diff --git a/Eigen/src/Eigenvalues/RealSchur.h b/Eigen/src/Eigenvalues/RealSchur.h
index 970500cd3..5cef6587b 100644
--- a/Eigen/src/Eigenvalues/RealSchur.h
+++ b/Eigen/src/Eigenvalues/RealSchur.h
@@ -408,28 +408,29 @@ inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& ex
   shiftInfo.coeffRef(1) = m_matT.coeff(iu - 1, iu - 1);
   shiftInfo.coeffRef(2) = m_matT.coeff(iu, iu - 1) * m_matT.coeff(iu - 1, iu);
 
-  // Wilkinson's original ad hoc shift
-  if (iter == 10) {
-    exshift += shiftInfo.coeff(0);
-    for (Index i = 0; i <= iu; ++i) m_matT.coeffRef(i, i) -= shiftInfo.coeff(0);
-    Scalar s = abs(m_matT.coeff(iu, iu - 1)) + abs(m_matT.coeff(iu - 1, iu - 2));
-    shiftInfo.coeffRef(0) = Scalar(0.75) * s;
-    shiftInfo.coeffRef(1) = Scalar(0.75) * s;
-    shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
-  }
-
-  // MATLAB's new ad hoc shift
-  if (iter == 30) {
-    Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
-    s = s * s + shiftInfo.coeff(2);
-    if (s > Scalar(0)) {
-      s = sqrt(s);
-      if (shiftInfo.coeff(1) < shiftInfo.coeff(0)) s = -s;
-      s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
-      s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
-      exshift += s;
-      for (Index i = 0; i <= iu; ++i) m_matT.coeffRef(i, i) -= s;
-      shiftInfo.setConstant(Scalar(0.964));
+  // Alternate exceptional shifting strategy every 16 iterations.
+  if (iter % 16 == 0) {
+    // Wilkinson's original ad hoc shift
+    if (iter % 32 != 0) {
+      exshift += shiftInfo.coeff(0);
+      for (Index i = 0; i <= iu; ++i) m_matT.coeffRef(i, i) -= shiftInfo.coeff(0);
+      Scalar s = abs(m_matT.coeff(iu, iu - 1)) + abs(m_matT.coeff(iu - 1, iu - 2));
+      shiftInfo.coeffRef(0) = Scalar(0.75) * s;
+      shiftInfo.coeffRef(1) = Scalar(0.75) * s;
+      shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
+    } else {
+      // MATLAB's new ad hoc shift
+      Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
+      s = s * s + shiftInfo.coeff(2);
+      if (s > Scalar(0)) {
+        s = sqrt(s);
+        if (shiftInfo.coeff(1) < shiftInfo.coeff(0)) s = -s;
+        s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
+        s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
+        exshift += s;
+        for (Index i = 0; i <= iu; ++i) m_matT.coeffRef(i, i) -= s;
+        shiftInfo.setConstant(Scalar(0.964));
+      }
     }
   }
 }
diff --git a/test/schur_real.cpp b/test/schur_real.cpp
index cd0be9251..4a9dd891f 100644
--- a/test/schur_real.cpp
+++ b/test/schur_real.cpp
@@ -97,6 +97,13 @@ void schur(int size = MatrixType::ColsAtCompileTime) {
   }
 }
 
+void test_bug2633() {
+  Eigen::MatrixXd A(4, 4);
+  A << 0, 0, 0, -2, 1, 0, 0, -0, 0, 1, 0, 2, 0, 0, 2, -0;
+  RealSchur<Eigen::MatrixXd> schur(A);
+  VERIFY(schur.info() == Eigen::Success);
+}
+
 EIGEN_DECLARE_TEST(schur_real) {
   CALL_SUBTEST_1((schur<Matrix4f>()));
   CALL_SUBTEST_2((schur<MatrixXd>(internal::random<int>(1, EIGEN_TEST_MAX_SIZE / 4))));
@@ -105,4 +112,6 @@ EIGEN_DECLARE_TEST(schur_real) {
 
   // Test problem size constructors
   CALL_SUBTEST_5(RealSchur<MatrixXf>(10));
+
+  CALL_SUBTEST_6((test_bug2633()));
 }
diff --git a/unsupported/Eigen/src/Polynomials/PolynomialSolver.h b/unsupported/Eigen/src/Polynomials/PolynomialSolver.h
index ec5bc54b2..8c0ce3b71 100644
--- a/unsupported/Eigen/src/Polynomials/PolynomialSolver.h
+++ b/unsupported/Eigen/src/Polynomials/PolynomialSolver.h
@@ -320,6 +320,7 @@ class PolynomialSolver : public PolynomialSolverBase<Scalar_, Deg_> {
       internal::companion<Scalar, Deg_> companion(poly);
       companion.balance();
       m_eigenSolver.compute(companion.denseMatrix());
+      eigen_assert(m_eigenSolver.info() == Eigen::Success);
       m_roots = m_eigenSolver.eigenvalues();
       // cleanup noise in imaginary part of real roots:
       // if the imaginary part is rather small compared to the real part