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 1.28E-16 3.08E-03 6
CGS NaN NaN 144 1
BiCGSTAB 1.29E-17 3.85E-03 6
GMRESm 2.55E-20 8.01E+15 1 X
QMR 2.58E-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 1.28E-16 3.08E-03 6
CGS NaN NaN 144 1
BiCGSTAB 1.29E-17 3.85E-03 6
GMRESm 1.27E-17 3.79E+13 1 X
QMR 2.58E-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 2.74E-18 1.24E-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 3.36E-16 1.08E-03 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 9.10E-22 6.18E-04 20
CGS NaN NaN 84 1
BiCGSTAB 6.78E-16 2.86E+12 74 X
GMRESm 8.93E-18 9.33E+11 1 X
QMR 4.12E-16 1.24E-03 20
-------------------------------------------------------
Order 27 SPD 3-d Poisson matrix (no preconditioning)
CG 2.83E-17 3.33E-03 4
Chebyshev 9.50E-16 1.41E-02 68
SOR 9.92E-16 2.50E-03 24
BiCG 2.83E-17 3.33E-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 4.15E-16 6.45E-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 4.74E-01 108 1
SOR 9.92E-16 2.50E-03 24
BiCG 2.41E-17 4.58E-03 4
CGS 4.09E-17 3.33E-03 4
BiCGSTAB 4.29E-18 4.99E-03 4
GMRESm 1.61E-17 5.25E+12 1 X
QMR 2.50E-16 3.74E-03 4
-------------------------------------------------------
Order 125 PDE1 nonsymmetric matrix (no preconditioning)
BiCG 2.13E-16 1.35E-03 65
CGS 8.11E-16 6.68E-03 91
BiCGSTAB 6.90E-16 5.12E-03 100
GMRESm 5.02E-16 1.30E+21 1 X
QMR 1.44E-14 1.41E-02 500 1
-------------------------------------------------------
Order 125 PDE1 nonsymmetric matrix (Jacobi preconditioning)
BiCG 7.38E-16 7.58E-02 60
CGS 3.76E-16 2.29E-02 71
BiCGSTAB 2.96E-16 4.05E-03 84
GMRESm 9.03E-26 4.17E+21 1 X
QMR 9.12E-14 6.65E-02 500 1
-------------------------------------------------------
Order 125 PDE2 nonsymmetric matrix (no preconditioning)
BiCG 7.58E-16 3.55E-02 31
CGS 9.46E-16 1.88E+00 40
BiCGSTAB 9.61E-03 1.36E+11 248 -10
GMRESm 3.44E-16 1.53E+22 1 X
QMR 1.03E-14 9.08E-02 500 1
-------------------------------------------------------
Order 125 PDE2 nonsymmetric matrix (Jacobi preconditioning)
BiCG 8.97E-16 4.95E-02 31
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 1.44E-14 9.14E-02 500 1
-------------------------------------------------------
Order 125 PDE3 nonsymmetric matrix (no preconditioning)
BiCG 1.18E-14 3.86E-03 500 1
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 6.97E-13 8.26E-03 500 1
-------------------------------------------------------
Order 125 PDE3 nonsymmetric matrix (Jacobi preconditioning)
BiCG 8.28E-16 2.34E-03 443
CGS 2.64E+08 3.61E+12 500 1
BiCGSTAB 4.69E+00 9.06E+11 106 -10
GMRESm 7.19E-19 4.64E+12 1 X
QMR 6.01E-14 2.22E-03 500 1
-------------------------------------------------------
Order 36 PDE4 nonsymmetric matrix (no preconditioning)
BiCG 1.10E-16 1.11E-02 42
CGS 9.41E-01 4.60E+11 144 1
BiCGSTAB 1.25E-01 6.84E+10 144 1
GMRESm 8.28E-22 1.43E+13 1 X
QMR 1.47E-14 8.35E-03 144 1
-------------------------------------------------------
Order 36 PDE4 nonsymmetric matrix (Jacobi preconditioning)
BiCG 1.10E-16 1.11E-02 42
CGS 9.41E-01 4.60E+11 144 1
BiCGSTAB 1.25E-01 6.84E+10 144 1
GMRESm 5.95E-18 1.64E+13 1 X
QMR 1.47E-14 8.35E-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.