# QMR

QMR is a short-recurrence version of GMRES for solving $Ax = b$ approximately for $x$ where $A$ is a linear operator and $b$ the right-hand side vector. $A$ may be non-symmetric.

## Usage

IterativeSolvers.qmr!Function
qmr!(x, A, b; kwargs...) -> x, [history]

Solves the problem $Ax = b$ with the Quasi-Minimal Residual (QMR) method.

Arguments

• x: Initial guess, will be updated in-place;
• A: linear operator;
• b: right-hand side.

Keywords

• initally_zero::Bool: If true assumes that iszero(x) so that one matrix-vector product can be saved when computing the initial residual vector;
• maxiter::Int = size(A, 2): maximum number of iterations;
• 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
• log::Bool: keep track of the residual norm in each iteration;
• verbose::Bool: print convergence information during the iteration.

Return values

if log is false

• x: approximate solution.

if log is true

• x: approximate solution;

• history: convergence history.

source

## Implementation details

QMR exploits the tridiagonal structure of the Hessenberg matrix. Although QMR is similar to GMRES, where instead of using the Arnoldi process, a pair of biorthogonal vector spaces $V$ and $W$ is constructed via the Lanczos process. It requires that the adjoint of $A$ adjoint(A) be available.

QMR enables the computation of $V$ and $W$ via a three-term recurrence. A three-term recurrence for the projection onto the solution vector can also be constructed from these values, using the portion of the last column of the Hessenberg matrix. Therefore we pre-allocate only eight vectors.

For more detail on the implementation see the original paper [Freund1990] or [Saad2003].

Tip

QMR can be used as an iterator via qmr_iterable!. This makes it possible to access the next, current, and previous Krylov basis vectors during the iteration.