Tests: Bound the conditioning of svd_min_norm's generated systems

libeigen/eigen!3424

Co-authored-by: Rasmus Munk Larsen <rmlarsen@gmail.com>

test/svd_common.h

diff --git a/test/svd_common.h b/test/svd_common.h
index f1e6a6f..e737a01 100644
--- a/test/svd_common.h
+++ b/test/svd_common.h
@@ -253,8 +253,10 @@
   MatrixType2 m2(rank, cols);
   m2.setRandom();
   if (SVD_FOR_MIN_NORM(MatrixType2)(m2).setThreshold(test_precision<Scalar>()).rank() != rank) {
-    // Ensure full row rank by making the leading square block diagonally dominant.
-    for (Index i = 0; i < rank; ++i) m2(i, i) += Scalar(1);
+    // With m2 = [B R] and s = 2 |m2|_F, sigma_min(m2 + s [I 0]) >= sigma_min(B + s I) >= s - |B|_2 >= |m2|_F (Weyl)
+    // and |m2 + s [I 0]|_2 <= 3 |m2|_F, so cond(m2) <= 3.
+    const Scalar shift = Scalar(2 * m2.norm());
+    m2.diagonal().array() += shift;
   }
 
   RhsType2 rhs2 = RhsType2::Random(rank);
@@ -277,7 +279,11 @@
   typedef Matrix<Scalar, RowsAtCompileTime3, 1> RhsType3;
   Index rows3 =
       ColsAtCompileTime == Dynamic ? internal::random<Index>(rank + 1, 2 * cols) : Index(ColsAtCompileTime + 1);
-  Matrix<Scalar, RowsAtCompileTime3, Dynamic> C = Matrix<Scalar, RowsAtCompileTime3, Dynamic>::Random(rows3, rank);
+  // Orthonormal columns give m3 the nonzero singular values of m2, so sigma_0 / sigma_{rank-1} = cond(m2) stays far
+  // below 1/threshold() = 1/(diagSize * eps), and solve() keeps every direction of m2.
+  using MatrixTypeC = Matrix<Scalar, RowsAtCompileTime3, Dynamic>;
+  const MatrixTypeC C =
+      HouseholderQR<MatrixTypeC>(MatrixTypeC::Random(rows3, rank)).householderQ() * MatrixTypeC::Identity(rows3, rank);
   MatrixType3 m3 = C * m2;
   RhsType3 rhs3 = C * rhs2;
   SVD_STATIC_OPTIONS(MatrixType3, Options) svd3(m3);