Page 253 - Compact Numerical Methods For Computers

240 Compact numerical methods for computers
Example 19.2. Surveying-data fitting
The output below illustrates the solution of a linear least-squares problem of the
type described in example 2.4. No weighting of the observations is employed here,
though in practice one would probably weight each height-difference observation
by some factor inversely related to the distance of the observation position from
the points whose height difference is measured. The problem given here was
generated by artificially perturbing the differences between the heights b=
T
(0, 100, 121, 96) . The quantities G printed are the residuals of the normal
equations.
RUN
SURVEYING LEAST SQUARES
# OF POINTS? 4
# OF OBSERVATIONS? 5
HEIGHT DIFF BETWEEN? 1 AND? 2=? -99.99
HEIGHT DIFF BETWEEN? 2 AND? 3=? -21.03
HEIGHT DIFF BETWEEN? 3 AND? 4=? 24.98
HEIGHT DIFF BETWEEN? 1 AND? 3=? -121.02
HEIGHT DIFF BETWEEN? 2 AND? 4=? 3.99
B( 1 )=-79.2575 G= 2.61738E-6
B( 2 )= 20.7375 G=-2.26933E-6
B( 3 )= 41.7575 G=-6.05617E-6
B( 4 )= 16.7625 G=-5.73596E-6
DIFF( 1 )=-99.995
DIFF( 2 )=-21.02
DIFF( 3 )= 24.995
DIFF( 4 )=-121.015
DIFF( 5 )= 3.97501
# MATRIX PRODUCTS= 4
HEIGHT FORM B(1)=0
2 99.995
3 121.015
4 96.02
The software diskette contains the data file EX24LSl.CNM which, used with the driver
DR24LS.PAS, will execute this example.

As a test of the method of conjugate gradients (algorithm 24) in solving
least-squares problems of the above type, a number of examples were generated
using all possible combinations of n heights. These heights were produced using a
pseudo-random-number generator which produces numbers in the interval (0,1).
All m=n*(n-1)/2 height differences were then computed and perturbed by
pseudo-random values formed from the output of the above-mentioned generator
minus 0·5 scaled to some range externally specified. Therefore if S1 is the scale
factor for the heights and S2 the scale factor for the perturbation and the function
RND(X) gives a number in (0,1), heights are computed using
S1*RND(X)

and perturbations on the height differences using
S2*[RND(X)-0·51.
Table 19.2 gives a summary of these calculations. It is important that the