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Inverse_SSE.h
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Inverse_SSE.h
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2001 Intel Corporation
// Copyright (C) 2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// 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/.
// The SSE code for the 4x4 float and double matrix inverse in this file
// comes from the following Intel's library:
// http://software.intel.com/en-us/articles/optimized-matrix-library-for-use-with-the-intel-pentiumr-4-processors-sse2-instructions/
//
// Here is the respective copyright and license statement:
//
// Copyright (c) 2001 Intel Corporation.
//
// Permition is granted to use, copy, distribute and prepare derivative works
// of this library for any purpose and without fee, provided, that the above
// copyright notice and this statement appear in all copies.
// Intel makes no representations about the suitability of this software for
// any purpose, and specifically disclaims all warranties.
// See LEGAL.TXT for all the legal information.
#ifndef EIGEN_INVERSE_SSE_H
#define EIGEN_INVERSE_SSE_H
namespace Eigen {
namespace internal {
template<typename MatrixType, typename ResultType>
struct compute_inverse_size4<Architecture::SSE, float, MatrixType, ResultType>
{
enum {
MatrixAlignment = bool(MatrixType::Flags&AlignedBit),
ResultAlignment = bool(ResultType::Flags&AlignedBit),
StorageOrdersMatch = (MatrixType::Flags&RowMajorBit) == (ResultType::Flags&RowMajorBit)
};
static void run(const MatrixType& matrix, ResultType& result)
{
EIGEN_ALIGN16 const unsigned int _Sign_PNNP[4] = { 0x00000000, 0x80000000, 0x80000000, 0x00000000 };
// Load the full matrix into registers
__m128 _L1 = matrix.template packet<MatrixAlignment>( 0);
__m128 _L2 = matrix.template packet<MatrixAlignment>( 4);
__m128 _L3 = matrix.template packet<MatrixAlignment>( 8);
__m128 _L4 = matrix.template packet<MatrixAlignment>(12);
// The inverse is calculated using "Divide and Conquer" technique. The
// original matrix is divide into four 2x2 sub-matrices. Since each
// register holds four matrix element, the smaller matrices are
// represented as a registers. Hence we get a better locality of the
// calculations.
__m128 A, B, C, D; // the four sub-matrices
if(!StorageOrdersMatch)
{
A = _mm_unpacklo_ps(_L1, _L2);
B = _mm_unpacklo_ps(_L3, _L4);
C = _mm_unpackhi_ps(_L1, _L2);
D = _mm_unpackhi_ps(_L3, _L4);
}
else
{
A = _mm_movelh_ps(_L1, _L2);
B = _mm_movehl_ps(_L2, _L1);
C = _mm_movelh_ps(_L3, _L4);
D = _mm_movehl_ps(_L4, _L3);
}
__m128 iA, iB, iC, iD, // partial inverse of the sub-matrices
DC, AB;
__m128 dA, dB, dC, dD; // determinant of the sub-matrices
__m128 det, d, d1, d2;
__m128 rd; // reciprocal of the determinant
// AB = A# * B
AB = _mm_mul_ps(_mm_shuffle_ps(A,A,0x0F), B);
AB = _mm_sub_ps(AB,_mm_mul_ps(_mm_shuffle_ps(A,A,0xA5), _mm_shuffle_ps(B,B,0x4E)));
// DC = D# * C
DC = _mm_mul_ps(_mm_shuffle_ps(D,D,0x0F), C);
DC = _mm_sub_ps(DC,_mm_mul_ps(_mm_shuffle_ps(D,D,0xA5), _mm_shuffle_ps(C,C,0x4E)));
// dA = |A|
dA = _mm_mul_ps(_mm_shuffle_ps(A, A, 0x5F),A);
dA = _mm_sub_ss(dA, _mm_movehl_ps(dA,dA));
// dB = |B|
dB = _mm_mul_ps(_mm_shuffle_ps(B, B, 0x5F),B);
dB = _mm_sub_ss(dB, _mm_movehl_ps(dB,dB));
// dC = |C|
dC = _mm_mul_ps(_mm_shuffle_ps(C, C, 0x5F),C);
dC = _mm_sub_ss(dC, _mm_movehl_ps(dC,dC));
// dD = |D|
dD = _mm_mul_ps(_mm_shuffle_ps(D, D, 0x5F),D);
dD = _mm_sub_ss(dD, _mm_movehl_ps(dD,dD));
// d = trace(AB*DC) = trace(A#*B*D#*C)
d = _mm_mul_ps(_mm_shuffle_ps(DC,DC,0xD8),AB);
// iD = C*A#*B
iD = _mm_mul_ps(_mm_shuffle_ps(C,C,0xA0), _mm_movelh_ps(AB,AB));
iD = _mm_add_ps(iD,_mm_mul_ps(_mm_shuffle_ps(C,C,0xF5), _mm_movehl_ps(AB,AB)));
// iA = B*D#*C
iA = _mm_mul_ps(_mm_shuffle_ps(B,B,0xA0), _mm_movelh_ps(DC,DC));
iA = _mm_add_ps(iA,_mm_mul_ps(_mm_shuffle_ps(B,B,0xF5), _mm_movehl_ps(DC,DC)));
// d = trace(AB*DC) = trace(A#*B*D#*C) [continue]
d = _mm_add_ps(d, _mm_movehl_ps(d, d));
d = _mm_add_ss(d, _mm_shuffle_ps(d, d, 1));
d1 = _mm_mul_ss(dA,dD);
d2 = _mm_mul_ss(dB,dC);
// iD = D*|A| - C*A#*B
iD = _mm_sub_ps(_mm_mul_ps(D,_mm_shuffle_ps(dA,dA,0)), iD);
// iA = A*|D| - B*D#*C;
iA = _mm_sub_ps(_mm_mul_ps(A,_mm_shuffle_ps(dD,dD,0)), iA);
// det = |A|*|D| + |B|*|C| - trace(A#*B*D#*C)
det = _mm_sub_ss(_mm_add_ss(d1,d2),d);
rd = _mm_div_ss(_mm_set_ss(1.0f), det);
// #ifdef ZERO_SINGULAR
// rd = _mm_and_ps(_mm_cmpneq_ss(det,_mm_setzero_ps()), rd);
// #endif
// iB = D * (A#B)# = D*B#*A
iB = _mm_mul_ps(D, _mm_shuffle_ps(AB,AB,0x33));
iB = _mm_sub_ps(iB, _mm_mul_ps(_mm_shuffle_ps(D,D,0xB1), _mm_shuffle_ps(AB,AB,0x66)));
// iC = A * (D#C)# = A*C#*D
iC = _mm_mul_ps(A, _mm_shuffle_ps(DC,DC,0x33));
iC = _mm_sub_ps(iC, _mm_mul_ps(_mm_shuffle_ps(A,A,0xB1), _mm_shuffle_ps(DC,DC,0x66)));
rd = _mm_shuffle_ps(rd,rd,0);
rd = _mm_xor_ps(rd, _mm_load_ps((float*)_Sign_PNNP));
// iB = C*|B| - D*B#*A
iB = _mm_sub_ps(_mm_mul_ps(C,_mm_shuffle_ps(dB,dB,0)), iB);
// iC = B*|C| - A*C#*D;
iC = _mm_sub_ps(_mm_mul_ps(B,_mm_shuffle_ps(dC,dC,0)), iC);
// iX = iX / det
iA = _mm_mul_ps(rd,iA);
iB = _mm_mul_ps(rd,iB);
iC = _mm_mul_ps(rd,iC);
iD = _mm_mul_ps(rd,iD);
result.template writePacket<ResultAlignment>( 0, _mm_shuffle_ps(iA,iB,0x77));
result.template writePacket<ResultAlignment>( 4, _mm_shuffle_ps(iA,iB,0x22));
result.template writePacket<ResultAlignment>( 8, _mm_shuffle_ps(iC,iD,0x77));
result.template writePacket<ResultAlignment>(12, _mm_shuffle_ps(iC,iD,0x22));
}
};
template<typename MatrixType, typename ResultType>
struct compute_inverse_size4<Architecture::SSE, double, MatrixType, ResultType>
{
enum {
MatrixAlignment = bool(MatrixType::Flags&AlignedBit),
ResultAlignment = bool(ResultType::Flags&AlignedBit),
StorageOrdersMatch = (MatrixType::Flags&RowMajorBit) == (ResultType::Flags&RowMajorBit)
};
static void run(const MatrixType& matrix, ResultType& result)
{
const __m128d _Sign_NP = _mm_castsi128_pd(_mm_set_epi32(0x0,0x0,0x80000000,0x0));
const __m128d _Sign_PN = _mm_castsi128_pd(_mm_set_epi32(0x80000000,0x0,0x0,0x0));
// The inverse is calculated using "Divide and Conquer" technique. The
// original matrix is divide into four 2x2 sub-matrices. Since each
// register of the matrix holds two element, the smaller matrices are
// consisted of two registers. Hence we get a better locality of the
// calculations.
// the four sub-matrices
__m128d A1, A2, B1, B2, C1, C2, D1, D2;
if(StorageOrdersMatch)
{
A1 = matrix.template packet<MatrixAlignment>( 0); B1 = matrix.template packet<MatrixAlignment>( 2);
A2 = matrix.template packet<MatrixAlignment>( 4); B2 = matrix.template packet<MatrixAlignment>( 6);
C1 = matrix.template packet<MatrixAlignment>( 8); D1 = matrix.template packet<MatrixAlignment>(10);
C2 = matrix.template packet<MatrixAlignment>(12); D2 = matrix.template packet<MatrixAlignment>(14);
}
else
{
__m128d tmp;
A1 = matrix.template packet<MatrixAlignment>( 0); C1 = matrix.template packet<MatrixAlignment>( 2);
A2 = matrix.template packet<MatrixAlignment>( 4); C2 = matrix.template packet<MatrixAlignment>( 6);
tmp = A1;
A1 = _mm_unpacklo_pd(A1,A2);
A2 = _mm_unpackhi_pd(tmp,A2);
tmp = C1;
C1 = _mm_unpacklo_pd(C1,C2);
C2 = _mm_unpackhi_pd(tmp,C2);
B1 = matrix.template packet<MatrixAlignment>( 8); D1 = matrix.template packet<MatrixAlignment>(10);
B2 = matrix.template packet<MatrixAlignment>(12); D2 = matrix.template packet<MatrixAlignment>(14);
tmp = B1;
B1 = _mm_unpacklo_pd(B1,B2);
B2 = _mm_unpackhi_pd(tmp,B2);
tmp = D1;
D1 = _mm_unpacklo_pd(D1,D2);
D2 = _mm_unpackhi_pd(tmp,D2);
}
__m128d iA1, iA2, iB1, iB2, iC1, iC2, iD1, iD2, // partial invese of the sub-matrices
DC1, DC2, AB1, AB2;
__m128d dA, dB, dC, dD; // determinant of the sub-matrices
__m128d det, d1, d2, rd;
// dA = |A|
dA = _mm_shuffle_pd(A2, A2, 1);
dA = _mm_mul_pd(A1, dA);
dA = _mm_sub_sd(dA, _mm_shuffle_pd(dA,dA,3));
// dB = |B|
dB = _mm_shuffle_pd(B2, B2, 1);
dB = _mm_mul_pd(B1, dB);
dB = _mm_sub_sd(dB, _mm_shuffle_pd(dB,dB,3));
// AB = A# * B
AB1 = _mm_mul_pd(B1, _mm_shuffle_pd(A2,A2,3));
AB2 = _mm_mul_pd(B2, _mm_shuffle_pd(A1,A1,0));
AB1 = _mm_sub_pd(AB1, _mm_mul_pd(B2, _mm_shuffle_pd(A1,A1,3)));
AB2 = _mm_sub_pd(AB2, _mm_mul_pd(B1, _mm_shuffle_pd(A2,A2,0)));
// dC = |C|
dC = _mm_shuffle_pd(C2, C2, 1);
dC = _mm_mul_pd(C1, dC);
dC = _mm_sub_sd(dC, _mm_shuffle_pd(dC,dC,3));
// dD = |D|
dD = _mm_shuffle_pd(D2, D2, 1);
dD = _mm_mul_pd(D1, dD);
dD = _mm_sub_sd(dD, _mm_shuffle_pd(dD,dD,3));
// DC = D# * C
DC1 = _mm_mul_pd(C1, _mm_shuffle_pd(D2,D2,3));
DC2 = _mm_mul_pd(C2, _mm_shuffle_pd(D1,D1,0));
DC1 = _mm_sub_pd(DC1, _mm_mul_pd(C2, _mm_shuffle_pd(D1,D1,3)));
DC2 = _mm_sub_pd(DC2, _mm_mul_pd(C1, _mm_shuffle_pd(D2,D2,0)));
// rd = trace(AB*DC) = trace(A#*B*D#*C)
d1 = _mm_mul_pd(AB1, _mm_shuffle_pd(DC1, DC2, 0));
d2 = _mm_mul_pd(AB2, _mm_shuffle_pd(DC1, DC2, 3));
rd = _mm_add_pd(d1, d2);
rd = _mm_add_sd(rd, _mm_shuffle_pd(rd, rd,3));
// iD = C*A#*B
iD1 = _mm_mul_pd(AB1, _mm_shuffle_pd(C1,C1,0));
iD2 = _mm_mul_pd(AB1, _mm_shuffle_pd(C2,C2,0));
iD1 = _mm_add_pd(iD1, _mm_mul_pd(AB2, _mm_shuffle_pd(C1,C1,3)));
iD2 = _mm_add_pd(iD2, _mm_mul_pd(AB2, _mm_shuffle_pd(C2,C2,3)));
// iA = B*D#*C
iA1 = _mm_mul_pd(DC1, _mm_shuffle_pd(B1,B1,0));
iA2 = _mm_mul_pd(DC1, _mm_shuffle_pd(B2,B2,0));
iA1 = _mm_add_pd(iA1, _mm_mul_pd(DC2, _mm_shuffle_pd(B1,B1,3)));
iA2 = _mm_add_pd(iA2, _mm_mul_pd(DC2, _mm_shuffle_pd(B2,B2,3)));
// iD = D*|A| - C*A#*B
dA = _mm_shuffle_pd(dA,dA,0);
iD1 = _mm_sub_pd(_mm_mul_pd(D1, dA), iD1);
iD2 = _mm_sub_pd(_mm_mul_pd(D2, dA), iD2);
// iA = A*|D| - B*D#*C;
dD = _mm_shuffle_pd(dD,dD,0);
iA1 = _mm_sub_pd(_mm_mul_pd(A1, dD), iA1);
iA2 = _mm_sub_pd(_mm_mul_pd(A2, dD), iA2);
d1 = _mm_mul_sd(dA, dD);
d2 = _mm_mul_sd(dB, dC);
// iB = D * (A#B)# = D*B#*A
iB1 = _mm_mul_pd(D1, _mm_shuffle_pd(AB2,AB1,1));
iB2 = _mm_mul_pd(D2, _mm_shuffle_pd(AB2,AB1,1));
iB1 = _mm_sub_pd(iB1, _mm_mul_pd(_mm_shuffle_pd(D1,D1,1), _mm_shuffle_pd(AB2,AB1,2)));
iB2 = _mm_sub_pd(iB2, _mm_mul_pd(_mm_shuffle_pd(D2,D2,1), _mm_shuffle_pd(AB2,AB1,2)));
// det = |A|*|D| + |B|*|C| - trace(A#*B*D#*C)
det = _mm_add_sd(d1, d2);
det = _mm_sub_sd(det, rd);
// iC = A * (D#C)# = A*C#*D
iC1 = _mm_mul_pd(A1, _mm_shuffle_pd(DC2,DC1,1));
iC2 = _mm_mul_pd(A2, _mm_shuffle_pd(DC2,DC1,1));
iC1 = _mm_sub_pd(iC1, _mm_mul_pd(_mm_shuffle_pd(A1,A1,1), _mm_shuffle_pd(DC2,DC1,2)));
iC2 = _mm_sub_pd(iC2, _mm_mul_pd(_mm_shuffle_pd(A2,A2,1), _mm_shuffle_pd(DC2,DC1,2)));
rd = _mm_div_sd(_mm_set_sd(1.0), det);
// #ifdef ZERO_SINGULAR
// rd = _mm_and_pd(_mm_cmpneq_sd(det,_mm_setzero_pd()), rd);
// #endif
rd = _mm_shuffle_pd(rd,rd,0);
// iB = C*|B| - D*B#*A
dB = _mm_shuffle_pd(dB,dB,0);
iB1 = _mm_sub_pd(_mm_mul_pd(C1, dB), iB1);
iB2 = _mm_sub_pd(_mm_mul_pd(C2, dB), iB2);
d1 = _mm_xor_pd(rd, _Sign_PN);
d2 = _mm_xor_pd(rd, _Sign_NP);
// iC = B*|C| - A*C#*D;
dC = _mm_shuffle_pd(dC,dC,0);
iC1 = _mm_sub_pd(_mm_mul_pd(B1, dC), iC1);
iC2 = _mm_sub_pd(_mm_mul_pd(B2, dC), iC2);
result.template writePacket<ResultAlignment>( 0, _mm_mul_pd(_mm_shuffle_pd(iA2, iA1, 3), d1)); // iA# / det
result.template writePacket<ResultAlignment>( 4, _mm_mul_pd(_mm_shuffle_pd(iA2, iA1, 0), d2));
result.template writePacket<ResultAlignment>( 2, _mm_mul_pd(_mm_shuffle_pd(iB2, iB1, 3), d1)); // iB# / det
result.template writePacket<ResultAlignment>( 6, _mm_mul_pd(_mm_shuffle_pd(iB2, iB1, 0), d2));
result.template writePacket<ResultAlignment>( 8, _mm_mul_pd(_mm_shuffle_pd(iC2, iC1, 3), d1)); // iC# / det
result.template writePacket<ResultAlignment>(12, _mm_mul_pd(_mm_shuffle_pd(iC2, iC1, 0), d2));
result.template writePacket<ResultAlignment>(10, _mm_mul_pd(_mm_shuffle_pd(iD2, iD1, 3), d1)); // iD# / det
result.template writePacket<ResultAlignment>(14, _mm_mul_pd(_mm_shuffle_pd(iD2, iD1, 0), d2));
}
};
} // end namespace internal
} // end namespace Eigen
#endif // EIGEN_INVERSE_SSE_H

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