From d6c9bf45e3f4d6885157c9d050838a9fc1b35686 Mon Sep 17 00:00:00 2001
From: Alexis Montoison <alexis.montoison@polymtl.ca>
Date: Fri, 31 Jul 2020 22:58:54 -0400
Subject: [PATCH] Explain benefits of matrix free operators

---
 docs/src/matrix-free.md | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/docs/src/matrix-free.md b/docs/src/matrix-free.md
index c8a8ed50b..f3ff62f32 100644
--- a/docs/src/matrix-free.md
+++ b/docs/src/matrix-free.md
@@ -112,3 +112,9 @@ opJ = LinearOperator(Float64, 3, 2, false, false, v -> J(v), w -> Jᵀ(w), u ->
 
 lsmr(opJ, -F(xk))
 ```
+
+Note that preconditioners can be also implemented as abstract operators.
+For instance, we could compute the Cholesky factorization of $M$ and $N$ and create linear operators that perform the forward and backsolves.
+
+Krylov methods combined with matrix free operators allow to reduce computation time and memory requirements considerably by avoiding building and storing the system matrix.
+In the field of partial differential equations, the implementation of high-performance matrix free operators and assembly free preconditioning is a subject of active research.