Documentation

Author: Pawel Latawiec

docs/make.jl

@@ -22,9 +22,11 @@ makedocs(
 			"BiCGStab(l)" => "linear_systems/bicgstabl.md",
 			"IDR(s)" => "linear_systems/idrs.md",
 			"Restarted GMRES" => "linear_systems/gmres.md",
+			"QMR" => "linear_systems/qmr.md"
 			"LSMR" => "linear_systems/lsmr.md",
 			"LSQR" => "linear_systems/lsqr.md",
 			"Stationary methods" => "linear_systems/stationary.md"
+			"LAL" => "linear_systems/lal.md"
 		],
 		"Eigenproblems" => [
 			"Power method" => "eigenproblems/power_method.md",

docs/src/linear_systems/lal.md

@@ -0,0 +1,26 @@
+# [Look-Ahead Lanczos Bi-Orthogonalization](@id LAL)
+
+The Look-ahead Lanczos Bi-orthogonalization process is a generalization of the Lanczos
+bi-orthoganilization (as used in, for instance, [QMR](@ref)). The look-ahead process detects
+(near-)singularities during the construction of the Lanczos iterates, known as break-down,
+and skips over them. This is particularly advantageous for linear systems with large
+null-spaces or eigenvalues close to 0.
+
+We provide an iterator interface for the Lanczos decomposition, following [^Freund1994],
+where the look-ahead Lanczos process is implemented as a two-term coupled recurrence.
+
+## Usage
+
+```@docs
+LookAheadLanczosDecomp
+```
+
+## 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