## TriCG

`Krylov.tricg`

— Function```
(x, y, stats) = tricg(A, b::AbstractVector{FC}, c::AbstractVector{FC};
M=I, N=I, ldiv::Bool=false,
spd::Bool=false, snd::Bool=false,
flip::Bool=false, τ::T=one(T),
ν::T=-one(T), atol::T=√eps(T),
rtol::T=√eps(T), itmax::Int=0,
timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
callback=solver->false, iostream::IO=kstdout)
```

`T`

is an `AbstractFloat`

such as `Float32`

, `Float64`

or `BigFloat`

. `FC`

is `T`

or `Complex{T}`

.

`(x, y, stats) = tricg(A, b, c, x0::AbstractVector, y0::AbstractVector; kwargs...)`

TriCG can be warm-started from initial guesses `x0`

and `y0`

where `kwargs`

are the same keyword arguments as above.

Given a matrix `A`

of dimension m × n, TriCG solves the Hermitian linear system

```
[ τE A ] [ x ] = [ b ]
[ Aᴴ νF ] [ y ] [ c ],
```

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.

By default, TriCG solves Hermitian and quasi-definite linear systems with τ = 1 and ν = -1.

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.

**Input arguments**

`A`

: a linear operator that models a matrix of dimension`m × n`

;`b`

: a vector of length`m`

;`c`

: a vector of length`n`

.

**Optional arguments**

`x0`

: a vector of length`m`

that represents an initial guess of the solution`x`

;`y0`

: a vector of length`n`

that represents an initial guess of the solution`y`

.

**Keyword arguments**

`M`

: linear operator that models a Hermitian positive-definite matrix of size`m`

used for centered preconditioning of the partitioned system;`N`

: linear operator that models a Hermitian positive-definite matrix of size`n`

used for centered preconditioning of the partitioned system;`ldiv`

: define whether the preconditioners use`ldiv!`

or`mul!`

;`spd`

: if`true`

, set`τ = 1`

and`ν = 1`

for Hermitian and positive-definite linear system;`snd`

: if`true`

, set`τ = -1`

and`ν = -1`

for Hermitian and negative-definite linear systems;`flip`

: if`true`

, set`τ = -1`

and`ν = 1`

for another known variant of Hermitian quasi-definite systems;`τ`

and`ν`

: diagonal scaling factors of the partitioned Hermitian linear system;`atol`

: absolute stopping tolerance based on the residual norm;`rtol`

: relative stopping tolerance based on the residual norm;`itmax`

: the maximum number of iterations. If`itmax=0`

, the default number of iterations is set to`m+n`

;`timemax`

: the time limit in seconds;`verbose`

: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every`verbose`

iterations;`history`

: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;`callback`

: function or functor called as`callback(solver)`

that returns`true`

if the Krylov method should terminate, and`false`

otherwise;`iostream`

: stream to which output is logged.

**Output arguments**

`x`

: a dense vector of length`m`

;`y`

: a dense vector of length`n`

;`stats`

: statistics collected on the run in a`SimpleStats`

structure.

**Reference**

- A. Montoison and D. Orban,
*TriCG and TriMR: Two Iterative Methods for Symmetric Quasi-Definite Systems*, SIAM Journal on Scientific Computing, 43(4), pp. 2502–2525, 2021.

`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`

.

## TriMR

`Krylov.trimr`

— Function```
(x, y, stats) = trimr(A, b::AbstractVector{FC}, c::AbstractVector{FC};
M=I, N=I, ldiv::Bool=false,
spd::Bool=false, snd::Bool=false,
flip::Bool=false, sp::Bool=false,
τ::T=one(T), ν::T=-one(T), atol::T=√eps(T),
rtol::T=√eps(T), itmax::Int=0,
timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
callback=solver->false, iostream::IO=kstdout)
```

`T`

is an `AbstractFloat`

such as `Float32`

, `Float64`

or `BigFloat`

. `FC`

is `T`

or `Complex{T}`

.

`(x, y, stats) = trimr(A, b, c, x0::AbstractVector, y0::AbstractVector; kwargs...)`

TriMR can be warm-started from initial guesses `x0`

and `y0`

where `kwargs`

are the same keyword arguments as above.

Given a matrix `A`

of dimension m × n, TriMR solves the symmetric linear system

```
[ τE A ] [ x ] = [ b ]
[ Aᴴ νF ] [ y ] [ c ],
```

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.

By default, TriMR solves symmetric and quasi-definite linear systems with τ = 1 and ν = -1.

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.

**Input arguments**

`A`

: a linear operator that models a matrix of dimension`m × n`

;`b`

: a vector of length`m`

;`c`

: a vector of length`n`

.

**Optional arguments**

`x0`

: a vector of length`m`

that represents an initial guess of the solution`x`

;`y0`

: a vector of length`n`

that represents an initial guess of the solution`y`

.

**Keyword arguments**

`M`

: linear operator that models a Hermitian positive-definite matrix of size`m`

used for centered preconditioning of the partitioned system;`N`

: linear operator that models a Hermitian positive-definite matrix of size`n`

used for centered preconditioning of the partitioned system;`ldiv`

: define whether the preconditioners use`ldiv!`

or`mul!`

;`spd`

: if`true`

, set`τ = 1`

and`ν = 1`

for Hermitian and positive-definite linear system;`snd`

: if`true`

, set`τ = -1`

and`ν = -1`

for Hermitian and negative-definite linear systems;`flip`

: if`true`

, set`τ = -1`

and`ν = 1`

for another known variant of Hermitian quasi-definite systems;`sp`

: if`true`

, set`τ = 1`

and`ν = 0`

for saddle-point systems;`τ`

and`ν`

: diagonal scaling factors of the partitioned Hermitian linear system;`atol`

: absolute stopping tolerance based on the residual norm;`rtol`

: relative stopping tolerance based on the residual norm;`itmax`

: the maximum number of iterations. If`itmax=0`

, the default number of iterations is set to`m+n`

;`timemax`

: the time limit in seconds;`verbose`

: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every`verbose`

iterations;`history`

: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;`callback`

: function or functor called as`callback(solver)`

that returns`true`

if the Krylov method should terminate, and`false`

otherwise;`iostream`

: stream to which output is logged.

**Output arguments**

`x`

: a dense vector of length`m`

;`y`

: a dense vector of length`n`

;`stats`

: statistics collected on the run in a`SimpleStats`

structure.

**Reference**

- A. Montoison and D. Orban,
*TriCG and TriMR: Two Iterative Methods for Symmetric Quasi-Definite Systems*, SIAM Journal on Scientific Computing, 43(4), pp. 2502–2525, 2021.

`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`

.