diff --git a/internal/ceres/cxsparse.h b/internal/ceres/cxsparse.h
index 14b1e01f9..2311f4a77 100644
--- a/internal/ceres/cxsparse.h
+++ b/internal/ceres/cxsparse.h
@@ -133,6 +133,7 @@ class CXSparse {
   int scratch_size_;
 };
 
+// Sparse cholesky factorization using CXSparse.
 class CXSparseCholesky : public SparseCholesky {
  public:
   // Factory
diff --git a/internal/ceres/linear_solver.h b/internal/ceres/linear_solver.h
index abd630fae..5f996a140 100644
--- a/internal/ceres/linear_solver.h
+++ b/internal/ceres/linear_solver.h
@@ -69,9 +69,15 @@ enum LinearSolverTerminationType {
   LINEAR_SOLVER_FATAL_ERROR
 };
 
+// This enum controls the fill-reducing ordering a sparse linear
+// algebra library should use before computing a sparse factorization
+// (usually Cholesky).
 enum OrderingType {
-  NATURAL,
-  AMD
+  NATURAL, // Do not re-order the matrix. This is useful when the
+           // matrix has been ordered using a fill-reducing ordering
+           // already.
+  AMD      // Use the Approximate Minimum Degree algorithm to re-order
+           // the matrix.
 };
 
 class LinearOperator;
diff --git a/internal/ceres/sparse_cholesky.h b/internal/ceres/sparse_cholesky.h
index 7c2ba9dbe..07d328d27 100644
--- a/internal/ceres/sparse_cholesky.h
+++ b/internal/ceres/sparse_cholesky.h
@@ -40,8 +40,8 @@
 namespace ceres {
 namespace internal {
 
-// An interface abstracts away the internal details of various sparse
-// linear algebra libraries and offers a simple API for solving
+// An interface that abstracts away the internal details of various
+// sparse linear algebra libraries and offers a simple API for solving
 // symmetric positive definite linear systems using a sparse Cholesky
 // factorization.
 //