From 8cc9232b9a58016eebf6c3c4a51143cdec89144e Mon Sep 17 00:00:00 2001
From: Gael Guennebaud <g.gael@free.fr>
Date: Thu, 11 Feb 2016 15:32:56 +0100
Subject: [PATCH] bug #774: fix a numerical issue producing unwanted
 reflections.

---
 Eigen/src/Geometry/Umeyama.h | 19 ++++---------------
 1 file changed, 4 insertions(+), 15 deletions(-)

diff --git a/Eigen/src/Geometry/Umeyama.h b/Eigen/src/Geometry/Umeyama.h
index 8d9b7a154..6943f719e 100644
--- a/Eigen/src/Geometry/Umeyama.h
+++ b/Eigen/src/Geometry/Umeyama.h
@@ -135,22 +135,11 @@ umeyama(const MatrixBase<Derived>& src, const MatrixBase<OtherDerived>& dst, boo
 
   // Eq. (39)
   VectorType S = VectorType::Ones(m);
-  if (sigma.determinant()<Scalar(0)) S(m-1) = Scalar(-1);
 
-  // Eq. (40) and (43)
-  const VectorType& d = svd.singularValues();
-  Index rank = 0; for (Index i=0; i<m; ++i) if (!internal::isMuchSmallerThan(d.coeff(i),d.coeff(0))) ++rank;
-  if (rank == m-1) {
-    if ( svd.matrixU().determinant() * svd.matrixV().determinant() > Scalar(0) ) {
-      Rt.block(0,0,m,m).noalias() = svd.matrixU()*svd.matrixV().transpose();
-    } else {
-      const Scalar s = S(m-1); S(m-1) = Scalar(-1);
-      Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
-      S(m-1) = s;
-    }
-  } else {
-    Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
-  }
+  if  ( svd.matrixU().determinant() * svd.matrixV().determinant() < 0 )
+    S(m-1) = -1;
+
+  Rt.block(0,0,m,m).noalias() = svd.matrixU() * S.asDiagonal() * svd.matrixV().transpose();
 
   if (with_scaling)
   {