Look-ahead Lanczos Biorthogonalization

Project.toml

@@ -3,6 +3,7 @@ uuid = "42fd0dbc-a981-5370-80f2-aaf504508153"
 version = "0.9.2"
 
 [deps]
+BlockDiagonals = "0a1fb500-61f7-11e9-3c65-f5ef3456f9f0"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
@@ -12,3 +13,4 @@ SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 [compat]
 RecipesBase = "0.6, 0.7, 0.8, 1.0"
 julia = "1.3"
+BlockDiagonals = "0.1"

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"
+			"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 +1,17 @@
+# [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)) [^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 (Freund, 1994),
+where the look-ahead Lanczos process is implemented as a two-term coupled recurrence.
+
+## Usage
+
+```@docs
+IterativeSolvers.LookAheadLanczosDecomp
+IterativeSolvers.LookAheadLanczosDecompLog
+```

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/IterativeSolvers.jl

@@ -13,6 +13,10 @@ include("history.jl")
 # Factorizations
 include("hessenberg.jl")
 
+# Krylov subspace methods
+include("limited_memory_matrices.jl")
+include("lal.jl")
+
 # Linear solvers
 include("stationary.jl")
 include("stationary_sparse.jl")