Page 34 - Compact Numerical Methods For Computers

24 Compact numerical methods for computers
Example 2.4. Surveying-data fitting

Consider the measurement of height differences by levelling (reading heights off a
vertical pole using a levelled telescope). This enables the difference between the
heights of two points i and j to be measured as
b , = h – h + r (2.29)
ij i j ij
where r is the error made in taking the measurement. If m height differences are
ij
measured, the best set of heights h is obtained as the solution to the least-squares
problem
T
minimise r r (2.30)
where
r = b – Ah

and each row of A has only two non-zero elements, 1 and -1, corresponding to
the indices of the two points involved in the height-difference measurement. Some-
times the error is defined as the weighted residual
r ij = [b – (h – h )]d ij
ij
j
i
where d is the horizontal distance between the two points (that is, the measure-
ij
ment error increases with distance).
A special feature of this particular problem is that the solution is only
determined to within a constant, h , because no origin for the height scale has
0
been specified. In many instances, only relative heights are important, as in a
study of subsidence of land. Nevertheless, the matrix A is rank-deficient, so any
method chosen to solve the problem as it has been presented should be capable of
finding a least-squares solution despite the singularity (see example 19.2).

2.4. THE INVERSE AND GENERALISED INVERSE OF A MATRIX
An important concept is that of the inverse of a square matrix A. It is defined as
-1
that square matrix, labelled A , such that
-1
-1
A A = AA = 1 n (2.31)
where 1 is the unit matrix of order n. The inverse exists only if A has full rank.
n
Algorithms exist which compute the inverse of a matrix explicitly, but these are of
value only if the matrix inverse itself is useful. These algorithms should not be
used, for instance, to solve equation (2.2) by means of the formal expression
- 1
x = A b (2.32)
since this is inefficient. Furthermore, the inverse of a matrix A can be computed
by setting the right-hand side b in equation (2.2) to the n successive columns of
the unit matrix 1 . Nonetheless, for positive definite symmetric matrices, this
n
monograph presents a very compact algorithm for the inverse in §8.2.
When A is rectangular, the concept of an inverse must be generalised. Corres-
+
ponding to (2.32) consider solving equation (2.14) by means of a matrix A , yet to
be defined, such that +
x = A b. (2.33)