fix LU and QR solve when rank==0, fix LLT when the matrix is purely 0

Eigen/src/Cholesky/LLT.h

@@ -136,7 +136,7 @@ void LLT<MatrixType>::compute(const MatrixType& a)
   for (int j = 1; j < size; ++j)
   {
     x = ei_real(a.coeff(j,j)) - m_matrix.row(j).start(j).squaredNorm();
-    if (x < cutoff)
+    if (x <= cutoff)
     {
       m_isPositiveDefinite = false;
       continue;

Eigen/src/LU/LU.h

@@ -515,9 +515,10 @@ bool LU<MatrixType>::solve(
         if(!ei_isMuchSmallerThan(c.coeff(row,col), biggest_in_c, m_precision))
           return false;
   }
-  m_lu.corner(TopLeft, m_rank, m_rank)
-      .template marked<UpperTriangular>()
-      .solveTriangularInPlace(c.corner(TopLeft, m_rank, c.cols()));
+  if(m_rank>0)
+    m_lu.corner(TopLeft, m_rank, m_rank)
+        .template marked<UpperTriangular>()
+        .solveTriangularInPlace(c.corner(TopLeft, m_rank, c.cols()));
 
   // Step 4
   result->resize(m_lu.cols(), b.cols());

Eigen/src/QR/QR.h

@@ -276,6 +276,9 @@ bool QR<MatrixType>::solve(
   // Q^T without explicitly forming matrixQ(). Investigate.
   *result = matrixQ().transpose()*b;
   
+  if(m_rank==0)
+    return result.isZero();
+
   if(!isSurjective())
   {
     // is result is in the image of R ?