diff --git a/gtsam/discrete/DiscreteFactorGraph.h b/gtsam/discrete/DiscreteFactorGraph.h
index 818eeda4e..e665ea88b 100644
--- a/gtsam/discrete/DiscreteFactorGraph.h
+++ b/gtsam/discrete/DiscreteFactorGraph.h
@@ -62,9 +62,17 @@ template<> struct EliminationTraits<DiscreteFactorGraph>
   typedef DiscreteBayesTree BayesTreeType;             ///< Type of Bayes tree
   typedef DiscreteJunctionTree JunctionTreeType;       ///< Type of Junction tree
   /// The default dense elimination function
-  static std::pair<boost::shared_ptr<ConditionalType>, boost::shared_ptr<FactorType> >
+  static std::pair<boost::shared_ptr<ConditionalType>,
+                   boost::shared_ptr<FactorType> >
   DefaultEliminate(const FactorGraphType& factors, const Ordering& keys) {
-    return EliminateDiscrete(factors, keys); }
+    return EliminateDiscrete(factors, keys);
+  }
+  /// The default ordering generation function
+  static Ordering DefaultOrderingFunc(
+      const FactorGraphType& graph,
+      boost::optional<const VariableIndex&> variableIndex) {
+    return Ordering::Colamd(*variableIndex);
+  }
 };
 
 /* ************************************************************************* */
diff --git a/gtsam/inference/EliminateableFactorGraph-inst.h b/gtsam/inference/EliminateableFactorGraph-inst.h
index 35e7505c9..bebce14cd 100644
--- a/gtsam/inference/EliminateableFactorGraph-inst.h
+++ b/gtsam/inference/EliminateableFactorGraph-inst.h
@@ -44,9 +44,16 @@ namespace gtsam {
       if (orderingType == Ordering::METIS) {
         Ordering computedOrdering = Ordering::Metis(asDerived());
         return eliminateSequential(computedOrdering, function, variableIndex);
-      } else {
+      } else if (orderingType == Ordering::COLAMD) {
         Ordering computedOrdering = Ordering::Colamd(*variableIndex);
         return eliminateSequential(computedOrdering, function, variableIndex);
+      } else if (orderingType == Ordering::NATURAL) {
+        Ordering computedOrdering = Ordering::Natural(asDerived());
+        return eliminateSequential(computedOrdering, function, variableIndex);
+      } else {
+        Ordering computedOrdering = EliminationTraitsType::DefaultOrderingFunc(
+            asDerived(), variableIndex);
+        return eliminateSequential(computedOrdering, function, variableIndex);
       }
     }
   }
@@ -100,9 +107,16 @@ namespace gtsam {
       if (orderingType == Ordering::METIS) {
         Ordering computedOrdering = Ordering::Metis(asDerived());
         return eliminateMultifrontal(computedOrdering, function, variableIndex);
-      } else {
+      } else if (orderingType == Ordering::COLAMD) {
         Ordering computedOrdering = Ordering::Colamd(*variableIndex);
         return eliminateMultifrontal(computedOrdering, function, variableIndex);
+      } else if (orderingType == Ordering::NATURAL) {
+        Ordering computedOrdering = Ordering::Natural(asDerived());
+        return eliminateMultifrontal(computedOrdering, function, variableIndex);
+      } else {
+        Ordering computedOrdering = EliminationTraitsType::DefaultOrderingFunc(
+            asDerived(), variableIndex);
+        return eliminateMultifrontal(computedOrdering, function, variableIndex);
       }
     }
   }
diff --git a/gtsam/linear/GaussianFactorGraph.h b/gtsam/linear/GaussianFactorGraph.h
index 0d5057aa8..c688eb13f 100644
--- a/gtsam/linear/GaussianFactorGraph.h
+++ b/gtsam/linear/GaussianFactorGraph.h
@@ -54,6 +54,12 @@ namespace gtsam {
     static std::pair<boost::shared_ptr<ConditionalType>, boost::shared_ptr<FactorType> >
       DefaultEliminate(const FactorGraphType& factors, const Ordering& keys) {
         return EliminatePreferCholesky(factors, keys); }
+    /// The default ordering generation function
+    static Ordering DefaultOrderingFunc(
+        const FactorGraphType& graph,
+        boost::optional<const VariableIndex&> variableIndex) {
+      return Ordering::Colamd(*variableIndex);
+    }
   };
 
   /* ************************************************************************* */
diff --git a/gtsam/symbolic/SymbolicFactorGraph.h b/gtsam/symbolic/SymbolicFactorGraph.h
index 36379fd83..8bb75cb97 100644
--- a/gtsam/symbolic/SymbolicFactorGraph.h
+++ b/gtsam/symbolic/SymbolicFactorGraph.h
@@ -46,6 +46,12 @@ namespace gtsam {
     static std::pair<boost::shared_ptr<ConditionalType>, boost::shared_ptr<FactorType> >
       DefaultEliminate(const FactorGraphType& factors, const Ordering& keys) {
         return EliminateSymbolic(factors, keys); }
+    /// The default ordering generation function
+    static Ordering DefaultOrderingFunc(
+        const FactorGraphType& graph,
+        boost::optional<const VariableIndex&> variableIndex) {
+      return Ordering::Colamd(*variableIndex);
+    }
   };
 
   /* ************************************************************************* */