Editing Conjugate gradient method (section)

=== Practical convergence ===

If initialized randomly, the first stage of iterations is often the fastest, as the error is eliminated within the Krylov subspace that initially reflects a smaller effective condition number. The second stage of convergence is typically well defined by the theoretical convergence bound with <math display="inline"> \sqrt{\kappa(\mathbf{A})}</math>, but may be super-linear, depending on a distribution of the spectrum of the matrix <math>A</math> and the spectral distribution of the error.<ref name="AG" /> In the last stage, the smallest attainable accuracy is reached and the convergence stalls or the method may even start diverging. In typical scientific computing applications in [[double-precision floating-point format]] for matrices of large sizes, the conjugate gradient method uses a stopping criterion with a tolerance that terminates the iterations during the first or second stage.