An implementation of USYMLQR

docs/make.jl

@@ -38,6 +38,7 @@ makedocs(
                            "MINARES" => "examples/minares.md",
                            "TriCG" => "examples/tricg.md",
                            "TriMR" => "examples/trimr.md",
+                           "USYMLQR" => "examples/usymlqr.md",
                            "BICGSTAB" => "examples/bicgstab.md",
                            "DQGMRES" => "examples/dqgmres.md",
                            "BLOCK-GMRES" => "examples/block_gmres.md",

docs/src/api.md

@@ -32,6 +32,7 @@ DqgmresSolver
 GmresSolver
 UsymlqSolver
 UsymqrSolver
+UsymlqrSolver
 TricgSolver
 TrimrSolver
 TrilqrSolver

docs/src/examples/usymlqr.md

@@ -0,0 +1,25 @@
+```@example usymlqr
+using Krylov, LinearOperators
+using LinearAlgebra, Printf, SparseArrays
+
+# Identity matrix.
+eye(n::Int) = sparse(1.0 * I, n, n)
+
+# Saddle-point systems
+n = m = 5
+A = [2^(i/j)*j + (-1)^(i-j) * n*(i-1) for i = 1:n, j = 1:n]
+b = ones(n)
+D = diagm(0 => [2.0 * i for i = 1:n])
+m, n = size(A)
+c = -b
+
+# [D   A] [x] = [b]
+# [Aᴴ  0] [y]   [c]
+opH = BlockDiagonalOperator(inv(D), eye(n))
+(x, y, stats) = usymlqr(A, b, c, M=D, ldiv=true)
+K = [D A; A' zeros(n,n)]
+B = [b; c]
+r = B - K * [x; y]
+resid = sqrt(dot(r, opH * r))
+@printf("USYMLQR: Relative residual: %8.1e\n", resid)
+```

docs/src/matrix_free.md

@@ -40,7 +40,7 @@ Some methods only require `A * v` products, whereas other ones also require `A'
 | CG, CR, CAR                                     | CGLS, CRLS, CGNE, CRMR                   |
 | SYMMLQ, CG-LANCZOS, MINRES, MINRES-QLP, MINARES | LSLQ, LSQR, LSMR, LNLQ, CRAIG, CRAIGMR   |
 | DIOM, FOM, DQGMRES, GMRES, FGMRES, BLOCK-GMRES  | BiLQ, QMR, BiLQR, USYMLQ, USYMQR, TriLQR |
-| CGS, BICGSTAB                                   | TriCG, TriMR                             |
+| CGS, BICGSTAB                                   | TriCG, TriMR, USYMLQR                    |
 | CG-LANCZOS-SHIFT                                | CGLS-LANCZOS-SHIFT                       |
 
 !!! info

docs/src/preconditioners.md

@@ -111,7 +111,7 @@ Methods concerned: [`CGNE`](@ref cgne), [`CRMR`](@ref crmr), [`LNLQ`](@ref lnlq)
 
 ### Saddle-point and symmetric quasi-definite systems
 
-[`TriCG`](@ref tricg) and [`TriMR`](@ref trimr) can take advantage of the structure of Hermitian systems $Kz = d$ with the 2x2 block structure
+[`TriCG`](@ref tricg), [`TriMR`](@ref trimr) and [`USYMLQR`](@ref usymlqr) can take advantage of the structure of Hermitian systems $Kz = d$ with the 2x2 block structure
 ```math
   \begin{bmatrix} \tau E & \phantom{-}A \\ A^H & \nu F \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} b \\ c \end{bmatrix},
 ```

docs/src/processes.md

@@ -294,7 +294,7 @@ T_{k,k+1} =
 
 The function [`saunders_simon_yip`](@ref saunders_simon_yip(::Any, ::AbstractVector{FC}, ::AbstractVector{FC}, ::Int) where FC <: (Union{Complex{T}, T} where T <: AbstractFloat)) returns $V_{k+1}$, $\beta_1$, $T_{k+1,k}$, $U_{k+1}$, $\bar{\gamma}_1$ and $T_{k,k+1}^H$.
 
-Related methods: [`USYMLQ`](@ref usymlq), [`USYMQR`](@ref usymqr), [`TriLQR`](@ref trilqr), [`TriCG`](@ref tricg) and [`TriMR`](@ref trimr).
+Related methods: [`USYMLQ`](@ref usymlq), [`USYMQR`](@ref usymqr), [`USYMLQR`](@ref usymlqr), [`TriLQR`](@ref trilqr), [`TriCG`](@ref tricg) and [`TriMR`](@ref trimr).
 
 !!! note
     The Saunders-Simon-Yip is equivalent to the block-Lanczos process applied to $\begin{bmatrix} 0 & A \\ A^H & 0 \end{bmatrix}$ with initial matrix $\begin{bmatrix} b & 0 \\ 0 & c \end{bmatrix}$.

docs/src/solvers/sp_sqd.md

@@ -15,3 +15,10 @@ tricg!
 trimr
 trimr!
 ```
+
+## USYMLQR
+
+```@docs
+usymlqr!
+usymlqr
+```

docs/src/storage.md

@@ -93,9 +93,9 @@ Each table summarizes the storage requirements of Krylov methods recommended to
 
 #### Saddle-point and Hermitian quasi-definite systems
 
-| Methods  | [`TriCG`](@ref tricg) | [`TriMR`](@ref trimr) |
-|:--------:|:---------------------:|:---------------------:|
-| Storage  | $6n + 6m$             | $8n + 8m$             |
+| Methods  | [`TriCG`](@ref tricg) | [`TriMR`](@ref trimr) | [`USYMLQR`](@ref usymlqr) |
+|:--------:|:---------------------:|:---------------------:|:-------------------------:|
+| Storage  | $6n + 6m$             | $8n + 8m$             | $7n + 6m$                 |
 
 #### Generalized saddle-point and non-Hermitian partitioned systems
 

src/Krylov.jl

@@ -34,6 +34,7 @@ include("gpmr.jl")
 
 include("usymlq.jl")
 include("usymqr.jl")
+include("usymlqr.jl")
 include("tricg.jl")
 include("trimr.jl")
 include("trilqr.jl")

src/block_krylov_solvers.jl

@@ -8,14 +8,14 @@ const BLOCK_KRYLOV_SOLVERS = Dict(:block_gmres => :BlockGmresSolver)
 abstract type BlockKrylovSolver{T,FC,SV,SM} end
 
 """
-Type for storing the vectors required by the in-place version of BLOCK-GMRES.
+Workspace for the in-place version of BLOCK-GMRES.
 
-The outer constructors
+The outer constructors:
 
     solver = BlockGmresSolver(m, n, p, memory, SV, SM)
     solver = BlockGmresSolver(A, B, memory = 5)
 
-may be used in order to create these vectors.
+can be used to initialize this workspace.
 `memory` is set to `div(n,p)` if the value given is larger than `div(n,p)`.
 `memory` is an optional argument in the second constructor.
 """

src/krylov_solve.jl

@@ -23,40 +23,41 @@ function solve! end
 
 # Krylov methods
 for (workspace, krylov, args, def_args, optargs, def_optargs, kwargs, def_kwargs) in [
-  (:LsmrSolver     , :lsmr            , args_lsmr            , def_args_lsmr            , ()                , ()                    , kwargs_lsmr            , def_kwargs_lsmr      )
-  (:CgsSolver      , :cgs             , args_cgs             , def_args_cgs             , optargs_cgs       , def_optargs_cgs       , kwargs_cgs             , def_kwargs_cgs       )
-  (:UsymlqSolver   , :usymlq          , args_usymlq          , def_args_usymlq          , optargs_usymlq    , def_optargs_usymlq    , kwargs_usymlq          , def_kwargs_usymlq    )
-  (:LnlqSolver     , :lnlq            , args_lnlq            , def_args_lnlq            , ()                , ()                    , kwargs_lnlq            , def_kwargs_lnlq      )
-  (:BicgstabSolver , :bicgstab        , args_bicgstab        , def_args_bicgstab        , optargs_bicgstab  , def_optargs_bicgstab  , kwargs_bicgstab        , def_kwargs_bicgstab  )
-  (:CrlsSolver     , :crls            , args_crls            , def_args_crls            , ()                , ()                    , kwargs_crls            , def_kwargs_crls      )
-  (:LsqrSolver     , :lsqr            , args_lsqr            , def_args_lsqr            , ()                , ()                    , kwargs_lsqr            , def_kwargs_lsqr      )
-  (:MinresSolver   , :minres          , args_minres          , def_args_minres          , optargs_minres    , def_optargs_minres    , kwargs_minres          , def_kwargs_minres    )
-  (:MinaresSolver  , :minares         , args_minares         , def_args_minares         , optargs_minares   , def_optargs_minares   , kwargs_minares         , def_kwargs_minares   )
-  (:CgneSolver     , :cgne            , args_cgne            , def_args_cgne            , ()                , ()                    , kwargs_cgne            , def_kwargs_cgne      )
-  (:DqgmresSolver  , :dqgmres         , args_dqgmres         , def_args_dqgmres         , optargs_dqgmres   , def_optargs_dqgmres   , kwargs_dqgmres         , def_kwargs_dqgmres   )
-  (:SymmlqSolver   , :symmlq          , args_symmlq          , def_args_symmlq          , optargs_symmlq    , def_optargs_symmlq    , kwargs_symmlq          , def_kwargs_symmlq    )
-  (:TrimrSolver    , :trimr           , args_trimr           , def_args_trimr           , optargs_trimr     , def_optargs_trimr     , kwargs_trimr           , def_kwargs_trimr     )
-  (:UsymqrSolver   , :usymqr          , args_usymqr          , def_args_usymqr          , optargs_usymqr    , def_optargs_usymqr    , kwargs_usymqr          , def_kwargs_usymqr    )
-  (:BilqrSolver    , :bilqr           , args_bilqr           , def_args_bilqr           , optargs_bilqr     , def_optargs_bilqr     , kwargs_bilqr           , def_kwargs_bilqr     )
-  (:CrSolver       , :cr              , args_cr              , def_args_cr              , optargs_cr        , def_optargs_cr        , kwargs_cr              , def_kwargs_cr        )
-  (:CarSolver      , :car             , args_car             , def_args_car             , optargs_car       , def_optargs_car       , kwargs_car             , def_kwargs_car       )
-  (:CraigmrSolver  , :craigmr         , args_craigmr         , def_args_craigmr         , ()                , ()                    , kwargs_craigmr         , def_kwargs_craigmr   )
-  (:TricgSolver    , :tricg           , args_tricg           , def_args_tricg           , optargs_tricg     , def_optargs_tricg     , kwargs_tricg           , def_kwargs_tricg     )
-  (:CraigSolver    , :craig           , args_craig           , def_args_craig           , ()                , ()                    , kwargs_craig           , def_kwargs_craig     )
-  (:DiomSolver     , :diom            , args_diom            , def_args_diom            , optargs_diom      , def_optargs_diom      , kwargs_diom            , def_kwargs_diom      )
-  (:LslqSolver     , :lslq            , args_lslq            , def_args_lslq            , ()                , ()                    , kwargs_lslq            , def_kwargs_lslq      )
-  (:TrilqrSolver   , :trilqr          , args_trilqr          , def_args_trilqr          , optargs_trilqr    , def_optargs_trilqr    , kwargs_trilqr          , def_kwargs_trilqr    )
-  (:CrmrSolver     , :crmr            , args_crmr            , def_args_crmr            , ()                , ()                    , kwargs_crmr            , def_kwargs_crmr      )
-  (:CgSolver       , :cg              , args_cg              , def_args_cg              , optargs_cg        , def_optargs_cg        , kwargs_cg              , def_kwargs_cg        )
-  (:CglsSolver     , :cgls            , args_cgls            , def_args_cgls            , ()                , ()                    , kwargs_cgls            , def_kwargs_cgls      )
-  (:CgLanczosSolver, :cg_lanczos      , args_cg_lanczos      , def_args_cg_lanczos      , optargs_cg_lanczos, def_optargs_cg_lanczos, kwargs_cg_lanczos      , def_kwargs_cg_lanczos)
-  (:BilqSolver     , :bilq            , args_bilq            , def_args_bilq            , optargs_bilq      , def_optargs_bilq      , kwargs_bilq            , def_kwargs_bilq      )
-  (:MinresQlpSolver, :minres_qlp      , args_minres_qlp      , def_args_minres_qlp      , optargs_minres_qlp, def_optargs_minres_qlp, kwargs_minres_qlp      , def_kwargs_minres_qlp)
-  (:QmrSolver      , :qmr             , args_qmr             , def_args_qmr             , optargs_qmr       , def_optargs_qmr       , kwargs_qmr             , def_kwargs_qmr       )
-  (:GmresSolver    , :gmres           , args_gmres           , def_args_gmres           , optargs_gmres     , def_optargs_gmres     , kwargs_gmres           , def_kwargs_gmres     )
-  (:FgmresSolver   , :fgmres          , args_fgmres          , def_args_fgmres          , optargs_fgmres    , def_optargs_fgmres    , kwargs_fgmres          , def_kwargs_fgmres    )
-  (:FomSolver      , :fom             , args_fom             , def_args_fom             , optargs_fom       , def_optargs_fom       , kwargs_fom             , def_kwargs_fom       )
-  (:GpmrSolver     , :gpmr            , args_gpmr            , def_args_gpmr            , optargs_gpmr      , def_optargs_gpmr      , kwargs_gpmr            , def_kwargs_gpmr      )
+  (:LsmrSolver     , :lsmr      , args_lsmr      , def_args_lsmr      , ()                , ()                    , kwargs_lsmr      , def_kwargs_lsmr      )
+  (:CgsSolver      , :cgs       , args_cgs       , def_args_cgs       , optargs_cgs       , def_optargs_cgs       , kwargs_cgs       , def_kwargs_cgs       )
+  (:UsymlqSolver   , :usymlq    , args_usymlq    , def_args_usymlq    , optargs_usymlq    , def_optargs_usymlq    , kwargs_usymlq    , def_kwargs_usymlq    )
+  (:LnlqSolver     , :lnlq      , args_lnlq      , def_args_lnlq      , ()                , ()                    , kwargs_lnlq      , def_kwargs_lnlq      )
+  (:BicgstabSolver , :bicgstab  , args_bicgstab  , def_args_bicgstab  , optargs_bicgstab  , def_optargs_bicgstab  , kwargs_bicgstab  , def_kwargs_bicgstab  )
+  (:CrlsSolver     , :crls      , args_crls      , def_args_crls      , ()                , ()                    , kwargs_crls      , def_kwargs_crls      )
+  (:LsqrSolver     , :lsqr      , args_lsqr      , def_args_lsqr      , ()                , ()                    , kwargs_lsqr      , def_kwargs_lsqr      )
+  (:MinresSolver   , :minres    , args_minres    , def_args_minres    , optargs_minres    , def_optargs_minres    , kwargs_minres    , def_kwargs_minres    )
+  (:MinaresSolver  , :minares   , args_minares   , def_args_minares   , optargs_minares   , def_optargs_minares   , kwargs_minares   , def_kwargs_minares   )
+  (:CgneSolver     , :cgne      , args_cgne      , def_args_cgne      , ()                , ()                    , kwargs_cgne      , def_kwargs_cgne      )
+  (:DqgmresSolver  , :dqgmres   , args_dqgmres   , def_args_dqgmres   , optargs_dqgmres   , def_optargs_dqgmres   , kwargs_dqgmres   , def_kwargs_dqgmres   )
+  (:SymmlqSolver   , :symmlq    , args_symmlq    , def_args_symmlq    , optargs_symmlq    , def_optargs_symmlq    , kwargs_symmlq    , def_kwargs_symmlq    )
+  (:TrimrSolver    , :trimr     , args_trimr     , def_args_trimr     , optargs_trimr     , def_optargs_trimr     , kwargs_trimr     , def_kwargs_trimr     )
+  (:UsymqrSolver   , :usymqr    , args_usymqr    , def_args_usymqr    , optargs_usymqr    , def_optargs_usymqr    , kwargs_usymqr    , def_kwargs_usymqr    )
+  (:BilqrSolver    , :bilqr     , args_bilqr     , def_args_bilqr     , optargs_bilqr     , def_optargs_bilqr     , kwargs_bilqr     , def_kwargs_bilqr     )
+  (:CrSolver       , :cr        , args_cr        , def_args_cr        , optargs_cr        , def_optargs_cr        , kwargs_cr        , def_kwargs_cr        )
+  (:CarSolver      , :car       , args_car       , def_args_car       , optargs_car       , def_optargs_car       , kwargs_car       , def_kwargs_car       )
+  (:CraigmrSolver  , :craigmr   , args_craigmr   , def_args_craigmr   , ()                , ()                    , kwargs_craigmr   , def_kwargs_craigmr   )
+  (:TricgSolver    , :tricg     , args_tricg     , def_args_tricg     , optargs_tricg     , def_optargs_tricg     , kwargs_tricg     , def_kwargs_tricg     )
+  (:CraigSolver    , :craig     , args_craig     , def_args_craig     , ()                , ()                    , kwargs_craig     , def_kwargs_craig     )
+  (:DiomSolver     , :diom      , args_diom      , def_args_diom      , optargs_diom      , def_optargs_diom      , kwargs_diom      , def_kwargs_diom      )
+  (:LslqSolver     , :lslq      , args_lslq      , def_args_lslq      , ()                , ()                    , kwargs_lslq      , def_kwargs_lslq      )
+  (:TrilqrSolver   , :trilqr    , args_trilqr    , def_args_trilqr    , optargs_trilqr    , def_optargs_trilqr    , kwargs_trilqr    , def_kwargs_trilqr    )
+  (:CrmrSolver     , :crmr      , args_crmr      , def_args_crmr      , ()                , ()                    , kwargs_crmr      , def_kwargs_crmr      )
+  (:CgSolver       , :cg        , args_cg        , def_args_cg        , optargs_cg        , def_optargs_cg        , kwargs_cg        , def_kwargs_cg        )
+  (:CglsSolver     , :cgls      , args_cgls      , def_args_cgls      , ()                , ()                    , kwargs_cgls      , def_kwargs_cgls      )
+  (:CgLanczosSolver, :cg_lanczos, args_cg_lanczos, def_args_cg_lanczos, optargs_cg_lanczos, def_optargs_cg_lanczos, kwargs_cg_lanczos, def_kwargs_cg_lanczos)
+  (:BilqSolver     , :bilq      , args_bilq      , def_args_bilq      , optargs_bilq      , def_optargs_bilq      , kwargs_bilq      , def_kwargs_bilq      )
+  (:MinresQlpSolver, :minres_qlp, args_minres_qlp, def_args_minres_qlp, optargs_minres_qlp, def_optargs_minres_qlp, kwargs_minres_qlp, def_kwargs_minres_qlp)
+  (:QmrSolver      , :qmr       , args_qmr       , def_args_qmr       , optargs_qmr       , def_optargs_qmr       , kwargs_qmr       , def_kwargs_qmr       )
+  (:GmresSolver    , :gmres     , args_gmres     , def_args_gmres     , optargs_gmres     , def_optargs_gmres     , kwargs_gmres     , def_kwargs_gmres     )
+  (:FgmresSolver   , :fgmres    , args_fgmres    , def_args_fgmres    , optargs_fgmres    , def_optargs_fgmres    , kwargs_fgmres    , def_kwargs_fgmres    )
+  (:FomSolver      , :fom       , args_fom       , def_args_fom       , optargs_fom       , def_optargs_fom       , kwargs_fom       , def_kwargs_fom       )
+  (:GpmrSolver     , :gpmr      , args_gpmr      , def_args_gpmr      , optargs_gpmr      , def_optargs_gpmr      , kwargs_gpmr      , def_kwargs_gpmr      )
+  (:UsymlqrSolver  , :usymlqr   , args_usymlqr   , def_args_usymlqr   , optargs_usymlqr   , def_optargs_usymlqr   , kwargs_usymlqr   , def_kwargs_usymlqr   )
   (:CgLanczosShiftSolver  , :cg_lanczos_shift  , args_cg_lanczos_shift  , def_args_cg_lanczos_shift  , (), (), kwargs_cg_lanczos_shift  , def_kwargs_cg_lanczos_shift  )
   (:CglsLanczosShiftSolver, :cgls_lanczos_shift, args_cgls_lanczos_shift, def_args_cgls_lanczos_shift, (), (), kwargs_cgls_lanczos_shift, def_kwargs_cgls_lanczos_shift)
 ]