Eigen  3.2.92
Inverse_SSE.h
1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2001 Intel Corporation
5 // Copyright (C) 2010 Gael Guennebaud <gael.guennebaud@inria.fr>
6 // Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
7 //
8 // This Source Code Form is subject to the terms of the Mozilla
9 // Public License v. 2.0. If a copy of the MPL was not distributed
10 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
11 
12 // The SSE code for the 4x4 float and double matrix inverse in this file
13 // comes from the following Intel's library:
14 // http://software.intel.com/en-us/articles/optimized-matrix-library-for-use-with-the-intel-pentiumr-4-processors-sse2-instructions/
15 //
16 // Here is the respective copyright and license statement:
17 //
18 // Copyright (c) 2001 Intel Corporation.
19 //
20 // Permition is granted to use, copy, distribute and prepare derivative works
21 // of this library for any purpose and without fee, provided, that the above
22 // copyright notice and this statement appear in all copies.
23 // Intel makes no representations about the suitability of this software for
24 // any purpose, and specifically disclaims all warranties.
25 // See LEGAL.TXT for all the legal information.
26 
27 #ifndef EIGEN_INVERSE_SSE_H
28 #define EIGEN_INVERSE_SSE_H
29 
30 namespace Eigen {
31 
32 namespace internal {
33 
34 template<typename MatrixType, typename ResultType>
35 struct compute_inverse_size4<Architecture::SSE, float, MatrixType, ResultType>
36 {
37  enum {
38  MatrixAlignment = traits<MatrixType>::Alignment,
39  ResultAlignment = traits<ResultType>::Alignment,
40  StorageOrdersMatch = (MatrixType::Flags&RowMajorBit) == (ResultType::Flags&RowMajorBit)
41  };
42  typedef typename conditional<(MatrixType::Flags&LinearAccessBit),MatrixType const &,typename MatrixType::PlainObject>::type ActualMatrixType;
43 
44  static void run(const MatrixType& mat, ResultType& result)
45  {
46  ActualMatrixType matrix(mat);
47  EIGEN_ALIGN16 const unsigned int _Sign_PNNP[4] = { 0x00000000, 0x80000000, 0x80000000, 0x00000000 };
48 
49  // Load the full matrix into registers
50  __m128 _L1 = matrix.template packet<MatrixAlignment>( 0);
51  __m128 _L2 = matrix.template packet<MatrixAlignment>( 4);
52  __m128 _L3 = matrix.template packet<MatrixAlignment>( 8);
53  __m128 _L4 = matrix.template packet<MatrixAlignment>(12);
54 
55  // The inverse is calculated using "Divide and Conquer" technique. The
56  // original matrix is divide into four 2x2 sub-matrices. Since each
57  // register holds four matrix element, the smaller matrices are
58  // represented as a registers. Hence we get a better locality of the
59  // calculations.
60 
61  __m128 A, B, C, D; // the four sub-matrices
62  if(!StorageOrdersMatch)
63  {
64  A = _mm_unpacklo_ps(_L1, _L2);
65  B = _mm_unpacklo_ps(_L3, _L4);
66  C = _mm_unpackhi_ps(_L1, _L2);
67  D = _mm_unpackhi_ps(_L3, _L4);
68  }
69  else
70  {
71  A = _mm_movelh_ps(_L1, _L2);
72  B = _mm_movehl_ps(_L2, _L1);
73  C = _mm_movelh_ps(_L3, _L4);
74  D = _mm_movehl_ps(_L4, _L3);
75  }
76 
77  __m128 iA, iB, iC, iD, // partial inverse of the sub-matrices
78  DC, AB;
79  __m128 dA, dB, dC, dD; // determinant of the sub-matrices
80  __m128 det, d, d1, d2;
81  __m128 rd; // reciprocal of the determinant
82 
83  // AB = A# * B
84  AB = _mm_mul_ps(_mm_shuffle_ps(A,A,0x0F), B);
85  AB = _mm_sub_ps(AB,_mm_mul_ps(_mm_shuffle_ps(A,A,0xA5), _mm_shuffle_ps(B,B,0x4E)));
86  // DC = D# * C
87  DC = _mm_mul_ps(_mm_shuffle_ps(D,D,0x0F), C);
88  DC = _mm_sub_ps(DC,_mm_mul_ps(_mm_shuffle_ps(D,D,0xA5), _mm_shuffle_ps(C,C,0x4E)));
89 
90  // dA = |A|
91  dA = _mm_mul_ps(_mm_shuffle_ps(A, A, 0x5F),A);
92  dA = _mm_sub_ss(dA, _mm_movehl_ps(dA,dA));
93  // dB = |B|
94  dB = _mm_mul_ps(_mm_shuffle_ps(B, B, 0x5F),B);
95  dB = _mm_sub_ss(dB, _mm_movehl_ps(dB,dB));
96 
97  // dC = |C|
98  dC = _mm_mul_ps(_mm_shuffle_ps(C, C, 0x5F),C);
99  dC = _mm_sub_ss(dC, _mm_movehl_ps(dC,dC));
100  // dD = |D|
101  dD = _mm_mul_ps(_mm_shuffle_ps(D, D, 0x5F),D);
102  dD = _mm_sub_ss(dD, _mm_movehl_ps(dD,dD));
103 
104  // d = trace(AB*DC) = trace(A#*B*D#*C)
105  d = _mm_mul_ps(_mm_shuffle_ps(DC,DC,0xD8),AB);
106 
107  // iD = C*A#*B
108  iD = _mm_mul_ps(_mm_shuffle_ps(C,C,0xA0), _mm_movelh_ps(AB,AB));
109  iD = _mm_add_ps(iD,_mm_mul_ps(_mm_shuffle_ps(C,C,0xF5), _mm_movehl_ps(AB,AB)));
110  // iA = B*D#*C
111  iA = _mm_mul_ps(_mm_shuffle_ps(B,B,0xA0), _mm_movelh_ps(DC,DC));
112  iA = _mm_add_ps(iA,_mm_mul_ps(_mm_shuffle_ps(B,B,0xF5), _mm_movehl_ps(DC,DC)));
113 
114  // d = trace(AB*DC) = trace(A#*B*D#*C) [continue]
115  d = _mm_add_ps(d, _mm_movehl_ps(d, d));
116  d = _mm_add_ss(d, _mm_shuffle_ps(d, d, 1));
117  d1 = _mm_mul_ss(dA,dD);
118  d2 = _mm_mul_ss(dB,dC);
119 
120  // iD = D*|A| - C*A#*B
121  iD = _mm_sub_ps(_mm_mul_ps(D,_mm_shuffle_ps(dA,dA,0)), iD);
122 
123  // iA = A*|D| - B*D#*C;
124  iA = _mm_sub_ps(_mm_mul_ps(A,_mm_shuffle_ps(dD,dD,0)), iA);
125 
126  // det = |A|*|D| + |B|*|C| - trace(A#*B*D#*C)
127  det = _mm_sub_ss(_mm_add_ss(d1,d2),d);
128  rd = _mm_div_ss(_mm_set_ss(1.0f), det);
129 
130 // #ifdef ZERO_SINGULAR
131 // rd = _mm_and_ps(_mm_cmpneq_ss(det,_mm_setzero_ps()), rd);
132 // #endif
133 
134  // iB = D * (A#B)# = D*B#*A
135  iB = _mm_mul_ps(D, _mm_shuffle_ps(AB,AB,0x33));
136  iB = _mm_sub_ps(iB, _mm_mul_ps(_mm_shuffle_ps(D,D,0xB1), _mm_shuffle_ps(AB,AB,0x66)));
137  // iC = A * (D#C)# = A*C#*D
138  iC = _mm_mul_ps(A, _mm_shuffle_ps(DC,DC,0x33));
139  iC = _mm_sub_ps(iC, _mm_mul_ps(_mm_shuffle_ps(A,A,0xB1), _mm_shuffle_ps(DC,DC,0x66)));
140 
141  rd = _mm_shuffle_ps(rd,rd,0);
142  rd = _mm_xor_ps(rd, _mm_load_ps((float*)_Sign_PNNP));
143 
144  // iB = C*|B| - D*B#*A
145  iB = _mm_sub_ps(_mm_mul_ps(C,_mm_shuffle_ps(dB,dB,0)), iB);
146 
147  // iC = B*|C| - A*C#*D;
148  iC = _mm_sub_ps(_mm_mul_ps(B,_mm_shuffle_ps(dC,dC,0)), iC);
149 
150  // iX = iX / det
151  iA = _mm_mul_ps(rd,iA);
152  iB = _mm_mul_ps(rd,iB);
153  iC = _mm_mul_ps(rd,iC);
154  iD = _mm_mul_ps(rd,iD);
155 
156  result.template writePacket<ResultAlignment>( 0, _mm_shuffle_ps(iA,iB,0x77));
157  result.template writePacket<ResultAlignment>( 4, _mm_shuffle_ps(iA,iB,0x22));
158  result.template writePacket<ResultAlignment>( 8, _mm_shuffle_ps(iC,iD,0x77));
159  result.template writePacket<ResultAlignment>(12, _mm_shuffle_ps(iC,iD,0x22));
160  }
161 
162 };
163 
164 template<typename MatrixType, typename ResultType>
165 struct compute_inverse_size4<Architecture::SSE, double, MatrixType, ResultType>
166 {
167  enum {
168  MatrixAlignment = traits<MatrixType>::Alignment,
169  ResultAlignment = traits<ResultType>::Alignment,
170  StorageOrdersMatch = (MatrixType::Flags&RowMajorBit) == (ResultType::Flags&RowMajorBit)
171  };
172  typedef typename conditional<(MatrixType::Flags&LinearAccessBit),MatrixType const &,typename MatrixType::PlainObject>::type ActualMatrixType;
173 
174  static void run(const MatrixType& mat, ResultType& result)
175  {
176  ActualMatrixType matrix(mat);
177  const __m128d _Sign_NP = _mm_castsi128_pd(_mm_set_epi32(0x0,0x0,0x80000000,0x0));
178  const __m128d _Sign_PN = _mm_castsi128_pd(_mm_set_epi32(0x80000000,0x0,0x0,0x0));
179 
180  // The inverse is calculated using "Divide and Conquer" technique. The
181  // original matrix is divide into four 2x2 sub-matrices. Since each
182  // register of the matrix holds two elements, the smaller matrices are
183  // consisted of two registers. Hence we get a better locality of the
184  // calculations.
185 
186  // the four sub-matrices
187  __m128d A1, A2, B1, B2, C1, C2, D1, D2;
188 
189  if(StorageOrdersMatch)
190  {
191  A1 = matrix.template packet<MatrixAlignment>( 0); B1 = matrix.template packet<MatrixAlignment>( 2);
192  A2 = matrix.template packet<MatrixAlignment>( 4); B2 = matrix.template packet<MatrixAlignment>( 6);
193  C1 = matrix.template packet<MatrixAlignment>( 8); D1 = matrix.template packet<MatrixAlignment>(10);
194  C2 = matrix.template packet<MatrixAlignment>(12); D2 = matrix.template packet<MatrixAlignment>(14);
195  }
196  else
197  {
198  __m128d tmp;
199  A1 = matrix.template packet<MatrixAlignment>( 0); C1 = matrix.template packet<MatrixAlignment>( 2);
200  A2 = matrix.template packet<MatrixAlignment>( 4); C2 = matrix.template packet<MatrixAlignment>( 6);
201  tmp = A1;
202  A1 = _mm_unpacklo_pd(A1,A2);
203  A2 = _mm_unpackhi_pd(tmp,A2);
204  tmp = C1;
205  C1 = _mm_unpacklo_pd(C1,C2);
206  C2 = _mm_unpackhi_pd(tmp,C2);
207 
208  B1 = matrix.template packet<MatrixAlignment>( 8); D1 = matrix.template packet<MatrixAlignment>(10);
209  B2 = matrix.template packet<MatrixAlignment>(12); D2 = matrix.template packet<MatrixAlignment>(14);
210  tmp = B1;
211  B1 = _mm_unpacklo_pd(B1,B2);
212  B2 = _mm_unpackhi_pd(tmp,B2);
213  tmp = D1;
214  D1 = _mm_unpacklo_pd(D1,D2);
215  D2 = _mm_unpackhi_pd(tmp,D2);
216  }
217 
218  __m128d iA1, iA2, iB1, iB2, iC1, iC2, iD1, iD2, // partial invese of the sub-matrices
219  DC1, DC2, AB1, AB2;
220  __m128d dA, dB, dC, dD; // determinant of the sub-matrices
221  __m128d det, d1, d2, rd;
222 
223  // dA = |A|
224  dA = _mm_shuffle_pd(A2, A2, 1);
225  dA = _mm_mul_pd(A1, dA);
226  dA = _mm_sub_sd(dA, _mm_shuffle_pd(dA,dA,3));
227  // dB = |B|
228  dB = _mm_shuffle_pd(B2, B2, 1);
229  dB = _mm_mul_pd(B1, dB);
230  dB = _mm_sub_sd(dB, _mm_shuffle_pd(dB,dB,3));
231 
232  // AB = A# * B
233  AB1 = _mm_mul_pd(B1, _mm_shuffle_pd(A2,A2,3));
234  AB2 = _mm_mul_pd(B2, _mm_shuffle_pd(A1,A1,0));
235  AB1 = _mm_sub_pd(AB1, _mm_mul_pd(B2, _mm_shuffle_pd(A1,A1,3)));
236  AB2 = _mm_sub_pd(AB2, _mm_mul_pd(B1, _mm_shuffle_pd(A2,A2,0)));
237 
238  // dC = |C|
239  dC = _mm_shuffle_pd(C2, C2, 1);
240  dC = _mm_mul_pd(C1, dC);
241  dC = _mm_sub_sd(dC, _mm_shuffle_pd(dC,dC,3));
242  // dD = |D|
243  dD = _mm_shuffle_pd(D2, D2, 1);
244  dD = _mm_mul_pd(D1, dD);
245  dD = _mm_sub_sd(dD, _mm_shuffle_pd(dD,dD,3));
246 
247  // DC = D# * C
248  DC1 = _mm_mul_pd(C1, _mm_shuffle_pd(D2,D2,3));
249  DC2 = _mm_mul_pd(C2, _mm_shuffle_pd(D1,D1,0));
250  DC1 = _mm_sub_pd(DC1, _mm_mul_pd(C2, _mm_shuffle_pd(D1,D1,3)));
251  DC2 = _mm_sub_pd(DC2, _mm_mul_pd(C1, _mm_shuffle_pd(D2,D2,0)));
252 
253  // rd = trace(AB*DC) = trace(A#*B*D#*C)
254  d1 = _mm_mul_pd(AB1, _mm_shuffle_pd(DC1, DC2, 0));
255  d2 = _mm_mul_pd(AB2, _mm_shuffle_pd(DC1, DC2, 3));
256  rd = _mm_add_pd(d1, d2);
257  rd = _mm_add_sd(rd, _mm_shuffle_pd(rd, rd,3));
258 
259  // iD = C*A#*B
260  iD1 = _mm_mul_pd(AB1, _mm_shuffle_pd(C1,C1,0));
261  iD2 = _mm_mul_pd(AB1, _mm_shuffle_pd(C2,C2,0));
262  iD1 = _mm_add_pd(iD1, _mm_mul_pd(AB2, _mm_shuffle_pd(C1,C1,3)));
263  iD2 = _mm_add_pd(iD2, _mm_mul_pd(AB2, _mm_shuffle_pd(C2,C2,3)));
264 
265  // iA = B*D#*C
266  iA1 = _mm_mul_pd(DC1, _mm_shuffle_pd(B1,B1,0));
267  iA2 = _mm_mul_pd(DC1, _mm_shuffle_pd(B2,B2,0));
268  iA1 = _mm_add_pd(iA1, _mm_mul_pd(DC2, _mm_shuffle_pd(B1,B1,3)));
269  iA2 = _mm_add_pd(iA2, _mm_mul_pd(DC2, _mm_shuffle_pd(B2,B2,3)));
270 
271  // iD = D*|A| - C*A#*B
272  dA = _mm_shuffle_pd(dA,dA,0);
273  iD1 = _mm_sub_pd(_mm_mul_pd(D1, dA), iD1);
274  iD2 = _mm_sub_pd(_mm_mul_pd(D2, dA), iD2);
275 
276  // iA = A*|D| - B*D#*C;
277  dD = _mm_shuffle_pd(dD,dD,0);
278  iA1 = _mm_sub_pd(_mm_mul_pd(A1, dD), iA1);
279  iA2 = _mm_sub_pd(_mm_mul_pd(A2, dD), iA2);
280 
281  d1 = _mm_mul_sd(dA, dD);
282  d2 = _mm_mul_sd(dB, dC);
283 
284  // iB = D * (A#B)# = D*B#*A
285  iB1 = _mm_mul_pd(D1, _mm_shuffle_pd(AB2,AB1,1));
286  iB2 = _mm_mul_pd(D2, _mm_shuffle_pd(AB2,AB1,1));
287  iB1 = _mm_sub_pd(iB1, _mm_mul_pd(_mm_shuffle_pd(D1,D1,1), _mm_shuffle_pd(AB2,AB1,2)));
288  iB2 = _mm_sub_pd(iB2, _mm_mul_pd(_mm_shuffle_pd(D2,D2,1), _mm_shuffle_pd(AB2,AB1,2)));
289 
290  // det = |A|*|D| + |B|*|C| - trace(A#*B*D#*C)
291  det = _mm_add_sd(d1, d2);
292  det = _mm_sub_sd(det, rd);
293 
294  // iC = A * (D#C)# = A*C#*D
295  iC1 = _mm_mul_pd(A1, _mm_shuffle_pd(DC2,DC1,1));
296  iC2 = _mm_mul_pd(A2, _mm_shuffle_pd(DC2,DC1,1));
297  iC1 = _mm_sub_pd(iC1, _mm_mul_pd(_mm_shuffle_pd(A1,A1,1), _mm_shuffle_pd(DC2,DC1,2)));
298  iC2 = _mm_sub_pd(iC2, _mm_mul_pd(_mm_shuffle_pd(A2,A2,1), _mm_shuffle_pd(DC2,DC1,2)));
299 
300  rd = _mm_div_sd(_mm_set_sd(1.0), det);
301 // #ifdef ZERO_SINGULAR
302 // rd = _mm_and_pd(_mm_cmpneq_sd(det,_mm_setzero_pd()), rd);
303 // #endif
304  rd = _mm_shuffle_pd(rd,rd,0);
305 
306  // iB = C*|B| - D*B#*A
307  dB = _mm_shuffle_pd(dB,dB,0);
308  iB1 = _mm_sub_pd(_mm_mul_pd(C1, dB), iB1);
309  iB2 = _mm_sub_pd(_mm_mul_pd(C2, dB), iB2);
310 
311  d1 = _mm_xor_pd(rd, _Sign_PN);
312  d2 = _mm_xor_pd(rd, _Sign_NP);
313 
314  // iC = B*|C| - A*C#*D;
315  dC = _mm_shuffle_pd(dC,dC,0);
316  iC1 = _mm_sub_pd(_mm_mul_pd(B1, dC), iC1);
317  iC2 = _mm_sub_pd(_mm_mul_pd(B2, dC), iC2);
318 
319  result.template writePacket<ResultAlignment>( 0, _mm_mul_pd(_mm_shuffle_pd(iA2, iA1, 3), d1)); // iA# / det
320  result.template writePacket<ResultAlignment>( 4, _mm_mul_pd(_mm_shuffle_pd(iA2, iA1, 0), d2));
321  result.template writePacket<ResultAlignment>( 2, _mm_mul_pd(_mm_shuffle_pd(iB2, iB1, 3), d1)); // iB# / det
322  result.template writePacket<ResultAlignment>( 6, _mm_mul_pd(_mm_shuffle_pd(iB2, iB1, 0), d2));
323  result.template writePacket<ResultAlignment>( 8, _mm_mul_pd(_mm_shuffle_pd(iC2, iC1, 3), d1)); // iC# / det
324  result.template writePacket<ResultAlignment>(12, _mm_mul_pd(_mm_shuffle_pd(iC2, iC1, 0), d2));
325  result.template writePacket<ResultAlignment>(10, _mm_mul_pd(_mm_shuffle_pd(iD2, iD1, 3), d1)); // iD# / det
326  result.template writePacket<ResultAlignment>(14, _mm_mul_pd(_mm_shuffle_pd(iD2, iD1, 0), d2));
327  }
328 };
329 
330 } // end namespace internal
331 
332 } // end namespace Eigen
333 
334 #endif // EIGEN_INVERSE_SSE_H
Definition: LDLT.h:16
const unsigned int RowMajorBit
Definition: Constants.h:61
Definition: Eigen_Colamd.h:54