
 DETAILS OF ITERATIVE TEMPLATES TEST:

    Univ. of Tennessee and Oak Ridge National Laboratory
    October 1, 1993
    Details of these algorithms are described in "Templates
    for the Solution of Linear Systems: Building Blocks for
    Iterative Methods", Barrett, Berry, Chan, Demmel, Donato,
    Dongarra, Eijkhout, Pozo, Romine, and van der Vorst,
    SIAM Publications, 1993.
    (ftp netlib2.cs.utk.edu; cd linalg; get templates.ps).


MACHINE PRECISION = 1.11E-16
CONVERGENCE TEST TOLERANCE = 1.00E-15


  For a detailed description of the following information,
  see the end of this file.

 ======================================================
            CONVERGENCE  NORMALIZED  NUM
   METHOD    CRITERION    RESIDUAL   ITER  INFO  FLAG
 ======================================================

  see the end of this file.
Order  36 SPD 2-d Poisson matrix (no preconditioning)

  CG        1.28E-16    3.08E-03       6
  Chebyshev 6.17E-11    3.29E+02     144      1
  SOR       2.18E-14    2.39E-01     144      1
  BiCG      9.09E-17    3.08E-03       6
  CGS            NaN         NaN     144      1
  BiCGSTAB  1.29E-17    3.85E-03       6
  GMRESm    4.00E-20    8.01E+15       1            X
  QMR       1.99E-16    5.40E-03       6
  Jacobi    3.02E-08    6.16E+05     144      1
 -------------------------------------------------------
Order  36 SPD 2-d Poisson matrix (Jacobi preconditioning)

  CG        1.28E-16    3.08E-03       6
  Chebyshev 1.20E-06    6.41E+06     144      1
  SOR       2.18E-14    2.39E-01     144      1
  BiCG      9.09E-17    3.08E-03       6
  CGS            NaN         NaN     144      1
  BiCGSTAB  1.29E-17    3.85E-03       6
  GMRESm    9.82E-18    3.79E+13       1            X
  QMR       1.99E-16    5.40E-03       6
 -------------------------------------------------------
Order   21 SPD Wathen matrix (no preconditioning)

  CG        2.74E-18    1.24E-03      26
  Chebyshev 1.52E-02    2.32E+10      84      1
  SOR       8.41E-11    3.28E+02      84      1
  BiCG      8.91E-18    1.39E-03      26
  CGS       4.07E-16    6.80E-02      27
  BiCGSTAB  1.24E-16    1.55E-03      27
  GMRESm    7.83E-16    2.95E+15       1            X
  QMR       2.98E-16    6.18E-04      26
 -------------------------------------------------------
Order   21 SPD Wathen matrix (Jacobi preconditioning)

  CG        9.10E-22    6.18E-04      20
  Chebyshev 5.58E-01    5.28E+11      84      1
  SOR       8.41E-11    3.28E+02      84      1
  BiCG      7.45E-22    1.08E-03      20
  CGS            NaN         NaN      84      1
  BiCGSTAB  6.78E-16    2.86E+12      74            X
  GMRESm    5.97E-18    9.33E+11       1            X
  QMR       6.72E-16    1.08E-03      20
 -------------------------------------------------------
Order  27 SPD 3-d Poisson matrix (no preconditioning)

  CG        2.83E-17    3.33E-03       4
  Chebyshev 9.50E-16    1.83E-02      68
  SOR       9.92E-16    2.50E-03      24
  BiCG      3.06E-17    2.91E-03       4
  CGS       1.34E-17    2.70E-03       4
  BiCGSTAB  1.42E-18    3.33E-03       4
  GMRESm    2.53E-17    4.50E+15       1            X
  QMR       4.77E-16    5.83E-03       4
  Jacobi    7.35E-16    1.33E-02      98
 -------------------------------------------------------
Order  27 SPD 3-d Poisson matrix (Jacobi preconditioning)

  CG        2.41E-17    4.58E-03       4
  Chebyshev 6.75E-14    4.79E-01     108      1
  SOR       9.92E-16    2.50E-03      24
  BiCG      2.70E-17    4.58E-03       4
  CGS       4.09E-17    3.33E-03       4
  BiCGSTAB  4.29E-18    4.99E-03       4
  GMRESm    3.31E-17    5.25E+12       1            X
  QMR       4.10E-16    4.99E-03       4
 -------------------------------------------------------
Order  125 PDE1 nonsymmetric matrix (no preconditioning)

  BiCG      1.03E-16    1.22E-03      62
  CGS       8.11E-16    6.68E-03      91
  BiCGSTAB  6.90E-16    5.12E-03     100
  GMRESm    3.45E-16    1.30E+21       1            X
  QMR       2.21E-15    1.08E-03     500      1
 -------------------------------------------------------
Order  125 PDE1 nonsymmetric matrix (Jacobi preconditioning)

  BiCG      8.62E-16    6.68E-02      61
  CGS       3.76E-16    2.29E-02      71
  BiCGSTAB  2.96E-16    4.05E-03      84
  GMRESm    9.85E-16    4.17E+21       1            X
  QMR       1.40E-13    1.42E-01     500      1
 -------------------------------------------------------
Order  125 PDE2 nonsymmetric matrix (no preconditioning)

  BiCG      2.06E-16    2.84E-02      33
  CGS       9.46E-16    1.88E+00      40
  BiCGSTAB  9.61E-03    1.36E+11     248    -10
  GMRESm    6.00E-16    1.53E+22       1            X
  QMR       1.26E-14    1.05E-01     500      1
 -------------------------------------------------------
Order  125 PDE2 nonsymmetric matrix (Jacobi preconditioning)

  BiCG      1.39E-16    4.96E-02      33
  CGS       1.68E-16    1.87E+00      37
  BiCGSTAB  2.62E-09    2.93E+04     500      1
  GMRESm    1.62E-19    1.18E+19       1            X
  QMR       2.76E-15    5.54E-02     500      1
 -------------------------------------------------------
Order  125 PDE3 nonsymmetric matrix (no preconditioning)

  BiCG      9.48E-16    2.47E-03     399
  CGS       8.28E+12    4.14E+12     500      1
  BiCGSTAB  1.11E+02    5.48E+11     116    -10
  GMRESm    2.88E-17    4.79E+15       1            X
  QMR       9.09E-14    3.02E-03     500      1
 -------------------------------------------------------
Order  125 PDE3 nonsymmetric matrix (Jacobi preconditioning)

  BiCG      9.51E-16    2.47E-03     447
  CGS       2.64E+08    3.61E+12     500      1
  BiCGSTAB  4.69E+00    9.06E+11     106    -10
  GMRESm    1.28E-19    4.75E+12       1            X
  QMR       1.10E-13    3.45E-03     500      1
 -------------------------------------------------------
Order   36 PDE4 nonsymmetric matrix (no preconditioning)

  BiCG      5.60E-16    5.56E-03      42
  CGS       9.41E-01    4.60E+11     144      1
  BiCGSTAB  1.25E-01    6.84E+10     144      1
  GMRESm    5.94E-23    1.42E+13       1            X
  QMR       2.20E-14    6.95E-03     144      1
 -------------------------------------------------------
Order   36 PDE4 nonsymmetric matrix (Jacobi preconditioning)

  BiCG      5.60E-16    5.56E-03      42
  CGS       9.41E-01    4.60E+11     144      1
  BiCGSTAB  1.25E-01    6.84E+10     144      1
  GMRESm    7.21E-17    6.76E+12       1            X
  QMR       2.20E-14    6.95E-03     144      1
 -------------------------------------------------------

 ======
 LEGEND:
 ======

    ==================
    SYSTEM DESCRIPTION
    ==================

    SPD matrices:

       WATH: "Wathen Matrix": consistent mass matrix
       F2SH: 2-d Poisson problem
       F3SH: 3-d Poisson problem

       PDE1: u_xx+u_yy+au_x+(a_x/2)u
             for a = 20exp[3.5(x**2+y**2 )]

    Nonsymmetric matrices:

       PDE2: u_xx+u_yy+u_zz+1000u_x
       PDE3  u_xx+u_yy+u_zz-10**5x**2(u_x+u_y+u_z )
       PDE4: u_xx+u_yy+u_zz+1000exp(xyz)(u_x+u_y-u_z)

    =====================
    CONVERGENCE CRITERION
    =====================

    Convergence criteria: residual as reported by the
    algorithm: ||AX - B|| / ||B||. Note that NaN may signify
    divergence of the residual to the point of numerical overflow.

    ===================
    NORMALIZED RESIDUAL
    ===================

    Normalized Residual: ||AX - B|| / (||A||||X||*N*TOL).
    This is an apostiori check of the iterated solution.

    ====
    INFO
    ====

    If this column is blank, then the algorithm claims to have
    found the solution to tolerance (i.e. INFO = 0).
    This should be verified by checking the normalizedresidual.

    Otherwise:

       = 1: Convergence not achieved given the maximum number of iterations.

       Input parameter errors:

       = -1: matrix dimension N < 0
       = -2: LDW < N
       = -3: Maximum number of iterations <= 0.
       = -4: For SOR: OMEGA not in interval (0,2)
             For GMRES: LDW2 < 2*RESTRT
       = -5: incorrect index request by uper level.
       = -6: incorrect job code from upper level.

       <= -10: Algorithm was terminated due to breakdown.
               See algorithm documentation for details.

    ====
    FLAG
    ====

       X: Algorithm has reported convergence, but
          approximate solution fails scaled
          residual check.

    =====
    NOTES
    =====

    GMRES: For the symmetric test matrices, the restart parameter is
    set to N. This should, theoretically, result in no restarting. For
    nonsymmetric testing the restart parameter is set to N / 2.

    Stationary methods:

    - Since the residual norm ||b-Ax|| is not available as part of
      the algorithm, the convergence criteria is different from the
      nonstationary methods. Here we use

         || X - X1 || / || X ||.

      That is, we compare the current approximated solution with the
      approximation from the previous step.

    - Since Jacobi and SOR do not use preconditioning,
      Jacobi is only iterated once per system, and SOR loops over
      different values for OMEGA (the first time through OMEGA = 1,
      i.e. the algorithm defaults to Gauss-Siedel). This explains the 
      different residual norms for SOR with the same matrix.

