();
   Index cols = A.cols();
   Index size = (std::min)(rows, cols);
 
   // X and Y are work space
   enum { StorageOrder = traits<MatrixType>::Flags & RowMajorBit };
   Matrix<Scalar,
          MatrixType::RowsAtCompileTime,
          Dynamic,
          StorageOrder,
          MatrixType::MaxRowsAtCompileTime> X(rows,maxBlockSize);
   Matrix<Scalar,
          MatrixType::ColsAtCompileTime,
          Dynamic,
          StorageOrder,
          MatrixType::MaxColsAtCompileTime> Y(cols,maxBlockSize);
   Index blockSize = (std::min)(maxBlockSize,size);
 
   Index k = 0;
   for(k = 0; k < size; k += blockSize)
   {
     Index bs = (std::min)(size-k,blockSize);  // actual size of the block
     Index brows = rows - k;                   // rows of the block
     Index bcols = cols - k;                   // columns of the block
 
     // partition the matrix A:
     // 
     //      | A00 A01 A02 |
     //      |             |
     // A  = | A10 A11 A12 |
     //      |             |
     //      | A20 A21 A22 |
     //
     // where A11 is a bs x bs diagonal block,
     // and let:
     //      | A11 A12 |
     //  B = |         |
     //      | A21 A22 |
 
     BlockType B = A.block(k,k,brows,bcols);
     
     // This stage performs the bidiagonalization of A11, A21, A12, and updating of A22.
     // Finally, the algorithm continue on the updated A22.
     //
     // However, if B is too small, or A22 empty, then let's use an unblocked strategy
     if(k+bs==cols || bcols<48) // somewhat arbitrary threshold
     {
       upperbidiagonalization_inplace_unblocked(B,
                                                &(bidiagonal.template diagonal<0>().coeffRef(k)),
                                                &(bidiagonal.template diagonal<1>().coeffRef(k)),
                                                X.data()
                                               );
       break; // We're done
     }
     else
     {
       upperbidiagonalization_blocked_helper<BlockType>( B,
                                                         &(bidiagonal.template diagonal<0>().coeffRef(k)),
                                                         &(bidiagonal.template diagonal<1>().coeffRef(k)),
                                                         bs,
                                                         X.topLeftCorner(brows,bs),
                                                         Y.topLeftCorner(bcols,bs)
                                                       );
     }
   }
 }
 
 template<typename _MatrixType>
 UpperBidiagonalization<_MatrixType>& UpperBidiagonalization<_MatrixType>::computeUnblocked(const _MatrixType& matrix)
 {
   Index rows = matrix.rows();
   Index cols = matrix.cols();
   EIGEN_ONLY_USED_FOR_DEBUG(cols);
 
   eigen_assert(rows >= cols && "UpperBidiagonalization is only for Arices satisfying rows>=cols.");
 
   m_householder = matrix;
 
   ColVectorType temp(rows);
 
   upperbidiagonalization_inplace_unblocked(m_householder,
                                            &(m_bidiagonal.template diagonal<0>().coeffRef(0)),
                                            &(m_bidiagonal.template diagonal<1>().coeffRef(0)),
                                            temp.data());
 
   m_isInitialized = true;
   return *this;
 }
 
 template<typename _MatrixType>
 UpperBidiagonalization<_MatrixType>& UpperBidiagonalization<_MatrixType>::compute(const _MatrixType& matrix)
 {
   Index rows = matrix.rows();
   Index cols = matrix.cols();
   EIGEN_ONLY_USED_FOR_DEBUG(rows);
   EIGEN_ONLY_USED_FOR_DEBUG(cols);
 
   eigen_assert(rows >= cols && "UpperBidiagonalization is only for Arices satisfying rows>=cols.");
 
   m_householder = matrix;
   upperbidiagonalization_inplace_blocked(m_householder, m_bidiagonal);
             
   m_isInitialized = true;
   return *this;
 }
 
 #if 0
 /** \return the Householder QR decomposition of \c *this.
   *
   * \sa class Bidiagonalization
   */
 template<typename Derived>
 const UpperBidiagonalization<typename MatrixBase<Derived>::PlainObject>
 MatrixBase<Derived>::bidiagonalization() const
 {
   return UpperBidiagonalization<PlainObject>(eval());
 }
 #endif
 
 } // end namespace internal
 
 } // end namespace Eigen
 
 #endif // EIGEN_BIDIAGONALIZATION_H

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