Improve docstrings and comments to describe getrf algo

Signed-off-by: Yuuichi Asahi <[email protected]>

batched/dense/impl/KokkosBatched_Getrf_Serial_Internal.hpp

@@ -126,9 +126,8 @@ KOKKOS_INLINE_FUNCTION int SerialGetrfInternalHost<Algo::Getrf::Unblocked>::invo
     // Use recursive code
     auto n1 = Kokkos::min(m, n) / 2;
 
-    //        [ A00 ]
-    // Factor [ --- ]
-    //        [ A10 ]
+    // Factor A0 = [[A00],
+    //              [A10]]
 
     // split A into two submatrices A = [A0, A1]
     auto A0    = Kokkos::subview(A, Kokkos::ALL, Kokkos::pair<int, int>(0, n1));
@@ -138,9 +137,8 @@ KOKKOS_INLINE_FUNCTION int SerialGetrfInternalHost<Algo::Getrf::Unblocked>::invo
 
     if (info == 0 && iinfo > 0) info = iinfo;
 
-    //                        [ A01 ]
-    // Apply interchanges to  [ --- ]
-    //                        [ A11 ]
+    // Apply interchanges to A1 = [[A01],
+    //                             [A11]]
 
     [[maybe_unused]] auto info_laswp = KokkosBatched::SerialLaswp<Direct::Forward>::invoke(ipiv0, A1);
 
@@ -202,7 +200,7 @@ KOKKOS_INLINE_FUNCTION int SerialGetrfInternalDevice<Algo::Getrf::Unblocked>::in
   if (m <= 0 || n <= 0) return 0;
 
   while (!stack.isEmpty()) {
-    // First of make a subview based on the current state
+    // Firstly, make a subview based on the current state
     int current[7];
     stack.pop(current);
 
@@ -280,13 +278,11 @@ KOKKOS_INLINE_FUNCTION int SerialGetrfInternalDevice<Algo::Getrf::Unblocked>::in
 
     } else if (state == 1) {
       // after first recursive call
-      //        [ A00 ]
-      // Factor [ --- ]
-      //        [ A10 ]
+      // Factor A0 = [[A00],
+      //              [A10]]
 
-      //                        [ A01 ]
-      // Apply interchanges to  [ --- ]
-      //                        [ A11 ]
+      // Apply interchanges to A1 = [[A01],
+      //                             [A11]]
       KokkosBatched::SerialLaswp<Direct::Forward>::invoke(ipiv0, A1);
 
       // Solve A00 * X = A01

batched/dense/src/KokkosBatched_Getrf.hpp

@@ -30,6 +30,20 @@ namespace KokkosBatched {
 ///   where P is a permutation matrix, L is lower triangular with unit
 ///   diagonal elements (lower trapezoidal if m > n), and U is upper
 ///   triangular (upper trapezoidal if m < n).
+///
+/// This is the recusive version of the algorithm. It divides the matrix
+/// into four submatrices:
+/// A = [[A00, A01],
+///      [A10, A11]]
+/// where A00 is a square matrix of size n0, A11 is a matrix of size n1 by n1
+/// with n0 = min(m, n) / 2 and n1 = n - n0.
+///
+/// This function calls itself to factorize A0 = [[A00],
+//                                                [A10]]
+/// do the swaps on A1 = [[A01],
+///                       [A11]]
+/// solve A01, update A11, then calls itself to factorize A11
+/// and do the swaps on A10.
 /// \tparam AViewType: Input type for the matrix, needs to be a 2D view
 /// \tparam PivViewType: Input type for the pivot indices, needs to be a 1D view
 ///