Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

Suppose that an approximation & to p is known. Then neglecting
terms of order 0 (E - %) ', we have
. Eq. (6.22) constitutes a system of N equations for the M parameter
changes pk - pko. The generalized matrix inversion provides a so-
lution to this system, irrespective of N = M, N < M, or N > M. If
the rank of the system matrix agi/apk is equal to Min (M, N), the
generalized matrix inversion provides in the case M = N (regular
system): the ordinary solrition,M < N (overconstrained system): the
least squares solution,M > N (underdetermined system): the
smallest correction vector p - %.
The generalized inverse exists also in the case when the rank of
agi/apk is smaller than Min(M, N).
After solving the system, the correction is applied to and this
vector in the next step serves as a new approximation to p, thus
starting an iterative scheme.
It is convenient to give all data the same variance ci2 thus
0'
defining as new data and matrix elements
thus weighing in a least squares solution the residuals according
to their accuracy which makes sense. Let
be the parameter correction vector. Then (6.22) reads
corresponds to the data kernels Gi(r) in the Backus-Gilbert
theory). In the generalized matrix inversion first G - is decomposed
into data eigenvectors u and parameter eigenvectors v :
-1 -j
G = U A
T
~
(6.26)
- =--