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diff --git a/extern/Eigen2/Eigen/src/QR/Tridiagonalization.h b/extern/Eigen2/Eigen/src/QR/Tridiagonalization.h new file mode 100644 index 00000000000..9ea39be717c --- /dev/null +++ b/extern/Eigen2/Eigen/src/QR/Tridiagonalization.h @@ -0,0 +1,431 @@ +// 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_TRIDIAGONALIZATION_H +#define EIGEN_TRIDIAGONALIZATION_H + +/** \ingroup QR_Module + * \nonstableyet + * + * \class Tridiagonalization + * + * \brief Trigiagonal decomposition of a selfadjoint matrix + * + * \param MatrixType the type of the matrix of which we are performing the tridiagonalization + * + * This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that: + * \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix. + * + * \sa MatrixBase::tridiagonalize() + */ +template<typename _MatrixType> class Tridiagonalization +{ + public: + + typedef _MatrixType MatrixType; + typedef typename MatrixType::Scalar Scalar; + typedef typename NumTraits<Scalar>::Real RealScalar; + typedef typename ei_packet_traits<Scalar>::type Packet; + + enum { + Size = MatrixType::RowsAtCompileTime, + SizeMinusOne = MatrixType::RowsAtCompileTime==Dynamic + ? Dynamic + : MatrixType::RowsAtCompileTime-1, + PacketSize = ei_packet_traits<Scalar>::size + }; + + typedef Matrix<Scalar, SizeMinusOne, 1> CoeffVectorType; + typedef Matrix<RealScalar, Size, 1> DiagonalType; + typedef Matrix<RealScalar, SizeMinusOne, 1> SubDiagonalType; + + typedef typename NestByValue<DiagonalCoeffs<MatrixType> >::RealReturnType DiagonalReturnType; + + typedef typename NestByValue<DiagonalCoeffs< + NestByValue<Block<MatrixType,SizeMinusOne,SizeMinusOne> > > >::RealReturnType SubDiagonalReturnType; + + /** This constructor initializes a Tridiagonalization object for + * further use with Tridiagonalization::compute() + */ + Tridiagonalization(int size = Size==Dynamic ? 2 : Size) + : m_matrix(size,size), m_hCoeffs(size-1) + {} + + Tridiagonalization(const MatrixType& matrix) + : m_matrix(matrix), + m_hCoeffs(matrix.cols()-1) + { + _compute(m_matrix, m_hCoeffs); + } + + /** Computes or re-compute the tridiagonalization for the matrix \a matrix. + * + * This method allows to re-use the allocated data. + */ + void compute(const MatrixType& matrix) + { + m_matrix = matrix; + m_hCoeffs.resize(matrix.rows()-1, 1); + _compute(m_matrix, m_hCoeffs); + } + + /** \returns the householder coefficients allowing to + * reconstruct the matrix Q from the packed data. + * + * \sa packedMatrix() + */ + inline CoeffVectorType householderCoefficients(void) const { return m_hCoeffs; } + + /** \returns the internal result of the decomposition. + * + * The returned matrix contains the following information: + * - the strict upper part is equal to the input matrix A + * - the diagonal and lower sub-diagonal represent the tridiagonal symmetric matrix (real). + * - the rest of the lower part contains the Householder vectors that, combined with + * Householder coefficients returned by householderCoefficients(), + * allows to reconstruct the matrix Q as follow: + * Q = H_{N-1} ... H_1 H_0 + * where the matrices H are the Householder transformations: + * H_i = (I - h_i * v_i * v_i') + * where h_i == householderCoefficients()[i] and v_i is a Householder vector: + * v_i = [ 0, ..., 0, 1, M(i+2,i), ..., M(N-1,i) ] + * + * See LAPACK for further details on this packed storage. + */ + inline const MatrixType& packedMatrix(void) const { return m_matrix; } + + MatrixType matrixQ(void) const; + MatrixType matrixT(void) const; + const DiagonalReturnType diagonal(void) const; + const SubDiagonalReturnType subDiagonal(void) const; + + static void decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ = true); + + private: + + static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs); + + static void _decomposeInPlace3x3(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ = true); + + protected: + MatrixType m_matrix; + CoeffVectorType m_hCoeffs; +}; + +/** \returns an expression of the diagonal vector */ +template<typename MatrixType> +const typename Tridiagonalization<MatrixType>::DiagonalReturnType +Tridiagonalization<MatrixType>::diagonal(void) const +{ + return m_matrix.diagonal().nestByValue().real(); +} + +/** \returns an expression of the sub-diagonal vector */ +template<typename MatrixType> +const typename Tridiagonalization<MatrixType>::SubDiagonalReturnType +Tridiagonalization<MatrixType>::subDiagonal(void) const +{ + int n = m_matrix.rows(); + return Block<MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1) + .nestByValue().diagonal().nestByValue().real(); +} + +/** constructs and returns the tridiagonal matrix T. + * Note that the matrix T is equivalent to the diagonal and sub-diagonal of the packed matrix. + * Therefore, it might be often sufficient to directly use the packed matrix, or the vector + * expressions returned by diagonal() and subDiagonal() instead of creating a new matrix. + */ +template<typename MatrixType> +typename Tridiagonalization<MatrixType>::MatrixType +Tridiagonalization<MatrixType>::matrixT(void) const +{ + // FIXME should this function (and other similar ones) rather take a matrix as argument + // and fill it ? (to avoid temporaries) + int n = m_matrix.rows(); + MatrixType matT = m_matrix; + matT.corner(TopRight,n-1, n-1).diagonal() = subDiagonal().template cast<Scalar>().conjugate(); + if (n>2) + { + matT.corner(TopRight,n-2, n-2).template part<UpperTriangular>().setZero(); + matT.corner(BottomLeft,n-2, n-2).template part<LowerTriangular>().setZero(); + } + return matT; +} + +#ifndef EIGEN_HIDE_HEAVY_CODE + +/** \internal + * Performs a tridiagonal decomposition of \a matA in place. + * + * \param matA the input selfadjoint matrix + * \param hCoeffs returned Householder coefficients + * + * The result is written in the lower triangular part of \a matA. + * + * Implemented from Golub's "Matrix Computations", algorithm 8.3.1. + * + * \sa packedMatrix() + */ +template<typename MatrixType> +void Tridiagonalization<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs) +{ + assert(matA.rows()==matA.cols()); + int n = matA.rows(); +// std::cerr << matA << "\n\n"; + for (int i = 0; i<n-2; ++i) + { + // let's consider the vector v = i-th column starting at position i+1 + + // start of the householder transformation + // squared norm of the vector v skipping the first element + RealScalar v1norm2 = matA.col(i).end(n-(i+2)).squaredNorm(); + + // FIXME comparing against 1 + if (ei_isMuchSmallerThan(v1norm2,static_cast<Scalar>(1))) + { + hCoeffs.coeffRef(i) = 0.; + } + else + { + Scalar v0 = matA.col(i).coeff(i+1); + RealScalar beta = ei_sqrt(ei_abs2(v0)+v1norm2); + if (ei_real(v0)>=0.) + beta = -beta; + matA.col(i).end(n-(i+2)) *= (Scalar(1)/(v0-beta)); + matA.col(i).coeffRef(i+1) = beta; + Scalar h = (beta - v0) / beta; + // end of the householder transformation + + // Apply similarity transformation to remaining columns, + // i.e., A = H' A H where H = I - h v v' and v = matA.col(i).end(n-i-1) + + matA.col(i).coeffRef(i+1) = 1; + + /* This is the initial algorithm which minimize operation counts and maximize + * the use of Eigen's expression. Unfortunately, the first matrix-vector product + * using Part<LowerTriangular|Selfadjoint> is very very slow */ + #ifdef EIGEN_NEVER_DEFINED + // matrix - vector product + hCoeffs.end(n-i-1) = (matA.corner(BottomRight,n-i-1,n-i-1).template part<LowerTriangular|SelfAdjoint>() + * (h * matA.col(i).end(n-i-1))).lazy(); + // simple axpy + hCoeffs.end(n-i-1) += (h * Scalar(-0.5) * matA.col(i).end(n-i-1).dot(hCoeffs.end(n-i-1))) + * matA.col(i).end(n-i-1); + // rank-2 update + //Block<MatrixType,Dynamic,1> B(matA,i+1,i,n-i-1,1); + matA.corner(BottomRight,n-i-1,n-i-1).template part<LowerTriangular>() -= + (matA.col(i).end(n-i-1) * hCoeffs.end(n-i-1).adjoint()).lazy() + + (hCoeffs.end(n-i-1) * matA.col(i).end(n-i-1).adjoint()).lazy(); + #endif + /* end initial algorithm */ + + /* If we still want to minimize operation count (i.e., perform operation on the lower part only) + * then we could provide the following algorithm for selfadjoint - vector product. However, a full + * matrix-vector product is still faster (at least for dynamic size, and not too small, did not check + * small matrices). The algo performs block matrix-vector and transposed matrix vector products. */ + #ifdef EIGEN_NEVER_DEFINED + int n4 = (std::max(0,n-4)/4)*4; + hCoeffs.end(n-i-1).setZero(); + for (int b=i+1; b<n4; b+=4) + { + // the ?x4 part: + hCoeffs.end(b-4) += + Block<MatrixType,Dynamic,4>(matA,b+4,b,n-b-4,4) * matA.template block<4,1>(b,i); + // the respective transposed part: + Block<CoeffVectorType,4,1>(hCoeffs, b, 0, 4,1) += + Block<MatrixType,Dynamic,4>(matA,b+4,b,n-b-4,4).adjoint() * Block<MatrixType,Dynamic,1>(matA,b+4,i,n-b-4,1); + // the 4x4 block diagonal: + Block<CoeffVectorType,4,1>(hCoeffs, b, 0, 4,1) += + (Block<MatrixType,4,4>(matA,b,b,4,4).template part<LowerTriangular|SelfAdjoint>() + * (h * Block<MatrixType,4,1>(matA,b,i,4,1))).lazy(); + } + #endif + // todo: handle the remaining part + /* end optimized selfadjoint - vector product */ + + /* Another interesting note: the above rank-2 update is much slower than the following hand written loop. + * After an analyze of the ASM, it seems GCC (4.2) generate poor code because of the Block. Moreover, + * if we remove the specialization of Block for Matrix then it is even worse, much worse ! */ + #ifdef EIGEN_NEVER_DEFINED + for (int j1=i+1; j1<n; ++j1) + for (int i1=j1; i1<n; ++i1) + matA.coeffRef(i1,j1) -= matA.coeff(i1,i)*ei_conj(hCoeffs.coeff(j1-1)) + + hCoeffs.coeff(i1-1)*ei_conj(matA.coeff(j1,i)); + #endif + /* end hand writen partial rank-2 update */ + + /* The current fastest implementation: the full matrix is used, no "optimization" to use/compute + * only half of the matrix. Custom vectorization of the inner col -= alpha X + beta Y such that access + * to col are always aligned. Once we support that in Assign, then the algorithm could be rewriten as + * a single compact expression. This code is therefore a good benchmark when will do that. */ + + // let's use the end of hCoeffs to store temporary values: + hCoeffs.end(n-i-1) = (matA.corner(BottomRight,n-i-1,n-i-1) * (h * matA.col(i).end(n-i-1))).lazy(); + // FIXME in the above expr a temporary is created because of the scalar multiple by h + + hCoeffs.end(n-i-1) += (h * Scalar(-0.5) * matA.col(i).end(n-i-1).dot(hCoeffs.end(n-i-1))) + * matA.col(i).end(n-i-1); + + const Scalar* EIGEN_RESTRICT pb = &matA.coeffRef(0,i); + const Scalar* EIGEN_RESTRICT pa = (&hCoeffs.coeffRef(0)) - 1; + for (int j1=i+1; j1<n; ++j1) + { + int starti = i+1; + int alignedEnd = starti; + if (PacketSize>1) + { + int alignedStart = (starti) + ei_alignmentOffset(&matA.coeffRef(starti,j1), n-starti); + alignedEnd = alignedStart + ((n-alignedStart)/PacketSize)*PacketSize; + + for (int i1=starti; i1<alignedStart; ++i1) + matA.coeffRef(i1,j1) -= matA.coeff(i1,i)*ei_conj(hCoeffs.coeff(j1-1)) + + hCoeffs.coeff(i1-1)*ei_conj(matA.coeff(j1,i)); + + Packet tmp0 = ei_pset1(hCoeffs.coeff(j1-1)); + Packet tmp1 = ei_pset1(matA.coeff(j1,i)); + Scalar* pc = &matA.coeffRef(0,j1); + for (int i1=alignedStart ; i1<alignedEnd; i1+=PacketSize) + ei_pstore(pc+i1,ei_psub(ei_pload(pc+i1), + ei_padd(ei_pmul(tmp0, ei_ploadu(pb+i1)), + ei_pmul(tmp1, ei_ploadu(pa+i1))))); + } + for (int i1=alignedEnd; i1<n; ++i1) + matA.coeffRef(i1,j1) -= matA.coeff(i1,i)*ei_conj(hCoeffs.coeff(j1-1)) + + hCoeffs.coeff(i1-1)*ei_conj(matA.coeff(j1,i)); + } + /* end optimized implementation */ + + // note: at that point matA(i+1,i+1) is the (i+1)-th element of the final diagonal + // note: the sequence of the beta values leads to the subdiagonal entries + matA.col(i).coeffRef(i+1) = beta; + + hCoeffs.coeffRef(i) = h; + } + } + if (NumTraits<Scalar>::IsComplex) + { + // Householder transformation on the remaining single scalar + int i = n-2; + Scalar v0 = matA.col(i).coeff(i+1); + RealScalar beta = ei_abs(v0); + if (ei_real(v0)>=0.) + beta = -beta; + matA.col(i).coeffRef(i+1) = beta; + if(ei_isMuchSmallerThan(beta, Scalar(1))) hCoeffs.coeffRef(i) = Scalar(0); + else hCoeffs.coeffRef(i) = (beta - v0) / beta; + } + else + { + hCoeffs.coeffRef(n-2) = 0; + } +} + +/** reconstructs and returns the matrix Q */ +template<typename MatrixType> +typename Tridiagonalization<MatrixType>::MatrixType +Tridiagonalization<MatrixType>::matrixQ(void) const +{ + int n = m_matrix.rows(); + MatrixType matQ = MatrixType::Identity(n,n); + for (int i = n-2; i>=0; i--) + { + Scalar tmp = m_matrix.coeff(i+1,i); + m_matrix.const_cast_derived().coeffRef(i+1,i) = 1; + + matQ.corner(BottomRight,n-i-1,n-i-1) -= + ((m_hCoeffs.coeff(i) * m_matrix.col(i).end(n-i-1)) * + (m_matrix.col(i).end(n-i-1).adjoint() * matQ.corner(BottomRight,n-i-1,n-i-1)).lazy()).lazy(); + + m_matrix.const_cast_derived().coeffRef(i+1,i) = tmp; + } + return matQ; +} + +/** Performs a full decomposition in place */ +template<typename MatrixType> +void Tridiagonalization<MatrixType>::decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ) +{ + int n = mat.rows(); + ei_assert(mat.cols()==n && diag.size()==n && subdiag.size()==n-1); + if (n==3 && (!NumTraits<Scalar>::IsComplex) ) + { + _decomposeInPlace3x3(mat, diag, subdiag, extractQ); + } + else + { + Tridiagonalization tridiag(mat); + diag = tridiag.diagonal(); + subdiag = tridiag.subDiagonal(); + if (extractQ) + mat = tridiag.matrixQ(); + } +} + +/** \internal + * Optimized path for 3x3 matrices. + * Especially useful for plane fitting. + */ +template<typename MatrixType> +void Tridiagonalization<MatrixType>::_decomposeInPlace3x3(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ) +{ + diag[0] = ei_real(mat(0,0)); + RealScalar v1norm2 = ei_abs2(mat(0,2)); + if (ei_isMuchSmallerThan(v1norm2, RealScalar(1))) + { + diag[1] = ei_real(mat(1,1)); + diag[2] = ei_real(mat(2,2)); + subdiag[0] = ei_real(mat(0,1)); + subdiag[1] = ei_real(mat(1,2)); + if (extractQ) + mat.setIdentity(); + } + else + { + RealScalar beta = ei_sqrt(ei_abs2(mat(0,1))+v1norm2); + RealScalar invBeta = RealScalar(1)/beta; + Scalar m01 = mat(0,1) * invBeta; + Scalar m02 = mat(0,2) * invBeta; + Scalar q = RealScalar(2)*m01*mat(1,2) + m02*(mat(2,2) - mat(1,1)); + diag[1] = ei_real(mat(1,1) + m02*q); + diag[2] = ei_real(mat(2,2) - m02*q); + subdiag[0] = beta; + subdiag[1] = ei_real(mat(1,2) - m01 * q); + if (extractQ) + { + mat(0,0) = 1; + mat(0,1) = 0; + mat(0,2) = 0; + mat(1,0) = 0; + mat(1,1) = m01; + mat(1,2) = m02; + mat(2,0) = 0; + mat(2,1) = m02; + mat(2,2) = -m01; + } + } +} + +#endif // EIGEN_HIDE_HEAVY_CODE + +#endif // EIGEN_TRIDIAGONALIZATION_H |