4 
PHOTOGRAMMETRIC ENGINEERING 
reached. On the other hand, especially in the 
case of a large block, it has the advantages 
that during the solution the required storage 
space is not increased and that for a given 
absolute density of control the required com 
putation time is only directly proportional to 
the number of photographs. Therefore, for 
very large blocks the iterative solution will 
be preferable. 
The principal iterative method of solution 
is the Gauss-Seidel method. Although the 
proof of the convergence of this method can 
be found in mathematical textbooks, several 
writers have deemed it necessary to include 
the proof in their paper. The asymmetry in 
herent in the method can be avoided by al 
ternately proceeding forth and back through 
the vector of unknowns. 
The rate of convergence is sufficiently fast 
only if a block-iterative procedure is used. 
The simplest submatrices for this purpose are 
those obtained from the above partitioning 
of the matrices in the Equations 2d and 2e. 
For large blocks with sparse control, one may 
then require from 50 to a few hundred itera 
tions. 
The unknowns in the normal equations can 
be arranged in groups according to the natural 
or to artificial strips such that the on-diagonal 
submatrix of each group is connected via a 
non-zero off-diagonal submatrix to the on- 
diagonal submatrices of only the preceding 
and the following group. If these much larger 
submatrices are used as units in the block- 
iterative procedure, the required number of 
iterations is of the order of ten [85]. However, 
the computations in each iteration step are 
then much more complicated. 
After a number of Gauss-Seidel iterations, 
one will observe that successive corrections 
to the obtained values of the unknowns tend 
to form a geometric series. This fact may be 
used to accelerate the convergence at that 
stage. 
A simple and safe, but still rather slow, 
accelerated method consists in computing the 
ratio r of the corrections in two successive 
iterations and multiplying the set of Gauss- 
Seidel corrections in each iteration by l-\-r 
or by l-\-r-\-r 2 . 
A much faster acceleration method consists 
in computing the sum of all following terms 
of the geometric series and using this sum as 
the correction. The set of Gauss-Seidel cor 
rections is then divided by 1 — r. This is 
Luysternik’s method [32]. If the Gauss- 
Seidel corrections do not form an exact geo 
metric series, the Luysternik acceleration 
produces more or less random deviations from 
the exact solution. A number of ordinary 
Gauss-Seidel iterations is then required before 
an acceleration procedure can again be used. 
A third acceleration method is called the 
block-successive overrelaxation method [33]. 
In theory, this method makes a sophisticated 
computation of an optimum acceleration 
factor possible. In practice, such a computa 
tion is too complicated. An educated guess as 
to the value of such a factor is therefore made 
and the sophistication consists mainly in the 
vocabulary that is used. This factor will be 
larger than one, and it must be smaller than 
two. 
A very different type of iterative solution 
is obtained if an orthogonalization method or 
a gradient iterative method is used. At the 
ITC, Kubik 129. 301 has found the method of 
conjugate gradients [34] to be the most useful 
one of these methods. In theory, if no round 
off errors are introduced, this method con 
verges to the exact solution after as many 
iterations as there are unknowns. Either the 
matrix of coefficients and the constant terms 
of the correction equations or those of the 
normal equations are stored and used in 
every iteration. Experience with a test pro 
gram showed that in general a satisfactory 
solution was obtained after some 60 itera 
tions. Accordingly, the method appears to 
be competitive with the Gauss-Seidel block- 
iterative method. 
6. THE ESSA-COAST AND GEODETIC SURVEY 
PROGRAM ([5]—[ 10]) 
From the beginning, the development of 
computer programs for strip- and block- 
adjustment at the Coast and Geodetic Survey 
has followed Schmid’s approach. 
In the present system, input for the block 
adjustment is provided by a set of programs 
or sub-programs for image coordinate refine 
ment, strip triangulation, polynomial strip 
transformation, and resection of photographs. 
In the block adjustment, the Equations 2d 
are formed. They are solved directly, by a 
procedure of Gaussian elimination (forward 
solution) and back substitution (back solu 
tion) which operates with the submatrices 
for each point and those for each photograph 
as units. With floating-point arithmetic and 
14-decimal digit word size, no round-off diffi 
culties are encountered even for the maxi 
mum size of block of about 180 photographs 
with six measured points across the centre of 
each photograph. 
The solution of the Equation 2d consists in 
corrections to the approximate values of the 
photograph parameters (that is, to three