## BiLQR

`Krylov.bilqr`

— Function```
(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]
[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`

.

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

`(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.

`Krylov.bilqr!`

— Function```
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`

.

## TriLQR

`Krylov.trilqr`

— Function```
(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]
[Aᴴ 0] [x] [c]
```

USYMLQ is used for solving primal system `Ax = b`

. USYMQR is used for solving dual system `Aᴴy = 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

`(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.

`Krylov.trilqr!`

— Function```
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`

.