Add method signDeterminant() to QR and related decompositions.

Eigen/src/QR/ColPivHouseholderQR.h

diff --git a/Eigen/src/QR/ColPivHouseholderQR.h b/Eigen/src/QR/ColPivHouseholderQR.h
index f1de6fd..bd0849d 100644
--- a/Eigen/src/QR/ColPivHouseholderQR.h
+++ b/Eigen/src/QR/ColPivHouseholderQR.h
@@ -238,6 +238,20 @@
    */
   typename MatrixType::RealScalar logAbsDeterminant() const;
 
+  /** \returns the sign of the determinant of the matrix of which
+   * *this is the QR decomposition. It has only linear complexity
+   * (that is, O(n) where n is the dimension of the square matrix)
+   * as the QR decomposition has already been computed.
+   *
+   * \note This is only for square matrices.
+   *
+   * \note This method is useful to work around the risk of overflow/underflow that's inherent
+   * to determinant computation.
+   *
+   * \sa determinant(), absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()
+   */
+  typename MatrixType::Scalar signDeterminant() const;
+
   /** \returns the rank of the matrix of which *this is the QR decomposition.
    *
    * \note This method has to determine which pivots should be considered nonzero.
@@ -424,26 +438,39 @@
 
 template <typename MatrixType, typename PermutationIndex>
 typename MatrixType::Scalar ColPivHouseholderQR<MatrixType, PermutationIndex>::determinant() const {
+  using Scalar = typename MatrixType::Scalar;
   eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
   Scalar detQ;
   internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
-  return m_qr.diagonal().prod() * detQ * Scalar(m_det_p);
+  return isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().prod() : Scalar(0);
 }
 
 template <typename MatrixType, typename PermutationIndex>
 typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType, PermutationIndex>::absDeterminant() const {
   using std::abs;
+  using RealScalar = typename MatrixType::RealScalar;
   eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
-  return abs(m_qr.diagonal().prod());
+  return isInjective() ? abs(m_qr.diagonal().prod()) : RealScalar(0);
 }
 
 template <typename MatrixType, typename PermutationIndex>
 typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType, PermutationIndex>::logAbsDeterminant() const {
+  using RealScalar = typename MatrixType::RealScalar;
   eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
-  return m_qr.diagonal().cwiseAbs().array().log().sum();
+  return isInjective() ? m_qr.diagonal().cwiseAbs().array().log().sum() : -NumTraits<RealScalar>::infinity();
+}
+
+template <typename MatrixType, typename PermutationIndex>
+typename MatrixType::Scalar ColPivHouseholderQR<MatrixType, PermutationIndex>::signDeterminant() const {
+  using Scalar = typename MatrixType::Scalar;
+  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
+  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
+  Scalar detQ;
+  internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
+  return isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().array().sign().prod() : Scalar(0);
 }
 
 /** Performs the QR factorization of the given matrix \a matrix. The result of

Eigen/src/QR/CompleteOrthogonalDecomposition.h

diff --git a/Eigen/src/QR/CompleteOrthogonalDecomposition.h b/Eigen/src/QR/CompleteOrthogonalDecomposition.h
index 8566e96..960ccb1 100644
--- a/Eigen/src/QR/CompleteOrthogonalDecomposition.h
+++ b/Eigen/src/QR/CompleteOrthogonalDecomposition.h
@@ -228,6 +228,21 @@
    */
   typename MatrixType::RealScalar logAbsDeterminant() const;
 
+  /** \returns the sign of the determinant of the
+   * matrix of which *this is the complete orthogonal decomposition. It has
+   * only linear complexity (that is, O(n) where n is the dimension of the
+   * square matrix) as the complete orthogonal decomposition has already been
+   * computed.
+   *
+   * \note This is only for square matrices.
+   *
+   * \note This method is useful to work around the risk of overflow/underflow
+   * that's inherent to determinant computation.
+   *
+   * \sa determinant(), absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()
+   */
+  typename MatrixType::Scalar signDeterminant() const;
+
   /** \returns the rank of the matrix of which *this is the complete orthogonal
    * decomposition.
    *
@@ -424,6 +439,11 @@
   return m_cpqr.logAbsDeterminant();
 }
 
+template <typename MatrixType, typename PermutationIndex>
+typename MatrixType::Scalar CompleteOrthogonalDecomposition<MatrixType, PermutationIndex>::signDeterminant() const {
+  return m_cpqr.signDeterminant();
+}
+
 /** Performs the complete orthogonal decomposition of the given matrix \a
  * matrix. The result of the factorization is stored into \c *this, and a
  * reference to \c *this is returned.

Eigen/src/QR/FullPivHouseholderQR.h

diff --git a/Eigen/src/QR/FullPivHouseholderQR.h b/Eigen/src/QR/FullPivHouseholderQR.h
index d93a5d1..0cadf23 100644
--- a/Eigen/src/QR/FullPivHouseholderQR.h
+++ b/Eigen/src/QR/FullPivHouseholderQR.h
@@ -248,6 +248,20 @@
    */
   typename MatrixType::RealScalar logAbsDeterminant() const;
 
+  /** \returns the sign of the determinant of the matrix of which
+   * *this is the QR decomposition. It has only linear complexity
+   * (that is, O(n) where n is the dimension of the square matrix)
+   * as the QR decomposition has already been computed.
+   *
+   * \note This is only for square matrices.
+   *
+   * \note This method is useful to work around the risk of overflow/underflow that's inherent
+   * to determinant computation.
+   *
+   * \sa determinant(), absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()
+   */
+  typename MatrixType::Scalar signDeterminant() const;
+
   /** \returns the rank of the matrix of which *this is the QR decomposition.
    *
    * \note This method has to determine which pivots should be considered nonzero.
@@ -421,26 +435,39 @@
 
 template <typename MatrixType, typename PermutationIndex>
 typename MatrixType::Scalar FullPivHouseholderQR<MatrixType, PermutationIndex>::determinant() const {
+  using Scalar = typename MatrixType::Scalar;
   eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
   Scalar detQ;
   internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
-  return m_qr.diagonal().prod() * detQ * Scalar(m_det_p);
+  return isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().prod() : Scalar(0);
 }
 
 template <typename MatrixType, typename PermutationIndex>
 typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType, PermutationIndex>::absDeterminant() const {
+  using RealScalar = typename MatrixType::RealScalar;
   using std::abs;
   eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
-  return abs(m_qr.diagonal().prod());
+  return isInjective() ? abs(m_qr.diagonal().prod()) : RealScalar(0);
 }
 
 template <typename MatrixType, typename PermutationIndex>
 typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType, PermutationIndex>::logAbsDeterminant() const {
+  using RealScalar = typename MatrixType::RealScalar;
   eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
-  return m_qr.diagonal().cwiseAbs().array().log().sum();
+  return isInjective() ? m_qr.diagonal().cwiseAbs().array().log().sum() : -NumTraits<RealScalar>::infinity();
+}
+
+template <typename MatrixType, typename PermutationIndex>
+typename MatrixType::Scalar FullPivHouseholderQR<MatrixType, PermutationIndex>::signDeterminant() const {
+  using Scalar = typename MatrixType::Scalar;
+  eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
+  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
+  Scalar detQ;
+  internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
+  return isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().array().sign().prod() : Scalar(0);
 }
 
 /** Performs the QR factorization of the given matrix \a matrix. The result of

Eigen/src/QR/HouseholderQR.h

diff --git a/Eigen/src/QR/HouseholderQR.h b/Eigen/src/QR/HouseholderQR.h
index 9e73672..e297372 100644
--- a/Eigen/src/QR/HouseholderQR.h
+++ b/Eigen/src/QR/HouseholderQR.h
@@ -187,6 +187,8 @@
    * \warning a determinant can be very big or small, so for matrices
    * of large enough dimension, there is a risk of overflow/underflow.
    * One way to work around that is to use logAbsDeterminant() instead.
+   * Also, do not rely on the determinant being exactly zero for testing
+   * singularity or rank-deficiency.
    *
    * \sa absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()
    */
@@ -202,6 +204,8 @@
    * \warning a determinant can be very big or small, so for matrices
    * of large enough dimension, there is a risk of overflow/underflow.
    * One way to work around that is to use logAbsDeterminant() instead.
+   * Also, do not rely on the determinant being exactly zero for testing
+   * singularity or rank-deficiency.
    *
    * \sa determinant(), logAbsDeterminant(), MatrixBase::determinant()
    */
@@ -217,10 +221,30 @@
    * \note This method is useful to work around the risk of overflow/underflow that's inherent
    * to determinant computation.
    *
+   * \warning Do not rely on the determinant being exactly zero for testing
+   * singularity or rank-deficiency.
+   *
    * \sa determinant(), absDeterminant(), MatrixBase::determinant()
    */
   typename MatrixType::RealScalar logAbsDeterminant() const;
 
+  /** \returns the sign of the determinant of the matrix of which
+   * *this is the QR decomposition. It has only linear complexity
+   * (that is, O(n) where n is the dimension of the square matrix)
+   * as the QR decomposition has already been computed.
+   *
+   * \note This is only for square matrices.
+   *
+   * \note This method is useful to work around the risk of overflow/underflow that's inherent
+   * to determinant computation.
+   *
+   * \warning Do not rely on the determinant being exactly zero for testing
+   * singularity or rank-deficiency.
+   *
+   * \sa determinant(), absDeterminant(), MatrixBase::determinant()
+   */
+  typename MatrixType::Scalar signDeterminant() const;
+
   inline Index rows() const { return m_qr.rows(); }
   inline Index cols() const { return m_qr.cols(); }
 
@@ -306,6 +330,15 @@
   return m_qr.diagonal().cwiseAbs().array().log().sum();
 }
 
+template <typename MatrixType>
+typename MatrixType::Scalar HouseholderQR<MatrixType>::signDeterminant() const {
+  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
+  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
+  Scalar detQ;
+  internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
+  return detQ * m_qr.diagonal().array().sign().prod();
+}
+
 namespace internal {
 
 /** \internal */

test/qr.cpp

diff --git a/test/qr.cpp b/test/qr.cpp
index de470ca..f7f6990 100644
--- a/test/qr.cpp
+++ b/test/qr.cpp
@@ -82,6 +82,7 @@
   m1 = m3 * m1 * m3.adjoint();
   qr.compute(m1);
   VERIFY_IS_APPROX(log(absdet), qr.logAbsDeterminant());
+  VERIFY_IS_APPROX(numext::sign(det), qr.signDeterminant());
   // This test is tricky if the determinant becomes too small.
   // Since we generate random numbers with magnitude range [0,1], the average determinant is 0.5^size
   RealScalar tol =
@@ -102,7 +103,7 @@
   VERIFY_RAISES_ASSERT(qr.householderQ())
   VERIFY_RAISES_ASSERT(qr.determinant())
   VERIFY_RAISES_ASSERT(qr.absDeterminant())
-  VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
+  VERIFY_RAISES_ASSERT(qr.signDeterminant())
 }
 
 EIGEN_DECLARE_TEST(qr) {

test/qr_colpivoting.cpp

diff --git a/test/qr_colpivoting.cpp b/test/qr_colpivoting.cpp
index 4f8711f..705573c 100644
--- a/test/qr_colpivoting.cpp
+++ b/test/qr_colpivoting.cpp
@@ -21,6 +21,7 @@
   Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);
 
   typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar RealScalar;
   typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
   MatrixType matrix;
   createRandomPIMatrixOfRank(rank, rows, cols, matrix);
@@ -56,6 +57,23 @@
 
   MatrixType pinv = cod.pseudoInverse();
   VERIFY_IS_APPROX(cod_solution, pinv * rhs);
+
+  // now construct a (square) matrix with prescribed determinant
+  Index size = internal::random<Index>(2, 20);
+  matrix.setZero(size, size);
+  for (int i = 0; i < size; i++) {
+    matrix(i, i) = internal::random<Scalar>();
+  }
+  Scalar det = matrix.diagonal().prod();
+  RealScalar absdet = abs(det);
+  CompleteOrthogonalDecomposition<MatrixType> cod2(matrix);
+  cod2.compute(matrix);
+  q = cod2.householderQ();
+  matrix = q * matrix * q.adjoint();
+  VERIFY_IS_APPROX(det, cod2.determinant());
+  VERIFY_IS_APPROX(absdet, cod2.absDeterminant());
+  VERIFY_IS_APPROX(numext::log(absdet), cod2.logAbsDeterminant());
+  VERIFY_IS_APPROX(numext::sign(det), cod2.signDeterminant());
 }
 
 template <typename MatrixType, int Cols2>
@@ -265,6 +283,7 @@
   VERIFY_IS_APPROX(det, qr.determinant());
   VERIFY_IS_APPROX(absdet, qr.absDeterminant());
   VERIFY_IS_APPROX(log(absdet), qr.logAbsDeterminant());
+  VERIFY_IS_APPROX(numext::sign(det), qr.signDeterminant());
 }
 
 template <typename MatrixType>
@@ -285,6 +304,7 @@
   VERIFY_RAISES_ASSERT(qr.determinant())
   VERIFY_RAISES_ASSERT(qr.absDeterminant())
   VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
+  VERIFY_RAISES_ASSERT(qr.signDeterminant())
 }
 
 template <typename MatrixType>
@@ -305,6 +325,7 @@
   VERIFY_RAISES_ASSERT(cod.determinant())
   VERIFY_RAISES_ASSERT(cod.absDeterminant())
   VERIFY_RAISES_ASSERT(cod.logAbsDeterminant())
+  VERIFY_RAISES_ASSERT(cod.signDeterminant())
 }
 
 EIGEN_DECLARE_TEST(qr_colpivoting) {

test/qr_fullpivoting.cpp

diff --git a/test/qr_fullpivoting.cpp b/test/qr_fullpivoting.cpp
index 71f3a51..2b6ecc5 100644
--- a/test/qr_fullpivoting.cpp
+++ b/test/qr_fullpivoting.cpp
@@ -105,6 +105,7 @@
   VERIFY_IS_APPROX(det, qr.determinant());
   VERIFY_IS_APPROX(absdet, qr.absDeterminant());
   VERIFY_IS_APPROX(log(absdet), qr.logAbsDeterminant());
+  VERIFY_IS_APPROX(numext::sign(det), qr.signDeterminant());
 }
 
 template <typename MatrixType>
@@ -125,6 +126,7 @@
   VERIFY_RAISES_ASSERT(qr.determinant())
   VERIFY_RAISES_ASSERT(qr.absDeterminant())
   VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
+  VERIFY_RAISES_ASSERT(qr.signDeterminant())
 }
 
 EIGEN_DECLARE_TEST(qr_fullpivoting) {