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* 4x4 inverse: revert to cofactors method

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* inverse tests: use createRandomMatrixOfRank, use more strict precision
* tests: createRandomMatrixOfRank: support 1x1 matrices
* determinant: nest the xpr
* Minor: add comment
This commit is contained in:
Benoit Jacob
2009-12-09 12:43:25 -05:00
parent f0315295e9
commit d2e44f2636
6 changed files with 43 additions and 121 deletions
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5
Eigen/src/Core/Minor.h
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@@ -1,7 +1,7 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2008 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
@@ -54,7 +54,8 @@ struct ei_traits<Minor<MatrixType> >
MaxColsAtCompileTime = (MatrixType::MaxColsAtCompileTime != Dynamic) ?
int(MatrixType::MaxColsAtCompileTime) - 1 : Dynamic,
Flags = _MatrixTypeNested::Flags & HereditaryBits,
CoeffReadCost = _MatrixTypeNested::CoeffReadCost
CoeffReadCost = _MatrixTypeNested::CoeffReadCost // minor is used typically on tiny matrices,
// where loops are unrolled and the 'if' evaluates at compile time
};
};

4
Eigen/src/LU/Determinant.h
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@@ -118,7 +118,9 @@ template<typename Derived>
inline typename ei_traits<Derived>::Scalar MatrixBase<Derived>::determinant() const
{
assert(rows() == cols());
return ei_determinant_impl<Derived>::run(derived());
typedef typename ei_nested<Derived,RowsAtCompileTime>::type Nested;
Nested nested(derived());
return ei_determinant_impl<typename ei_cleantype<Nested>::type>::run(nested);
}
#endif // EIGEN_DETERMINANT_H

101
Eigen/src/LU/Inverse.h
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@@ -182,93 +182,28 @@ struct ei_compute_inverse_and_det_with_check<MatrixType, ResultType, 3>
*** Size 4 implementation ***
****************************/
template<typename MatrixType, typename ResultType>
void ei_compute_inverse_size4_helper(const MatrixType& matrix, ResultType& result)
{
/* Let's split M into four 2x2 blocks:
* (P Q)
* (R S)
* If P is invertible, with inverse denoted by P_inverse, and if
* (S - R*P_inverse*Q) is also invertible, then the inverse of M is
* (P' Q')
* (R' S')
* where
* S' = (S - R*P_inverse*Q)^(-1)
* P' = P1 + (P1*Q) * S' *(R*P_inverse)
* Q' = -(P_inverse*Q) * S'
* R' = -S' * (R*P_inverse)
*/
typedef Block<ResultType,2,2> XprBlock22;
typedef typename MatrixBase<XprBlock22>::PlainMatrixType Block22;
Block22 P_inverse;
ei_compute_inverse<XprBlock22, Block22>::run(matrix.template block<2,2>(0,0), P_inverse);
const Block22 Q = matrix.template block<2,2>(0,2);
const Block22 P_inverse_times_Q = P_inverse * Q;
const XprBlock22 R = matrix.template block<2,2>(2,0);
const Block22 R_times_P_inverse = R * P_inverse;
const Block22 R_times_P_inverse_times_Q = R_times_P_inverse * Q;
const XprBlock22 S = matrix.template block<2,2>(2,2);
const Block22 X = S - R_times_P_inverse_times_Q;
Block22 Y;
ei_compute_inverse<Block22, Block22>::run(X, Y);
result.template block<2,2>(2,2) = Y;
result.template block<2,2>(2,0) = - Y * R_times_P_inverse;
const Block22 Z = P_inverse_times_Q * Y;
result.template block<2,2>(0,2) = - Z;
result.template block<2,2>(0,0) = P_inverse + Z * R_times_P_inverse;
}
template<typename MatrixType, typename ResultType>
struct ei_compute_inverse<MatrixType, ResultType, 4>
{
static inline void run(const MatrixType& _matrix, ResultType& result)
static inline void run(const MatrixType& matrix, ResultType& result)
{
typedef typename ResultType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
// we will do row permutations on the matrix. This copy should have negligible cost.
// if not, consider working in-place on the matrix (const-cast it, but then undo the permutations
// to nevertheless honor constness)
typename MatrixType::PlainMatrixType matrix(_matrix);
// let's extract from the 2 first colums a 2x2 block whose determinant is as big as possible.
int good_row0, good_row1, good_i;
Matrix<RealScalar,6,1> absdet;
// any 2x2 block with determinant above this threshold will be considered good enough.
// The magic value 1e-1 here comes from experimentation. The bigger it is, the higher the precision,
// the slower the computation. This value 1e-1 gives precision almost as good as the brutal cofactors
// algorithm, both in average and in worst-case precision.
RealScalar d = (matrix.col(0).squaredNorm()+matrix.col(1).squaredNorm()) * RealScalar(1e-1);
#define ei_inv_size4_helper_macro(i,row0,row1) \
absdet[i] = ei_abs(matrix.coeff(row0,0)*matrix.coeff(row1,1) \
- matrix.coeff(row0,1)*matrix.coeff(row1,0)); \
if(absdet[i] > d) { good_row0=row0; good_row1=row1; goto good; }
ei_inv_size4_helper_macro(0,0,1)
ei_inv_size4_helper_macro(1,0,2)
ei_inv_size4_helper_macro(2,0,3)
ei_inv_size4_helper_macro(3,1,2)
ei_inv_size4_helper_macro(4,1,3)
ei_inv_size4_helper_macro(5,2,3)
// no 2x2 block has determinant bigger than the threshold. So just take the one that
// has the biggest determinant
absdet.maxCoeff(&good_i);
good_row0 = good_i <= 2 ? 0 : good_i <= 4 ? 1 : 2;
good_row1 = good_i <= 2 ? good_i+1 : good_i <= 4 ? good_i-1 : 3;
// now good_row0 and good_row1 are correctly set
good:
// do row permutations to move this 2x2 block to the top
matrix.row(0).swap(matrix.row(good_row0));
matrix.row(1).swap(matrix.row(good_row1));
// now applying our helper function is numerically stable
ei_compute_inverse_size4_helper(matrix, result);
// Since we did row permutations on the original matrix, we need to do column permutations
// in the reverse order on the inverse
result.col(1).swap(result.col(good_row1));
result.col(0).swap(result.col(good_row0));
result.coeffRef(0,0) = matrix.minor(0,0).determinant();
result.coeffRef(1,0) = -matrix.minor(0,1).determinant();
result.coeffRef(2,0) = matrix.minor(0,2).determinant();
result.coeffRef(3,0) = -matrix.minor(0,3).determinant();
result.coeffRef(0,2) = matrix.minor(2,0).determinant();
result.coeffRef(1,2) = -matrix.minor(2,1).determinant();
result.coeffRef(2,2) = matrix.minor(2,2).determinant();
result.coeffRef(3,2) = -matrix.minor(2,3).determinant();
result.coeffRef(0,1) = -matrix.minor(1,0).determinant();
result.coeffRef(1,1) = matrix.minor(1,1).determinant();
result.coeffRef(2,1) = -matrix.minor(1,2).determinant();
result.coeffRef(3,1) = matrix.minor(1,3).determinant();
result.coeffRef(0,3) = -matrix.minor(3,0).determinant();
result.coeffRef(1,3) = matrix.minor(3,1).determinant();
result.coeffRef(2,3) = -matrix.minor(3,2).determinant();
result.coeffRef(3,3) = matrix.minor(3,3).determinant();
result /= (matrix.col(0).cwise()*result.row(0).transpose()).sum();
}
};