diff options
Diffstat (limited to 'extern/Eigen2/Eigen/src/QR/Tridiagonalization.h')
| -rw-r--r-- | extern/Eigen2/Eigen/src/QR/Tridiagonalization.h | 431 |
1 files changed, 0 insertions, 431 deletions
diff --git a/extern/Eigen2/Eigen/src/QR/Tridiagonalization.h b/extern/Eigen2/Eigen/src/QR/Tridiagonalization.h deleted file mode 100644 index 9ea39be717c..00000000000 --- a/extern/Eigen2/Eigen/src/QR/Tridiagonalization.h +++ /dev/null @@ -1,431 +0,0 @@ -// 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 |