Add documentation

Author: Pawel Latawiec

docs/make.jl

@@ -22,7 +22,8 @@ makedocs(
 			"BiCGStab(l)" => "linear_systems/bicgstabl.md",
 			"IDR(s)" => "linear_systems/idrs.md",
 			"Restarted GMRES" => "linear_systems/gmres.md",
-			"QMR" => "linear_systems/qmr.md",
+			"QMR" => "linear_systems/qmr.md"
+			"LALQMR" => "linear_systems/lalqmr.md"
 			"LSMR" => "linear_systems/lsmr.md",
 			"LSQR" => "linear_systems/lsqr.md",
 			"Stationary methods" => "linear_systems/stationary.md",

docs/src/linear_systems/lalqmr.md

@@ -0,0 +1,28 @@
+# [LALQMR](@id LALQMR)
+
+Look-ahead Lanczos Quasi-minimal Residual (LALQMR) is the look-ahead variant of [QMR](@ref) for solving $Ax = b$ approximately for $x$ where $A$ is a linear operator and $b$ the right-hand side vector. $A$ may be non-symmetric. The Krylov subspace is generated via the [Look-ahead Lanczos process](@LAL)
+
+## Usage
+
+```@docs
+lalqmr
+lalqmr!
+```
+
+## Implementation details
+This implementation of LALQMR follows [^Freund1994] and [^Freund1993], where we generate the Krylov subspace via Lanczos bi-orthogonalization based on the matrix `A` and its transpose. In the regular Lanczos process, we may encounter singularities during the construction of the Krylov basis. The look-ahead process avoids this by building blocks instead, avoiding the singularities. Therefore, the LALQMR technique is well-suited towards problematic linear systems. Typically, most systems will not have blocks of size larger than 4 made, and most iterations are "regular", or blocks of size 1 [^Freund1994].
+
+For more detail on the implementation see the original paper [^Freund1994]. This implementation follows the paper, with the exception that if a block of maximum size is reached during the Lanczos process, the block is closed instead of restarted.
+
+For a solution vector of size `n`, the memory allocated will be `(12 + max_memory * 5) * n`.
+
+!!! tip
+    LALQMR can be used as an [iterator](@ref Iterators) via `lalqmr_iterable!`. This makes it possible to access the next, current, and previous Krylov basis vectors during the iteration.
+
+## References
+[^Freund1993]:
+Freund, R. W., Gutknecht, M. H., & Nachtigal, N. M. (1993). An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices. SIAM Journal on Scientific Computing, 14(1), 137–158. https://doi.org/10.1137/0914009
+[^Freund1994]:
+Freund, R. W., & Nachtigal, N. M. (1994). An Implementation of the QMR Method Based on Coupled Two-Term Recurrences. SIAM Journal on Scientific Computing, 15(2), 313–337. https://doi.org/10.1137/0915022
+[^Freund1990]:
+Freund, W. R., & Nachtigal, N. M. (1990). QMR : for a Quasi-Minimal Residual Linear Method Systems. (December).

docs/src/linear_systems/qmr.md

@@ -1,6 +1,6 @@
 # [QMR](@id QMR)
 
-QMR is a short-recurrence version of GMRES for solving $Ax = b$ approximately for $x$ where $A$ is a linear operator and $b$ the right-hand side vector. $A$ may be non-symmetric.
+QMR is a short-recurrence version of GMRES for solving $Ax = b$ approximately for $x$ where $A$ is a linear operator and $b$ the right-hand side vector. $A$ may be non-symmetric. This is a simplified version of [LALQMR](@ref)
 
 ## Usage