312 17 Datum transformation problems
order means the elimination order of the variables. However, using the Groebner
basis method requires 3017.69 seconds!
We have chosen different 7-equation combinations from the original 9 equa-
tions and we changed also the monomial order during the processing (see Table
17.5 for a few chosen sequences). The Dixon resultant solution was indifferent to
these changes, the running time always being ca. 0.016 seconds. However, the run-
ning time for the reduced Groebner basis solution was dependent on the chosen
7-equation combination and changes also with the order of variables to be elim-
inated (see Table 17.5). In case of four chosen sequences (from the 36 possible
choices), one notices considerable change in the required computational time, see
for example the last column in Table 17.5. For example, for the combination of
(f
2
, f
3
, f
4
, f
5
, f
6
, f
8
, f
9
), the computation time was 50 minutes using the monomial
order {a, b, c} and only 0.484 seconds if we changed the monomial order to {b, c, a}!
With Dixon resultant, the running time was only 0.016 seconds, independent from
the order of variables. The Groebner basis approach is clearly affected by both
(a) the c ombinatorial sequence and
(b) the monomial order.
These two factors are undesirable since users are not often privy to the optimal
sequence and order during data processing. However, there is a third important
factor having strong influence on the performance of the Groebner basis com-
putation: this is the elimination order. In general, using MonomialOrder − >
EliminationOrder can ensure the b e st behavior, (Lichtblau, Priv. Comm.).
Table 17.5. Running times (seconds ) for cases of different sequences and order of the
variables to be eliminated using reduced Groebner basis with relative coordinates
Sequence f
1
, f
2
, f
3
, f
4
, f
1
, f
2
, f
3
, f
4
, f
1
, f
2
, f
3
, f
4
, f
2
, f
3
, f
4
, f
5
,
Order of variables f
6
, f
7
, f
8
f
5
, f
8
, f
9
f
5
, f
7
, f
9
f
6
, f
8
, f
9
a,b,c 0.219 0.688 0.985 3017.69
a,c,b 0.36 35.921 0.672 2601.75
b,a,c 0.25 0.922 0.765 3172.48
b,c,a 163.765 0.547 33.985 0.484
c,a,b 0.719 52.328 0.562 1831.42
c,b,a 174.547 0.75 47.797 0.532
The Dixon resultant therefore proved to be faster in this case , and very robust
in that it is insensitive to the order of variables, unlike the Groebner basis. This
feature can be very important from a practical point of view, because in the case
of Groebner basis, the user should find the proper combinatorial sequence and
monomial order via a trial-error method. For the sequence (f
2
, f
3
, f
4
, f
5
, f
6
, f
8
, f
9
)
in Table 17.5 for example, only two orders can provide a solution in a reasonable
time from the six possible orders of the variables.
As a test, three Hungarian points in the ETRS89 system (x
1
, y
1
, z
1
, . . . , z
3
)
and in the local Hungarian system HD72 (Hungarian Datum 1972) (X
1
, Y
1
, Z
1
, . . . ,