Eigen  3.3.1
TriangularSolverMatrix.h
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 
13 namespace Eigen {
14 
15 namespace internal {
16 
17 // if the rhs is row major, let's transpose the product
18 template <typename Scalar, typename Index, int Side, int Mode, bool Conjugate, int TriStorageOrder>
19 struct 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  */
38 template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
39 struct 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 };
47 template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
48 EIGEN_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 
56  typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> TriMapper;
57  typedef blas_data_mapper<Scalar, Index, ColMajor> OtherMapper;
58  TriMapper tri(_tri, triStride);
59  OtherMapper other(_other, otherStride);
60 
61  typedef gebp_traits<Scalar,Scalar> Traits;
62 
63  enum {
64  SmallPanelWidth = EIGEN_PLAIN_ENUM_MAX(Traits::mr,Traits::nr),
65  IsLower = (Mode&Lower) == Lower
66  };
67 
68  Index kc = blocking.kc(); // cache block size along the K direction
69  Index mc = (std::min)(size,blocking.mc()); // cache block size along the M direction
70 
71  std::size_t sizeA = kc*mc;
72  std::size_t sizeB = kc*cols;
73 
74  ei_declare_aligned_stack_constructed_variable(Scalar, blockA, sizeA, blocking.blockA());
75  ei_declare_aligned_stack_constructed_variable(Scalar, blockB, sizeB, blocking.blockB());
76 
77  conj_if<Conjugate> conj;
78  gebp_kernel<Scalar, Scalar, Index, OtherMapper, Traits::mr, Traits::nr, Conjugate, false> gebp_kernel;
79  gemm_pack_lhs<Scalar, Index, TriMapper, Traits::mr, Traits::LhsProgress, TriStorageOrder> pack_lhs;
80  gemm_pack_rhs<Scalar, Index, OtherMapper, Traits::nr, ColMajor, false, true> pack_rhs;
81 
82  // the goal here is to subdivise the Rhs panels such that we keep some cache
83  // coherence when accessing the rhs elements
84  std::ptrdiff_t l1, l2, l3;
85  manage_caching_sizes(GetAction, &l1, &l2, &l3);
86  Index subcols = cols>0 ? l2/(4 * sizeof(Scalar) * std::max<Index>(otherStride,size)) : 0;
87  subcols = std::max<Index>((subcols/Traits::nr)*Traits::nr, Traits::nr);
88 
89  for(Index k2=IsLower ? 0 : size;
90  IsLower ? k2<size : k2>0;
91  IsLower ? k2+=kc : k2-=kc)
92  {
93  const Index actual_kc = (std::min)(IsLower ? size-k2 : k2, kc);
94 
95  // We have selected and packed a big horizontal panel R1 of rhs. Let B be the packed copy of this panel,
96  // and R2 the remaining part of rhs. The corresponding vertical panel of lhs is split into
97  // A11 (the triangular part) and A21 the remaining rectangular part.
98  // Then the high level algorithm is:
99  // - B = R1 => general block copy (done during the next step)
100  // - R1 = A11^-1 B => tricky part
101  // - update B from the new R1 => actually this has to be performed continuously during the above step
102  // - R2 -= A21 * B => GEPP
103 
104  // The tricky part: compute R1 = A11^-1 B while updating B from R1
105  // The idea is to split A11 into multiple small vertical panels.
106  // Each panel can be split into a small triangular part T1k which is processed without optimization,
107  // and the remaining small part T2k which is processed using gebp with appropriate block strides
108  for(Index j2=0; j2<cols; j2+=subcols)
109  {
110  Index actual_cols = (std::min)(cols-j2,subcols);
111  // for each small vertical panels [T1k^T, T2k^T]^T of lhs
112  for (Index k1=0; k1<actual_kc; k1+=SmallPanelWidth)
113  {
114  Index actualPanelWidth = std::min<Index>(actual_kc-k1, SmallPanelWidth);
115  // tr solve
116  for (Index k=0; k<actualPanelWidth; ++k)
117  {
118  // TODO write a small kernel handling this (can be shared with trsv)
119  Index i = IsLower ? k2+k1+k : k2-k1-k-1;
120  Index rs = actualPanelWidth - k - 1; // remaining size
121  Index s = TriStorageOrder==RowMajor ? (IsLower ? k2+k1 : i+1)
122  : IsLower ? i+1 : i-rs;
123 
124  Scalar a = (Mode & UnitDiag) ? Scalar(1) : Scalar(1)/conj(tri(i,i));
125  for (Index j=j2; j<j2+actual_cols; ++j)
126  {
127  if (TriStorageOrder==RowMajor)
128  {
129  Scalar b(0);
130  const Scalar* l = &tri(i,s);
131  Scalar* r = &other(s,j);
132  for (Index i3=0; i3<k; ++i3)
133  b += conj(l[i3]) * r[i3];
134 
135  other(i,j) = (other(i,j) - b)*a;
136  }
137  else
138  {
139  Scalar b = (other(i,j) *= a);
140  Scalar* r = &other(s,j);
141  const Scalar* l = &tri(s,i);
142  for (Index i3=0;i3<rs;++i3)
143  r[i3] -= b * conj(l[i3]);
144  }
145  }
146  }
147 
148  Index lengthTarget = actual_kc-k1-actualPanelWidth;
149  Index startBlock = IsLower ? k2+k1 : k2-k1-actualPanelWidth;
150  Index blockBOffset = IsLower ? k1 : lengthTarget;
151 
152  // update the respective rows of B from other
153  pack_rhs(blockB+actual_kc*j2, other.getSubMapper(startBlock,j2), actualPanelWidth, actual_cols, actual_kc, blockBOffset);
154 
155  // GEBP
156  if (lengthTarget>0)
157  {
158  Index startTarget = IsLower ? k2+k1+actualPanelWidth : k2-actual_kc;
159 
160  pack_lhs(blockA, tri.getSubMapper(startTarget,startBlock), actualPanelWidth, lengthTarget);
161 
162  gebp_kernel(other.getSubMapper(startTarget,j2), blockA, blockB+actual_kc*j2, lengthTarget, actualPanelWidth, actual_cols, Scalar(-1),
163  actualPanelWidth, actual_kc, 0, blockBOffset);
164  }
165  }
166  }
167 
168  // R2 -= A21 * B => GEPP
169  {
170  Index start = IsLower ? k2+kc : 0;
171  Index end = IsLower ? size : k2-kc;
172  for(Index i2=start; i2<end; i2+=mc)
173  {
174  const Index actual_mc = (std::min)(mc,end-i2);
175  if (actual_mc>0)
176  {
177  pack_lhs(blockA, tri.getSubMapper(i2, IsLower ? k2 : k2-kc), actual_kc, actual_mc);
178 
179  gebp_kernel(other.getSubMapper(i2, 0), blockA, blockB, actual_mc, actual_kc, cols, Scalar(-1), -1, -1, 0, 0);
180  }
181  }
182  }
183  }
184  }
185 
186 /* Optimized triangular solver with multiple left hand sides and the trinagular matrix on the right
187  */
188 template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
189 struct triangular_solve_matrix<Scalar,Index,OnTheRight,Mode,Conjugate,TriStorageOrder,ColMajor>
190 {
191  static EIGEN_DONT_INLINE void run(
192  Index size, Index otherSize,
193  const Scalar* _tri, Index triStride,
194  Scalar* _other, Index otherStride,
195  level3_blocking<Scalar,Scalar>& blocking);
196 };
197 template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
198 EIGEN_DONT_INLINE void triangular_solve_matrix<Scalar,Index,OnTheRight,Mode,Conjugate,TriStorageOrder,ColMajor>::run(
199  Index size, Index otherSize,
200  const Scalar* _tri, Index triStride,
201  Scalar* _other, Index otherStride,
202  level3_blocking<Scalar,Scalar>& blocking)
203  {
204  Index rows = otherSize;
205 
206  typedef blas_data_mapper<Scalar, Index, ColMajor> LhsMapper;
207  typedef const_blas_data_mapper<Scalar, Index, TriStorageOrder> RhsMapper;
208  LhsMapper lhs(_other, otherStride);
209  RhsMapper rhs(_tri, triStride);
210 
211  typedef gebp_traits<Scalar,Scalar> Traits;
212  enum {
213  RhsStorageOrder = TriStorageOrder,
214  SmallPanelWidth = EIGEN_PLAIN_ENUM_MAX(Traits::mr,Traits::nr),
215  IsLower = (Mode&Lower) == Lower
216  };
217 
218  Index kc = blocking.kc(); // cache block size along the K direction
219  Index mc = (std::min)(rows,blocking.mc()); // cache block size along the M direction
220 
221  std::size_t sizeA = kc*mc;
222  std::size_t sizeB = kc*size;
223 
224  ei_declare_aligned_stack_constructed_variable(Scalar, blockA, sizeA, blocking.blockA());
225  ei_declare_aligned_stack_constructed_variable(Scalar, blockB, sizeB, blocking.blockB());
226 
227  conj_if<Conjugate> conj;
228  gebp_kernel<Scalar, Scalar, Index, LhsMapper, Traits::mr, Traits::nr, false, Conjugate> gebp_kernel;
229  gemm_pack_rhs<Scalar, Index, RhsMapper, Traits::nr, RhsStorageOrder> pack_rhs;
230  gemm_pack_rhs<Scalar, Index, RhsMapper, Traits::nr, RhsStorageOrder,false,true> pack_rhs_panel;
231  gemm_pack_lhs<Scalar, Index, LhsMapper, Traits::mr, Traits::LhsProgress, ColMajor, false, true> pack_lhs_panel;
232 
233  for(Index k2=IsLower ? size : 0;
234  IsLower ? k2>0 : k2<size;
235  IsLower ? k2-=kc : k2+=kc)
236  {
237  const Index actual_kc = (std::min)(IsLower ? k2 : size-k2, kc);
238  Index actual_k2 = IsLower ? k2-actual_kc : k2 ;
239 
240  Index startPanel = IsLower ? 0 : k2+actual_kc;
241  Index rs = IsLower ? actual_k2 : size - actual_k2 - actual_kc;
242  Scalar* geb = blockB+actual_kc*actual_kc;
243 
244  if (rs>0) pack_rhs(geb, rhs.getSubMapper(actual_k2,startPanel), actual_kc, rs);
245 
246  // triangular packing (we only pack the panels off the diagonal,
247  // neglecting the blocks overlapping the diagonal
248  {
249  for (Index j2=0; j2<actual_kc; j2+=SmallPanelWidth)
250  {
251  Index actualPanelWidth = std::min<Index>(actual_kc-j2, SmallPanelWidth);
252  Index actual_j2 = actual_k2 + j2;
253  Index panelOffset = IsLower ? j2+actualPanelWidth : 0;
254  Index panelLength = IsLower ? actual_kc-j2-actualPanelWidth : j2;
255 
256  if (panelLength>0)
257  pack_rhs_panel(blockB+j2*actual_kc,
258  rhs.getSubMapper(actual_k2+panelOffset, actual_j2),
259  panelLength, actualPanelWidth,
260  actual_kc, panelOffset);
261  }
262  }
263 
264  for(Index i2=0; i2<rows; i2+=mc)
265  {
266  const Index actual_mc = (std::min)(mc,rows-i2);
267 
268  // triangular solver kernel
269  {
270  // for each small block of the diagonal (=> vertical panels of rhs)
271  for (Index j2 = IsLower
272  ? (actual_kc - ((actual_kc%SmallPanelWidth) ? Index(actual_kc%SmallPanelWidth)
273  : Index(SmallPanelWidth)))
274  : 0;
275  IsLower ? j2>=0 : j2<actual_kc;
276  IsLower ? j2-=SmallPanelWidth : j2+=SmallPanelWidth)
277  {
278  Index actualPanelWidth = std::min<Index>(actual_kc-j2, SmallPanelWidth);
279  Index absolute_j2 = actual_k2 + j2;
280  Index panelOffset = IsLower ? j2+actualPanelWidth : 0;
281  Index panelLength = IsLower ? actual_kc - j2 - actualPanelWidth : j2;
282 
283  // GEBP
284  if(panelLength>0)
285  {
286  gebp_kernel(lhs.getSubMapper(i2,absolute_j2),
287  blockA, blockB+j2*actual_kc,
288  actual_mc, panelLength, actualPanelWidth,
289  Scalar(-1),
290  actual_kc, actual_kc, // strides
291  panelOffset, panelOffset); // offsets
292  }
293 
294  // unblocked triangular solve
295  for (Index k=0; k<actualPanelWidth; ++k)
296  {
297  Index j = IsLower ? absolute_j2+actualPanelWidth-k-1 : absolute_j2+k;
298 
299  Scalar* r = &lhs(i2,j);
300  for (Index k3=0; k3<k; ++k3)
301  {
302  Scalar b = conj(rhs(IsLower ? j+1+k3 : absolute_j2+k3,j));
303  Scalar* a = &lhs(i2,IsLower ? j+1+k3 : absolute_j2+k3);
304  for (Index i=0; i<actual_mc; ++i)
305  r[i] -= a[i] * b;
306  }
307  if((Mode & UnitDiag)==0)
308  {
309  Scalar b = conj(rhs(j,j));
310  for (Index i=0; i<actual_mc; ++i)
311  r[i] /= b;
312  }
313  }
314 
315  // pack the just computed part of lhs to A
316  pack_lhs_panel(blockA, LhsMapper(_other+absolute_j2*otherStride+i2, otherStride),
317  actualPanelWidth, actual_mc,
318  actual_kc, j2);
319  }
320  }
321 
322  if (rs>0)
323  gebp_kernel(lhs.getSubMapper(i2, startPanel), blockA, geb,
324  actual_mc, actual_kc, rs, Scalar(-1),
325  -1, -1, 0, 0);
326  }
327  }
328  }
329 
330 } // end namespace internal
331 
332 } // end namespace Eigen
333 
334 #endif // EIGEN_TRIANGULAR_SOLVER_MATRIX_H
Definition: Constants.h:320
Definition: Constants.h:335
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_conjugate_op< typename Derived::Scalar >, const Derived > conj(const Eigen::ArrayBase< Derived > &x)
Namespace containing all symbols from the Eigen library.
Definition: Core:287
Definition: Constants.h:204
Definition: Constants.h:208
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:33
Definition: Constants.h:333
Definition: Constants.h:206
Definition: Eigen_Colamd.h:50
Definition: Constants.h:322