From a946feda72c74e27f1778f8a27e63772cc75bd37 Mon Sep 17 00:00:00 2001
From: Wolfgang Bangerth <bangerth@colostate.edu>
Date: Fri, 15 Jul 2022 14:40:58 -0600
Subject: [PATCH] Expand a discussion in step-58.

---
 doc/doxygen/references.bib       | 53 ++++++++++++++++++++++++++++++++
 examples/step-58/doc/results.dox | 24 +++++++++++++--
 2 files changed, 74 insertions(+), 3 deletions(-)

diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib
index 6ee00d582f..9ddbbef63a 100644
--- a/doc/doxygen/references.bib
+++ b/doc/doxygen/references.bib
@@ -611,6 +611,59 @@
 }
 
 
+% ------------------------------------
+% Step 58
+% ------------------------------------
+
+@article{Fraysse2005,
+  author = { Frayss\'e, Val\'erie and Giraud, Luc and Gratton, Serge and Langou, Julien },
+  title = { Algorithm 842: A set of GMRES routines for real and complex arithmetics on high performance computers },
+  journal = { ACM Trans. Math. Softw. },
+  year = { 2005 },
+  volume = { 31 },
+  issue = { 2 },
+  pages = { 228--238 },
+  doi = {10.1145/1067967.1067970},
+  url = {http://doi.org/10.1145/1067967.1067970},
+}
+
+
+@article{Day2001,
+  author = { Day, David and Heroux, Michael A. },
+  title = { Solving Complex-Valued Linear Systems via Equivalent Real Formulations },
+  journal = { SIAM J. Sci. Comput. },
+  year = { 2001 },
+  volume = { 23 },
+  issue = { 2 },
+  pages = { 480--498 },
+  doi = {10.1137/S1064827500372262},
+  url = {http://doi.org/10.1137/S1064827500372262},
+}
+
+@article{Axelsson2014,
+  author = { Axelsson, Owe and Neytcheva, Maya and Ahmad, Bashir },
+  title = { A comparison of iterative methods to solve complex valued linear algebraic systems },
+  journal = { Numer Algor },
+  year = { 2014 },
+  volume = { 66 },
+  issue = { 4 },
+  pages = { 811--841 },
+  doi = {10.1007/s11075-013-9764-1},
+  url = {http://doi.org/10.1007/s11075-013-9764-1},
+}
+
+@article{Liao2016,
+  author = { Liao, Li-Dan and Zhang, Guo-Feng },
+  title = { Preconditioning of complex linear systems from the Helmholtz equation },
+  journal = { Computers \&amp; Mathematics with Applications },
+  year = { 2016 },
+  volume = { 72 },
+  issue = { 9 },
+  pages = { 2473--2485 },
+  doi = {10.1016/j.camwa.2016.09.004},
+  url = {http://doi.org/10.1016/j.camwa.2016.09.004},
+}
+
 % ------------------------------------
 % Step 61
 % ------------------------------------
diff --git a/examples/step-58/doc/results.dox b/examples/step-58/doc/results.dox
index c4f515b360..0126023bc3 100644
--- a/examples/step-58/doc/results.dox
+++ b/examples/step-58/doc/results.dox
@@ -176,10 +176,28 @@ know that we could do far better:
   of these in other tutorial programs. In all cases, they need to be
   accompanied by good preconditioners. For the current case, one
   could in principle use GMRES -- a method that does not require
-  any specific properties of the matrix -- but would be better
-  advised to implement an iterative scheme that exploits the one
+  any specific properties of the matrix -- as the outer solver but
+  at least at the time of writing this sentence (in 2022), the
+  SolverGMRES class can only handle real-valued linear systems.
+  This can be overcome by implementing a variation of GMRES that
+  can deal with complex-valued matrices and vectors, see for
+  example
+  @cite Fraysse2005 .
+  Even better would be to implement an iterative scheme that exploits the one
   structural feature we know is true for this problem: That the matrix
-  is complex-symmetric (albeit not Hermitian).
+  is complex-symmetric (albeit not Hermitian), for which a literature
+  search would probably find schemes as well.
+
+- A different strategy towards iterative solvers would be to break the
+  linear system into a $2\times 2$ block system of real and imaginary
+  components, like we did in step-29. This would then enable using
+  real-valued iterative solvers on the outer level (e.g., the existing
+  GMRES implementation), but one would have to come up with preconditioners
+  that exploit the block structure. There is, again, literature on the
+  topic, of which we simply point out a non-representative sample:
+  @cite Axelsson2014 ,
+  @cite Day2001 ,
+  @cite Liao2016 .
 
 
 <h4> Boundary conditions </h4>
-- 
2.39.5