diff options
Diffstat (limited to 'Eigen/src/SVD/UpperBidiagonalization.h')
| -rw-r--r-- | Eigen/src/SVD/UpperBidiagonalization.h | 326 |
1 files changed, 295 insertions, 31 deletions
diff --git a/Eigen/src/SVD/UpperBidiagonalization.h b/Eigen/src/SVD/UpperBidiagonalization.h index 587de37a5..0b1460894 100644 --- a/Eigen/src/SVD/UpperBidiagonalization.h +++ b/Eigen/src/SVD/UpperBidiagonalization.h @@ -2,6 +2,7 @@ // 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 @@ -28,15 +29,15 @@ template<typename _MatrixType> class UpperBidiagonalization }; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; - typedef typename MatrixType::Index Index; + 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> BidiagonalType; + 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, - CwiseUnaryOp<internal::scalar_conjugate_op<Scalar>, const Diagonal<const MatrixType,0> > + const typename internal::remove_all<typename Diagonal<const MatrixType,0>::ConjugateReturnType>::type > HouseholderUSequenceType; typedef HouseholderSequence< const typename internal::remove_all<typename MatrixType::ConjugateReturnType>::type, @@ -52,7 +53,7 @@ template<typename _MatrixType> class UpperBidiagonalization */ UpperBidiagonalization() : m_householder(), m_bidiagonal(), m_isInitialized(false) {} - UpperBidiagonalization(const MatrixType& matrix) + explicit UpperBidiagonalization(const MatrixType& matrix) : m_householder(matrix.rows(), matrix.cols()), m_bidiagonal(matrix.cols(), matrix.cols()), m_isInitialized(false) @@ -61,6 +62,7 @@ template<typename _MatrixType> class UpperBidiagonalization } UpperBidiagonalization& compute(const MatrixType& matrix); + UpperBidiagonalization& computeUnblocked(const MatrixType& matrix); const MatrixType& householder() const { return m_householder; } const BidiagonalType& bidiagonal() const { return m_bidiagonal; } @@ -85,45 +87,307 @@ template<typename _MatrixType> class UpperBidiagonalization bool m_isInitialized; }; -template<typename _MatrixType> -UpperBidiagonalization<_MatrixType>& UpperBidiagonalization<_MatrixType>::compute(const _MatrixType& matrix) +// 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) { - Index rows = matrix.rows(); - Index cols = matrix.cols(); - - eigen_assert(rows >= cols && "UpperBidiagonalization is only for matrices satisfying rows>=cols."); - - m_householder = matrix; + typedef typename MatrixType::Scalar Scalar; - ColVectorType temp(rows); + 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 m_householder - m_householder.col(k).tail(remainingRows) - .makeHouseholderInPlace(m_householder.coeffRef(k,k), - m_bidiagonal.template diagonal<0>().coeffRef(k)); - // apply householder transform to remaining part of m_householder on the left - m_householder.bottomRightCorner(remainingRows, remainingCols) - .applyHouseholderOnTheLeft(m_householder.col(k).tail(remainingRows-1), - m_householder.coeff(k,k), - temp.data()); + // 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).transpose(), 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; + 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 begining 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 begining 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) = 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); - // construct right householder transform in-place in m_householder - m_householder.row(k).tail(remainingCols) - .makeHouseholderInPlace(m_householder.coeffRef(k,k+1), - m_bidiagonal.template diagonal<1>().coeffRef(k)); - // apply householder transform to remaining part of m_householder on the left - m_householder.bottomRightCorner(remainingRows-1, remainingCols) - .applyHouseholderOnTheRight(m_householder.row(k).tail(remainingCols-1).transpose(), - m_householder.coeff(k,k+1), - temp.data()); + // 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; } |