BDCSVD.h
No OneTemporary
Actions

Subscribers

None

File Metadata

Created: Sat, Feb 22, 17:58

BDCSVD.h
View Options

	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra.
	//
	// We used the "A Divide-And-Conquer Algorithm for the Bidiagonal SVD"
	// research report written by Ming Gu and Stanley C.Eisenstat
	// The code variable names correspond to the names they used in their
	// report
	//
	// Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
	// Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
	// Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
	// Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
	// Copyright (C) 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
	// Copyright (C) 2014-2017 Gael Guennebaud <gael.guennebaud@inria.fr>
	//
	// 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_BDCSVD_H
	#define EIGEN_BDCSVD_H
	// #define EIGEN_BDCSVD_DEBUG_VERBOSE
	// #define EIGEN_BDCSVD_SANITY_CHECKS

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	#undef eigen_internal_assert
	#define eigen_internal_assert(X) assert(X);
	#endif

	namespace Eigen {

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	IOFormat bdcsvdfmt(8, 0, ", ", "\n", " [", "]");
	#endif

	template<typename _MatrixType> class BDCSVD;

	namespace internal {

	template<typename _MatrixType>
	struct traits<BDCSVD<_MatrixType> >
	: traits<_MatrixType>
	{
	typedef _MatrixType MatrixType;
	};

	} // end namespace internal


	/** \ingroup SVD_Module
	*
	*
	* \class BDCSVD
	*
	* \brief class Bidiagonal Divide and Conquer SVD
	*
	* \tparam _MatrixType the type of the matrix of which we are computing the SVD decomposition
	*
	* This class first reduces the input matrix to bi-diagonal form using class UpperBidiagonalization,
	* and then performs a divide-and-conquer diagonalization. Small blocks are diagonalized using class JacobiSVD.
	* You can control the switching size with the setSwitchSize() method, default is 16.
	* For small matrice (<16), it is thus preferable to directly use JacobiSVD. For larger ones, BDCSVD is highly
	* recommended and can several order of magnitude faster.
	*
	* \warning this algorithm is unlikely to provide accurate result when compiled with unsafe math optimizations.
	* For instance, this concerns Intel's compiler (ICC), which performs such optimization by default unless
	* you compile with the \c -fp-model \c precise option. Likewise, the \c -ffast-math option of GCC or clang will
	* significantly degrade the accuracy.
	*
	* \sa class JacobiSVD
	*/
	template<typename _MatrixType>
	class BDCSVD : public SVDBase<BDCSVD<_MatrixType> >
	{
	typedef SVDBase<BDCSVD> Base;

	public:
	using Base::rows;
	using Base::cols;
	using Base::computeU;
	using Base::computeV;

	typedef _MatrixType MatrixType;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
	typedef typename NumTraits<RealScalar>::Literal Literal;
	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 typename Base::MatrixUType MatrixUType;
	typedef typename Base::MatrixVType MatrixVType;
	typedef typename Base::SingularValuesType SingularValuesType;

	typedef Matrix<Scalar, Dynamic, Dynamic, ColMajor> MatrixX;
	typedef Matrix<RealScalar, Dynamic, Dynamic, ColMajor> MatrixXr;
	typedef Matrix<RealScalar, Dynamic, 1> VectorType;
	typedef Array<RealScalar, Dynamic, 1> ArrayXr;
	typedef Array<Index,1,Dynamic> ArrayXi;
	typedef Ref<ArrayXr> ArrayRef;
	typedef Ref<ArrayXi> IndicesRef;

	/** \brief Default Constructor.
	*
	* The default constructor is useful in cases in which the user intends to
	* perform decompositions via BDCSVD::compute(const MatrixType&).
	*/
	BDCSVD() : m_algoswap(16), m_isTranspose(false), m_compU(false), m_compV(false), m_numIters(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 BDCSVD()
	*/
	BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0)
	: m_algoswap(16), m_numIters(0)
	{
	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.
	*/
	BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
	: m_algoswap(16), m_numIters(0)
	{
	compute(matrix, computationOptions);
	}

	~BDCSVD()
	{
	}

	/** \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.
	*/
	BDCSVD& 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).
	*/
	BDCSVD& compute(const MatrixType& matrix)
	{
	return compute(matrix, this->m_computationOptions);
	}

	void setSwitchSize(int s)
	{
	eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3");
	m_algoswap = s;
	}

	private:
	void allocate(Index rows, Index cols, unsigned int computationOptions);
	void divide(Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift);
	void computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V);
	void computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, VectorType& singVals, ArrayRef shifts, ArrayRef mus);
	void perturbCol0(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat);
	void computeSingVecs(const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V);
	void deflation43(Index firstCol, Index shift, Index i, Index size);
	void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size);
	void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift);
	template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
	void copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naivev);
	void structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1);
	static RealScalar secularEq(RealScalar x, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift);

	protected:
	MatrixXr m_naiveU, m_naiveV;
	MatrixXr m_computed;
	Index m_nRec;
	ArrayXr m_workspace;
	ArrayXi m_workspaceI;
	int m_algoswap;
	bool m_isTranspose, m_compU, m_compV;

	using Base::m_singularValues;
	using Base::m_diagSize;
	using Base::m_computeFullU;
	using Base::m_computeFullV;
	using Base::m_computeThinU;
	using Base::m_computeThinV;
	using Base::m_matrixU;
	using Base::m_matrixV;
	using Base::m_info;
	using Base::m_isInitialized;
	using Base::m_nonzeroSingularValues;

	public:
	int m_numIters;
	}; //end class BDCSVD


	// Method to allocate and initialize matrix and attributes
	template<typename MatrixType>
	void BDCSVD<MatrixType>::allocate(Eigen::Index rows, Eigen::Index cols, unsigned int computationOptions)
	{
	m_isTranspose = (cols > rows);

	if (Base::allocate(rows, cols, computationOptions))
	return;

	m_computed = MatrixXr::Zero(m_diagSize + 1, m_diagSize );
	m_compU = computeV();
	m_compV = computeU();
	if (m_isTranspose)
	std::swap(m_compU, m_compV);

	if (m_compU) m_naiveU = MatrixXr::Zero(m_diagSize + 1, m_diagSize + 1 );
	else m_naiveU = MatrixXr::Zero(2, m_diagSize + 1 );

	if (m_compV) m_naiveV = MatrixXr::Zero(m_diagSize, m_diagSize);

	m_workspace.resize((m_diagSize+1)(m_diagSize+1)3);
	m_workspaceI.resize(3*m_diagSize);
	}// end allocate

	template<typename MatrixType>
	BDCSVD<MatrixType>& BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions)
	{
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "\n\n\n======================================================================================================================\n\n\n";
	#endif
	allocate(matrix.rows(), matrix.cols(), computationOptions);
	using std::abs;

	const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();

	//**** step -1 - If the problem is too small, directly falls back to JacobiSVD and return
	if(matrix.cols() < m_algoswap)
	{
	// FIXME this line involves temporaries
	JacobiSVD<MatrixType> jsvd(matrix,computationOptions);
	m_isInitialized = true;
	m_info = jsvd.info();
	if (m_info == Success \|\| m_info == NoConvergence) {
	if(computeU()) m_matrixU = jsvd.matrixU();
	if(computeV()) m_matrixV = jsvd.matrixV();
	m_singularValues = jsvd.singularValues();
	m_nonzeroSingularValues = jsvd.nonzeroSingularValues();
	}
	return *this;
	}

	//**** step 0 - Copy the input matrix and apply scaling to reduce over/under-flows
	RealScalar scale = matrix.cwiseAbs().template maxCoeff<PropagateNaN>();
	if (!(numext::isfinite)(scale)) {
	m_isInitialized = true;
	m_info = InvalidInput;
	return *this;
	}

	if(scale==Literal(0)) scale = Literal(1);
	MatrixX copy;
	if (m_isTranspose) copy = matrix.adjoint()/scale;
	else copy = matrix/scale;

	//**** step 1 - Bidiagonalization
	// FIXME this line involves temporaries
	internal::UpperBidiagonalization<MatrixX> bid(copy);

	//**** step 2 - Divide & Conquer
	m_naiveU.setZero();
	m_naiveV.setZero();
	// FIXME this line involves a temporary matrix
	m_computed.topRows(m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose();
	m_computed.template bottomRows<1>().setZero();
	divide(0, m_diagSize - 1, 0, 0, 0);
	if (m_info != Success && m_info != NoConvergence) {
	m_isInitialized = true;
	return *this;
	}

	//**** step 3 - Copy singular values and vectors
	for (int i=0; i<m_diagSize; i++)
	{
	RealScalar a = abs(m_computed.coeff(i, i));
	m_singularValues.coeffRef(i) = a * scale;
	if (a<considerZero)
	{
	m_nonzeroSingularValues = i;
	m_singularValues.tail(m_diagSize - i - 1).setZero();
	break;
	}
	else if (i == m_diagSize - 1)
	{
	m_nonzeroSingularValues = i + 1;
	break;
	}
	}

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	// std::cout << "m_naiveU\n" << m_naiveU << "\n\n";
	// std::cout << "m_naiveV\n" << m_naiveV << "\n\n";
	#endif
	if(m_isTranspose) copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU);
	else copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV);

	m_isInitialized = true;
	return *this;
	}// end compute


	template<typename MatrixType>
	template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
	void BDCSVD<MatrixType>::copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naiveV)
	{
	// Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa
	if (computeU())
	{
	Index Ucols = m_computeThinU ? m_diagSize : householderU.cols();
	m_matrixU = MatrixX::Identity(householderU.cols(), Ucols);
	m_matrixU.topLeftCorner(m_diagSize, m_diagSize) = naiveV.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
	householderU.applyThisOnTheLeft(m_matrixU); // FIXME this line involves a temporary buffer
	}
	if (computeV())
	{
	Index Vcols = m_computeThinV ? m_diagSize : householderV.cols();
	m_matrixV = MatrixX::Identity(householderV.cols(), Vcols);
	m_matrixV.topLeftCorner(m_diagSize, m_diagSize) = naiveU.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
	householderV.applyThisOnTheLeft(m_matrixV); // FIXME this line involves a temporary buffer
	}
	}

	/** \internal
	* Performs A = A * B exploiting the special structure of the matrix A. Splitting A as:
	* A = [A1]
	* [A2]
	* such that A1.rows()==n1, then we assume that at least half of the columns of A1 and A2 are zeros.
	* We can thus pack them prior to the the matrix product. However, this is only worth the effort if the matrix is large
	* enough.
	*/
	template<typename MatrixType>
	void BDCSVD<MatrixType>::structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1)
	{
	Index n = A.rows();
	if(n>100)
	{
	// If the matrices are large enough, let's exploit the sparse structure of A by
	// splitting it in half (wrt n1), and packing the non-zero columns.
	Index n2 = n - n1;
	Map<MatrixXr> A1(m_workspace.data() , n1, n);
	Map<MatrixXr> A2(m_workspace.data()+ n1*n, n2, n);
	Map<MatrixXr> B1(m_workspace.data()+ n*n, n, n);
	Map<MatrixXr> B2(m_workspace.data()+2nn, n, n);
	Index k1=0, k2=0;
	for(Index j=0; j<n; ++j)
	{
	if( (A.col(j).head(n1).array()!=Literal(0)).any() )
	{
	A1.col(k1) = A.col(j).head(n1);
	B1.row(k1) = B.row(j);
	++k1;
	}
	if( (A.col(j).tail(n2).array()!=Literal(0)).any() )
	{
	A2.col(k2) = A.col(j).tail(n2);
	B2.row(k2) = B.row(j);
	++k2;
	}
	}

	A.topRows(n1).noalias() = A1.leftCols(k1) * B1.topRows(k1);
	A.bottomRows(n2).noalias() = A2.leftCols(k2) * B2.topRows(k2);
	}
	else
	{
	Map<MatrixXr,Aligned> tmp(m_workspace.data(),n,n);
	tmp.noalias() = A*B;
	A = tmp;
	}
	}

	// The divide algorithm is done "in place", we are always working on subsets of the same matrix. The divide methods takes as argument the
	// place of the submatrix we are currently working on.

	//@param firstCol : The Index of the first column of the submatrix of m_computed and for m_naiveU;
	//@param lastCol : The Index of the last column of the submatrix of m_computed and for m_naiveU;
	// lastCol + 1 - firstCol is the size of the submatrix.
	//@param firstRowW : The Index of the first row of the matrix W that we are to change. (see the reference paper section 1 for more information on W)
	//@param firstRowW : Same as firstRowW with the column.
	//@param shift : Each time one takes the left submatrix, one must add 1 to the shift. Why? Because! We actually want the last column of the U submatrix
	// to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper.
	template<typename MatrixType>
	void BDCSVD<MatrixType>::divide(Eigen::Index firstCol, Eigen::Index lastCol, Eigen::Index firstRowW, Eigen::Index firstColW, Eigen::Index shift)
	{
	// requires rows = cols + 1;
	using std::pow;
	using std::sqrt;
	using std::abs;
	const Index n = lastCol - firstCol + 1;
	const Index k = n/2;
	const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
	RealScalar alphaK;
	RealScalar betaK;
	RealScalar r0;
	RealScalar lambda, phi, c0, s0;
	VectorType l, f;
	// We use the other algorithm which is more efficient for small
	// matrices.
	if (n < m_algoswap)
	{
	// FIXME this line involves temporaries
	JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), ComputeFullU \| (m_compV ? ComputeFullV : 0));
	m_info = b.info();
	if (m_info != Success && m_info != NoConvergence) return;
	if (m_compU)
	m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = b.matrixU();
	else
	{
	m_naiveU.row(0).segment(firstCol, n + 1).real() = b.matrixU().row(0);
	m_naiveU.row(1).segment(firstCol, n + 1).real() = b.matrixU().row(n);
	}
	if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = b.matrixV();
	m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
	m_computed.diagonal().segment(firstCol + shift, n) = b.singularValues().head(n);
	return;
	}
	// We use the divide and conquer algorithm
	alphaK = m_computed(firstCol + k, firstCol + k);
	betaK = m_computed(firstCol + k + 1, firstCol + k);
	// The divide must be done in that order in order to have good results. Divide change the data inside the submatrices
	// and the divide of the right submatrice reads one column of the left submatrice. That's why we need to treat the
	// right submatrix before the left one.
	divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift);
	if (m_info != Success && m_info != NoConvergence) return;
	divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
	if (m_info != Success && m_info != NoConvergence) return;

	if (m_compU)
	{
	lambda = m_naiveU(firstCol + k, firstCol + k);
	phi = m_naiveU(firstCol + k + 1, lastCol + 1);
	}
	else
	{
	lambda = m_naiveU(1, firstCol + k);
	phi = m_naiveU(0, lastCol + 1);
	}
	r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda)) + abs(betaK * phi) * abs(betaK * phi));
	if (m_compU)
	{
	l = m_naiveU.row(firstCol + k).segment(firstCol, k);
	f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1);
	}
	else
	{
	l = m_naiveU.row(1).segment(firstCol, k);
	f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1);
	}
	if (m_compV) m_naiveV(firstRowW+k, firstColW) = Literal(1);
	if (r0<considerZero)
	{
	c0 = Literal(1);
	s0 = Literal(0);
	}
	else
	{
	c0 = alphaK * lambda / r0;
	s0 = betaK * phi / r0;
	}

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(m_naiveU.allFinite());
	assert(m_naiveV.allFinite());
	assert(m_computed.allFinite());
	#endif

	if (m_compU)
	{
	MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1));
	// we shiftW Q1 to the right
	for (Index i = firstCol + k - 1; i >= firstCol; i--)
	m_naiveU.col(i + 1).segment(firstCol, k + 1) = m_naiveU.col(i).segment(firstCol, k + 1);
	// we shift q1 at the left with a factor c0
	m_naiveU.col(firstCol).segment( firstCol, k + 1) = (q1 * c0);
	// last column = q1 * - s0
	m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) = (q1 * ( - s0));
	// first column = q2 * s0
	m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) = m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) * s0;
	// q2 *= c0
	m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0;
	}
	else
	{
	RealScalar q1 = m_naiveU(0, firstCol + k);
	// we shift Q1 to the right
	for (Index i = firstCol + k - 1; i >= firstCol; i--)
	m_naiveU(0, i + 1) = m_naiveU(0, i);
	// we shift q1 at the left with a factor c0
	m_naiveU(0, firstCol) = (q1 * c0);
	// last column = q1 * - s0
	m_naiveU(0, lastCol + 1) = (q1 * ( - s0));
	// first column = q2 * s0
	m_naiveU(1, firstCol) = m_naiveU(1, lastCol + 1) *s0;
	// q2 *= c0
	m_naiveU(1, lastCol + 1) *= c0;
	m_naiveU.row(1).segment(firstCol + 1, k).setZero();
	m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero();
	}

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(m_naiveU.allFinite());
	assert(m_naiveV.allFinite());
	assert(m_computed.allFinite());
	#endif

	m_computed(firstCol + shift, firstCol + shift) = r0;
	m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) = alphaK * l.transpose().real();
	m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) = betaK * f.transpose().real();

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	ArrayXr tmp1 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues();
	#endif
	// Second part: try to deflate singular values in combined matrix
	deflation(firstCol, lastCol, k, firstRowW, firstColW, shift);
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	ArrayXr tmp2 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues();
	std::cout << "\n\nj1 = " << tmp1.transpose().format(bdcsvdfmt) << "\n";
	std::cout << "j2 = " << tmp2.transpose().format(bdcsvdfmt) << "\n\n";
	std::cout << "err: " << ((tmp1-tmp2).abs()>1e-12*tmp2.abs()).transpose() << "\n";
	static int count = 0;
	std::cout << "# " << ++count << "\n\n";
	assert((tmp1-tmp2).matrix().norm() < 1e-14*tmp2.matrix().norm());
	// assert(count<681);
	// assert(((tmp1-tmp2).abs()<1e-13*tmp2.abs()).all());
	#endif

	// Third part: compute SVD of combined matrix
	MatrixXr UofSVD, VofSVD;
	VectorType singVals;
	computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD);

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(UofSVD.allFinite());
	assert(VofSVD.allFinite());
	#endif

	if (m_compU)
	structured_update(m_naiveU.block(firstCol, firstCol, n + 1, n + 1), UofSVD, (n+2)/2);
	else
	{
	Map<Matrix<RealScalar,2,Dynamic>,Aligned> tmp(m_workspace.data(),2,n+1);
	tmp.noalias() = m_naiveU.middleCols(firstCol, n+1) * UofSVD;
	m_naiveU.middleCols(firstCol, n + 1) = tmp;
	}

	if (m_compV) structured_update(m_naiveV.block(firstRowW, firstColW, n, n), VofSVD, (n+1)/2);

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(m_naiveU.allFinite());
	assert(m_naiveV.allFinite());
	assert(m_computed.allFinite());
	#endif

	m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero();
	m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals;
	}// end divide

	// Compute SVD of m_computed.block(firstCol, firstCol, n + 1, n); this block only has non-zeros in
	// the first column and on the diagonal and has undergone deflation, so diagonal is in increasing
	// order except for possibly the (0,0) entry. The computed SVD is stored U, singVals and V, except
	// that if m_compV is false, then V is not computed. Singular values are sorted in decreasing order.
	//
	// TODO Opportunities for optimization: better root finding algo, better stopping criterion, better
	// handling of round-off errors, be consistent in ordering
	// For instance, to solve the secular equation using FMM, see http://www.stat.uchicago.edu/~lekheng/courses/302/classics/greengard-rokhlin.pdf
	template <typename MatrixType>
	void BDCSVD<MatrixType>::computeSVDofM(Eigen::Index firstCol, Eigen::Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V)
	{
	const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
	using std::abs;
	ArrayRef col0 = m_computed.col(firstCol).segment(firstCol, n);
	m_workspace.head(n) = m_computed.block(firstCol, firstCol, n, n).diagonal();
	ArrayRef diag = m_workspace.head(n);
	diag(0) = Literal(0);

	// Allocate space for singular values and vectors
	singVals.resize(n);
	U.resize(n+1, n+1);
	if (m_compV) V.resize(n, n);

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	if (col0.hasNaN() \|\| diag.hasNaN())
	std::cout << "\n\nHAS NAN\n\n";
	#endif

	// Many singular values might have been deflated, the zero ones have been moved to the end,
	// but others are interleaved and we must ignore them at this stage.
	// To this end, let's compute a permutation skipping them:
	Index actual_n = n;
	while(actual_n>1 && diag(actual_n-1)==Literal(0)) {--actual_n; eigen_internal_assert(col0(actual_n)==Literal(0)); }
	Index m = 0; // size of the deflated problem
	for(Index k=0;k<actual_n;++k)
	if(abs(col0(k))>considerZero)
	m_workspaceI(m++) = k;
	Map<ArrayXi> perm(m_workspaceI.data(),m);

	Map<ArrayXr> shifts(m_workspace.data()+1*n, n);
	Map<ArrayXr> mus(m_workspace.data()+2*n, n);
	Map<ArrayXr> zhat(m_workspace.data()+3*n, n);

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "computeSVDofM using:\n";
	std::cout << " z: " << col0.transpose() << "\n";
	std::cout << " d: " << diag.transpose() << "\n";
	#endif

	// Compute singVals, shifts, and mus
	computeSingVals(col0, diag, perm, singVals, shifts, mus);

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << " j: " << (m_computed.block(firstCol, firstCol, n, n)).jacobiSvd().singularValues().transpose().reverse() << "\n\n";
	std::cout << " sing-val: " << singVals.transpose() << "\n";
	std::cout << " mu: " << mus.transpose() << "\n";
	std::cout << " shift: " << shifts.transpose() << "\n";

	{
	std::cout << "\n\n mus: " << mus.head(actual_n).transpose() << "\n\n";
	std::cout << " check1 (expect0) : " << ((singVals.array()-(shifts+mus)) / singVals.array()).head(actual_n).transpose() << "\n\n";
	assert((((singVals.array()-(shifts+mus)) / singVals.array()).head(actual_n) >= 0).all());
	std::cout << " check2 (>0) : " << ((singVals.array()-diag) / singVals.array()).head(actual_n).transpose() << "\n\n";
	assert((((singVals.array()-diag) / singVals.array()).head(actual_n) >= 0).all());
	}
	#endif

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(singVals.allFinite());
	assert(mus.allFinite());
	assert(shifts.allFinite());
	#endif

	// Compute zhat
	perturbCol0(col0, diag, perm, singVals, shifts, mus, zhat);
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << " zhat: " << zhat.transpose() << "\n";
	#endif

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(zhat.allFinite());
	#endif

	computeSingVecs(zhat, diag, perm, singVals, shifts, mus, U, V);

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "U^T U: " << (U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() << "\n";
	std::cout << "V^T V: " << (V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() << "\n";
	#endif

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(m_naiveU.allFinite());
	assert(m_naiveV.allFinite());
	assert(m_computed.allFinite());
	assert(U.allFinite());
	assert(V.allFinite());
	// assert((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() < 100NumTraits<RealScalar>::epsilon() n);
	// assert((V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 100NumTraits<RealScalar>::epsilon() n);
	#endif

	// Because of deflation, the singular values might not be completely sorted.
	// Fortunately, reordering them is a O(n) problem
	for(Index i=0; i<actual_n-1; ++i)
	{
	if(singVals(i)>singVals(i+1))
	{
	using std::swap;
	swap(singVals(i),singVals(i+1));
	U.col(i).swap(U.col(i+1));
	if(m_compV) V.col(i).swap(V.col(i+1));
	}
	}

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	{
	bool singular_values_sorted = (((singVals.segment(1,actual_n-1)-singVals.head(actual_n-1))).array() >= 0).all();
	if(!singular_values_sorted)
	std::cout << "Singular values are not sorted: " << singVals.segment(1,actual_n).transpose() << "\n";
	assert(singular_values_sorted);
	}
	#endif

	// Reverse order so that singular values in increased order
	// Because of deflation, the zeros singular-values are already at the end
	singVals.head(actual_n).reverseInPlace();
	U.leftCols(actual_n).rowwise().reverseInPlace();
	if (m_compV) V.leftCols(actual_n).rowwise().reverseInPlace();

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	JacobiSVD<MatrixXr> jsvd(m_computed.block(firstCol, firstCol, n, n) );
	std::cout << " * j: " << jsvd.singularValues().transpose() << "\n\n";
	std::cout << " * sing-val: " << singVals.transpose() << "\n";
	// std::cout << " * err: " << ((jsvd.singularValues()-singVals)>1e-13*singVals.norm()).transpose() << "\n";
	#endif
	}

	template <typename MatrixType>
	typename BDCSVD<MatrixType>::RealScalar BDCSVD<MatrixType>::secularEq(RealScalar mu, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift)
	{
	Index m = perm.size();
	RealScalar res = Literal(1);
	for(Index i=0; i<m; ++i)
	{
	Index j = perm(i);
	// The following expression could be rewritten to involve only a single division,
	// but this would make the expression more sensitive to overflow.
	res += (col0(j) / (diagShifted(j) - mu)) * (col0(j) / (diag(j) + shift + mu));
	}
	return res;

	}

	template <typename MatrixType>
	void BDCSVD<MatrixType>::computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm,
	VectorType& singVals, ArrayRef shifts, ArrayRef mus)
	{
	using std::abs;
	using std::swap;
	using std::sqrt;

	Index n = col0.size();
	Index actual_n = n;
	// Note that here actual_n is computed based on col0(i)==0 instead of diag(i)==0 as above
	// because 1) we have diag(i)==0 => col0(i)==0 and 2) if col0(i)==0, then diag(i) is already a singular value.
	while(actual_n>1 && col0(actual_n-1)==Literal(0)) --actual_n;

	for (Index k = 0; k < n; ++k)
	{
	if (col0(k) == Literal(0) \|\| actual_n==1)
	{
	// if col0(k) == 0, then entry is deflated, so singular value is on diagonal
	// if actual_n==1, then the deflated problem is already diagonalized
	singVals(k) = k==0 ? col0(0) : diag(k);
	mus(k) = Literal(0);
	shifts(k) = k==0 ? col0(0) : diag(k);
	continue;
	}

	// otherwise, use secular equation to find singular value
	RealScalar left = diag(k);
	RealScalar right; // was: = (k != actual_n-1) ? diag(k+1) : (diag(actual_n-1) + col0.matrix().norm());
	if(k==actual_n-1)
	right = (diag(actual_n-1) + col0.matrix().norm());
	else
	{
	// Skip deflated singular values,
	// recall that at this stage we assume that z[j]!=0 and all entries for which z[j]==0 have been put aside.
	// This should be equivalent to using perm[]
	Index l = k+1;
	while(col0(l)==Literal(0)) { ++l; eigen_internal_assert(l<actual_n); }
	right = diag(l);
	}

	// first decide whether it's closer to the left end or the right end
	RealScalar mid = left + (right-left) / Literal(2);
	RealScalar fMid = secularEq(mid, col0, diag, perm, diag, Literal(0));
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "right-left = " << right-left << "\n";
	// std::cout << "fMid = " << fMid << " " << secularEq(mid-left, col0, diag, perm, ArrayXr(diag-left), left)
	// << " " << secularEq(mid-right, col0, diag, perm, ArrayXr(diag-right), right) << "\n";
	std::cout << " = " << secularEq(left+RealScalar(0.000001)*(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.1) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.2) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.3) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.4) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.49) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.5) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.51) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.6) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.7) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.8) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.9) *(right-left), col0, diag, perm, diag, 0)
	<< " " << secularEq(left+RealScalar(0.999999)*(right-left), col0, diag, perm, diag, 0) << "\n";
	#endif
	RealScalar shift = (k == actual_n-1 \|\| fMid > Literal(0)) ? left : right;

	// measure everything relative to shift
	Map<ArrayXr> diagShifted(m_workspace.data()+4*n, n);
	diagShifted = diag - shift;

	if(k!=actual_n-1)
	{
	// check that after the shift, f(mid) is still negative:
	RealScalar midShifted = (right - left) / RealScalar(2);
	if(shift==right)
	midShifted = -midShifted;
	RealScalar fMidShifted = secularEq(midShifted, col0, diag, perm, diagShifted, shift);
	if(fMidShifted>0)
	{
	// fMid was erroneous, fix it:
	shift = fMidShifted > Literal(0) ? left : right;
	diagShifted = diag - shift;
	}
	}

	// initial guess
	RealScalar muPrev, muCur;
	if (shift == left)
	{
	muPrev = (right - left) * RealScalar(0.1);
	if (k == actual_n-1) muCur = right - left;
	else muCur = (right - left) * RealScalar(0.5);
	}
	else
	{
	muPrev = -(right - left) * RealScalar(0.1);
	muCur = -(right - left) * RealScalar(0.5);
	}

	RealScalar fPrev = secularEq(muPrev, col0, diag, perm, diagShifted, shift);
	RealScalar fCur = secularEq(muCur, col0, diag, perm, diagShifted, shift);
	if (abs(fPrev) < abs(fCur))
	{
	swap(fPrev, fCur);
	swap(muPrev, muCur);
	}

	// rational interpolation: fit a function of the form a / mu + b through the two previous
	// iterates and use its zero to compute the next iterate
	bool useBisection = fPrev*fCur>Literal(0);
	while (fCur!=Literal(0) && abs(muCur - muPrev) > Literal(8) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(muCur), abs(muPrev)) && abs(fCur - fPrev)>NumTraits<RealScalar>::epsilon() && !useBisection)
	{
	++m_numIters;

	// Find a and b such that the function f(mu) = a / mu + b matches the current and previous samples.
	RealScalar a = (fCur - fPrev) / (Literal(1)/muCur - Literal(1)/muPrev);
	RealScalar b = fCur - a / muCur;
	// And find mu such that f(mu)==0:
	RealScalar muZero = -a/b;
	RealScalar fZero = secularEq(muZero, col0, diag, perm, diagShifted, shift);

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert((numext::isfinite)(fZero));
	#endif

	muPrev = muCur;
	fPrev = fCur;
	muCur = muZero;
	fCur = fZero;

	if (shift == left && (muCur < Literal(0) \|\| muCur > right - left)) useBisection = true;
	if (shift == right && (muCur < -(right - left) \|\| muCur > Literal(0))) useBisection = true;
	if (abs(fCur)>abs(fPrev)) useBisection = true;
	}

	// fall back on bisection method if rational interpolation did not work
	if (useBisection)
	{
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "useBisection for k = " << k << ", actual_n = " << actual_n << "\n";
	#endif
	RealScalar leftShifted, rightShifted;
	if (shift == left)
	{
	// to avoid overflow, we must have mu > max(real_min, \|z(k)\|/sqrt(real_max)),
	// the factor 2 is to be more conservative
	leftShifted = numext::maxi<RealScalar>( (std::numeric_limits<RealScalar>::min)(), Literal(2) * abs(col0(k)) / sqrt((std::numeric_limits<RealScalar>::max)()) );

	// check that we did it right:
	eigen_internal_assert( (numext::isfinite)( (col0(k)/leftShifted)*(col0(k)/(diag(k)+shift+leftShifted)) ) );
	// I don't understand why the case k==0 would be special there:
	// if (k == 0) rightShifted = right - left; else
	rightShifted = (k==actual_n-1) ? right : ((right - left) * RealScalar(0.51)); // theoretically we can take 0.5, but let's be safe
	}
	else
	{
	leftShifted = -(right - left) * RealScalar(0.51);
	if(k+1<n)
	rightShifted = -numext::maxi<RealScalar>( (std::numeric_limits<RealScalar>::min)(), abs(col0(k+1)) / sqrt((std::numeric_limits<RealScalar>::max)()) );
	else
	rightShifted = -(std::numeric_limits<RealScalar>::min)();
	}

	RealScalar fLeft = secularEq(leftShifted, col0, diag, perm, diagShifted, shift);
	eigen_internal_assert(fLeft<Literal(0));

	#if defined EIGEN_INTERNAL_DEBUGGING \|\| defined EIGEN_BDCSVD_SANITY_CHECKS
	RealScalar fRight = secularEq(rightShifted, col0, diag, perm, diagShifted, shift);
	#endif

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	if(!(numext::isfinite)(fLeft))
	std::cout << "f(" << leftShifted << ") =" << fLeft << " ; " << left << " " << shift << " " << right << "\n";
	assert((numext::isfinite)(fLeft));

	if(!(numext::isfinite)(fRight))
	std::cout << "f(" << rightShifted << ") =" << fRight << " ; " << left << " " << shift << " " << right << "\n";
	// assert((numext::isfinite)(fRight));
	#endif

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	if(!(fLeft * fRight<0))
	{
	std::cout << "f(leftShifted) using leftShifted=" << leftShifted << " ; diagShifted(1:10):" << diagShifted.head(10).transpose() << "\n ; "
	<< "left==shift=" << bool(left==shift) << " ; left-shift = " << (left-shift) << "\n";
	std::cout << "k=" << k << ", " << fLeft << " * " << fRight << " == " << fLeft * fRight << " ; "
	<< "[" << left << " .. " << right << "] -> [" << leftShifted << " " << rightShifted << "], shift=" << shift
	<< " , f(right)=" << secularEq(0, col0, diag, perm, diagShifted, shift)
	<< " == " << secularEq(right, col0, diag, perm, diag, 0) << " == " << fRight << "\n";
	}
	#endif
	eigen_internal_assert(fLeft * fRight < Literal(0));

	if(fLeft<Literal(0))
	{
	while (rightShifted - leftShifted > Literal(2) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(leftShifted), abs(rightShifted)))
	{
	RealScalar midShifted = (leftShifted + rightShifted) / Literal(2);
	fMid = secularEq(midShifted, col0, diag, perm, diagShifted, shift);
	eigen_internal_assert((numext::isfinite)(fMid));

	if (fLeft * fMid < Literal(0))
	{
	rightShifted = midShifted;
	}
	else
	{
	leftShifted = midShifted;
	fLeft = fMid;
	}
	}
	muCur = (leftShifted + rightShifted) / Literal(2);
	}
	else
	{
	// We have a problem as shifting on the left or right give either a positive or negative value
	// at the middle of [left,right]...
	// Instead fo abbording or entering an infinite loop,
	// let's just use the middle as the estimated zero-crossing:
	muCur = (right - left) * RealScalar(0.5);
	if(shift == right)
	muCur = -muCur;
	}
	}

	singVals[k] = shift + muCur;
	shifts[k] = shift;
	mus[k] = muCur;

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	if(k+1<n)
	std::cout << "found " << singVals[k] << " == " << shift << " + " << muCur << " from " << diag(k) << " .. " << diag(k+1) << "\n";
	#endif
	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(k==0 \|\| singVals[k]>=singVals[k-1]);
	assert(singVals[k]>=diag(k));
	#endif

	// perturb singular value slightly if it equals diagonal entry to avoid division by zero later
	// (deflation is supposed to avoid this from happening)
	// - this does no seem to be necessary anymore -
	// if (singVals[k] == left) singVals[k] *= 1 + NumTraits<RealScalar>::epsilon();
	// if (singVals[k] == right) singVals[k] *= 1 - NumTraits<RealScalar>::epsilon();
	}
	}


	// zhat is perturbation of col0 for which singular vectors can be computed stably (see Section 3.1)
	template <typename MatrixType>
	void BDCSVD<MatrixType>::perturbCol0
	(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals,
	const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat)
	{
	using std::sqrt;
	Index n = col0.size();
	Index m = perm.size();
	if(m==0)
	{
	zhat.setZero();
	return;
	}
	Index lastIdx = perm(m-1);
	// The offset permits to skip deflated entries while computing zhat
	for (Index k = 0; k < n; ++k)
	{
	if (col0(k) == Literal(0)) // deflated
	zhat(k) = Literal(0);
	else
	{
	// see equation (3.6)
	RealScalar dk = diag(k);
	RealScalar prod = (singVals(lastIdx) + dk) * (mus(lastIdx) + (shifts(lastIdx) - dk));
	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	if(prod<0) {
	std::cout << "k = " << k << " ; z(k)=" << col0(k) << ", diag(k)=" << dk << "\n";
	std::cout << "prod = " << "(" << singVals(lastIdx) << " + " << dk << ") * (" << mus(lastIdx) << " + (" << shifts(lastIdx) << " - " << dk << "))" << "\n";
	std::cout << " = " << singVals(lastIdx) + dk << " * " << mus(lastIdx) + (shifts(lastIdx) - dk) << "\n";
	}
	assert(prod>=0);
	#endif

	for(Index l = 0; l<m; ++l)
	{
	Index i = perm(l);
	if(i!=k)
	{
	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	if(i>=k && (l==0 \|\| l-1>=m))
	{
	std::cout << "Error in perturbCol0\n";
	std::cout << " " << k << "/" << n << " " << l << "/" << m << " " << i << "/" << n << " ; " << col0(k) << " " << diag(k) << " " << "\n";
	std::cout << " " <<diag(i) << "\n";
	Index j = (i<k /\|\| l==0/) ? i : perm(l-1);
	std::cout << " " << "j=" << j << "\n";
	}
	#endif
	Index j = i<k ? i : perm(l-1);
	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	if(!(dk!=Literal(0) \|\| diag(i)!=Literal(0)))
	{
	std::cout << "k=" << k << ", i=" << i << ", l=" << l << ", perm.size()=" << perm.size() << "\n";
	}
	assert(dk!=Literal(0) \|\| diag(i)!=Literal(0));
	#endif
	prod = ((singVals(j)+dk) / ((diag(i)+dk))) ((mus(j)+(shifts(j)-dk)) / ((diag(i)-dk)));
	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(prod>=0);
	#endif
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	if(i!=k && numext::abs(((singVals(j)+dk)(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)(diag(i)-dk)) - 1) > 0.9 )
	std::cout << " " << ((singVals(j)+dk)(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)(diag(i)-dk)) << " == (" << (singVals(j)+dk) << " * " << (mus(j)+(shifts(j)-dk))
	<< ") / (" << (diag(i)+dk) << " * " << (diag(i)-dk) << ")\n";
	#endif
	}
	}
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "zhat(" << k << ") = sqrt( " << prod << ") ; " << (singVals(lastIdx) + dk) << " * " << mus(lastIdx) + shifts(lastIdx) << " - " << dk << "\n";
	#endif
	RealScalar tmp = sqrt(prod);
	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert((numext::isfinite)(tmp));
	#endif
	zhat(k) = col0(k) > Literal(0) ? RealScalar(tmp) : RealScalar(-tmp);
	}
	}
	}

	// compute singular vectors
	template <typename MatrixType>
	void BDCSVD<MatrixType>::computeSingVecs
	(const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals,
	const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V)
	{
	Index n = zhat.size();
	Index m = perm.size();

	for (Index k = 0; k < n; ++k)
	{
	if (zhat(k) == Literal(0))
	{
	U.col(k) = VectorType::Unit(n+1, k);
	if (m_compV) V.col(k) = VectorType::Unit(n, k);
	}
	else
	{
	U.col(k).setZero();
	for(Index l=0;l<m;++l)
	{
	Index i = perm(l);
	U(i,k) = zhat(i)/(((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k]));
	}
	U(n,k) = Literal(0);
	U.col(k).normalize();

	if (m_compV)
	{
	V.col(k).setZero();
	for(Index l=1;l<m;++l)
	{
	Index i = perm(l);
	V(i,k) = diag(i) * zhat(i) / (((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k]));
	}
	V(0,k) = Literal(-1);
	V.col(k).normalize();
	}
	}
	}
	U.col(n) = VectorType::Unit(n+1, n);
	}


	// page 12_13
	// i >= 1, di almost null and zi non null.
	// We use a rotation to zero out zi applied to the left of M
	template <typename MatrixType>
	void BDCSVD<MatrixType>::deflation43(Eigen::Index firstCol, Eigen::Index shift, Eigen::Index i, Eigen::Index size)
	{
	using std::abs;
	using std::sqrt;
	using std::pow;
	Index start = firstCol + shift;
	RealScalar c = m_computed(start, start);
	RealScalar s = m_computed(start+i, start);
	RealScalar r = numext::hypot(c,s);
	if (r == Literal(0))
	{
	m_computed(start+i, start+i) = Literal(0);
	return;
	}
	m_computed(start,start) = r;
	m_computed(start+i, start) = Literal(0);
	m_computed(start+i, start+i) = Literal(0);

	JacobiRotation<RealScalar> J(c/r,-s/r);
	if (m_compU) m_naiveU.middleRows(firstCol, size+1).applyOnTheRight(firstCol, firstCol+i, J);
	else m_naiveU.applyOnTheRight(firstCol, firstCol+i, J);
	}// end deflation 43


	// page 13
	// i,j >= 1, i!=j and \|di - dj\| < epsilon * norm2(M)
	// We apply two rotations to have zj = 0;
	// TODO deflation44 is still broken and not properly tested
	template <typename MatrixType>
	void BDCSVD<MatrixType>::deflation44(Eigen::Index firstColu , Eigen::Index firstColm, Eigen::Index firstRowW, Eigen::Index firstColW, Eigen::Index i, Eigen::Index j, Eigen::Index size)
	{
	using std::abs;
	using std::sqrt;
	using std::conj;
	using std::pow;
	RealScalar c = m_computed(firstColm+i, firstColm);
	RealScalar s = m_computed(firstColm+j, firstColm);
	RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s));
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "deflation 4.4: " << i << "," << j << " -> " << c << " " << s << " " << r << " ; "
	<< m_computed(firstColm + i-1, firstColm) << " "
	<< m_computed(firstColm + i, firstColm) << " "
	<< m_computed(firstColm + i+1, firstColm) << " "
	<< m_computed(firstColm + i+2, firstColm) << "\n";
	std::cout << m_computed(firstColm + i-1, firstColm + i-1) << " "
	<< m_computed(firstColm + i, firstColm+i) << " "
	<< m_computed(firstColm + i+1, firstColm+i+1) << " "
	<< m_computed(firstColm + i+2, firstColm+i+2) << "\n";
	#endif
	if (r==Literal(0))
	{
	m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
	return;
	}
	c/=r;
	s/=r;
	m_computed(firstColm + i, firstColm) = r;
	m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i);
	m_computed(firstColm + j, firstColm) = Literal(0);

	JacobiRotation<RealScalar> J(c,-s);
	if (m_compU) m_naiveU.middleRows(firstColu, size+1).applyOnTheRight(firstColu + i, firstColu + j, J);
	else m_naiveU.applyOnTheRight(firstColu+i, firstColu+j, J);
	if (m_compV) m_naiveV.middleRows(firstRowW, size).applyOnTheRight(firstColW + i, firstColW + j, J);
	}// end deflation 44


	// acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive]
	template <typename MatrixType>
	void BDCSVD<MatrixType>::deflation(Eigen::Index firstCol, Eigen::Index lastCol, Eigen::Index k, Eigen::Index firstRowW, Eigen::Index firstColW, Eigen::Index shift)
	{
	using std::sqrt;
	using std::abs;
	const Index length = lastCol + 1 - firstCol;

	Block<MatrixXr,Dynamic,1> col0(m_computed, firstCol+shift, firstCol+shift, length, 1);
	Diagonal<MatrixXr> fulldiag(m_computed);
	VectorBlock<Diagonal<MatrixXr>,Dynamic> diag(fulldiag, firstCol+shift, length);

	const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
	RealScalar maxDiag = diag.tail((std::max)(Index(1),length-1)).cwiseAbs().maxCoeff();
	RealScalar epsilon_strict = numext::maxi<RealScalar>(considerZero,NumTraits<RealScalar>::epsilon() * maxDiag);
	RealScalar epsilon_coarse = Literal(8) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(col0.cwiseAbs().maxCoeff(), maxDiag);

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(m_naiveU.allFinite());
	assert(m_naiveV.allFinite());
	assert(m_computed.allFinite());
	#endif

	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "\ndeflate:" << diag.head(k+1).transpose() << " \| " << diag.segment(k+1,length-k-1).transpose() << "\n";
	#endif

	//condition 4.1
	if (diag(0) < epsilon_coarse)
	{
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "deflation 4.1, because " << diag(0) << " < " << epsilon_coarse << "\n";
	#endif
	diag(0) = epsilon_coarse;
	}

	//condition 4.2
	for (Index i=1;i<length;++i)
	if (abs(col0(i)) < epsilon_strict)
	{
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "deflation 4.2, set z(" << i << ") to zero because " << abs(col0(i)) << " < " << epsilon_strict << " (diag(" << i << ")=" << diag(i) << ")\n";
	#endif
	col0(i) = Literal(0);
	}

	//condition 4.3
	for (Index i=1;i<length; i++)
	if (diag(i) < epsilon_coarse)
	{
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "deflation 4.3, cancel z(" << i << ")=" << col0(i) << " because diag(" << i << ")=" << diag(i) << " < " << epsilon_coarse << "\n";
	#endif
	deflation43(firstCol, shift, i, length);
	}

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(m_naiveU.allFinite());
	assert(m_naiveV.allFinite());
	assert(m_computed.allFinite());
	#endif
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "to be sorted: " << diag.transpose() << "\n\n";
	std::cout << " : " << col0.transpose() << "\n\n";
	#endif
	{
	// Check for total deflation
	// If we have a total deflation, then we have to consider col0(0)==diag(0) as a singular value during sorting
	bool total_deflation = (col0.tail(length-1).array()<considerZero).all();

	// Sort the diagonal entries, since diag(1:k-1) and diag(k:length) are already sorted, let's do a sorted merge.
	// First, compute the respective permutation.
	Index *permutation = m_workspaceI.data();
	{
	permutation[0] = 0;
	Index p = 1;

	// Move deflated diagonal entries at the end.
	for(Index i=1; i<length; ++i)
	if(abs(diag(i))<considerZero)
	permutation[p++] = i;

	Index i=1, j=k+1;
	for( ; p < length; ++p)
	{
	if (i > k) permutation[p] = j++;
	else if (j >= length) permutation[p] = i++;
	else if (diag(i) < diag(j)) permutation[p] = j++;
	else permutation[p] = i++;
	}
	}

	// If we have a total deflation, then we have to insert diag(0) at the right place
	if(total_deflation)
	{
	for(Index i=1; i<length; ++i)
	{
	Index pi = permutation[i];
	if(abs(diag(pi))<considerZero \|\| diag(0)<diag(pi))
	permutation[i-1] = permutation[i];
	else
	{
	permutation[i-1] = 0;
	break;
	}
	}
	}

	// Current index of each col, and current column of each index
	Index *realInd = m_workspaceI.data()+length;
	Index realCol = m_workspaceI.data()+2length;

	for(int pos = 0; pos< length; pos++)
	{
	realCol[pos] = pos;
	realInd[pos] = pos;
	}

	for(Index i = total_deflation?0:1; i < length; i++)
	{
	const Index pi = permutation[length - (total_deflation ? i+1 : i)];
	const Index J = realCol[pi];

	using std::swap;
	// swap diagonal and first column entries:
	swap(diag(i), diag(J));
	if(i!=0 && J!=0) swap(col0(i), col0(J));

	// change columns
	if (m_compU) m_naiveU.col(firstCol+i).segment(firstCol, length + 1).swap(m_naiveU.col(firstCol+J).segment(firstCol, length + 1));
	else m_naiveU.col(firstCol+i).segment(0, 2) .swap(m_naiveU.col(firstCol+J).segment(0, 2));
	if (m_compV) m_naiveV.col(firstColW + i).segment(firstRowW, length).swap(m_naiveV.col(firstColW + J).segment(firstRowW, length));

	//update real pos
	const Index realI = realInd[i];
	realCol[realI] = J;
	realCol[pi] = i;
	realInd[J] = realI;
	realInd[i] = pi;
	}
	}
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "sorted: " << diag.transpose().format(bdcsvdfmt) << "\n";
	std::cout << " : " << col0.transpose() << "\n\n";
	#endif

	//condition 4.4
	{
	Index i = length-1;
	while(i>0 && (abs(diag(i))<considerZero \|\| abs(col0(i))<considerZero)) --i;
	for(; i>1;--i)
	if( (diag(i) - diag(i-1)) < NumTraits<RealScalar>::epsilon()*maxDiag )
	{
	#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
	std::cout << "deflation 4.4 with i = " << i << " because " << diag(i) << " - " << diag(i-1) << " == " << (diag(i) - diag(i-1)) << " < " << NumTraits<RealScalar>::epsilon()/diag(i)*/maxDiag << "\n";
	#endif
	eigen_internal_assert(abs(diag(i) - diag(i-1))<epsilon_coarse && " diagonal entries are not properly sorted");
	deflation44(firstCol, firstCol + shift, firstRowW, firstColW, i-1, i, length);
	}
	}

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	for(Index j=2;j<length;++j)
	assert(diag(j-1)<=diag(j) \|\| abs(diag(j))<considerZero);
	#endif

	#ifdef EIGEN_BDCSVD_SANITY_CHECKS
	assert(m_naiveU.allFinite());
	assert(m_naiveV.allFinite());
	assert(m_computed.allFinite());
	#endif
	}//end deflation

	/** \svd_module
	*
	* \return the singular value decomposition of \c *this computed by Divide & Conquer algorithm
	*
	* \sa class BDCSVD
	*/
	template<typename Derived>
	BDCSVD<typename MatrixBase<Derived>::PlainObject>
	MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const
	{
	return BDCSVD<PlainObject>(*this, computationOptions);
	}

	} // end namespace Eigen

	#endif

BDCSVD.hNo OneTemporaryActions

File Metadata

BDCSVD.hView Options

Event Timeline

BDCSVD.h
No OneTemporary
Actions

BDCSVD.h
View Options