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SYMMLQ

Krylov.symmlq — Function
(x, stats) = symmlq(A, b::AbstractVector{FC}; window::Int=0,
                    M=I, λ::T=zero(T), transfer_to_cg::Bool=true,
                    λest::T=zero(T), atol::T=√eps(T), rtol::T=√eps(T),
                    etol::T=√eps(T), itmax::Int=0, conlim::T=1/√eps(T),
                    verbose::Int=0, history::Bool=false,
                    ldiv::Bool=false, callback=solver->false)

T is an AbstractFloat such as Float32, Float64 or BigFloat. FC is T or Complex{T}.

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.

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.

SYMMLQ can be warm-started from an initial guess x0 with

(x, stats) = symmlq(A, b, x0; kwargs...)

where kwargs are the same keyword arguments as above.

The callback is called as callback(solver) and should return true if the main loop should terminate, and false otherwise.

Reference

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Krylov.symmlq! — Function
solver = symmlq!(solver::SymmlqSolver, A, b; kwargs...)
solver = symmlq!(solver::SymmlqSolver, A, b, x0; kwargs...)

where kwargs are keyword arguments of symmlq.

See SymmlqSolver for more details about the solver.

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MINRES

Krylov.minres — Function
(x, stats) = minres(A, b::AbstractVector{FC};
                    M=I, λ::T=zero(T), atol::T=√eps(T)/100,
                    rtol::T=√eps(T)/100, ratol :: T=zero(T), 
                    rrtol :: T=zero(T), etol::T=√eps(T),
                    window::Int=5, itmax::Int=0,
                    conlim::T=1/√eps(T), verbose::Int=0,
                    history::Bool=false, ldiv::Bool=false,
                    callback=solver->false)

T is an AbstractFloat such as Float32, Float64 or BigFloat. FC is T or Complex{T}.

Solve the shifted linear least-squares problem

minimize ‖b - (A + λI)x‖₂²

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.

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 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.

MINRES can be warm-started from an initial guess x0 with

(x, stats) = minres(A, b, x0; kwargs...)

where kwargs are the same keyword arguments as above.

The callback is called as callback(solver) and should return true if the main loop should terminate, and false otherwise.

Reference

source
Krylov.minres! — Function
solver = minres!(solver::MinresSolver, A, b; kwargs...)
solver = minres!(solver::MinresSolver, A, b, x0; kwargs...)

where kwargs are keyword arguments of minres.

See MinresSolver for more details about the solver.

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MINRES-QLP

Krylov.minres_qlp — Function
(x, stats) = minres_qlp(A, b::AbstractVector{FC};
                        M=I, atol::T=√eps(T), rtol::T=√eps(T),
                        ctol::T=√eps(T), λ::T=zero(T), itmax::Int=0,
                        verbose::Int=0, history::Bool=false,
                        ldiv::Bool=false, callback=solver->false)

T is an AbstractFloat such as Float32, Float64 or BigFloat. 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. 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. M also indicates the weighted norm in which residuals are measured.

MINRES-QLP can be warm-started from an initial guess x0 with

(x, stats) = minres_qlp(A, b, x0; kwargs...)

where kwargs are the same keyword arguments as above.

The callback is called as callback(solver) and should return true if the main loop should terminate, and false otherwise.

References

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Krylov.minres_qlp! — Function
solver = minres_qlp!(solver::MinresQlpSolver, A, b; kwargs...)
solver = minres_qlp!(solver::MinresQlpSolver, A, b, x0; kwargs...)

where kwargs are keyword arguments of minres_qlp.

See MinresQlpSolver for more details about the solver.

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