* 4x4 inverse: revert to cofactors method

* inverse tests: use createRandomMatrixOfRank, use more strict precision
* tests: createRandomMatrixOfRank: support 1x1 matrices
* determinant: nest the xpr
* Minor: add comment

Eigen/src/Core/Minor.h

@@ -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
   };
 };
 

Eigen/src/LU/Determinant.h

@@ -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

Eigen/src/LU/Inverse.h

@@ -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;
+    result.coeffRef(0,0) = matrix.minor(0,0).determinant();
-    typedef typename MatrixType::RealScalar RealScalar;
+    result.coeffRef(1,0) = -matrix.minor(0,1).determinant();
-
+    result.coeffRef(2,0) = matrix.minor(0,2).determinant();
-    // we will do row permutations on the matrix. This copy should have negligible cost.
+    result.coeffRef(3,0) = -matrix.minor(0,3).determinant();
-    // if not, consider working in-place on the matrix (const-cast it, but then undo the permutations
+    result.coeffRef(0,2) = matrix.minor(2,0).determinant();
-    // to nevertheless honor constness)
+    result.coeffRef(1,2) = -matrix.minor(2,1).determinant();
-    typename MatrixType::PlainMatrixType matrix(_matrix);
+    result.coeffRef(2,2) = matrix.minor(2,2).determinant();
-
+    result.coeffRef(3,2) = -matrix.minor(2,3).determinant();
-    // let's extract from the 2 first colums a 2x2 block whose determinant is as big as possible.
+    result.coeffRef(0,1) = -matrix.minor(1,0).determinant();
-    int good_row0, good_row1, good_i;
+    result.coeffRef(1,1) = matrix.minor(1,1).determinant();
-    Matrix<RealScalar,6,1> absdet;
+    result.coeffRef(2,1) = -matrix.minor(1,2).determinant();
-
+    result.coeffRef(3,1) = matrix.minor(1,3).determinant();
-    // any 2x2 block with determinant above this threshold will be considered good enough.
+    result.coeffRef(0,3) = -matrix.minor(3,0).determinant();
-    // The magic value 1e-1 here comes from experimentation. The bigger it is, the higher the precision,
+    result.coeffRef(1,3) = matrix.minor(3,1).determinant();
-    // the slower the computation. This value 1e-1 gives precision almost as good as the brutal cofactors
+    result.coeffRef(2,3) = -matrix.minor(3,2).determinant();
-    // algorithm, both in average and in worst-case precision.
+    result.coeffRef(3,3) = matrix.minor(3,3).determinant();
-    RealScalar d = (matrix.col(0).squaredNorm()+matrix.col(1).squaredNorm()) * RealScalar(1e-1);
+    result /= (matrix.col(0).cwise()*result.row(0).transpose()).sum();
-    #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));
   }
 };
 

test/inverse.cpp

@@ -38,18 +38,11 @@ template<typename MatrixType> void inverse(const MatrixType& m)
   typedef typename NumTraits<Scalar>::Real RealScalar;
   typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
 
-  MatrixType m1 = MatrixType::Random(rows, cols),
+  MatrixType m1(rows, cols),
              m2(rows, cols),
              mzero = MatrixType::Zero(rows, cols),
              identity = MatrixType::Identity(rows, rows);
-
+  createRandomMatrixOfRank(rows,rows,rows,m1);
-  if (ei_is_same_type<RealScalar,float>::ret)
-  {
-    // let's build a more stable to inverse matrix
-    MatrixType a = MatrixType::Random(rows,cols);
-    m1 += m1 * m1.adjoint() + a * a.adjoint();
-  }
-
   m2 = m1.inverse();
   VERIFY_IS_APPROX(m1, m2.inverse() );
 

test/main.h

@@ -353,12 +353,25 @@ void createRandomMatrixOfRank(int desired_rank, int rows, int cols, MatrixType&
   typedef Matrix<Scalar, Rows, Rows> MatrixAType;
   typedef Matrix<Scalar, Cols, Cols> MatrixBType;
 
+  if(desired_rank == 0)
+  {
+    m.setZero(rows,cols);
+    return;
+  }
+
+  if(desired_rank == 1)
+  {
+    m = VectorType::Random(rows) * VectorType::Random(cols).transpose();
+    return;
+  }
+
   MatrixAType a = MatrixAType::Random(rows,rows);
   MatrixType d = MatrixType::Identity(rows,cols);
   MatrixBType  b = MatrixBType::Random(cols,cols);
 
   // set the diagonal such that only desired_rank non-zero entries reamain
   const int diag_size = std::min(d.rows(),d.cols());
+  if(diag_size != desired_rank)
     d.diagonal().segment(desired_rank, diag_size-desired_rank) = VectorType::Zero(diag_size-desired_rank);
 
   HouseholderQR<MatrixAType> qra(a);

test/prec_inverse_4x4.cpp

@@ -26,28 +26,6 @@
 #include <Eigen/LU>
 #include <algorithm>
 
-Matrix4f inverse(const Matrix4f& m)
-{
-  Matrix4f r;
-  r(0,0) = m.minor(0,0).determinant();
-  r(1,0) = -m.minor(0,1).determinant();
-  r(2,0) = m.minor(0,2).determinant();
-  r(3,0) = -m.minor(0,3).determinant();
-  r(0,2) = m.minor(2,0).determinant();
-  r(1,2) = -m.minor(2,1).determinant();
-  r(2,2) = m.minor(2,2).determinant();
-  r(3,2) = -m.minor(2,3).determinant();
-  r(0,1) = -m.minor(1,0).determinant();
-  r(1,1) = m.minor(1,1).determinant();
-  r(2,1) = -m.minor(1,2).determinant();
-  r(3,1) = m.minor(1,3).determinant();
-  r(0,3) = -m.minor(3,0).determinant();
-  r(1,3) = m.minor(3,1).determinant();
-  r(2,3) = -m.minor(3,2).determinant();
-  r(3,3) = m.minor(3,3).determinant();
-  return r / (m(0,0)*r(0,0) + m(1,0)*r(0,1) + m(2,0)*r(0,2) + m(3,0)*r(0,3));
-}
-
 template<typename MatrixType> void inverse_permutation_4x4()
 {
   typedef typename MatrixType::Scalar Scalar;
@@ -79,7 +57,7 @@ template<typename MatrixType> void inverse_general_4x4(int repeat)
     do {
       m = MatrixType::Random();
       absdet = ei_abs(m.determinant());
-    } while(absdet < 10 * epsilon<Scalar>());
+    } while(absdet < epsilon<Scalar>());
     MatrixType inv = m.inverse();
     double error = double( (m*inv-MatrixType::Identity()).norm() * absdet / epsilon<Scalar>() );
     error_sum += error;
@@ -89,8 +67,8 @@ template<typename MatrixType> void inverse_general_4x4(int repeat)
   double error_avg = error_sum / repeat;
   EIGEN_DEBUG_VAR(error_avg);
   EIGEN_DEBUG_VAR(error_max);
-  VERIFY(error_avg < (NumTraits<Scalar>::IsComplex ? 8.4 : 1.4) );
+  VERIFY(error_avg < (NumTraits<Scalar>::IsComplex ? 8.0 : 1.0));
-  VERIFY(error_max < (NumTraits<Scalar>::IsComplex ? 160.0 : 75.) );
+  VERIFY(error_max < (NumTraits<Scalar>::IsComplex ? 64.0 : 16.0));
 }
 
 void test_prec_inverse_4x4()