(x, y, stats) = bilqr(A, b::AbstractVector{FC}, c::AbstractVector{FC};
                      atol::T=√eps(T), rtol::T=√eps(T), transfer_to_bicg::Bool=true,
                      itmax::Int=0, verbose::Int=0, history::Bool=false,
                      callback=solver->false)

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

Combine BiLQ and QMR to solve adjoint systems.

[0  A] [y] = [b]
[Aᵀ 0] [x]   [c]

The relation bᵀc ≠ 0 must be satisfied. BiLQ is used for solving primal system Ax = b. QMR is used for solving dual system Aᵀy = c.

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.

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

(x, y, stats) = bilqr(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

• A. Montoison and D. Orban, BiLQ: An Iterative Method for Nonsymmetric Linear Systems with a Quasi-Minimum Error Property, SIAM Journal on Matrix Analysis and Applications, 41(3), pp. 1145–1166, 2020.

solver = bilqr!(solver::BilqrSolver, A, b, c; kwargs...)
solver = bilqr!(solver::BilqrSolver, A, b, c, x0, y0; kwargs...)

where kwargs are keyword arguments of bilqr.

See BilqrSolver for more details about the solver.

(x, y, stats) = trilqr(A, b::AbstractVector{FC}, c::AbstractVector{FC};
                       atol::T=√eps(T), rtol::T=√eps(T), transfer_to_usymcg::Bool=true,
                       itmax::Int=0, verbose::Int=0, history::Bool=false,
                       callback=solver->false)

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

Combine USYMLQ and USYMQR to solve adjoint systems.

[0  A] [y] = [b]
[Aᵀ 0] [x]   [c]

USYMLQ is used for solving primal system Ax = b. USYMQR is used for solving dual system Aᵀy = c.

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.

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

(x, y, stats) = trilqr(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

• A. Montoison and D. Orban, BiLQ: An Iterative Method for Nonsymmetric Linear Systems with a Quasi-Minimum Error Property, SIAM Journal on Matrix Analysis and Applications, 41(3), pp. 1145–1166, 2020.

solver = trilqr!(solver::TrilqrSolver, A, b, c; kwargs...)
solver = trilqr!(solver::TrilqrSolver, A, b, c, x0, y0; kwargs...)

where kwargs are keyword arguments of trilqr.

See TrilqrSolver for more details about the solver.