Edit on GitHub

TriCG

Krylov.tricg — Function
(x, y, stats) = tricg(A, b::AbstractVector{FC}, c::AbstractVector{FC};
                      M=I, N=I, atol::T=√eps(T), rtol::T=√eps(T),
                      spd::Bool=false, snd::Bool=false, flip::Bool=false,
                      τ::T=one(T), ν::T=-one(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}.

TriCG solves the symmetric linear system

[ τE    A ] [ x ] = [ b ]
[  Aᵀ  νF ] [ y ]   [ c ],

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.

By default, TriCG solves symmetric and quasi-definite linear systems with τ = 1 and ν = -1. If flip = true, TriCG solves another known variant of SQD systems where τ = -1 and ν = 1. If spd = true, τ = ν = 1 and the associated symmetric and positive definite linear system is solved. If snd = true, τ = ν = -1 and the associated symmetric and negative definite linear system is solved. τ and ν are also keyword arguments that can be directly modified for more specific problems.

TriCG is based on the preconditioned orthogonal tridiagonalization process and its relation with the preconditioned block-Lanczos process.

[ M   0 ]
[ 0   N ]

indicates the weighted norm in which residuals are measured. It's the Euclidean norm when M and N are identity operators.

TriCG stops when itmax iterations are reached or when ‖rₖ‖ ≤ atol + ‖r₀‖ * rtol. atol is an absolute tolerance and rtol is a relative tolerance.

Additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every verbose iterations.

TriCG can be warm-started from initial guesses x0 and y0 with the method

(x, y, stats) = tricg(A, b, c, x0, y0; 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.tricg! — Function
solver = tricg!(solver::TricgSolver, A, b, c; kwargs...)
solver = tricg!(solver::TricgSolver, A, b, c, x0, y0; kwargs...)

where kwargs are keyword arguments of tricg.

See TricgSolver for more details about the solver.

source

TriMR

Krylov.trimr — Function
(x, y, stats) = trimr(A, b::AbstractVector{FC}, c::AbstractVector{FC};
                      M=I, N=I, atol::T=√eps(T), rtol::T=√eps(T),
                      spd::Bool=false, snd::Bool=false, flip::Bool=false, sp::Bool=false,
                      τ::T=one(T), ν::T=-one(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}.

TriMR solves the symmetric linear system

[ τE    A ] [ x ] = [ b ]
[  Aᵀ  νF ] [ y ]   [ c ],

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.

By default, TriMR solves symmetric and quasi-definite linear systems with τ = 1 and ν = -1. If flip = true, TriMR solves another known variant of SQD systems where τ = -1 and ν = 1. If spd = true, τ = ν = 1 and the associated symmetric and positive definite linear system is solved. If snd = true, τ = ν = -1 and the associated symmetric and negative definite linear system is solved. If sp = true, τ = 1, ν = 0 and the associated saddle-point linear system is solved. τ and ν are also keyword arguments that can be directly modified for more specific problems.

TriMR is based on the preconditioned orthogonal tridiagonalization process and its relation with the preconditioned block-Lanczos process.

[ M   0 ]
[ 0   N ]

indicates the weighted norm in which residuals are measured. It's the Euclidean norm when M and N are identity operators.

TriMR stops when itmax iterations are reached or when ‖rₖ‖ ≤ atol + ‖r₀‖ * rtol. atol is an absolute tolerance and rtol is a relative tolerance.

Additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every verbose iterations.

TriMR can be warm-started from initial guesses x0 and y0 with the method

(x, y, stats) = trimr(A, b, c, x0, y0; 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.trimr! — Function
solver = trimr!(solver::TrimrSolver, A, b, c; kwargs...)
solver = trimr!(solver::TrimrSolver, A, b, c, x0, y0; kwargs...)

where kwargs are keyword arguments of trimr.

See TrimrSolver for more details about the solver.

source