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backport 4x4 inverse changes:
- use cofactors - use Intel's SSE code in the float case
This commit is contained in:
Benoit Jacob
parent
b581cb870c
commit
4262117f84
@ -75,93 +75,147 @@ void ei_compute_inverse_in_size3_case(const MatrixType& matrix, MatrixType* resu
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result->coeffRef(2, 2) = matrix.minor(2,2).determinant() * invdet;
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}
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template<typename MatrixType>
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bool ei_compute_inverse_in_size4_case_helper(const MatrixType& matrix, MatrixType* result)
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template<typename MatrixType, typename Scalar = typename MatrixType::Scalar>
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struct ei_compute_inverse_in_size4_case
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{
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/* Let's split M into four 2x2 blocks:
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* (P Q)
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* (R S)
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* If P is invertible, with inverse denoted by P_inverse, and if
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* (S - R*P_inverse*Q) is also invertible, then the inverse of M is
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* (P' Q')
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* (R' S')
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* where
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* S' = (S - R*P_inverse*Q)^(-1)
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* P' = P1 + (P1*Q) * S' *(R*P_inverse)
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* Q' = -(P_inverse*Q) * S'
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* R' = -S' * (R*P_inverse)
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*/
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typedef Block<MatrixType,2,2> XprBlock22;
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typedef typename MatrixBase<XprBlock22>::PlainMatrixType Block22;
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Block22 P_inverse;
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ei_compute_inverse_in_size2_case(matrix.template block<2,2>(0,0), &P_inverse);
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const Block22 Q = matrix.template block<2,2>(0,2);
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const Block22 P_inverse_times_Q = P_inverse * Q;
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const XprBlock22 R = matrix.template block<2,2>(2,0);
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const Block22 R_times_P_inverse = R * P_inverse;
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const Block22 R_times_P_inverse_times_Q = R_times_P_inverse * Q;
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const XprBlock22 S = matrix.template block<2,2>(2,2);
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const Block22 X = S - R_times_P_inverse_times_Q;
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Block22 Y;
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ei_compute_inverse_in_size2_case(X, &Y);
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result->template block<2,2>(2,2) = Y;
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result->template block<2,2>(2,0) = - Y * R_times_P_inverse;
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const Block22 Z = P_inverse_times_Q * Y;
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result->template block<2,2>(0,2) = - Z;
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result->template block<2,2>(0,0) = P_inverse + Z * R_times_P_inverse;
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return true;
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}
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template<typename MatrixType>
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void ei_compute_inverse_in_size4_case(const MatrixType& _matrix, MatrixType* result)
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static void run(const MatrixType& matrix, MatrixType& result)
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{
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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// we will do row permutations on the matrix. This copy should have negligible cost.
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// if not, consider working in-place on the matrix (const-cast it, but then undo the permutations
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// to nevertheless honor constness)
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typename MatrixType::PlainMatrixType matrix(_matrix);
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// let's extract from the 2 first colums a 2x2 block whose determinant is as big as possible.
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int good_row0, good_row1, good_i;
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Matrix<RealScalar,6,1> absdet;
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// any 2x2 block with determinant above this threshold will be considered good enough.
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// The magic value 1e-1 here comes from experimentation. The bigger it is, the higher the precision,
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// the slower the computation. This value 1e-1 gives precision almost as good as the brutal cofactors
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// algorithm, both in average and in worst-case precision.
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RealScalar d = (matrix.col(0).squaredNorm()+matrix.col(1).squaredNorm()) * RealScalar(1e-1);
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#define ei_inv_size4_helper_macro(i,row0,row1) \
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absdet[i] = ei_abs(matrix.coeff(row0,0)*matrix.coeff(row1,1) \
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- matrix.coeff(row0,1)*matrix.coeff(row1,0)); \
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if(absdet[i] > d) { good_row0=row0; good_row1=row1; goto good; }
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ei_inv_size4_helper_macro(0,0,1)
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ei_inv_size4_helper_macro(1,0,2)
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ei_inv_size4_helper_macro(2,0,3)
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ei_inv_size4_helper_macro(3,1,2)
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ei_inv_size4_helper_macro(4,1,3)
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ei_inv_size4_helper_macro(5,2,3)
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// no 2x2 block has determinant bigger than the threshold. So just take the one that
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// has the biggest determinant
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absdet.maxCoeff(&good_i);
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good_row0 = good_i <= 2 ? 0 : good_i <= 4 ? 1 : 2;
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good_row1 = good_i <= 2 ? good_i+1 : good_i <= 4 ? good_i-1 : 3;
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// now good_row0 and good_row1 are correctly set
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good:
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// do row permutations to move this 2x2 block to the top
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matrix.row(0).swap(matrix.row(good_row0));
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matrix.row(1).swap(matrix.row(good_row1));
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// now applying our helper function is numerically stable
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ei_compute_inverse_in_size4_case_helper(matrix, result);
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// Since we did row permutations on the original matrix, we need to do column permutations
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// in the reverse order on the inverse
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result->col(1).swap(result->col(good_row1));
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result->col(0).swap(result->col(good_row0));
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result.coeffRef(0,0) = matrix.minor(0,0).determinant();
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result.coeffRef(1,0) = -matrix.minor(0,1).determinant();
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result.coeffRef(2,0) = matrix.minor(0,2).determinant();
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result.coeffRef(3,0) = -matrix.minor(0,3).determinant();
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result.coeffRef(0,2) = matrix.minor(2,0).determinant();
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result.coeffRef(1,2) = -matrix.minor(2,1).determinant();
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result.coeffRef(2,2) = matrix.minor(2,2).determinant();
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result.coeffRef(3,2) = -matrix.minor(2,3).determinant();
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result.coeffRef(0,1) = -matrix.minor(1,0).determinant();
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result.coeffRef(1,1) = matrix.minor(1,1).determinant();
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result.coeffRef(2,1) = -matrix.minor(1,2).determinant();
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result.coeffRef(3,1) = matrix.minor(1,3).determinant();
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result.coeffRef(0,3) = -matrix.minor(3,0).determinant();
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result.coeffRef(1,3) = matrix.minor(3,1).determinant();
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result.coeffRef(2,3) = -matrix.minor(3,2).determinant();
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result.coeffRef(3,3) = matrix.minor(3,3).determinant();
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result /= (matrix.col(0).cwise()*result.row(0).transpose()).sum();
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}
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};
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#ifdef EIGEN_VECTORIZE_SSE
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template<typename MatrixType>
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struct ei_compute_inverse_in_size4_case<MatrixType, float>
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{
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static void run(const MatrixType& matrix, MatrixType& result)
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{
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// Variables (Streaming SIMD Extensions registers) which will contain cofactors and, later, the
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// lines of the inverted matrix.
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__m128 minor0, minor1, minor2, minor3;
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// Variables which will contain the lines of the reference matrix and, later (after the transposition),
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// the columns of the original matrix.
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__m128 row0, row1, row2, row3;
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// Temporary variables and the variable that will contain the matrix determinant.
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__m128 det, tmp1;
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// Matrix transposition
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const float *src = matrix.data();
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tmp1 = _mm_loadh_pi(_mm_castpd_ps(_mm_load_sd((double*)src)), (__m64*)(src+ 4));
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row1 = _mm_loadh_pi(_mm_castpd_ps(_mm_load_sd((double*)(src+8))), (__m64*)(src+12));
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row0 = _mm_shuffle_ps(tmp1, row1, 0x88);
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row1 = _mm_shuffle_ps(row1, tmp1, 0xDD);
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tmp1 = _mm_loadh_pi(_mm_castpd_ps(_mm_load_sd((double*)(src+ 2))), (__m64*)(src+ 6));
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row3 = _mm_loadh_pi(_mm_castpd_ps(_mm_load_sd((double*)(src+10))), (__m64*)(src+14));
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row2 = _mm_shuffle_ps(tmp1, row3, 0x88);
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row3 = _mm_shuffle_ps(row3, tmp1, 0xDD);
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// Cofactors calculation. Because in the process of cofactor computation some pairs in three-
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// element products are repeated, it is not reasonable to load these pairs anew every time. The
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// values in the registers with these pairs are formed using shuffle instruction. Cofactors are
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// calculated row by row (4 elements are placed in 1 SP FP SIMD floating point register).
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tmp1 = _mm_mul_ps(row2, row3);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0xB1);
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minor0 = _mm_mul_ps(row1, tmp1);
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minor1 = _mm_mul_ps(row0, tmp1);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0x4E);
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minor0 = _mm_sub_ps(_mm_mul_ps(row1, tmp1), minor0);
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minor1 = _mm_sub_ps(_mm_mul_ps(row0, tmp1), minor1);
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minor1 = _mm_shuffle_ps(minor1, minor1, 0x4E);
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// -----------------------------------------------
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tmp1 = _mm_mul_ps(row1, row2);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0xB1);
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minor0 = _mm_add_ps(_mm_mul_ps(row3, tmp1), minor0);
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minor3 = _mm_mul_ps(row0, tmp1);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0x4E);
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minor0 = _mm_sub_ps(minor0, _mm_mul_ps(row3, tmp1));
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minor3 = _mm_sub_ps(_mm_mul_ps(row0, tmp1), minor3);
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minor3 = _mm_shuffle_ps(minor3, minor3, 0x4E);
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// -----------------------------------------------
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tmp1 = _mm_mul_ps(_mm_shuffle_ps(row1, row1, 0x4E), row3);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0xB1);
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row2 = _mm_shuffle_ps(row2, row2, 0x4E);
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minor0 = _mm_add_ps(_mm_mul_ps(row2, tmp1), minor0);
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minor2 = _mm_mul_ps(row0, tmp1);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0x4E);
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minor0 = _mm_sub_ps(minor0, _mm_mul_ps(row2, tmp1));
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minor2 = _mm_sub_ps(_mm_mul_ps(row0, tmp1), minor2);
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minor2 = _mm_shuffle_ps(minor2, minor2, 0x4E);
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// -----------------------------------------------
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tmp1 = _mm_mul_ps(row0, row1);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0xB1);
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minor2 = _mm_add_ps(_mm_mul_ps(row3, tmp1), minor2);
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minor3 = _mm_sub_ps(_mm_mul_ps(row2, tmp1), minor3);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0x4E);
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minor2 = _mm_sub_ps(_mm_mul_ps(row3, tmp1), minor2);
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minor3 = _mm_sub_ps(minor3, _mm_mul_ps(row2, tmp1));
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// -----------------------------------------------
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tmp1 = _mm_mul_ps(row0, row3);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0xB1);
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minor1 = _mm_sub_ps(minor1, _mm_mul_ps(row2, tmp1));
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minor2 = _mm_add_ps(_mm_mul_ps(row1, tmp1), minor2);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0x4E);
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minor1 = _mm_add_ps(_mm_mul_ps(row2, tmp1), minor1);
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minor2 = _mm_sub_ps(minor2, _mm_mul_ps(row1, tmp1));
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// -----------------------------------------------
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tmp1 = _mm_mul_ps(row0, row2);
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0xB1);
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minor1 = _mm_add_ps(_mm_mul_ps(row3, tmp1), minor1);
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minor3 = _mm_sub_ps(minor3, _mm_mul_ps(row1, tmp1));
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tmp1 = _mm_shuffle_ps(tmp1, tmp1, 0x4E);
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minor1 = _mm_sub_ps(minor1, _mm_mul_ps(row3, tmp1));
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minor3 = _mm_add_ps(_mm_mul_ps(row1, tmp1), minor3);
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// Evaluation of determinant and its reciprocal value. In the original Intel document,
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// 1/det was evaluated using a fast rcpps command with subsequent approximation using
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// the Newton-Raphson algorithm. Here, we go for a IEEE-compliant division instead,
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// so as to not compromise precision at all.
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det = _mm_mul_ps(row0, minor0);
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det = _mm_add_ps(_mm_shuffle_ps(det, det, 0x4E), det);
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det = _mm_add_ss(_mm_shuffle_ps(det, det, 0xB1), det);
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// tmp1= _mm_rcp_ss(det);
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// det= _mm_sub_ss(_mm_add_ss(tmp1, tmp1), _mm_mul_ss(det, _mm_mul_ss(tmp1, tmp1)));
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det = _mm_div_ss(_mm_set_ss(1.0f), det); // <--- yay, one original line not copied from Intel
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det = _mm_shuffle_ps(det, det, 0x00);
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// warning, Intel's variable naming is very confusing: now 'det' is 1/det !
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// Multiplication of cofactors by 1/det. Storing the inverse matrix to the address in pointer src.
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minor0 = _mm_mul_ps(det, minor0);
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float *dst = result.data();
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_mm_storel_pi((__m64*)(dst), minor0);
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_mm_storeh_pi((__m64*)(dst+2), minor0);
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minor1 = _mm_mul_ps(det, minor1);
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_mm_storel_pi((__m64*)(dst+4), minor1);
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_mm_storeh_pi((__m64*)(dst+6), minor1);
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minor2 = _mm_mul_ps(det, minor2);
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_mm_storel_pi((__m64*)(dst+ 8), minor2);
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_mm_storeh_pi((__m64*)(dst+10), minor2);
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minor3 = _mm_mul_ps(det, minor3);
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_mm_storel_pi((__m64*)(dst+12), minor3);
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_mm_storeh_pi((__m64*)(dst+14), minor3);
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}
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};
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#endif
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/***********************************************
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*** Part 2 : selector and MatrixBase methods ***
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@ -210,7 +264,7 @@ struct ei_compute_inverse<MatrixType, 4>
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{
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static inline void run(const MatrixType& matrix, MatrixType* result)
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{
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ei_compute_inverse_in_size4_case(matrix, result);
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ei_compute_inverse_in_size4_case<MatrixType>::run(matrix, *result);
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}
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};
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@ -45,7 +45,6 @@ template<typename MatrixType> void inverse_permutation_4x4()
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{
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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double error_max = 0.;
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Vector4i indices(0,1,2,3);
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for(int i = 0; i < 24; ++i)
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{
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@ -56,12 +55,9 @@ template<typename MatrixType> void inverse_permutation_4x4()
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m(indices(3),3) = 1;
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MatrixType inv = m.inverse();
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double error = double( (m*inv-MatrixType::Identity()).norm() / epsilon<Scalar>() );
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error_max = std::max(error_max, error);
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VERIFY(error == 0.0);
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std::next_permutation(indices.data(),indices.data()+4);
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}
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std::cerr << "inverse_permutation_4x4, Scalar = " << type_name<Scalar>() << std::endl;
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EIGEN_DEBUG_VAR(error_max);
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VERIFY(error_max < 1. );
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}
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template<typename MatrixType> void inverse_general_4x4(int repeat)
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@ -86,8 +82,8 @@ template<typename MatrixType> void inverse_general_4x4(int repeat)
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double error_avg = error_sum / repeat;
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EIGEN_DEBUG_VAR(error_avg);
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EIGEN_DEBUG_VAR(error_max);
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VERIFY(error_avg < (NumTraits<Scalar>::IsComplex ? 8.4 : 1.4) );
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VERIFY(error_max < (NumTraits<Scalar>::IsComplex ? 160.0 : 75.) );
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VERIFY(error_avg < (NumTraits<Scalar>::IsComplex ? 8.0 : 1.0));
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VERIFY(error_max < (NumTraits<Scalar>::IsComplex ? 64.0 : 20.0));
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}
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void test_prec_inverse_4x4()
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