5.1.7 iparm(_JT_ stop_) - stopping test

• Current index: 7
• Intent: In
• Default: _JT_ stop_ relchg_

  Variable	Meaning	Value
  _JT_ stop_ relchg_	$${\frac{\Vert x - x_{old} \Vert }{\Vert x\Vert }}$$	0
  _JT_ stop_ axb_	$${\frac{\Vert r\Vert }{\Vert A\Vert \Vert x\Vert + \Vert b\Vert }}$$	1
  _JT_ stop_ b_	$${\frac{\Vert r\Vert }{\Vert b\Vert }}$$	2
  _JT_ stop_ x_	$${\frac{\Vert r\Vert }{\Vert x\Vert }}$$	3
  _JT_ stop_ r0_	$${\frac{\Vert r\Vert }{\Vert r_0 \Vert }}$$	4
  _JT_ stop_ r_	| r|	5

  where A, x, and b are the coefficient matrix, current estimate of the solution vector, and source vector of the linear system being solved, Ax = b. When iparm(_JT_ stop_) is positive, the vector r is whatever value of the residual, r = b - Ax, is normally available during the course of the iteration for the method chosen. When iparm(_JT_ stop_) is negative, b - Ax is explicitly computed and used in the convergence test.

  The vector r₀ is the initial residual, i.e.the residual computed using the initial guess for x, or r₀ = b - Ax₀.

• Recommendations (but please also read the Notes):

  Method	iparm(_JT_ stop_)
  CG	2
  GMRES	2
  BCGS	-2
  TFQMR	-2
  Jacobi	0
  SOR	0

• Further notes:
  • The norm to use is set by iparm(_JT_ norm_).

  • _JT_ stop_ relchg_ is fine for the stationary methods (Jacobi, SOR, SSOR) since:
    • The residual is not directly available.
    • These methods converge nearly monotonically anyway.

    However, this test is completely inappropriate and dangerous for the nonstationary methods. It is the default only because it is the only test that ``works'' without extra work for all methods.

  • _JT_ stop_ b_ is probably the best for most situations. However, it can lead to difficulties when | A|| x| > > | b|. In that case, use _JT_ stop_ axb_.

  • Normally, _JT_ stop_ axb_ and _JT_ stop_ x_ increase the cost of GMRES significantly, since a new iterate is normally not computed at each iteration, but only every iparm(_JT_ nold_) iterations. In order to minimize this extra cost, JT_ GMRES forces calculation of new iterates for the first five iterations, and thereafter uses the last available iterate in convergence tests.

  • Note that if _JT_ stop_ axb_ is used, the norm of the coefficient is needed. This value is calculated unless the storage format is unknown (ia(_JT_storage_)= _JT_ storage_ unknown_), in which case the value must be passed in as rparm(_JT_ anorm_).

  • _JT_ stop_ r0_ is in general not recommended, as it depends too much on the initial guess. Also note that when the initial guess of x = 0 is used, this test is equivalent to _JT_ stop_ b_.

  • _JT_ stop_ r_ should be used carefully.

  • A stopping test based on the true residual (i.e.a value of iparm(_JT_ stop_) < 0), is recommended for BCGS and TFQMR simply to protect against false convergence. However, it adds significant overhead to each iteration.

    A more efficient way to protect against false convergence is to use a stopping test based on the recursively-generated residual (i.e.a value of iparm(_JT_ stop_) > 0), and then compute the true residual and corresponding error estimate after ``convergence''. If the error estimate based on the true residual does not meet the desired convergence criterion, the the final iterate can be used as the initial guess for additional iterations with the same (or a different) method.

  • For the stationary iterative methods (Jacobi, SOR, SSOR), the true residual is always used when iparm(_JT_ stop_) $\neq$ 0, since an estimated residual isn't built up during the iteration. So, e.g., iparm(_JT_ stop_) = _JT_ stop_ axb_ is the same as iparm(_JT_ stop_) = -_JT_ stop_ axb_. This adds quite a bit of expense to each iteration, but is useful if you want to plot the residual history.