Add the size of each linear system in the docstring

src/bicgstab.jl

@@ -24,7 +24,7 @@ export bicgstab, bicgstab!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the square linear system Ax = b using BICGSTAB.
+Solve the square linear system Ax = b of size n using BICGSTAB.
 BICGSTAB requires two initial vectors `b` and `c`.
 The relation `bᴴc ≠ 0` must be satisfied and by default `c = b`.
 

src/bilq.jl

@@ -21,7 +21,7 @@ export bilq, bilq!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the square linear system Ax = b using BiLQ.
+Solve the square linear system Ax = b of size n using BiLQ.
 
 BiLQ is based on the Lanczos biorthogonalization process and requires two initial vectors `b` and `c`.
 The relation `bᴴc ≠ 0` must be satisfied and by default `c = b`.

src/bilqr.jl

@@ -27,8 +27,8 @@ Combine BiLQ and QMR to solve adjoint systems.
     [Aᴴ 0] [x]   [c]
 
 The relation `bᴴc ≠ 0` must be satisfied.
-BiLQ is used for solving primal system `Ax = b`.
-QMR is used for solving dual system `Aᴴy = c`.
+BiLQ is used for solving primal system `Ax = b` of size n.
+QMR is used for solving dual system `Aᴴy = c` of size n.
 
 An option gives the possibility of transferring from the BiLQ point to the
 BiCG point, when it exists. The transfer is based on the residual norm.

src/cg.jl

@@ -26,16 +26,15 @@ export cg, cg!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-The conjugate gradient method to solve the symmetric linear system Ax = b.
+The conjugate gradient method to solve the Hermitian linear system Ax = b of size n.
 
 The method does _not_ abort if A is not definite.
 
 A preconditioner M may be provided in the form of a linear operator and is
-assumed to be symmetric and positive definite.
+assumed to be Hermitian and positive definite.
 M also indicates the weighted norm in which residuals are measured.
 
-If `itmax=0`, the default number of iterations is set to `2 * n`,
-with `n = length(b)`.
+If `itmax=0`, the default number of iterations is set to `2 * n`.
 
 CG can be warm-started from an initial guess `x0` with
 

src/cg_lanczos.jl

@@ -23,14 +23,12 @@ export cg_lanczos, cg_lanczos!
 `FC` is `T` or `Complex{T}`.
 
 The Lanczos version of the conjugate gradient method to solve the
-symmetric linear system
-
-    Ax = b
+Hermitian linear system Ax = b of size n.
 
 The method does _not_ abort if A is not definite.
 
 A preconditioner M may be provided in the form of a linear operator and is
-assumed to be hermitian and positive definite.
+assumed to be Hermitian and positive definite.
 
 CG-LANCZOS can be warm-started from an initial guess `x0` with
 

src/cg_lanczos_shift.jl

@@ -27,12 +27,12 @@ export cg_lanczos_shift, cg_lanczos_shift!
 The Lanczos version of the conjugate gradient method to solve a family
 of shifted systems
 
-    (A + αI) x = b  (α = α₁, ..., αₙ)
+    (A + αI) x = b  (α = α₁, ..., αₚ)
 
-The method does _not_ abort if A + αI is not definite.
+of size n. The method does _not_ abort if A + αI is not definite.
 
 A preconditioner M may be provided in the form of a linear operator and is
-assumed to be hermitian and positive definite.
+assumed to be Hermitian and positive definite.
 
 The callback is called as `callback(solver)` and should return `true` if the main loop should terminate,
 and `false` otherwise.
@@ -41,11 +41,11 @@ and `false` otherwise.
 
 * `A`: a linear operator that models a Hermitian matrix of dimension n;
 * `b`: a vector of length n;
-* `shifts`: a vector of length nshifts.
+* `shifts`: a vector of length p.
 
 #### Output arguments
 
-* `x`: a vector of nshifts dense vectors, each one of length n;
+* `x`: a vector of p dense vectors, each one of length n;
 * `stats`: statistics collected on the run in a [`LanczosShiftStats`](@ref) structure.
 
 #### References

src/cgls.jl

@@ -42,7 +42,7 @@ Solve the regularized linear least-squares problem
 
     minimize ‖b - Ax‖₂² + λ‖x‖₂²
 
-using the Conjugate Gradient (CG) method, where λ ≥ 0 is a regularization
+of size n × m using the Conjugate Gradient (CG) method, where λ ≥ 0 is a regularization
 parameter. This method is equivalent to applying CG to the normal equations
 
     (AᴴA + λI) x = Aᴴb

src/cgne.jl

@@ -42,7 +42,7 @@ Solve the consistent linear system
 
     Ax + √λs = b
 
-using the Conjugate Gradient (CG) method, where λ ≥ 0 is a regularization
+of size n × m using the Conjugate Gradient (CG) method, where λ ≥ 0 is a regularization
 parameter. This method is equivalent to applying CG to the normal equations
 of the second kind
 

src/cgs.jl

@@ -19,7 +19,7 @@ export cgs, cgs!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the consistent linear system Ax = b using conjugate gradient squared algorithm.
+Solve the consistent linear system Ax = b of size n using CGS.
 CGS requires two initial vectors `b` and `c`.
 The relation `bᴴc ≠ 0` must be satisfied and by default `c = b`.
 

src/cr.jl

@@ -23,16 +23,16 @@ export cr, cr!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-A truncated version of Stiefel’s Conjugate Residual method to solve the symmetric linear system Ax = b or the least-squares problem min ‖b - Ax‖.
-The matrix A must be positive semi-definite.
+A truncated version of Stiefel’s Conjugate Residual method to solve the Hermitian linear system Ax = b
+of size n or the least-squares problem min ‖b - Ax‖ if A is singular.
+The matrix A must be Hermitian semi-definite.
 
-A preconditioner M may be provided in the form of a linear operator and is assumed to be symmetric and positive definite.
+A preconditioner M may be provided in the form of a linear operator and is assumed to be Hermitian and positive definite.
 M also indicates the weighted norm in which residuals are measured.
 
 In a linesearch context, 'linesearch' must be set to 'true'.
 
-If `itmax=0`, the default number of iterations is set to `2 * n`,
-with `n = length(b)`.
+If `itmax=0`, the default number of iterations is set to `2 * n`.
 
 CR can be warm-started from an initial guess `x0` with
 

src/craig.jl

@@ -47,7 +47,7 @@ Find the least-norm solution of the consistent linear system
 
     Ax + λ²y = b
 
-using the Golub-Kahan implementation of Craig's method, where λ ≥ 0 is a
+of size n × m using the Golub-Kahan implementation of Craig's method, where λ ≥ 0 is a
 regularization parameter. This method is equivalent to CGNE but is more
 stable.
 

src/craigmr.jl

@@ -40,7 +40,7 @@ Solve the consistent linear system
 
     Ax + λ²y = b
 
-using the CRAIGMR method, where λ ≥ 0 is a regularization parameter.
+of size n × m using the CRAIGMR method, where λ ≥ 0 is a regularization parameter.
 This method is equivalent to applying the Conjugate Residuals method
 to the normal equations of the second kind
 

src/crls.jl

@@ -34,8 +34,8 @@ Solve the linear least-squares problem
 
     minimize ‖b - Ax‖₂² + λ‖x‖₂²
 
-using the Conjugate Residuals (CR) method. This method is equivalent to
-applying MINRES to the normal equations
+of size n × m using the Conjugate Residuals (CR) method.
+This method is equivalent to applying MINRES to the normal equations
 
     (AᴴA + λI) x = Aᴴb.
 

src/crmr.jl

@@ -40,7 +40,7 @@ Solve the consistent linear system
 
     Ax + √λs = b
 
-using the Conjugate Residual (CR) method, where λ ≥ 0 is a regularization
+of size n × m using the Conjugate Residual (CR) method, where λ ≥ 0 is a regularization
 parameter. This method is equivalent to applying CR to the normal equations
 of the second kind
 

src/diom.jl

@@ -20,7 +20,7 @@ export diom, diom!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the consistent linear system Ax = b using DIOM.
+Solve the consistent linear system Ax = b of size n using DIOM.
 
 DIOM only orthogonalizes the new vectors of the Krylov basis against the `memory` most recent vectors.
 If CG is well defined on `Ax = b` and `memory = 2`, DIOM is theoretically equivalent to CG.

src/dqgmres.jl

@@ -20,7 +20,7 @@ export dqgmres, dqgmres!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the consistent linear system Ax = b using DQGMRES.
+Solve the consistent linear system Ax = b of size n using DQGMRES.
 
 DQGMRES algorithm is based on the incomplete Arnoldi orthogonalization process
 and computes a sequence of approximate solutions with the quasi-minimal residual property.

src/fgmres.jl

@@ -20,7 +20,7 @@ export fgmres, fgmres!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the linear system Ax = b using FGMRES.
+Solve the linear system Ax = b of size n using FGMRES.
 
 FGMRES computes a sequence of approximate solutions with minimum residual.
 FGMRES is a variant of GMRES that allows changes in the right preconditioner at each iteration.

src/fom.jl

@@ -20,7 +20,7 @@ export fom, fom!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the linear system Ax = b using FOM.
+Solve the linear system Ax = b of size n using FOM.
 
 FOM algorithm is based on the Arnoldi process and a Galerkin condition.
 

src/gmres.jl

@@ -20,7 +20,7 @@ export gmres, gmres!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the linear system Ax = b using GMRES.
+Solve the linear system Ax = b of size n using GMRES.
 
 GMRES algorithm is based on the Arnoldi process and computes a sequence of approximate solutions with the minimum residual.
 

src/gpmr.jl

@@ -27,7 +27,7 @@ GPMR solves the unsymmetric partitioned linear system
     [ λI   A ] [ x ] = [ b ]
     [  B  μI ] [ y ]   [ c ],
 
-where λ and μ are real or complex numbers.
+of size (n+m) × (n+m) where λ and μ are real or complex numbers.
 `A` can have any shape and `B` has the shape of `Aᴴ`.
 `A`, `B`, `b` and `c` must be all nonzero.
 

src/lnlq.jl

@@ -38,7 +38,7 @@ Find the least-norm solution of the consistent linear system
 
     Ax + λ²y = b
 
-using the LNLQ method, where λ ≥ 0 is a regularization parameter.
+of size n × m using the LNLQ method, where λ ≥ 0 is a regularization parameter.
 
 For a system in the form Ax = b, LNLQ method is equivalent to applying
 SYMMLQ to AAᴴy = b and recovering x = Aᴴy but is more stable.

src/lslq.jl

@@ -38,7 +38,7 @@ Solve the regularized linear least-squares problem
 
     minimize ‖b - Ax‖₂² + λ²‖x‖₂²
 
-using the LSLQ method, where λ ≥ 0 is a regularization parameter.
+of size n × m using the LSLQ method, where λ ≥ 0 is a regularization parameter.
 LSLQ is formally equivalent to applying SYMMLQ to the normal equations
 
     (AᴴA + λ²I) x = Aᴴb

src/lsmr.jl

@@ -43,7 +43,7 @@ Solve the regularized linear least-squares problem
 
     minimize ‖b - Ax‖₂² + λ²‖x‖₂²
 
-using the LSMR method, where λ ≥ 0 is a regularization parameter.
+of size n × m using the LSMR method, where λ ≥ 0 is a regularization parameter.
 LSMR is formally equivalent to applying MINRES to the normal equations
 
     (AᴴA + λ²I) x = Aᴴb

src/lsqr.jl

@@ -42,7 +42,7 @@ Solve the regularized linear least-squares problem
 
     minimize ‖b - Ax‖₂² + λ²‖x‖₂²
 
-using the LSQR method, where λ ≥ 0 is a regularization parameter.
+of size n × m using the LSQR method, where λ ≥ 0 is a regularization parameter.
 LSQR is formally equivalent to applying CG to the normal equations
 
     (AᴴA + λ²I) x = Aᴴb

src/minres.jl

@@ -3,7 +3,7 @@
 #
 #  minimize ‖Ax - b‖₂
 #
-# where A is square and symmetric.
+# where A is Hermitian.
 #
 # MINRES is formally equivalent to applying the conjugate residuals method
 # to Ax = b when A is positive definite, but is more general and also applies
@@ -43,8 +43,8 @@ or the shifted linear system
 
     (A + λI) x = b
 
-using the MINRES method, where λ ≥ 0 is a shift parameter,
-where A is square and symmetric.
+of size n using the MINRES method, where λ ≥ 0 is a shift parameter,
+where A is Hermitian.
 
 MINRES is formally equivalent to applying CR to Ax=b when A is positive
 definite, but is typically more stable and also applies to the case where
@@ -53,7 +53,7 @@ A is indefinite.
 MINRES produces monotonic residuals ‖r‖₂ and optimality residuals ‖Aᴴr‖₂.
 
 A preconditioner M may be provided in the form of a linear operator and is
-assumed to be symmetric and positive definite.
+assumed to be Hermitian and positive definite.
 
 MINRES can be warm-started from an initial guess `x0` with
 

src/minres_qlp.jl

@@ -27,11 +27,11 @@ export minres_qlp, minres_qlp!
 `FC` is `T` or `Complex{T}`.
 
 MINRES-QLP is the only method based on the Lanczos process that returns the minimum-norm
-solution on singular inconsistent systems (A + λI)x = b, where λ is a shift parameter.
+solution on singular inconsistent systems (A + λI)x = b of size n, where λ is a shift parameter.
 It is significantly more complex but can be more reliable than MINRES when A is ill-conditioned.
 
 A preconditioner M may be provided in the form of a linear operator and is
-assumed to be symmetric and positive definite.
+assumed to be Hermitian and positive definite.
 M also indicates the weighted norm in which residuals are measured.
 
 MINRES-QLP can be warm-started from an initial guess `x0` with

src/qmr.jl

@@ -29,7 +29,7 @@ export qmr, qmr!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the square linear system Ax = b using QMR.
+Solve the square linear system Ax = b of size n using QMR.
 
 QMR is based on the Lanczos biorthogonalization process and requires two initial vectors `b` and `c`.
 The relation `bᴴc ≠ 0` must be satisfied and by default `c = b`.

src/symmlq.jl

@@ -1,5 +1,5 @@
 # An implementation of SYMMLQ for the solution of the
-# linear system Ax = b, where A is square and symmetric.
+# linear system Ax = b, where A is Hermitian.
 #
 # This implementation follows the original implementation by
 # Michael Saunders described in
@@ -27,13 +27,13 @@ Solve the shifted linear system
 
     (A + λI) x = b
 
-using the SYMMLQ method, where λ is a shift parameter,
-and A is square and symmetric.
+of size n using the SYMMLQ method, where λ is a shift parameter,
+and A is Hermitian.
 
 SYMMLQ produces monotonic errors ‖x* - x‖₂.
 
 A preconditioner M may be provided in the form of a linear operator and is
-assumed to be symmetric and positive definite.
+assumed to be Hermitian and positive definite.
 
 SYMMLQ can be warm-started from an initial guess `x0` with
 

src/tricg.jl

@@ -27,7 +27,7 @@ TriCG solves the symmetric linear system
     [ τE    A ] [ x ] = [ b ]
     [  Aᴴ  νF ] [ y ]   [ c ],
 
-where τ and ν are real numbers, E = M⁻¹ ≻ 0 and F = N⁻¹ ≻ 0.
+of size (n+m) × (n+m) where τ and ν are real numbers, E = M⁻¹ ≻ 0 and F = N⁻¹ ≻ 0.
 `b` and `c` must both be nonzero.
 TriCG could breakdown if `τ = 0` or `ν = 0`.
 It's recommended to use TriMR in these cases.

src/trilqr.jl

@@ -26,8 +26,8 @@ Combine USYMLQ and USYMQR to solve adjoint systems.
     [0  A] [y] = [b]
     [Aᴴ 0] [x]   [c]
 
-USYMLQ is used for solving primal system `Ax = b`.
-USYMQR is used for solving dual system `Aᴴy = c`.
+USYMLQ is used for solving primal system `Ax = b` of size n.
+USYMQR is used for solving dual system `Aᴴy = c` of size m.
 
 An option gives the possibility of transferring from the USYMLQ point to the
 USYMCG point, when it exists. The transfer is based on the residual norm.

src/trimr.jl

@@ -27,7 +27,7 @@ TriMR solves the symmetric linear system
     [ τE    A ] [ x ] = [ b ]
     [  Aᴴ  νF ] [ y ]   [ c ],
 
-where τ and ν are real numbers, E = M⁻¹ ≻ 0, F = N⁻¹ ≻ 0.
+of size (n+m) × (n+m) where τ and ν are real numbers, E = M⁻¹ ≻ 0, F = N⁻¹ ≻ 0.
 `b` and `c` must both be nonzero.
 TriMR handles saddle-point systems (`τ = 0` or `ν = 0`) and adjoint systems (`τ = 0` and `ν = 0`) without any risk of breakdown.
 

src/usymlq.jl

@@ -28,7 +28,7 @@ export usymlq, usymlq!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the linear system Ax = b using the USYMLQ method.
+Solve the linear system Ax = b of size n × m using the USYMLQ method.
 
 USYMLQ is based on the orthogonal tridiagonalization process and requires two initial nonzero vectors `b` and `c`.
 The vector `c` is only used to initialize the process and a default value can be `b` or `Aᴴb` depending on the shape of `A`.

src/usymqr.jl

@@ -28,7 +28,7 @@ export usymqr, usymqr!
 `T` is an `AbstractFloat` such as `Float32`, `Float64` or `BigFloat`.
 `FC` is `T` or `Complex{T}`.
 
-Solve the linear system Ax = b using USYMQR.
+Solve the linear system Ax = b of size n × m using USYMQR.
 
 USYMQR is based on the orthogonal tridiagonalization process and requires two initial nonzero vectors `b` and `c`.
 The vector `c` is only used to initialize the process and a default value can be `b` or `Aᴴb` depending on the shape of `A`.