
 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
 ======================================================

Order  36 SPD 2-d Poisson matrix (no preconditioning)

  CG        1.35E-16    3.08E-03       6
  Chebyshev 6.17E-11    3.29E+02     144      1
  SOR       2.18E-14    2.39E-01     144      1
  BiCG      1.35E-16    3.08E-03       6
  CGS       8.17E-16    4.63E-03       6
  BiCGSTAB  2.60E-17    3.08E-03       6
  GMRESm    9.70E-16    7.71E-03       6
  QMR       2.72E-16    3.08E-03       6
  Jacobi    3.02E-08    6.16E+05     144      1
 -------------------------------------------------------
Order  36 SPD 2-d Poisson matrix (Jacobi preconditioning)

  CG        1.35E-16    3.08E-03       6
  Chebyshev 1.20E-06    6.41E+06     144      1
  SOR       2.18E-14    2.39E-01     144      1
  BiCG      1.35E-16    3.08E-03       6
  CGS       8.17E-16    4.63E-03       6
  BiCGSTAB  2.60E-17    3.08E-03       6
  GMRESm    2.43E-16    7.71E-03       6
  QMR       2.72E-16    3.08E-03       6
 -------------------------------------------------------
Order   21 SPD Wathen matrix (no preconditioning)

  CG        8.25E-18    1.52E-03      26
  Chebyshev 1.52E-02    2.32E+10      84      1
  SOR       8.41E-11    3.28E+02      84      1
  BiCG      8.25E-18    1.52E-03      26
  CGS       4.03E-16    3.51E-02      27
  BiCGSTAB  1.32E-16    1.10E-03      29
  GMRESm    7.97E-16    8.35E-04      35
  QMR       4.42E-16    7.49E-04      26
 -------------------------------------------------------
Order   21 SPD Wathen matrix (Jacobi preconditioning)

  CG        8.34E-22    1.31E-03      20
  Chebyshev 5.58E-01    5.28E+11      84      1
  SOR       8.41E-11    3.28E+02      84      1
  BiCG      8.34E-22    1.31E-03      20
  CGS       2.42E+25    7.90E+12      84      1
  BiCGSTAB  2.80E-13    2.21E+12      84      1
  GMRESm    1.93E-17    5.21E-04      17
  QMR       2.67E-16    3.62E-04      20
 -------------------------------------------------------
Order  27 SPD 3-d Poisson matrix (no preconditioning)

  CG        2.00E-17    2.91E-03       4
  Chebyshev 9.50E-16    1.25E-02      68
  SOR       1.08E-16    2.91E-03      25
  BiCG      2.00E-17    2.91E-03       4
  CGS       1.34E-17    2.50E-03       4
  BiCGSTAB  1.96E-18    2.50E-03       4
  GMRESm    1.14E-16    3.33E-03       4
  QMR       3.69E-16    4.16E-03       4
  Jacobi    7.35E-16    1.33E-02      98
 -------------------------------------------------------
Order  27 SPD 3-d Poisson matrix (Jacobi preconditioning)

  CG        2.47E-17    3.33E-03       4
  Chebyshev 6.75E-14    4.80E-01     108      1
  SOR       1.08E-16    2.91E-03      25
  BiCG      2.47E-17    3.33E-03       4
  CGS       9.51E-18    3.74E-03       4
  BiCGSTAB  1.62E-18    3.74E-03       4
  GMRESm    6.35E-17    4.37E-03       4
  QMR       4.53E-16    6.66E-03       4
 -------------------------------------------------------
Order  125 PDE1 nonsymmetric matrix (no preconditioning)

  BiCG      1.82E-16    1.38E-02      65
  CGS       1.96E-16    1.05E-02      96
  BiCGSTAB  2.12E-16    1.32E-02     102
  GMRESm    9.18E-16    1.29E-03      27
  QMR       1.56E-15    1.78E-03     500      1
 -------------------------------------------------------
Order  125 PDE1 nonsymmetric matrix (Jacobi preconditioning)

  BiCG      2.27E-16    3.86E-02      61
  CGS       1.48E-16    5.82E-03      69
  BiCGSTAB  9.77E-16    3.02E-03      85
  GMRESm    1.44E-16    1.52E-02      27
  QMR       1.66E-14    8.35E-02     500      1
 -------------------------------------------------------
Order  125 PDE2 nonsymmetric matrix (no preconditioning)

  BiCG      2.32E-16    3.85E-02      31
  CGS       7.53E-16    1.56E+00      45
  BiCGSTAB  1.44E-11    1.75E+02     500      1
  GMRESm    5.75E-16    7.00E-03      25
  QMR       6.99E-15    6.88E-02     500      1
 -------------------------------------------------------
Order  125 PDE2 nonsymmetric matrix (Jacobi preconditioning)

  BiCG      4.39E-16    3.94E-02      31
  CGS       2.88E-02    2.13E+11     500      1
  BiCGSTAB  8.27E-06    1.06E+08     500      1
  GMRESm    3.13E-16    7.95E-03      25
  QMR       4.16E-15    4.58E-02     500      1
 -------------------------------------------------------
Order  125 PDE3 nonsymmetric matrix (no preconditioning)

  BiCG      8.06E-16    3.88E-03     445
  CGS       4.99E+05    5.32E+12     500      1
  BiCGSTAB  2.19E+01    4.48E+11      58    -10
  GMRESm    1.41E-01    3.37E+07     500      1
  QMR       1.21E-13    4.56E-03     500      1
 -------------------------------------------------------
Order  125 PDE3 nonsymmetric matrix (Jacobi preconditioning)

  BiCG      7.49E-16    3.13E-03     410
  CGS       5.87E+12    4.90E+12     500      1
  BiCGSTAB  5.25E+00    1.43E+12     234    -10
  GMRESm    2.66E-02    3.08E+07     500      1
  QMR       1.36E-13    5.81E-03     500      1
 -------------------------------------------------------
Order   36 PDE4 nonsymmetric matrix (no preconditioning)

  BiCG      1.81E-16    1.08E-02      42
  CGS       6.81E-02    3.03E+10     144      1
  BiCGSTAB  5.07E-02    2.90E+10     144      1
  GMRESm       -NaNQ        NaNQ     144      1
  QMR       2.08E-14    1.02E-02     144      1
 -------------------------------------------------------
Order   36 PDE4 nonsymmetric matrix (Jacobi preconditioning)

  BiCG      1.81E-16    1.08E-02      42
  CGS       6.81E-02    3.03E+10     144      1
  BiCGSTAB  5.07E-02    2.90E+10     144      1
  GMRESm       -NaNQ        NaNQ     144      1
  QMR       2.08E-14    1.02E-02     144      1
 -------------------------------------------------------

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

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

    SPD matrices:

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

    Nonsymmetric matrices:

       PDE1: u_xx+u_yy+au_x+(a_x/2)u
             for a = 20exp[3.5(x**2+y**2 )]
       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. Note that if a
    NaN occurs in the convergence criterion (discussed above,
    a NaN will be seen here also.

    ====
    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 normalized residual.

    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

       <= -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.

