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 = 5.96E-08
CONVERGENCE TEST TOLERANCE = 1.00E-07
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 5.56E-08 1.66E-02 6
Chebyshev 8.79E-08 3.73E-02 97
SOR 8.66E-08 1.45E-02 72
BiCG 5.56E-08 1.66E-02 6
CGS 2.16E-05 1.22E+00 122
BiCGSTAB 1.42E-08 1.86E-02 6
GMRESm 2.15E-08 8.01E+07 1 X
QMR 9.93E-08 2.07E-02 6
Jacobi 9.21E-08 2.90E-02 136
-------------------------------------------------------
Order 36 SPD 2-d Poisson matrix (Jacobi preconditioning)
CG 5.56E-08 1.66E-02 6
Chebyshev 1.20E-06 1.16E-01 144 1
SOR 8.66E-08 1.45E-02 72
BiCG 5.56E-08 1.66E-02 6
CGS 2.16E-05 1.22E+00 122
BiCGSTAB 1.42E-08 1.86E-02 6
GMRESm 9.12E-09 3.79E+05 1 X
QMR 9.93E-08 2.07E-02 6
-------------------------------------------------------
Order 21 SPD Wathen matrix (no preconditioning)
CG 8.47E-08 6.64E-03 23
Chebyshev 1.52E-02 2.32E+02 84 1
SOR 9.65E-08 4.15E-03 56
BiCG 8.47E-08 6.64E-03 23
CGS 1.79E-08 3.17E-01 48
BiCGSTAB 4.07E-08 1.16E-02 21
GMRESm 1.37E-08 2.95E+07 1 X
QMR 2.99E-07 4.98E-03 84 1
-------------------------------------------------------
Order 21 SPD Wathen matrix (Jacobi preconditioning)
CG 4.36E-08 5.81E-03 15
Chebyshev 5.58E-01 5.28E+03 84 1
SOR 9.65E-08 4.15E-03 56
BiCG 4.36E-08 5.81E-03 15
CGS 3.73E+10 7.76E+04 84 1
BiCGSTAB 1.14E-08 3.95E+04 51 X
GMRESm 8.16E-08 9.33E+03 1 X
QMR 1.06E-07 9.96E-03 84 1
-------------------------------------------------------
Order 27 SPD 3-d Poisson matrix (no preconditioning)
CG 1.16E-08 8.94E-03 4
Chebyshev 7.01E-08 2.68E-02 30
SOR 9.87E-08 1.79E-02 12
BiCG 1.16E-08 8.94E-03 4
CGS 5.67E-09 1.79E-02 4
BiCGSTAB 6.69E-10 1.12E-02 4
GMRESm 1.50E-08 4.50E+07 1 X
QMR 4.08E-08 1.79E-02 4
Jacobi 6.94E-08 1.12E-02 47
-------------------------------------------------------
Order 27 SPD 3-d Poisson matrix (Jacobi preconditioning)
CG 8.26E-08 1.12E-02 4
Chebyshev 8.94E-08 1.01E-01 51
SOR 9.87E-08 1.79E-02 12
BiCG 8.26E-08 1.12E-02 4
CGS 1.87E-08 1.79E-02 4
BiCGSTAB 4.67E-10 2.01E-02 4
GMRESm 2.65E-08 5.25E+04 1 X
QMR 1.42E-07 2.01E-02 108 1
-------------------------------------------------------
Order 36 PDE4 nonsymmetric matrix (no preconditioning)
BiCG 5.31E-08 3.36E-02 48
CGS 5.49E-01 1.86E+03 144 1
BiCGSTAB 3.05E+00 2.56E+04 103 -10
GMRESm 7.36E-09 1.42E+05 1 X
QMR 7.59E-06 4.11E-02 144 1
-------------------------------------------------------
Order 36 PDE4 nonsymmetric matrix (Jacobi preconditioning)
BiCG 5.31E-08 3.36E-02 48
CGS 5.49E-01 1.86E+03 144 1
BiCGSTAB 3.05E+00 2.56E+04 103 -10
GMRESm 3.89E-08 8.80E+04 1 X
QMR 7.59E-06 4.11E-02 144 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.