| // This file is part of Eigen, a lightweight C++ template library |
| // for linear algebra. |
| // |
| // Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr> |
| // Modifications Copyright (C) 2022 Intel Corporation |
| // |
| // 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 |
| // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. |
| |
| #ifndef EIGEN_TRIANGULAR_SOLVER_MATRIX_H |
| #define EIGEN_TRIANGULAR_SOLVER_MATRIX_H |
| |
| // IWYU pragma: private |
| #include "../InternalHeaderCheck.h" |
| |
| namespace Eigen { |
| |
| namespace internal { |
| |
| template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride, |
| bool Specialized> |
| struct trsmKernelL { |
| // Generic Implementation of triangular solve for triangular matrix on left and multiple rhs. |
| // Handles non-packed matrices. |
| static void kernel(Index size, Index otherSize, const Scalar* _tri, Index triStride, Scalar* _other, Index otherIncr, |
| Index otherStride); |
| }; |
| |
| template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride, |
| bool Specialized> |
| struct trsmKernelR { |
| // Generic Implementation of triangular solve for triangular matrix on right and multiple lhs. |
| // Handles non-packed matrices. |
| static void kernel(Index size, Index otherSize, const Scalar* _tri, Index triStride, Scalar* _other, Index otherIncr, |
| Index otherStride); |
| }; |
| |
| template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride, |
| bool Specialized> |
| EIGEN_STRONG_INLINE void trsmKernelL<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, |
| Specialized>::kernel(Index size, Index otherSize, const Scalar* _tri, |
| Index triStride, Scalar* _other, Index otherIncr, |
| Index otherStride) { |
| typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> TriMapper; |
| typedef blas_data_mapper<Scalar, Index, ColMajor, Unaligned, OtherInnerStride> OtherMapper; |
| TriMapper tri(_tri, triStride); |
| OtherMapper other(_other, otherStride, otherIncr); |
| |
| enum { IsLower = (Mode & Lower) == Lower }; |
| conj_if<Conjugate> conj; |
| |
| // tr solve |
| for (Index k = 0; k < size; ++k) { |
| // TODO write a small kernel handling this (can be shared with trsv) |
| Index i = IsLower ? k : -k - 1; |
| Index rs = size - k - 1; // remaining size |
| Index s = TriStorageOrder == RowMajor ? (IsLower ? 0 : i + 1) : IsLower ? i + 1 : i - rs; |
| |
| Scalar a = (Mode & UnitDiag) ? Scalar(1) : Scalar(Scalar(1)/conj(tri(i,i))); |
| for (Index j = 0; j < otherSize; ++j) { |
| if (TriStorageOrder == RowMajor) { |
| Scalar b(0); |
| const Scalar* l = &tri(i, s); |
| typename OtherMapper::LinearMapper r = other.getLinearMapper(s, j); |
| for (Index i3 = 0; i3 < k; ++i3) b += conj(l[i3]) * r(i3); |
| |
| other(i, j) = (other(i, j) - b) * a; |
| } else { |
| Scalar& otherij = other(i, j); |
| otherij *= a; |
| Scalar b = otherij; |
| typename OtherMapper::LinearMapper r = other.getLinearMapper(s, j); |
| typename TriMapper::LinearMapper l = tri.getLinearMapper(s, i); |
| for (Index i3 = 0; i3 < rs; ++i3) r(i3) -= b * conj(l(i3)); |
| } |
| } |
| } |
| } |
| |
| template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride, |
| bool Specialized> |
| EIGEN_STRONG_INLINE void trsmKernelR<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, |
| Specialized>::kernel(Index size, Index otherSize, const Scalar* _tri, |
| Index triStride, Scalar* _other, Index otherIncr, |
| Index otherStride) { |
| typedef typename NumTraits<Scalar>::Real RealScalar; |
| typedef blas_data_mapper<Scalar, Index, ColMajor, Unaligned, OtherInnerStride> LhsMapper; |
| typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> RhsMapper; |
| LhsMapper lhs(_other, otherStride, otherIncr); |
| RhsMapper rhs(_tri, triStride); |
| |
| enum { RhsStorageOrder = TriStorageOrder, IsLower = (Mode & Lower) == Lower }; |
| conj_if<Conjugate> conj; |
| |
| for (Index k = 0; k < size; ++k) { |
| Index j = IsLower ? size - k - 1 : k; |
| |
| typename LhsMapper::LinearMapper r = lhs.getLinearMapper(0, j); |
| for (Index k3 = 0; k3 < k; ++k3) { |
| Scalar b = conj(rhs(IsLower ? j + 1 + k3 : k3, j)); |
| typename LhsMapper::LinearMapper a = lhs.getLinearMapper(0, IsLower ? j + 1 + k3 : k3); |
| for (Index i = 0; i < otherSize; ++i) r(i) -= a(i) * b; |
| } |
| if ((Mode & UnitDiag) == 0) { |
| Scalar inv_rjj = RealScalar(1) / conj(rhs(j, j)); |
| for (Index i = 0; i < otherSize; ++i) r(i) *= inv_rjj; |
| } |
| } |
| } |
| |
| // if the rhs is row major, let's transpose the product |
| template <typename Scalar, typename Index, int Side, int Mode, bool Conjugate, int TriStorageOrder, |
| int OtherInnerStride> |
| struct triangular_solve_matrix<Scalar, Index, Side, Mode, Conjugate, TriStorageOrder, RowMajor, OtherInnerStride> { |
| static void run(Index size, Index cols, const Scalar* tri, Index triStride, Scalar* _other, Index otherIncr, |
| Index otherStride, level3_blocking<Scalar, Scalar>& blocking) { |
| triangular_solve_matrix< |
| Scalar, Index, Side == OnTheLeft ? OnTheRight : OnTheLeft, (Mode & UnitDiag) | ((Mode & Upper) ? Lower : Upper), |
| NumTraits<Scalar>::IsComplex && Conjugate, TriStorageOrder == RowMajor ? ColMajor : RowMajor, ColMajor, |
| OtherInnerStride>::run(size, cols, tri, triStride, _other, otherIncr, otherStride, blocking); |
| } |
| }; |
| |
| /* Optimized triangular solver with multiple right hand side and the triangular matrix on the left |
| */ |
| template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride> |
| struct triangular_solve_matrix<Scalar, Index, OnTheLeft, Mode, Conjugate, TriStorageOrder, ColMajor, OtherInnerStride> { |
| static EIGEN_DONT_INLINE void run(Index size, Index otherSize, const Scalar* _tri, Index triStride, Scalar* _other, |
| Index otherIncr, Index otherStride, level3_blocking<Scalar, Scalar>& blocking); |
| }; |
| |
| template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride> |
| EIGEN_DONT_INLINE void triangular_solve_matrix<Scalar, Index, OnTheLeft, Mode, Conjugate, TriStorageOrder, ColMajor, |
| OtherInnerStride>::run(Index size, Index otherSize, const Scalar* _tri, |
| Index triStride, Scalar* _other, Index otherIncr, |
| Index otherStride, |
| level3_blocking<Scalar, Scalar>& blocking) { |
| Index cols = otherSize; |
| |
| std::ptrdiff_t l1, l2, l3; |
| manage_caching_sizes(GetAction, &l1, &l2, &l3); |
| |
| #if defined(EIGEN_VECTORIZE_AVX512) && EIGEN_USE_AVX512_TRSM_L_KERNELS && EIGEN_ENABLE_AVX512_NOCOPY_TRSM_L_CUTOFFS |
| EIGEN_IF_CONSTEXPR( |
| (OtherInnerStride == 1 && (std::is_same<Scalar, float>::value || std::is_same<Scalar, double>::value))) { |
| // Very rough cutoffs to determine when to call trsm w/o packing |
| // For small problem sizes trsmKernel compiled with clang is generally faster. |
| // TODO: Investigate better heuristics for cutoffs. |
| double L2Cap = 0.5; // 50% of L2 size |
| if (size < avx512_trsm_cutoff<Scalar>(l2, cols, L2Cap)) { |
| trsmKernelL<Scalar, Index, Mode, Conjugate, TriStorageOrder, 1, /*Specialized=*/true>::kernel( |
| size, cols, _tri, triStride, _other, 1, otherStride); |
| return; |
| } |
| } |
| #endif |
| |
| typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> TriMapper; |
| typedef blas_data_mapper<Scalar, Index, ColMajor, Unaligned, OtherInnerStride> OtherMapper; |
| TriMapper tri(_tri, triStride); |
| OtherMapper other(_other, otherStride, otherIncr); |
| |
| typedef gebp_traits<Scalar, Scalar> Traits; |
| |
| enum { SmallPanelWidth = plain_enum_max(Traits::mr, Traits::nr), IsLower = (Mode & Lower) == Lower }; |
| |
| Index kc = blocking.kc(); // cache block size along the K direction |
| Index mc = (std::min)(size, blocking.mc()); // cache block size along the M direction |
| |
| std::size_t sizeA = kc * mc; |
| std::size_t sizeB = kc * cols; |
| |
| ei_declare_aligned_stack_constructed_variable(Scalar, blockA, sizeA, blocking.blockA()); |
| ei_declare_aligned_stack_constructed_variable(Scalar, blockB, sizeB, blocking.blockB()); |
| |
| gebp_kernel<Scalar, Scalar, Index, OtherMapper, Traits::mr, Traits::nr, Conjugate, false> gebp_kernel; |
| gemm_pack_lhs<Scalar, Index, TriMapper, Traits::mr, Traits::LhsProgress, typename Traits::LhsPacket4Packing, |
| TriStorageOrder> |
| pack_lhs; |
| gemm_pack_rhs<Scalar, Index, OtherMapper, Traits::nr, ColMajor, false, true> pack_rhs; |
| |
| // the goal here is to subdivise the Rhs panels such that we keep some cache |
| // coherence when accessing the rhs elements |
| Index subcols = cols > 0 ? l2 / (4 * sizeof(Scalar) * std::max<Index>(otherStride, size)) : 0; |
| subcols = std::max<Index>((subcols / Traits::nr) * Traits::nr, Traits::nr); |
| |
| for (Index k2 = IsLower ? 0 : size; IsLower ? k2 < size : k2 > 0; IsLower ? k2 += kc : k2 -= kc) { |
| const Index actual_kc = (std::min)(IsLower ? size - k2 : k2, kc); |
| |
| // We have selected and packed a big horizontal panel R1 of rhs. Let B be the packed copy of this panel, |
| // and R2 the remaining part of rhs. The corresponding vertical panel of lhs is split into |
| // A11 (the triangular part) and A21 the remaining rectangular part. |
| // Then the high level algorithm is: |
| // - B = R1 => general block copy (done during the next step) |
| // - R1 = A11^-1 B => tricky part |
| // - update B from the new R1 => actually this has to be performed continuously during the above step |
| // - R2 -= A21 * B => GEPP |
| |
| // The tricky part: compute R1 = A11^-1 B while updating B from R1 |
| // The idea is to split A11 into multiple small vertical panels. |
| // Each panel can be split into a small triangular part T1k which is processed without optimization, |
| // and the remaining small part T2k which is processed using gebp with appropriate block strides |
| for (Index j2 = 0; j2 < cols; j2 += subcols) { |
| Index actual_cols = (std::min)(cols - j2, subcols); |
| // for each small vertical panels [T1k^T, T2k^T]^T of lhs |
| for (Index k1 = 0; k1 < actual_kc; k1 += SmallPanelWidth) { |
| Index actualPanelWidth = std::min<Index>(actual_kc - k1, SmallPanelWidth); |
| // tr solve |
| { |
| Index i = IsLower ? k2 + k1 : k2 - k1; |
| #if defined(EIGEN_VECTORIZE_AVX512) && EIGEN_USE_AVX512_TRSM_L_KERNELS |
| EIGEN_IF_CONSTEXPR( |
| (OtherInnerStride == 1 && (std::is_same<Scalar, float>::value || std::is_same<Scalar, double>::value))) { |
| i = IsLower ? k2 + k1 : k2 - k1 - actualPanelWidth; |
| } |
| #endif |
| trsmKernelL<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, /*Specialized=*/true>::kernel( |
| actualPanelWidth, actual_cols, _tri + i + (i)*triStride, triStride, |
| _other + i * OtherInnerStride + j2 * otherStride, otherIncr, otherStride); |
| } |
| |
| Index lengthTarget = actual_kc - k1 - actualPanelWidth; |
| Index startBlock = IsLower ? k2 + k1 : k2 - k1 - actualPanelWidth; |
| Index blockBOffset = IsLower ? k1 : lengthTarget; |
| |
| // update the respective rows of B from other |
| pack_rhs(blockB + actual_kc * j2, other.getSubMapper(startBlock, j2), actualPanelWidth, actual_cols, actual_kc, |
| blockBOffset); |
| |
| // GEBP |
| if (lengthTarget > 0) { |
| Index startTarget = IsLower ? k2 + k1 + actualPanelWidth : k2 - actual_kc; |
| |
| pack_lhs(blockA, tri.getSubMapper(startTarget, startBlock), actualPanelWidth, lengthTarget); |
| |
| gebp_kernel(other.getSubMapper(startTarget, j2), blockA, blockB + actual_kc * j2, lengthTarget, |
| actualPanelWidth, actual_cols, Scalar(-1), actualPanelWidth, actual_kc, 0, blockBOffset); |
| } |
| } |
| } |
| |
| // R2 -= A21 * B => GEPP |
| { |
| Index start = IsLower ? k2 + kc : 0; |
| Index end = IsLower ? size : k2 - kc; |
| for (Index i2 = start; i2 < end; i2 += mc) { |
| const Index actual_mc = (std::min)(mc, end - i2); |
| if (actual_mc > 0) { |
| pack_lhs(blockA, tri.getSubMapper(i2, IsLower ? k2 : k2 - kc), actual_kc, actual_mc); |
| |
| gebp_kernel(other.getSubMapper(i2, 0), blockA, blockB, actual_mc, actual_kc, cols, Scalar(-1), -1, -1, 0, 0); |
| } |
| } |
| } |
| } |
| } |
| |
| /* Optimized triangular solver with multiple left hand sides and the triangular matrix on the right |
| */ |
| template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride> |
| struct triangular_solve_matrix<Scalar, Index, OnTheRight, Mode, Conjugate, TriStorageOrder, ColMajor, |
| OtherInnerStride> { |
| static EIGEN_DONT_INLINE void run(Index size, Index otherSize, const Scalar* _tri, Index triStride, Scalar* _other, |
| Index otherIncr, Index otherStride, level3_blocking<Scalar, Scalar>& blocking); |
| }; |
| |
| template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder, int OtherInnerStride> |
| EIGEN_DONT_INLINE void triangular_solve_matrix<Scalar, Index, OnTheRight, Mode, Conjugate, TriStorageOrder, ColMajor, |
| OtherInnerStride>::run(Index size, Index otherSize, const Scalar* _tri, |
| Index triStride, Scalar* _other, Index otherIncr, |
| Index otherStride, |
| level3_blocking<Scalar, Scalar>& blocking) { |
| Index rows = otherSize; |
| |
| #if defined(EIGEN_VECTORIZE_AVX512) && EIGEN_USE_AVX512_TRSM_R_KERNELS && EIGEN_ENABLE_AVX512_NOCOPY_TRSM_R_CUTOFFS |
| EIGEN_IF_CONSTEXPR( |
| (OtherInnerStride == 1 && (std::is_same<Scalar, float>::value || std::is_same<Scalar, double>::value))) { |
| // TODO: Investigate better heuristics for cutoffs. |
| std::ptrdiff_t l1, l2, l3; |
| manage_caching_sizes(GetAction, &l1, &l2, &l3); |
| double L2Cap = 0.5; // 50% of L2 size |
| if (size < avx512_trsm_cutoff<Scalar>(l2, rows, L2Cap)) { |
| trsmKernelR<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, /*Specialized=*/true>::kernel( |
| size, rows, _tri, triStride, _other, 1, otherStride); |
| return; |
| } |
| } |
| #endif |
| |
| typedef blas_data_mapper<Scalar, Index, ColMajor, Unaligned, OtherInnerStride> LhsMapper; |
| typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> RhsMapper; |
| LhsMapper lhs(_other, otherStride, otherIncr); |
| RhsMapper rhs(_tri, triStride); |
| |
| typedef gebp_traits<Scalar, Scalar> Traits; |
| enum { |
| RhsStorageOrder = TriStorageOrder, |
| SmallPanelWidth = plain_enum_max(Traits::mr, Traits::nr), |
| IsLower = (Mode & Lower) == Lower |
| }; |
| |
| Index kc = blocking.kc(); // cache block size along the K direction |
| Index mc = (std::min)(rows, blocking.mc()); // cache block size along the M direction |
| |
| std::size_t sizeA = kc * mc; |
| std::size_t sizeB = kc * size; |
| |
| ei_declare_aligned_stack_constructed_variable(Scalar, blockA, sizeA, blocking.blockA()); |
| ei_declare_aligned_stack_constructed_variable(Scalar, blockB, sizeB, blocking.blockB()); |
| |
| gebp_kernel<Scalar, Scalar, Index, LhsMapper, Traits::mr, Traits::nr, false, Conjugate> gebp_kernel; |
| gemm_pack_rhs<Scalar, Index, RhsMapper, Traits::nr, RhsStorageOrder> pack_rhs; |
| gemm_pack_rhs<Scalar, Index, RhsMapper, Traits::nr, RhsStorageOrder, false, true> pack_rhs_panel; |
| gemm_pack_lhs<Scalar, Index, LhsMapper, Traits::mr, Traits::LhsProgress, typename Traits::LhsPacket4Packing, ColMajor, |
| false, true> |
| pack_lhs_panel; |
| |
| for (Index k2 = IsLower ? size : 0; IsLower ? k2 > 0 : k2 < size; IsLower ? k2 -= kc : k2 += kc) { |
| const Index actual_kc = (std::min)(IsLower ? k2 : size - k2, kc); |
| Index actual_k2 = IsLower ? k2 - actual_kc : k2; |
| |
| Index startPanel = IsLower ? 0 : k2 + actual_kc; |
| Index rs = IsLower ? actual_k2 : size - actual_k2 - actual_kc; |
| Scalar* geb = blockB + actual_kc * actual_kc; |
| |
| if (rs > 0) pack_rhs(geb, rhs.getSubMapper(actual_k2, startPanel), actual_kc, rs); |
| |
| // triangular packing (we only pack the panels off the diagonal, |
| // neglecting the blocks overlapping the diagonal |
| { |
| for (Index j2 = 0; j2 < actual_kc; j2 += SmallPanelWidth) { |
| Index actualPanelWidth = std::min<Index>(actual_kc - j2, SmallPanelWidth); |
| Index actual_j2 = actual_k2 + j2; |
| Index panelOffset = IsLower ? j2 + actualPanelWidth : 0; |
| Index panelLength = IsLower ? actual_kc - j2 - actualPanelWidth : j2; |
| |
| if (panelLength > 0) |
| pack_rhs_panel(blockB + j2 * actual_kc, rhs.getSubMapper(actual_k2 + panelOffset, actual_j2), panelLength, |
| actualPanelWidth, actual_kc, panelOffset); |
| } |
| } |
| |
| for (Index i2 = 0; i2 < rows; i2 += mc) { |
| const Index actual_mc = (std::min)(mc, rows - i2); |
| |
| // triangular solver kernel |
| { |
| // for each small block of the diagonal (=> vertical panels of rhs) |
| for (Index j2 = IsLower ? (actual_kc - ((actual_kc % SmallPanelWidth) ? Index(actual_kc % SmallPanelWidth) |
| : Index(SmallPanelWidth))) |
| : 0; |
| IsLower ? j2 >= 0 : j2 < actual_kc; IsLower ? j2 -= SmallPanelWidth : j2 += SmallPanelWidth) { |
| Index actualPanelWidth = std::min<Index>(actual_kc - j2, SmallPanelWidth); |
| Index absolute_j2 = actual_k2 + j2; |
| Index panelOffset = IsLower ? j2 + actualPanelWidth : 0; |
| Index panelLength = IsLower ? actual_kc - j2 - actualPanelWidth : j2; |
| |
| // GEBP |
| if (panelLength > 0) { |
| gebp_kernel(lhs.getSubMapper(i2, absolute_j2), blockA, blockB + j2 * actual_kc, actual_mc, panelLength, |
| actualPanelWidth, Scalar(-1), actual_kc, actual_kc, // strides |
| panelOffset, panelOffset); // offsets |
| } |
| |
| { |
| // unblocked triangular solve |
| trsmKernelR<Scalar, Index, Mode, Conjugate, TriStorageOrder, OtherInnerStride, |
| /*Specialized=*/true>::kernel(actualPanelWidth, actual_mc, |
| _tri + absolute_j2 + absolute_j2 * triStride, triStride, |
| _other + i2 * OtherInnerStride + absolute_j2 * otherStride, |
| otherIncr, otherStride); |
| } |
| // pack the just computed part of lhs to A |
| pack_lhs_panel(blockA, lhs.getSubMapper(i2, absolute_j2), actualPanelWidth, actual_mc, actual_kc, j2); |
| } |
| } |
| |
| if (rs > 0) |
| gebp_kernel(lhs.getSubMapper(i2, startPanel), blockA, geb, actual_mc, actual_kc, rs, Scalar(-1), -1, -1, 0, 0); |
| } |
| } |
| } |
| } // end namespace internal |
| |
| } // end namespace Eigen |
| |
| #endif // EIGEN_TRIANGULAR_SOLVER_MATRIX_H |
