Norm computation

[John R. Wicks]

I recently peformed solved a linear system of very high dimension
distributed over 32 Mac XServe's.  I was rather surprised by the performance
statistics it reported, given below.  In particular, how can VecNorm be so
much more expensive than VecDot, since VecNorm should simply involve taking
a single square root of a dot product.

--- Event Stage 2: LinearSolve

MatMult               19 1.0 1.8057e+01 1.7 1.19e+08 1.7 1.9e+04 5.5e+04
0.0e+00  2 17 49 49  0  16 17 50 50  0  2214
MatMultTranspose      19 1.0 1.6234e+01 2.1 1.73e+08 2.1 1.9e+04 5.5e+04
0.0e+00  2 18 49 49  0  11 18 50 50  0  2601
MatSolve              20 1.0 1.2656e+01 1.5 1.45e+08 1.5 0.0e+00 0.0e+00
0.0e+00  1 18  0  0  0  10 18  0  0  0  3200
MatSolveTranspos      20 1.0 1.3608e+01 1.5 1.40e+08 1.5 0.0e+00 0.0e+00
0.0e+00  2 18  0  0  0  11 18  0  0  0  2976
MatLUFactorNum         1 1.0 1.9609e+01 6.5 2.71e+08 9.3 0.0e+00 0.0e+00
0.0e+00  1 14  0  0  0  10 14  0  0  0  1635
MatILUFactorSym        1 1.0 5.3393e+00 3.9 0.00e+00 0.0 0.0e+00 0.0e+00
1.0e+00  1  0  0  0  1   4  0  0  0  2     0
MatGetRowIJ            1 1.0 1.7881e-05 1.6 0.00e+00 0.0 0.0e+00 0.0e+00
0.0e+00  0  0  0  0  0   0  0  0  0  0     0
MatGetOrdering         1 1.0 2.4659e-01 1.6 0.00e+00 0.0 0.0e+00 0.0e+00
2.0e+00  0  0  0  0  3   0  0  0  0  3     0
VecDot                38 1.0 1.2653e+01 1.9 4.24e+07 2.2 0.0e+00 0.0e+00
3.8e+01  1  4  0  0 49  10  4  0  0 62   710
VecNorm               20 1.0 3.5348e+01 4.0 2.05e+07 6.0 0.0e+00 0.0e+00
2.0e+01  4  2  0  0 26  25  2  0  0 33   134
VecCopy                4 1.0 2.7451e-01 4.3 0.00e+00 0.0 0.0e+00 0.0e+00
0.0e+00  0  0  0  0  0   0  0  0  0  0     0
VecSet                79 1.0 8.9448e-01 1.9 0.00e+00 0.0 0.0e+00 0.0e+00
0.0e+00  0  0  0  0  0   1  0  0  0  0     0
VecAXPY               57 1.0 3.2704e+00 2.2 2.33e+08 1.5 0.0e+00 0.0e+00
0.0e+00  0  6  0  0  0   2  6  0  0  0  4118
VecAYPX               36 1.0 1.5667e+00 1.7 2.28e+08 1.3 0.0e+00 0.0e+00
0.0e+00  0  4  0  0  0   1  4  0  0  0  5430
VecScatterBegin       38 1.0 1.1066e+00 2.1 0.00e+00 0.0 3.8e+04 5.5e+04
0.0e+00  0  0 97 98  0   1  0100100  0     0
VecScatterEnd         38 1.0 1.5381e+01 2.0 0.00e+00 0.0 0.0e+00 0.0e+00
0.0e+00  2  0  0  0  0  12  0  0  0  0     0
KSPSetup               2 1.0 8.2719e-01 1.4 0.00e+00 0.0 0.0e+00 0.0e+00
0.0e+00  0  0  0  0  0   1  0  0  0  0     0
KSPSolve              20 1.0 1.2881e+01 1.5 1.42e+08 1.5 0.0e+00 0.0e+00
0.0e+00  1 18  0  0  0  10 18  0  0  0  3144
PCSetUp                2 1.0 2.5356e+01 5.5 1.96e+08 8.7 0.0e+00 0.0e+00
3.0e+00  2 14  0  0  4  13 14  0  0  5  1265
PCApply               40 1.0 4.7115e+01 2.1 1.59e+08 2.4 0.0e+00 0.0e+00
3.0e+00  5 49  0  0  4  35 49  0  0  5  2400