Example Solve with by the conjugate gradient method. Let then . Also, note that c is the reciprocal of the condition number of A , estimated using the L 2 -norm. For ill-conditioned A , the Conjugate Gradient Method is very slow. For ill-conditioned A , the Conjugate Gradient Method is very slow.
Careful analysis is required to choose a Q .
In some cases, for the preconditioned conjugate gradient method, one could reduce convergence cost to . In preconditioned Conjugate Gradient the recast is actually not computed or used. Let be the spectral radius for the Since Gauss-Seidel can decrease as much as Jacobi. Now, let be the spectral radius for the SOR solution. Then when optimal. so SOR is approximately faster than either Jacobi or Gauss Seidel Methods.