1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
5//
6// This Source Code Form is subject to the terms of the Mozilla
7// Public License v. 2.0. If a copy of the MPL was not distributed
8// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10#ifndef EIGEN_TRIANGULAR_SOLVER_MATRIX_H
11#define EIGEN_TRIANGULAR_SOLVER_MATRIX_H
12
13namespace Eigen {
14
15namespace internal {
16
17// if the rhs is row major, let's transpose the product
18template <typename Scalar, typename Index, int Side, int Mode, bool Conjugate, int TriStorageOrder>
19struct triangular_solve_matrix<Scalar,Index,Side,Mode,Conjugate,TriStorageOrder,RowMajor>
20{
21  static void run(
22    Index size, Index cols,
23    const Scalar*  tri, Index triStride,
24    Scalar* _other, Index otherStride,
25    level3_blocking<Scalar,Scalar>& blocking)
26  {
27    triangular_solve_matrix<
28      Scalar, Index, Side==OnTheLeft?OnTheRight:OnTheLeft,
29      (Mode&UnitDiag) | ((Mode&Upper) ? Lower : Upper),
30      NumTraits<Scalar>::IsComplex && Conjugate,
31      TriStorageOrder==RowMajor ? ColMajor : RowMajor, ColMajor>
32      ::run(size, cols, tri, triStride, _other, otherStride, blocking);
33  }
34};
35
36/* Optimized triangular solver with multiple right hand side and the triangular matrix on the left
37 */
38template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
39struct triangular_solve_matrix<Scalar,Index,OnTheLeft,Mode,Conjugate,TriStorageOrder,ColMajor>
40{
41  static EIGEN_DONT_INLINE void run(
42    Index size, Index otherSize,
43    const Scalar* _tri, Index triStride,
44    Scalar* _other, Index otherStride,
45    level3_blocking<Scalar,Scalar>& blocking);
46};
47template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
48EIGEN_DONT_INLINE void triangular_solve_matrix<Scalar,Index,OnTheLeft,Mode,Conjugate,TriStorageOrder,ColMajor>::run(
49    Index size, Index otherSize,
50    const Scalar* _tri, Index triStride,
51    Scalar* _other, Index otherStride,
52    level3_blocking<Scalar,Scalar>& blocking)
53  {
54    Index cols = otherSize;
55    const_blas_data_mapper<Scalar, Index, TriStorageOrder> tri(_tri,triStride);
56    blas_data_mapper<Scalar, Index, ColMajor> other(_other,otherStride);
57
58    typedef gebp_traits<Scalar,Scalar> Traits;
59    enum {
60      SmallPanelWidth   = EIGEN_PLAIN_ENUM_MAX(Traits::mr,Traits::nr),
61      IsLower = (Mode&Lower) == Lower
62    };
63
64    Index kc = blocking.kc();                   // cache block size along the K direction
65    Index mc = (std::min)(size,blocking.mc());  // cache block size along the M direction
66
67    std::size_t sizeA = kc*mc;
68    std::size_t sizeB = kc*cols;
69    std::size_t sizeW = kc*Traits::WorkSpaceFactor;
70
71    ei_declare_aligned_stack_constructed_variable(Scalar, blockA, sizeA, blocking.blockA());
72    ei_declare_aligned_stack_constructed_variable(Scalar, blockB, sizeB, blocking.blockB());
73    ei_declare_aligned_stack_constructed_variable(Scalar, blockW, sizeW, blocking.blockW());
74
75    conj_if<Conjugate> conj;
76    gebp_kernel<Scalar, Scalar, Index, Traits::mr, Traits::nr, Conjugate, false> gebp_kernel;
77    gemm_pack_lhs<Scalar, Index, Traits::mr, Traits::LhsProgress, TriStorageOrder> pack_lhs;
78    gemm_pack_rhs<Scalar, Index, Traits::nr, ColMajor, false, true> pack_rhs;
79
80    // the goal here is to subdivise the Rhs panels such that we keep some cache
81    // coherence when accessing the rhs elements
82    std::ptrdiff_t l1, l2;
83    manage_caching_sizes(GetAction, &l1, &l2);
84    Index subcols = cols>0 ? l2/(4 * sizeof(Scalar) * otherStride) : 0;
85    subcols = std::max<Index>((subcols/Traits::nr)*Traits::nr, Traits::nr);
86
87    for(Index k2=IsLower ? 0 : size;
88        IsLower ? k2<size : k2>0;
89        IsLower ? k2+=kc : k2-=kc)
90    {
91      const Index actual_kc = (std::min)(IsLower ? size-k2 : k2, kc);
92
93      // We have selected and packed a big horizontal panel R1 of rhs. Let B be the packed copy of this panel,
94      // and R2 the remaining part of rhs. The corresponding vertical panel of lhs is split into
95      // A11 (the triangular part) and A21 the remaining rectangular part.
96      // Then the high level algorithm is:
97      //  - B = R1                    => general block copy (done during the next step)
98      //  - R1 = A11^-1 B             => tricky part
99      //  - update B from the new R1  => actually this has to be performed continuously during the above step
100      //  - R2 -= A21 * B             => GEPP
101
102      // The tricky part: compute R1 = A11^-1 B while updating B from R1
103      // The idea is to split A11 into multiple small vertical panels.
104      // Each panel can be split into a small triangular part T1k which is processed without optimization,
105      // and the remaining small part T2k which is processed using gebp with appropriate block strides
106      for(Index j2=0; j2<cols; j2+=subcols)
107      {
108        Index actual_cols = (std::min)(cols-j2,subcols);
109        // for each small vertical panels [T1k^T, T2k^T]^T of lhs
110        for (Index k1=0; k1<actual_kc; k1+=SmallPanelWidth)
111        {
112          Index actualPanelWidth = std::min<Index>(actual_kc-k1, SmallPanelWidth);
113          // tr solve
114          for (Index k=0; k<actualPanelWidth; ++k)
115          {
116            // TODO write a small kernel handling this (can be shared with trsv)
117            Index i  = IsLower ? k2+k1+k : k2-k1-k-1;
118            Index s  = IsLower ? k2+k1 : i+1;
119            Index rs = actualPanelWidth - k - 1; // remaining size
120
121            Scalar a = (Mode & UnitDiag) ? Scalar(1) : Scalar(1)/conj(tri(i,i));
122            for (Index j=j2; j<j2+actual_cols; ++j)
123            {
124              if (TriStorageOrder==RowMajor)
125              {
126                Scalar b(0);
127                const Scalar* l = &tri(i,s);
128                Scalar* r = &other(s,j);
129                for (Index i3=0; i3<k; ++i3)
130                  b += conj(l[i3]) * r[i3];
131
132                other(i,j) = (other(i,j) - b)*a;
133              }
134              else
135              {
136                Index s = IsLower ? i+1 : i-rs;
137                Scalar b = (other(i,j) *= a);
138                Scalar* r = &other(s,j);
139                const Scalar* l = &tri(s,i);
140                for (Index i3=0;i3<rs;++i3)
141                  r[i3] -= b * conj(l[i3]);
142              }
143            }
144          }
145
146          Index lengthTarget = actual_kc-k1-actualPanelWidth;
147          Index startBlock   = IsLower ? k2+k1 : k2-k1-actualPanelWidth;
148          Index blockBOffset = IsLower ? k1 : lengthTarget;
149
150          // update the respective rows of B from other
151          pack_rhs(blockB+actual_kc*j2, &other(startBlock,j2), otherStride, actualPanelWidth, actual_cols, actual_kc, blockBOffset);
152
153          // GEBP
154          if (lengthTarget>0)
155          {
156            Index startTarget  = IsLower ? k2+k1+actualPanelWidth : k2-actual_kc;
157
158            pack_lhs(blockA, &tri(startTarget,startBlock), triStride, actualPanelWidth, lengthTarget);
159
160            gebp_kernel(&other(startTarget,j2), otherStride, blockA, blockB+actual_kc*j2, lengthTarget, actualPanelWidth, actual_cols, Scalar(-1),
161                        actualPanelWidth, actual_kc, 0, blockBOffset, blockW);
162          }
163        }
164      }
165
166      // R2 -= A21 * B => GEPP
167      {
168        Index start = IsLower ? k2+kc : 0;
169        Index end   = IsLower ? size : k2-kc;
170        for(Index i2=start; i2<end; i2+=mc)
171        {
172          const Index actual_mc = (std::min)(mc,end-i2);
173          if (actual_mc>0)
174          {
175            pack_lhs(blockA, &tri(i2, IsLower ? k2 : k2-kc), triStride, actual_kc, actual_mc);
176
177            gebp_kernel(_other+i2, otherStride, blockA, blockB, actual_mc, actual_kc, cols, Scalar(-1), -1, -1, 0, 0, blockW);
178          }
179        }
180      }
181    }
182  }
183
184/* Optimized triangular solver with multiple left hand sides and the trinagular matrix on the right
185 */
186template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
187struct triangular_solve_matrix<Scalar,Index,OnTheRight,Mode,Conjugate,TriStorageOrder,ColMajor>
188{
189  static EIGEN_DONT_INLINE void run(
190    Index size, Index otherSize,
191    const Scalar* _tri, Index triStride,
192    Scalar* _other, Index otherStride,
193    level3_blocking<Scalar,Scalar>& blocking);
194};
195template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
196EIGEN_DONT_INLINE void triangular_solve_matrix<Scalar,Index,OnTheRight,Mode,Conjugate,TriStorageOrder,ColMajor>::run(
197    Index size, Index otherSize,
198    const Scalar* _tri, Index triStride,
199    Scalar* _other, Index otherStride,
200    level3_blocking<Scalar,Scalar>& blocking)
201  {
202    Index rows = otherSize;
203    const_blas_data_mapper<Scalar, Index, TriStorageOrder> rhs(_tri,triStride);
204    blas_data_mapper<Scalar, Index, ColMajor> lhs(_other,otherStride);
205
206    typedef gebp_traits<Scalar,Scalar> Traits;
207    enum {
208      RhsStorageOrder   = TriStorageOrder,
209      SmallPanelWidth   = EIGEN_PLAIN_ENUM_MAX(Traits::mr,Traits::nr),
210      IsLower = (Mode&Lower) == Lower
211    };
212
213    Index kc = blocking.kc();                   // cache block size along the K direction
214    Index mc = (std::min)(rows,blocking.mc());  // cache block size along the M direction
215
216    std::size_t sizeA = kc*mc;
217    std::size_t sizeB = kc*size;
218    std::size_t sizeW = kc*Traits::WorkSpaceFactor;
219
220    ei_declare_aligned_stack_constructed_variable(Scalar, blockA, sizeA, blocking.blockA());
221    ei_declare_aligned_stack_constructed_variable(Scalar, blockB, sizeB, blocking.blockB());
222    ei_declare_aligned_stack_constructed_variable(Scalar, blockW, sizeW, blocking.blockW());
223
224    conj_if<Conjugate> conj;
225    gebp_kernel<Scalar,Scalar, Index, Traits::mr, Traits::nr, false, Conjugate> gebp_kernel;
226    gemm_pack_rhs<Scalar, Index, Traits::nr,RhsStorageOrder> pack_rhs;
227    gemm_pack_rhs<Scalar, Index, Traits::nr,RhsStorageOrder,false,true> pack_rhs_panel;
228    gemm_pack_lhs<Scalar, Index, Traits::mr, Traits::LhsProgress, ColMajor, false, true> pack_lhs_panel;
229
230    for(Index k2=IsLower ? size : 0;
231        IsLower ? k2>0 : k2<size;
232        IsLower ? k2-=kc : k2+=kc)
233    {
234      const Index actual_kc = (std::min)(IsLower ? k2 : size-k2, kc);
235      Index actual_k2 = IsLower ? k2-actual_kc : k2 ;
236
237      Index startPanel = IsLower ? 0 : k2+actual_kc;
238      Index rs = IsLower ? actual_k2 : size - actual_k2 - actual_kc;
239      Scalar* geb = blockB+actual_kc*actual_kc;
240
241      if (rs>0) pack_rhs(geb, &rhs(actual_k2,startPanel), triStride, actual_kc, rs);
242
243      // triangular packing (we only pack the panels off the diagonal,
244      // neglecting the blocks overlapping the diagonal
245      {
246        for (Index j2=0; j2<actual_kc; j2+=SmallPanelWidth)
247        {
248          Index actualPanelWidth = std::min<Index>(actual_kc-j2, SmallPanelWidth);
249          Index actual_j2 = actual_k2 + j2;
250          Index panelOffset = IsLower ? j2+actualPanelWidth : 0;
251          Index panelLength = IsLower ? actual_kc-j2-actualPanelWidth : j2;
252
253          if (panelLength>0)
254          pack_rhs_panel(blockB+j2*actual_kc,
255                         &rhs(actual_k2+panelOffset, actual_j2), triStride,
256                         panelLength, actualPanelWidth,
257                         actual_kc, panelOffset);
258        }
259      }
260
261      for(Index i2=0; i2<rows; i2+=mc)
262      {
263        const Index actual_mc = (std::min)(mc,rows-i2);
264
265        // triangular solver kernel
266        {
267          // for each small block of the diagonal (=> vertical panels of rhs)
268          for (Index j2 = IsLower
269                      ? (actual_kc - ((actual_kc%SmallPanelWidth) ? Index(actual_kc%SmallPanelWidth)
270                                                                  : Index(SmallPanelWidth)))
271                      : 0;
272               IsLower ? j2>=0 : j2<actual_kc;
273               IsLower ? j2-=SmallPanelWidth : j2+=SmallPanelWidth)
274          {
275            Index actualPanelWidth = std::min<Index>(actual_kc-j2, SmallPanelWidth);
276            Index absolute_j2 = actual_k2 + j2;
277            Index panelOffset = IsLower ? j2+actualPanelWidth : 0;
278            Index panelLength = IsLower ? actual_kc - j2 - actualPanelWidth : j2;
279
280            // GEBP
281            if(panelLength>0)
282            {
283              gebp_kernel(&lhs(i2,absolute_j2), otherStride,
284                          blockA, blockB+j2*actual_kc,
285                          actual_mc, panelLength, actualPanelWidth,
286                          Scalar(-1),
287                          actual_kc, actual_kc, // strides
288                          panelOffset, panelOffset, // offsets
289                          blockW);  // workspace
290            }
291
292            // unblocked triangular solve
293            for (Index k=0; k<actualPanelWidth; ++k)
294            {
295              Index j = IsLower ? absolute_j2+actualPanelWidth-k-1 : absolute_j2+k;
296
297              Scalar* r = &lhs(i2,j);
298              for (Index k3=0; k3<k; ++k3)
299              {
300                Scalar b = conj(rhs(IsLower ? j+1+k3 : absolute_j2+k3,j));
301                Scalar* a = &lhs(i2,IsLower ? j+1+k3 : absolute_j2+k3);
302                for (Index i=0; i<actual_mc; ++i)
303                  r[i] -= a[i] * b;
304              }
305              Scalar b = (Mode & UnitDiag) ? Scalar(1) : Scalar(1)/conj(rhs(j,j));
306              for (Index i=0; i<actual_mc; ++i)
307                r[i] *= b;
308            }
309
310            // pack the just computed part of lhs to A
311            pack_lhs_panel(blockA, _other+absolute_j2*otherStride+i2, otherStride,
312                           actualPanelWidth, actual_mc,
313                           actual_kc, j2);
314          }
315        }
316
317        if (rs>0)
318          gebp_kernel(_other+i2+startPanel*otherStride, otherStride, blockA, geb,
319                      actual_mc, actual_kc, rs, Scalar(-1),
320                      -1, -1, 0, 0, blockW);
321      }
322    }
323  }
324
325} // end namespace internal
326
327} // end namespace Eigen
328
329#endif // EIGEN_TRIANGULAR_SOLVER_MATRIX_H
330