partition of the set X is made so that 
every equation concerns only variables of 
two or one (index 'i' can correspond to 
'j') sets of the partition. The meaning of 
this assumption will be more manifest in 
the next paragraph; now, let' s define the 
graph G-(V,E) for structuring the matrix. 
Definition 1.1. Under the assumption of 
the hypothesis 1.1, let G-(V,E) be the 
graph defined in the oT Towing way: 
-V= (ni,ns, n nn}; 
5 = ((nı,n;)[3 X;, X; and f(X,,X4)-0 ]. 
Every set X; is represented in the graph 
by the node ni and all equations 
considering the variables of X; and Xj by 
the edge (nj, ny y. 
Once the graph is built, the next step is 
to number its nodes and edges and, using 
this order, to rearrange variables and 
equations respectively. Formally this 
operation corresponds to define two 
functions: 
V2 Vo-7»5 41,2,... D) V(ni)=j 
Zar BE ==> [1,2,.-..,} E((ni,nj))=k 
where 'n' is the number of nodes and 'm' 
the number of edges. 
The classic approach also structures 
matrices in a similar way, but it 
associates one node to every variable and 
an edge to every not null coefficient of 
the matrix. The proposed definition 1.1 of 
the graph implies two advantages. The 
first - one. is the. reduction of ‚the 
dimension of the sets V and E; infact the 
classic approach creates a .biunivocal 
correspondence between one node and one 
variable, so that BALIPSP while 
definition 1.1 associates one node to one 
set X; of the variables. In this case |V| 
is usually less then |X| and only in the 
worst case |V|=|E|. The difference becomes 
more evident if we evaluate the dimension 
of the set E because one edge includes all 
the equations between two sets of the 
partition (see the table of the next 
paragraph). 
The second advantage regards the 
linearized systems of equations. In this 
case the solution is obtained at the end 
of an interactive process in which many 
different linear equation systems are 
solved. Chosen an initial point, the first 
computed solution is considered a better 
approximation of the not linear system 
and so it's used for another linearization 
that produces a new linear system to 
solve. In the classic approach the graph 
must be rebuilt and the nodes renumbered 
at every step because set E changes. On 
the contrary using the graph defined at 
1.1, functions V() and E() have to be 
fixed just one time, at the beginning. 
This is due to the fact that the graph is 
defined without any assumption on the 
linear system; not only, hypothesis 1.1 
doesn't require any information on the 
type of the equation (i.e. integral, 
differential, trigonometric,..); the only 
important thing is which variables are 
related by the equations; consequently, 
the graph is not related to the linear 
system but, more properly, to the nature 
of the problem. 
102 
1.2 Block adjustment application 
Let us consider the block adjustment 
relationships, by independent models, for 
the coordinates of the same tie point 
relating to two different models. 
Xh,i Tx,i Xp, j Tx,3 
MRilYn,i|*tlTy,i| 7 A3Rj Yn3|- Tod ^9 
Zn,i 2,1 h,j Tz,j 
where the vector (Xn, irYn,irZn,4]* contains 
the model coordinates of the h tie point 
in model 'i'; the vector (T,,irTy,irTz,i) 
contains the three translation parameters 
of model 'i'; R, is the rotation matrix of 
model 'i' and, M inaliv, '^;' is the scale 
factor. The same parameters can be defined 
for model 'j'. The relationship contains 
fourteen variables: 
-O Me RIT. i Tv. ir 
- x ‚x PE Re (Ty 3rTy, 37 Tz, 
representing ho ‘sévèn known parameters 
associated to each model. 
For control points the following equations 
can be defined: 
Xn Xni Tx,i 
Yn = M Ri Yh,i + Ty,i 
Zn Zh,i Tz,i 
where the vector (Xy,Y,,25)* contains the 
ground coordinates of the nt control 
point. In this or the variables are: 
- seven (A;,Q;,®; T eielsoi'la i). Li. Che 
control point is Fo both in altimetry 
and planimetry. 
T. 
‚i 
- eight (,,0,,9;,K, 1 Tx,1rTy, 1: Ty, ir Zn) if 
the control point is’ known only in 
planimetry. 
- nine (51,0, 9, , Ki, T,, irTy 
the control point is 
altimetry. 
Now, suppose to index all the 
points in the following way; 
(Tg, irXn»; Yn) if 
OW only in 
control 
- from 1 to p all points known in 
altimetry; 
- from p*1 to q those one known in 
planimetry; 
- from q*1 to r those one known in 
planimetry and altimetry. 
Tf 
2 Mj - Duis í91, Ki, Ty 1; Ty, 17 Tz,1) i-1..n 
- Nj = {X;,¥5}  i=1...p, 
Ny = (21 } i=p+l...q, 
then, the set of variables X can be 
written in the following way: 
xe(lcass Mio NUUG, 
Note that if all points are known both in 
altimetry and planimetry, the previous 
expression reduces to: 
1.-p =p+1..g N;) 
X = "n s Mi 
that is to say, the set of variables 
corresponds only to the orientation 
parameters of the models. Known points in 
planimetry or in altimetry add to the 
linear system one or two new variables. 
It' s manifest that: 
a) (My,.., My, N,,....,Nq) is a partition 
of X; 
and 
E2 
alt 
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