<s>
Least-squares	B-Algorithm
adjustment	I-Algorithm
is	O
a	O
model	O
for	O
the	O
solution	O
of	O
an	O
overdetermined	O
system	O
of	O
equations	O
based	O
on	O
the	O
principle	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
of	O
observation	O
residuals	O
.	O
</s>
<s>
It	O
is	O
used	O
extensively	O
in	O
the	O
disciplines	O
of	O
surveying	O
,	O
geodesy	O
,	O
and	O
photogrammetry	B-Application
—	O
the	O
field	O
of	O
geomatics	O
,	O
collectively	O
.	O
</s>
<s>
There	O
are	O
three	O
forms	O
of	O
least	B-Algorithm
squares	I-Algorithm
adjustment	I-Algorithm
:	O
parametric	O
,	O
conditional	O
,	O
and	O
combined	O
:	O
</s>
<s>
One	O
can	O
proceed	O
to	O
Taylor	O
series	O
expansion	O
of	O
the	O
equations	O
,	O
which	O
results	O
in	O
the	O
Jacobians	O
or	O
design	B-Algorithm
matrices	I-Algorithm
:	O
the	O
first	O
one	O
,	O
</s>
<s>
In	O
the	O
parametric	O
adjustment	O
,	O
the	O
second	O
design	B-Algorithm
matrix	I-Algorithm
is	O
an	O
identity	O
,	O
B	O
=	O
-I	O
,	O
and	O
the	O
misclosure	O
vector	O
can	O
be	O
interpreted	O
as	O
the	O
pre-fit	O
residuals	O
,	O
,	O
so	O
the	O
system	O
simplifies	O
to	O
:	O
</s>
<s>
which	O
is	O
in	O
the	O
form	O
of	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
.	O
</s>
<s>
In	O
the	O
conditional	O
adjustment	O
,	O
the	O
first	O
design	B-Algorithm
matrix	I-Algorithm
is	O
null	O
,	O
A	O
=	O
0	O
.	O
</s>
<s>
For	O
the	O
more	O
general	O
cases	O
,	O
Lagrange	O
multipliers	O
are	O
introduced	O
to	O
relate	O
the	O
two	O
Jacobian	O
matrices	O
and	O
transform	O
the	O
constrained	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
problem	O
into	O
an	O
unconstrained	O
one	O
(	O
albeit	O
a	O
larger	O
one	O
)	O
.	O
</s>
<s>
Given	O
the	O
matrices	O
and	O
vectors	O
above	O
,	O
their	O
solution	O
is	O
found	O
via	O
standard	O
least-squares	B-Algorithm
methods	I-Algorithm
;	O
e.g.	O
,	O
forming	O
the	O
normal	B-Algorithm
matrix	I-Algorithm
and	O
applying	O
Cholesky	O
decomposition	O
,	O
applying	O
the	O
QR	O
factorization	O
directly	O
to	O
the	O
Jacobian	O
matrix	O
,	O
iterative	B-Algorithm
methods	I-Algorithm
for	O
very	O
large	O
systems	O
,	O
etc	O
.	O
</s>
<s>
Helmert	O
)	O
,	O
is	O
related	O
to	O
the	O
errors-in-variables	B-Algorithm
models	I-Algorithm
and	O
total	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
If	O
rank	O
deficiency	O
is	O
encountered	O
,	O
it	O
can	O
often	O
be	O
rectified	O
by	O
the	O
inclusion	O
of	O
additional	O
equations	O
imposing	O
constraints	O
on	O
the	O
parameters	O
and/or	O
observations	O
,	O
leading	O
to	O
constrained	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
