# BiCGStab(l)

BiCGStab(l) solves the problem $Ax = b$ approximately for $x$ where $A$ is a general, linear operator and $b$ the right-hand side vector. The methods combines BiCG with $l$ GMRES iterations, resulting in a short-reccurence iteration. As a result the memory is fixed as well as the computational costs per iteration.

## Usage

`IterativeSolvers.bicgstabl`

— Function`bicgstabl(A, b, l; kwargs...) -> x, [history]`

Same as `bicgstabl!`

, but allocates a solution vector `x`

initialized with zeros.

`IterativeSolvers.bicgstabl!`

— Function`bicgstabl!(x, A, b, l; kwargs...) -> x, [history]`

**Arguments**

`A`

: linear operator;`b`

: right hand side (vector);`l::Int = 2`

: Number of GMRES steps.

**Keywords**

`max_mv_products::Int = size(A, 2)`

: maximum number of matrix vector products.

For BiCGStab(l) this is a less dubious term than "number of iterations";

`Pl = Identity()`

: left preconditioner of the method;`abstol::Real = zero(real(eltype(b)))`

,`reltol::Real = sqrt(eps(real(eltype(b))))`

: absolute and relative tolerance for the stopping condition`|r_k| ≤ max(reltol * |r_0|, abstol)`

, where`r_k ≈ A * x_k - b`

is the approximate residual in the`k`

th iteration;Note - The true residual norm is never computed during the iterations, only an approximation;
- If a left preconditioner is given, the stopping condition is based on the
*preconditioned residual*.

**Return values**

**if log is false**

`x`

: approximate solution.

**if log is true**

`x`

: approximate solution;`history`

: convergence history.

## Implementation details

The method is based on the original article ^{[Sleijpen1993]}, but does not implement later improvements. The normal equations arising from the GMRES steps are solved without orthogonalization. Hence the method should only be reliable for relatively small values of $l$.

The `r`

and `u`

factors are pre-allocated as matrices of size $n \times (l + 1)$, so that BLAS2 methods can be used. Also the random shadow residual is pre-allocated as a vector. Hence the storage costs are approximately $2l + 3$ vectors.

BiCGStabl(l) can be used as an iterator.

- Sleijpen1993Sleijpen, Gerard LG, and Diederik R. Fokkema. "BiCGstab(l) for linear equations involving unsymmetric matrices with complex spectrum." Electronic Transactions on Numerical Analysis 1.11 (1993): 2000.