Each Krylov method is able to call a callback function as callback(solver) at each iteration. The callback should return true if the main loop should terminate, and false otherwise. If the method terminated because of the callback, the output status will be "user-requested exit". For example, if the user defines minres_callback(solver::MinresSolver), it can be passed to the solver using

(x, stats) = minres(A, b, callback = minres_callback)

If you need to write a callback that uses variables that are not in a KrylovSolver, use a closure:

function custom_stopping_condition(solver::KrylovSolver, A, b, r, tol)
  mul!(r, A, solver.x)
  r .-= b               # r := b - Ax
  bool = norm(r) ≤ tol  # tolerance based on the 2-norm of the residual
  return bool

cg_callback(solver) = custom_stopping_condition(solver, A, b, r, tol)
(x, stats) = cg(A, b, callback = cg_callback)

Alternatively, use a structure and make it callable:

mutable struct CallbackWorkspace{T}

function (workspace::CallbackWorkspace)(solver::KrylovSolver)
  mul!(workspace.r, workspace.A, solver.x)
  workspace.r .-= workspace.b
  bool = norm(workspace.r) ≤ workspace.tol
  return bool

bicgstab_callback = CallbackWorkspace(A, b, r, tol)
(x, stats) = bicgstab(A, b, callback = bicgstab_callback)

Although the main goal of a callback is to add new stopping conditions, it can also retrieve information from the workspace of a Krylov method along the iterations. We now illustrate how to store all iterates $x_k$ of the GMRES method.

S = Krylov.ktypeof(b)
global X = S[]  # Storage for GMRES iterates

function gmres_callback(solver)
  z = solver.z
  k = solver.inner_iter
  nr = sum(1:k)
  V = solver.V
  R = solver.R
  y = copy(z)

  # Solve Rk * yk = zk
  for i = k : -1 : 1
    pos = nr + i - k
    for j = k : -1 : i+1
      y[i] = y[i] - R[pos] * y[j]
      pos = pos - j + 1
    y[i] = y[i] / R[pos]

  # xk = Vk * yk
  xk = sum(V[i] * y[i] for i = 1:k)
  push!(X, xk)

  return false  # We don't want to add new stopping conditions

(x, stats) = gmres(A, b, callback = gmres_callback)