Fix LAL docs

Author: Pawel Latawiec

docs/make.jl

@@ -22,10 +22,10 @@ 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",
 			"LSMR" => "linear_systems/lsmr.md",
 			"LSQR" => "linear_systems/lsqr.md",
-			"Stationary methods" => "linear_systems/stationary.md"
+			"Stationary methods" => "linear_systems/stationary.md",
 			"LAL" => "linear_systems/lal.md"
 		],
 		"Eigenproblems" => [

docs/src/linear_systems/lal.md

@@ -1,26 +1,17 @@
 # [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
+bi-orthoganilization (as used in, for instance, [QMR](@ref)) [^Freund1993], [^Freund1994]. 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],
+We provide an iterator interface for the Lanczos decomposition (Freund, 1994),
 where the look-ahead Lanczos process is implemented as a two-term coupled recurrence.
 
 ## Usage
 
 ```@docs
-LookAheadLanczosDecomp
+IterativeSolvers.LookAheadLanczosDecomp
+IterativeSolvers.LookAheadLanczosDecompLog
 ```
-
-## 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

docs/src/linear_systems/qmr.md

@@ -10,17 +10,9 @@ qmr!
 ```
 
 ## Implementation details
-QMR exploits the tridiagonal structure of the Hessenberg matrix. Although QMR is similar to GMRES, where instead of using the Arnoldi process, a pair of biorthogonal vector spaces $V$ and $W$ is constructed via the Lanczos process. It requires that the adjoint of $A$ `adjoint(A)` be available.
+QMR exploits the tridiagonal structure of the Hessenberg matrix. Although QMR is similar to GMRES, where instead of using the Arnoldi process, a pair of biorthogonal vector spaces $V$ and $W$ is constructed via the Lanczos process. It requires that the adjoint of $A$ `adjoint(A)` be available [^Saad2003], [^Freund1990].
 
 QMR enables the computation of $V$ and $W$ via a three-term recurrence. A three-term recurrence for the projection onto the solution vector can also be constructed from these values, using the portion of the last column of the Hessenberg matrix. Therefore we pre-allocate only eight vectors.
 
-For more detail on the implementation see the original paper [^Freund1990] or [^Saad2003].
-
 !!! tip
     QMR can be used as an [iterator](@ref Iterators) via `qmr_iterable!`. This makes it possible to access the next, current, and previous Krylov basis vectors during the iteration.
-
-[^Saad2003]:
-    Saad, Y. (2003). Interactive method for sparse linear system.
-[^Freund1990]:
-    Freund, W. R., & Nachtigal, N. M. (1990). QMR : for a Quasi-Minimal
-    Residual Linear Method Systems. (December).

src/lal.jl

@@ -7,7 +7,7 @@ import BlockDiagonals
 """
     LookAheadLanczosDecompOptions
 
-Options for [`LookAheadLanczosDecomp`](@ref).
+Options for [`IterativeSolvers.LookAheadLanczosDecomp`](@ref).
 
 # Fields
 - `max_iter`: Maximum number of iterations
@@ -25,7 +25,7 @@ end
 """
     LookAheadLanczosDecompLog
 
-Log for [`LookAheadLanczosDecomp`](@ref). In particular, logs the sizes of blocks constructed in the P-Q and V-W sequences.
+Log for [`IterativeSolvers.LookAheadLanczosDecomp`](@ref). In particular, logs the sizes of blocks constructed in the P-Q and V-W sequences.
 """
 struct LookAheadLanczosDecompLog
     PQ_block_count::Dict{Int, Int}
@@ -115,7 +115,7 @@ end
 """
     LookAheadLanczosDecomp(A, v, w; kwargs...)
 
-Provides an iterable which constructs basis vectors for a Krylov subspace generated by `A` given by two initial vectors `v` and `w`. This implementation follows [^Freund1994], where a coupled two-term recurrence is used to generate both a V-W and a P-Q sequence. Following the reference, the Lanczos sequence is generated by `A` and `transpose(A)`.
+Provides an iterable which constructs basis vectors for a Krylov subspace generated by `A` given by two initial vectors `v` and `w` [^Freund1993]. This implementation follows [^Freund1994], where a coupled two-term recurrence is used to generate both a V-W and a P-Q sequence. Following the reference, the Lanczos sequence is generated by `A` and `transpose(A)`.
 
 # Arguments
 - `A`: Operator used to construct Krylov subspace
@@ -127,12 +127,13 @@ Provides an iterable which constructs basis vectors for a Krylov subspace genera
 - `max_iter=size(A, 2)`: Maximum number of iterations 
 - `max_block_size=4`: Maximum look-ahead block size to construct. Following [^Freund1994], it is rare for blocks to go beyond size 3. This pre-allocates the block storage for the computation to size `(max_block_size, length(v))`. If a block would be built that exceeds this size, the estimate of `norm(A)` is adjusted to allow the block to close.
 - `max_memory=4`: Maximum memory to store the sequence vectors. This may be greater than the block size.
-- `log=false`: Flag determining whether to log history in a [`LookAheadLanczosDecompLog`](@ref)
+- `log=false`: Flag determining whether to log history in a [`IterativeSolvers.LookAheadLanczosDecompLog`](@ref)
 - `verbose=false`: Flag determining verbosity of output during 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
 """