TriangularSolverMatrix.h
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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(
23  const Scalar* tri, Index triStride,
24  Scalar* _other, Index otherStride,
26  {
29  (Mode&UnitDiag) | ((Mode&Upper) ? Lower : Upper),
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>
40 {
41  static EIGEN_DONT_INLINE void run(
42  Index size, Index otherSize,
43  const Scalar* _tri, Index triStride,
44  Scalar* _other, Index otherStride,
46 };
47 template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
49  Index size, Index otherSize,
50  const Scalar* _tri, Index triStride,
51  Scalar* _other, Index otherStride,
53  {
54  Index cols = otherSize;
55 
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 
76 
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 triangular matrix on the right
187  */
188 template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
190 {
191  static EIGEN_DONT_INLINE void run(
192  Index size, Index otherSize,
193  const Scalar* _tri, Index triStride,
194  Scalar* _other, Index otherStride,
196 };
197 template <typename Scalar, typename Index, int Mode, bool Conjugate, int TriStorageOrder>
199  Index size, Index otherSize,
200  const Scalar* _tri, Index triStride,
201  Scalar* _other, Index otherStride,
203  {
204  Index rows = otherSize;
205  typedef typename NumTraits<Scalar>::Real RealScalar;
206 
209  LhsMapper lhs(_other, otherStride);
210  RhsMapper rhs(_tri, triStride);
211 
212  typedef gebp_traits<Scalar,Scalar> Traits;
213  enum {
214  RhsStorageOrder = TriStorageOrder,
215  SmallPanelWidth = EIGEN_PLAIN_ENUM_MAX(Traits::mr,Traits::nr),
216  IsLower = (Mode&Lower) == Lower
217  };
218 
219  Index kc = blocking.kc(); // cache block size along the K direction
220  Index mc = (std::min)(rows,blocking.mc()); // cache block size along the M direction
221 
222  std::size_t sizeA = kc*mc;
223  std::size_t sizeB = kc*size;
224 
225  ei_declare_aligned_stack_constructed_variable(Scalar, blockA, sizeA, blocking.blockA());
226  ei_declare_aligned_stack_constructed_variable(Scalar, blockB, sizeB, blocking.blockB());
227 
233 
234  for(Index k2=IsLower ? size : 0;
235  IsLower ? k2>0 : k2<size;
236  IsLower ? k2-=kc : k2+=kc)
237  {
238  const Index actual_kc = (std::min)(IsLower ? k2 : size-k2, kc);
239  Index actual_k2 = IsLower ? k2-actual_kc : k2 ;
240 
241  Index startPanel = IsLower ? 0 : k2+actual_kc;
242  Index rs = IsLower ? actual_k2 : size - actual_k2 - actual_kc;
243  Scalar* geb = blockB+actual_kc*actual_kc;
244 
245  if (rs>0) pack_rhs(geb, rhs.getSubMapper(actual_k2,startPanel), actual_kc, rs);
246 
247  // triangular packing (we only pack the panels off the diagonal,
248  // neglecting the blocks overlapping the diagonal
249  {
250  for (Index j2=0; j2<actual_kc; j2+=SmallPanelWidth)
251  {
252  Index actualPanelWidth = std::min<Index>(actual_kc-j2, SmallPanelWidth);
253  Index actual_j2 = actual_k2 + j2;
254  Index panelOffset = IsLower ? j2+actualPanelWidth : 0;
255  Index panelLength = IsLower ? actual_kc-j2-actualPanelWidth : j2;
256 
257  if (panelLength>0)
258  pack_rhs_panel(blockB+j2*actual_kc,
259  rhs.getSubMapper(actual_k2+panelOffset, actual_j2),
260  panelLength, actualPanelWidth,
261  actual_kc, panelOffset);
262  }
263  }
264 
265  for(Index i2=0; i2<rows; i2+=mc)
266  {
267  const Index actual_mc = (std::min)(mc,rows-i2);
268 
269  // triangular solver kernel
270  {
271  // for each small block of the diagonal (=> vertical panels of rhs)
272  for (Index j2 = IsLower
273  ? (actual_kc - ((actual_kc%SmallPanelWidth) ? Index(actual_kc%SmallPanelWidth)
274  : Index(SmallPanelWidth)))
275  : 0;
276  IsLower ? j2>=0 : j2<actual_kc;
277  IsLower ? j2-=SmallPanelWidth : j2+=SmallPanelWidth)
278  {
279  Index actualPanelWidth = std::min<Index>(actual_kc-j2, SmallPanelWidth);
280  Index absolute_j2 = actual_k2 + j2;
281  Index panelOffset = IsLower ? j2+actualPanelWidth : 0;
282  Index panelLength = IsLower ? actual_kc - j2 - actualPanelWidth : j2;
283 
284  // GEBP
285  if(panelLength>0)
286  {
287  gebp_kernel(lhs.getSubMapper(i2,absolute_j2),
288  blockA, blockB+j2*actual_kc,
289  actual_mc, panelLength, actualPanelWidth,
290  Scalar(-1),
291  actual_kc, actual_kc, // strides
292  panelOffset, panelOffset); // offsets
293  }
294 
295  // unblocked triangular solve
296  for (Index k=0; k<actualPanelWidth; ++k)
297  {
298  Index j = IsLower ? absolute_j2+actualPanelWidth-k-1 : absolute_j2+k;
299 
300  Scalar* r = &lhs(i2,j);
301  for (Index k3=0; k3<k; ++k3)
302  {
303  Scalar b = conj(rhs(IsLower ? j+1+k3 : absolute_j2+k3,j));
304  Scalar* a = &lhs(i2,IsLower ? j+1+k3 : absolute_j2+k3);
305  for (Index i=0; i<actual_mc; ++i)
306  r[i] -= a[i] * b;
307  }
308  if((Mode & UnitDiag)==0)
309  {
310  Scalar inv_rjj = RealScalar(1)/conj(rhs(j,j));
311  for (Index i=0; i<actual_mc; ++i)
312  r[i] *= inv_rjj;
313  }
314  }
315 
316  // pack the just computed part of lhs to A
317  pack_lhs_panel(blockA, LhsMapper(_other+absolute_j2*otherStride+i2, otherStride),
318  actualPanelWidth, actual_mc,
319  actual_kc, j2);
320  }
321  }
322 
323  if (rs>0)
324  gebp_kernel(lhs.getSubMapper(i2, startPanel), blockA, geb,
325  actual_mc, actual_kc, rs, Scalar(-1),
326  -1, -1, 0, 0);
327  }
328  }
329  }
330 
331 } // end namespace internal
332 
333 } // end namespace Eigen
334 
335 #endif // EIGEN_TRIANGULAR_SOLVER_MATRIX_H
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