Subject: Re: [ublas] [RFC PATCH] ublas: improved dense matrix multiplication performance
From: Michael Lehn (michael.lehn_at_[hidden])
Date: 2016-02-15 13:54:32
Thanks Imre,
and again sorry for the late response. The next weeks I hopefully have more time, so let me know
if there is something I can do.
But at least I had some time to run some benchmarks comparing the LU-factorization (and therefore
indirectly the GEMM and TRSM) with Intel MKL:
http://www.mathematik.uni-ulm.de/~lehn/test_ublas/session8/page01.html
It also contains some first comparison with other C++ libraries. I started with Eigen. I also started with
a simple BLAZE implementation of the LU algorithms. As I wrote in a previous post, I really believe that
C++ has a great potential for HPC.
Cheers,
Michael
On 13 Feb 2016, at 14:43, Imre Palik <imre_palik_at_[hidden]> wrote:
> This patch includes the gemm implementation from Michael Lehn to
> boost::ublas.
>
> This modifies the workings of ublas::prod() and ublas::axppy_prod()
> to use gemm() above a certain matrix size.
>
> This patch only contains the basic architecture, and a generic c++
> implementation. All the architecture, or compiler specific stuff
> will go to follow-up patches.
>
> Signed-off-by: Imre Palik <imre_palik_at_[hidden]>
> Cc: Michael Lehn <michael.lehn_at_[hidden]>
> ---
> include/boost/numeric/ublas/detail/gemm.hpp | 279 ++++++++++++++++++++++
> include/boost/numeric/ublas/matrix_expression.hpp | 59 ++++-
> include/boost/numeric/ublas/operation.hpp | 87 ++++++-
> 3 files changed, 407 insertions(+), 18 deletions(-)
> create mode 100644 include/boost/numeric/ublas/detail/gemm.hpp
>
> diff --git a/include/boost/numeric/ublas/detail/gemm.hpp b/include/boost/numeric/ublas/detail/gemm.hpp
> new file mode 100644
> index 0000000..cb4a343
> --- /dev/null
> +++ b/include/boost/numeric/ublas/detail/gemm.hpp
> @@ -0,0 +1,279 @@
> +//
> +// Copyright (c) 2016
> +// Michael Lehn, Imre Palik
> +//
> +// Distributed under the Boost Software License, Version 1.0. (See
> +// accompanying file LICENSE_1_0.txt or copy at
> +// http://www.boost.org/LICENSE_1_0.txt)
> +
> +#ifndef _BOOST_UBLAS_GEMM_
> +#define _BOOST_UBLAS_GEMM_
> +
> +#include <boost/type_traits/common_type.hpp>
> +#include <boost/align/aligned_alloc.hpp>
> +#include <boost/align/assume_aligned.hpp>
> +#include <boost/static_assert.hpp>
> +
> +namespace boost { namespace numeric { namespace ublas { namespace detail {
> +
> + template <typename T>
> + struct prod_block_size {
> + static const unsigned mc = 256;
> + static const unsigned kc = 512; // stripe length
> + static const unsigned nc = 4092;
> + static const unsigned mr = 4; // stripe width for lhs
> + static const unsigned nr = 12; // stripe width for rhs
> + static const unsigned align = 64; // align temporary arrays to this boundary
> + static const unsigned limit = 26; // Use gemm from this size
> + BOOST_STATIC_ASSERT_MSG(mc>0 && kc>0 && nc>0 && mr>0 && nr>0, "Invalid block size.");
> + BOOST_STATIC_ASSERT_MSG(mc % mr == 0, "MC must be a multiple of MR.");
> + BOOST_STATIC_ASSERT_MSG(nc % nr == 0, "NC must be a multiple of NR.");
> + };
> +
> + template <typename E>
> + void
> + gescal(const typename E::value_type &alpha, matrix_expression<E> &X)
> + {
> + typedef typename E::size_type size_type;
> +
> + for (size_type i=0; i<X().size1(); ++i) {
> + for (size_type j=0; j<X().size2(); ++j) {
> + X()(i,j) *= alpha;
> + }
> + }
> + }
> +
> + template <typename Index, typename T>
> + void
> + geaxpy(Index m, Index n, const T &alpha,
> + const T *X, Index incRowX, Index incColX,
> + T *Y, Index incRowY, Index incColY)
> + {
> + for (Index j=0; j<n; ++j) {
> + for (Index i=0; i<m; ++i) {
> + Y[i*incRowY+j*incColY] += alpha*X[i*incRowX+j*incColX];
> + }
> + }
> + }
> +
> + template <typename Index, typename TX>
> + void
> + gescal(Index m, Index n,
> + const TX &alpha,
> + TX *X, Index incRowX, Index incColX)
> + {
> + for (Index j=0; j<n; ++j) {
> + for (Index i=0; i<m; ++i) {
> + X[i*incRowX+j*incColX] *= alpha;
> + }
> + }
> + }
> +
> + //-- Micro Kernel --------------------------------------------------------------
> + template <typename Index, typename T, typename TC, typename BlockSize>
> + void
> + ugemm(Index kc, TC alpha, const T *A, const T *B,
> + TC beta, TC *C, Index incRowC, Index incColC)
> + {
> + BOOST_ALIGN_ASSUME_ALIGNED(A, BlockSize::align);
> + BOOST_ALIGN_ASSUME_ALIGNED(B, BlockSize::align);
> + static const Index MR = BlockSize::mr;
> + static const Index NR = BlockSize::nr;
> + typename boost::aligned_storage<sizeof(T[MR*NR]),alignof(BlockSize::align)>::type Pa;
> + T *P = reinterpret_cast<T*>(Pa.address());
> + for (unsigned c = 0; c < MR * NR; c++)
> + P[c] = 0;
> +
> + for (Index l=0; l<kc; ++l) {
> + for (Index i=0; i<MR; ++i) {
> + for (Index j=0; j<NR; ++j) {
> + P[i* NR+j] += A[i]*B[j];
> + }
> + }
> + A += MR;
> + B += NR;
> + }
> +
> + if (alpha!=TC(1)) {
> + for (Index i=0; i<MR; ++i) {
> + for (Index j=0; j<NR; ++j) {
> + P[i*NR+j] *= alpha;
> + }
> + }
> + }
> +
> + if (beta == TC(0)) {
> + for (Index i=0; i<MR; ++i) {
> + for (Index j=0; j<NR; ++j) {
> + C[i*incRowC+j*incColC] = P[i*NR+j];
> + }
> + }
> + } else {
> + for (Index i=0; i<MR; ++i) {
> + for (Index j=0; j<NR; ++j) {
> + C[i*incRowC+j*incColC] *= beta;
> + C[i*incRowC+j*incColC] += P[i*NR+j];
> + }
> + }
> + }
> + }
> +
> + //-- Macro Kernel --------------------------------------------------------------
> + template <typename Index, typename T, typename TC, typename BlockSize>
> + void
> + mgemm(Index mc, Index nc, Index kc, TC alpha,
> + const T *A, const T *B, TC beta,
> + TC *C, Index incRowC, Index incColC)
> + {
> + static const Index MR = BlockSize::mr;
> + static const Index NR = BlockSize::nr;
> + const Index mp = (mc+MR-1) / MR;
> + const Index np = (nc+NR-1) / NR;
> + const Index mr_ = mc % MR;
> + const Index nr_ = nc % NR;
> +
> + for (Index j=0; j<np; ++j) {
> + const Index nr = (j!=np-1 || nr_==0) ? NR : nr_;
> + T C_[MR*NR];
> +
> + for (Index i=0; i<mp; ++i) {
> + const Index mr = (i!=mp-1 || mr_==0) ? MR : mr_;
> +
> + if (mr==MR && nr==NR) {
> + ugemm<Index, T, TC, BlockSize>(kc, alpha,
> + &A[i*kc*MR], &B[j*kc*NR],
> + beta,
> + &C[i*MR*incRowC+j*NR*incColC],
> + incRowC, incColC);
> + } else {
> + std::fill_n(C_, MR*NR, T(0));
> + ugemm<Index, T, TC, BlockSize>(kc, alpha,
> + &A[i*kc*MR], &B[j*kc*NR],
> + T(0),
> + C_, NR, Index(1));
> + gescal(mr, nr, beta,
> + &C[i*MR*incRowC+j*NR*incColC],
> + incRowC, incColC);
> + geaxpy(mr, nr, T(1), C_, NR, Index(1),
> + &C[i*MR*incRowC+j*NR*incColC],
> + incRowC, incColC);
> + }
> + }
> + }
> + }
> +
> + //-- Packing blocks ------------------------------------------------------------
> + template <typename E, typename T, typename BlockSize>
> + void
> + pack_A(const matrix_expression<E> &A, T *p)
> + {
> + typedef typename E::size_type size_type;
> +
> + const size_type mc = A ().size1();
> + const size_type kc = A ().size2();
> + static const size_type MR = BlockSize::mr;
> + const size_type mp = (mc+MR-1) / MR;
> +
> + for (size_type j=0; j<kc; ++j) {
> + for (size_type l=0; l<mp; ++l) {
> + for (size_type i0=0; i0<MR; ++i0) {
> + size_type i = l*MR + i0;
> + size_type nu = l*MR*kc + j*MR + i0;
> + p[nu] = (i<mc) ? A()(i,j) : T(0);
> + }
> + }
> + }
> + }
> +
> + template <typename E, typename T, typename BlockSize>
> + void
> + pack_B(const matrix_expression<E> &B, T *p)
> + {
> + typedef typename E::size_type size_type;
> +
> + const size_type kc = B ().size1();
> + const size_type nc = B ().size2();
> + static const size_type NR = BlockSize::nr;
> + const size_type np = (nc+NR-1) / NR;
> +
> + for (size_type l=0; l<np; ++l) {
> + for (size_type j0=0; j0<NR; ++j0) {
> + for (size_type i=0; i<kc; ++i) {
> + size_type j = l*NR+j0;
> + size_type nu = l*NR*kc + i*NR + j0;
> + p[nu] = (j<nc) ? B()(i,j) : T(0);
> + }
> + }
> + }
> + }
> +
> + //-- Frame routine -------------------------------------------------------------
> + template <typename E1, typename E2, typename E3, typename BlockSize>
> + void
> + gemm(typename E3::value_type alpha, const matrix_expression<E1> &e1,
> + const matrix_expression<E2> &e2,
> + typename E3::value_type beta, matrix_expression<E3> &e3)
> + {
> + typedef typename E3::size_type size_type;
> + typedef typename E1::value_type value_type1;
> + typedef typename E2::value_type value_type2;
> + typedef typename E3::value_type value_type3;
> + typedef typename common_type<value_type1, value_type2>::type value_type_i;
> +
> + static const size_type MC = BlockSize::mc;
> + static const size_type NC = BlockSize::nc;
> +
> + const size_type m = BOOST_UBLAS_SAME (e3 ().size1 (), e1 ().size1 ());
> + const size_type n = BOOST_UBLAS_SAME (e3 ().size2 (), e2 ().size2 ());
> + const size_type k = BOOST_UBLAS_SAME (e1 ().size2 (), e2 ().size1 ());
> +
> + static const size_type KC = BlockSize::kc;
> + const size_type mb = (m+MC-1) / MC;
> + const size_type nb = (n+NC-1) / NC;
> + const size_type kb = (k+KC-1) / KC;
> + const size_type mc_ = m % MC;
> + const size_type nc_ = n % NC;
> + const size_type kc_ = k % KC;
> +
> + value_type3 *C_ = &e3()(0,0);
> + const size_type incRowC = &e3()(1,0) - &e3()(0,0);
> + const size_type incColC = &e3()(0,1) - &e3()(0,0);
> + value_type_i *A =
> + static_cast<value_type_i *>(boost::alignment::aligned_alloc(BlockSize::align,
> + sizeof(value_type_i) * MC*KC));
> + value_type_i *B =
> + static_cast<value_type_i *>(boost::alignment::aligned_alloc(BlockSize::align,
> + sizeof(value_type_i) * KC*NC));
> +
> + if (alpha==value_type3(0) || k==0) {
> + gescal(beta, e3);
> + return;
> + }
> +
> + for (size_type j=0; j<nb; ++j) {
> + size_type nc = (j!=nb-1 || nc_==0) ? NC : nc_;
> +
> + for (size_type l=0; l<kb; ++l) {
> + size_type kc = (l!=kb-1 || kc_==0) ? KC : kc_;
> + value_type3 beta_ = (l==0) ? beta : value_type3(1);
> +
> + const matrix_range<const E2> Bs = subrange(e2(), l*KC, l*KC+kc, j*NC, j*NC+nc);
> + pack_B<matrix_range<const E2>, value_type_i, BlockSize>(Bs, B);
> +
> + for (size_type i=0; i<mb; ++i) {
> + size_type mc = (i!=mb-1 || mc_==0) ? MC : mc_;
> +
> + const matrix_range<const E1> As = subrange(e1(), i*MC, i*MC+mc, l*KC, l*KC+kc);
> + pack_A<matrix_range<const E1>, value_type_i, BlockSize>(As, A);
> +
> + mgemm<size_type, value_type_i, value_type3, BlockSize>(mc, nc, kc, alpha, A, B, beta_,
> + &C_[i*MC*incRowC+j*NC*incColC],
> + incRowC, incColC);
> + }
> + }
> + }
> + boost::alignment::aligned_free(A);
> + boost::alignment::aligned_free(B);
> + }
> +}}}}
> +#endif
> diff --git a/include/boost/numeric/ublas/matrix_expression.hpp b/include/boost/numeric/ublas/matrix_expression.hpp
> index a363130..22bdb44 100644
> --- a/include/boost/numeric/ublas/matrix_expression.hpp
> +++ b/include/boost/numeric/ublas/matrix_expression.hpp
> @@ -14,6 +14,7 @@
> #define _BOOST_UBLAS_MATRIX_EXPRESSION_
>
> #include <boost/numeric/ublas/vector_expression.hpp>
> +#include <boost/numeric/ublas/detail/gemm.hpp>
>
> // Expression templates based on ideas of Todd Veldhuizen and Geoffrey Furnish
> // Iterators based on ideas of Jeremy Siek
> @@ -5460,20 +5461,40 @@ namespace boost { namespace numeric { namespace ublas {
> expression2_closure_type e2_;
> };
>
> + namespace detail {
> + template<class E1, class E2, class P, bool s>
> + struct binary_calculate_result_type;
> +
> + template<class E1, class E2, class P>
> + struct binary_calculate_result_type<E1, E2, P, false> {
> + typedef matrix_matrix_binary<E1, E2, matrix_matrix_prod<E1, E2, P> > result_type;
> + };
> +
> + // TODO: should elaborate on this for some dense types.
> + template<class E1, class E2, class P>
> + struct binary_calculate_result_type<E1, E2, P, true> {
> + typedef matrix<P> result_type;
> + };
> + }
> +
> template<class T1, class E1, class T2, class E2>
> struct matrix_matrix_binary_traits {
> - typedef unknown_storage_tag storage_category;
> + // typedef unknown_storage_tag storage_category;
> + typedef typename storage_restrict_traits<typename E1::storage_category, typename E2::storage_category>::storage_category storage_category;
> typedef unknown_orientation_tag orientation_category;
> typedef typename promote_traits<T1, T2>::promote_type promote_type;
> typedef matrix_matrix_binary<E1, E2, matrix_matrix_prod<E1, E2, promote_type> > expression_type;
> #ifndef BOOST_UBLAS_SIMPLE_ET_DEBUG
> - typedef expression_type result_type;
> + // typedef expression_type result_type;
> + typedef typename detail::binary_calculate_result_type<E1, E2, promote_type, boost::is_base_of<dense_proxy_tag, storage_category>::value>::result_type result_type;
> #else
> typedef typename E1::matrix_temporary_type result_type;
> #endif
> };
>
> - template<class E1, class E2>
> + template<class E1, class E2,
> + typename B = detail::prod_block_size<typename common_type<typename E1::value_type,
> + typename E2::value_type>::type> >
> BOOST_UBLAS_INLINE
> typename matrix_matrix_binary_traits<typename E1::value_type, E1,
> typename E2::value_type, E2>::result_type
> @@ -5481,13 +5502,41 @@ namespace boost { namespace numeric { namespace ublas {
> const matrix_expression<E2> &e2,
> unknown_storage_tag,
> unknown_orientation_tag) {
> +
> typedef typename matrix_matrix_binary_traits<typename E1::value_type, E1,
> typename E2::value_type, E2>::expression_type expression_type;
> return expression_type (e1 (), e2 ());
> }
>
> + template<class E1, class E2,
> + typename B = detail::prod_block_size<typename common_type<typename E1::value_type,
> + typename E2::value_type>::type> >
> + BOOST_UBLAS_INLINE
> + typename matrix_matrix_binary_traits<typename E1::value_type, E1,
> + typename E2::value_type, E2>::result_type
> + prod (const matrix_expression<E1> &e1,
> + const matrix_expression<E2> &e2,
> + dense_proxy_tag,
> + unknown_orientation_tag) {
> + typedef typename matrix_matrix_binary_traits<typename E1::value_type, E1,
> + typename E2::value_type, E2>::expression_type expression_type;
> + typedef typename E1::matrix_temporary_type result_type;
> + typedef typename result_type::value_type result_value;
> +
> + if (e1 ().size1() < B::limit || e2 ().size2() < B::limit) {
> + return expression_type (e1 (), e2 ());
> + } else {
> + result_type rv(e1 ().size1(), e2 ().size2());
> + detail::gemm<E1, E2, result_type, B>(result_value(1), e1, e2,
> + result_value(0), rv);
> + return rv;
> + }
> + }
> +
> // Dispatcher
> - template<class E1, class E2>
> + template<class E1, class E2,
> + typename B = detail::prod_block_size<typename common_type<typename E1::value_type,
> + typename E2::value_type>::type> >
> BOOST_UBLAS_INLINE
> typename matrix_matrix_binary_traits<typename E1::value_type, E1,
> typename E2::value_type, E2>::result_type
> @@ -5498,7 +5547,7 @@ namespace boost { namespace numeric { namespace ublas {
> typename E2::value_type, E2>::storage_category storage_category;
> typedef typename matrix_matrix_binary_traits<typename E1::value_type, E1,
> typename E2::value_type, E2>::orientation_category orientation_category;
> - return prod (e1, e2, storage_category (), orientation_category ());
> + return prod<E1, E2, B> (e1, e2, storage_category (), orientation_category ());
> }
>
> template<class E1, class E2>
> diff --git a/include/boost/numeric/ublas/operation.hpp b/include/boost/numeric/ublas/operation.hpp
> index 64657cc..80bfab6 100644
> --- a/include/boost/numeric/ublas/operation.hpp
> +++ b/include/boost/numeric/ublas/operation.hpp
> @@ -14,7 +14,7 @@
> #define _BOOST_UBLAS_OPERATION_
>
> #include <boost/numeric/ublas/matrix_proxy.hpp>
> -
> +#include <boost/numeric/ublas/detail/gemm.hpp>
> /** \file operation.hpp
> * \brief This file contains some specialized products.
> */
> @@ -637,13 +637,45 @@ namespace boost { namespace numeric { namespace ublas {
> return m;
> }
>
> - // Dispatcher
> - template<class M, class E1, class E2, class TRI>
> + template<class M, class E1, class E2,
> + typename B = detail::prod_block_size<typename common_type<typename E1::value_type,
> + typename E2::value_type>::type> >
> + BOOST_UBLAS_INLINE
> + M&
> + axpy_prod (const matrix_expression<E1> &e1,
> + const matrix_expression<E2> &e2,
> + M &m, full,
> + dense_proxy_tag, bool init = true)
> + {
> + typedef typename M::value_type value_type;
> +
> + if (m.size1() < B::limit || m.size2() < B::limit) {
> + typedef typename M::storage_category storage_category;
> + typedef typename M::orientation_category orientation_category;
> +
> + if (init)
> + m.assign (zero_matrix<value_type> (e1 ().size1 (), e2 ().size2 ()));
> + return axpy_prod (e1, e2, m, full (), storage_category (),
> + orientation_category ());
> + } else {
> +
> +
> + detail::gemm<E1, E2, M, B>(value_type(1), e1, e2,
> + value_type(init? 0 : 1), m);
> + return m;
> + }
> + }
> +
> + // Dispatchers
> + template<class M, class E1, class E2, class TRI, class S,
> + typename B = detail::prod_block_size<typename common_type<typename E1::value_type,
> + typename E2::value_type>::type> >
> BOOST_UBLAS_INLINE
> M &
> axpy_prod (const matrix_expression<E1> &e1,
> const matrix_expression<E2> &e2,
> - M &m, TRI, bool init = true) {
> + M &m, TRI,
> + S, bool init = true) {
> typedef typename M::value_type value_type;
> typedef typename M::storage_category storage_category;
> typedef typename M::orientation_category orientation_category;
> @@ -651,9 +683,31 @@ namespace boost { namespace numeric { namespace ublas {
>
> if (init)
> m.assign (zero_matrix<value_type> (e1 ().size1 (), e2 ().size2 ()));
> - return axpy_prod (e1, e2, m, triangular_restriction (), storage_category (), orientation_category ());
> +
> + return axpy_prod (e1, e2, m, triangular_restriction (), storage_category (),
> + orientation_category ());
> }
> - template<class M, class E1, class E2, class TRI>
> +
> +
> + template<class M, class E1, class E2, class TRI,
> + typename B = detail::prod_block_size<typename common_type<typename E1::value_type,
> + typename E2::value_type>::type> >
> + BOOST_UBLAS_INLINE
> + M &
> + axpy_prod (const matrix_expression<E1> &e1,
> + const matrix_expression<E2> &e2,
> + M &m, TRI, bool init = true) {
> + typedef typename M::value_type value_type;
> + typedef typename M::storage_category storage_category;
> + typedef typename M::orientation_category orientation_category;
> + typedef TRI triangular_restriction;
> +
> + return axpy_prod<M, E1, E2, TRI, B> (e1, e2, m, triangular_restriction (),
> + storage_category (), init);
> + }
> + template<class M, class E1, class E2, class TRI,
> + typename B = detail::prod_block_size<typename common_type<typename E1::value_type,
> + typename E2::value_type>::type> >
> BOOST_UBLAS_INLINE
> M
> axpy_prod (const matrix_expression<E1> &e1,
> @@ -663,7 +717,8 @@ namespace boost { namespace numeric { namespace ublas {
> typedef TRI triangular_restriction;
>
> matrix_type m (e1 ().size1 (), e2 ().size2 ());
> - return axpy_prod (e1, e2, m, triangular_restriction (), true);
> + return axpy_prod<M, E1, E2, TRI, B> (e1, e2, m, triangular_restriction (),
> + true);
> }
>
> /** \brief computes <tt>M += A X</tt> or <tt>M = A X</tt> in an
> @@ -690,7 +745,9 @@ namespace boost { namespace numeric { namespace ublas {
> \param E1 type of a matrix expression \c A
> \param E2 type of a matrix expression \c X
> */
> - template<class M, class E1, class E2>
> + template<class M, class E1, class E2,
> + typename B = detail::prod_block_size<typename common_type<typename E1::value_type,
> + typename E2::value_type>::type> >
> BOOST_UBLAS_INLINE
> M &
> axpy_prod (const matrix_expression<E1> &e1,
> @@ -700,11 +757,15 @@ namespace boost { namespace numeric { namespace ublas {
> typedef typename M::storage_category storage_category;
> typedef typename M::orientation_category orientation_category;
>
> - if (init)
> - m.assign (zero_matrix<value_type> (e1 ().size1 (), e2 ().size2 ()));
> - return axpy_prod (e1, e2, m, full (), storage_category (), orientation_category ());
> + // if (init)
> + // m.assign (zero_matrix<value_type> (e1 ().size1 (), e2 ().size2 ()));
> + return axpy_prod<M, E1, E2, B> (e1, e2, m, full (), storage_category (),
> + init);
> }
> - template<class M, class E1, class E2>
> +
> + template<class M, class E1, class E2,
> + typename B = detail::prod_block_size<typename common_type<typename E1::value_type,
> + typename E2::value_type>::type> >
> BOOST_UBLAS_INLINE
> M
> axpy_prod (const matrix_expression<E1> &e1,
> @@ -712,7 +773,7 @@ namespace boost { namespace numeric { namespace ublas {
> typedef M matrix_type;
>
> matrix_type m (e1 ().size1 (), e2 ().size2 ());
> - return axpy_prod (e1, e2, m, full (), true);
> + return axpy_prod<M, E1, E2, B> (e1, e2, m, full (), true);
> }
>
>
> --
> 1.9.1
>