all updating is 
cal instabilities 
normal matrix 
sign matrix is 
or sequentially 
an on-line 
vector can be 
ess by back 
accomodate 
A version of 
ntleman (1973) 
, reduces the 
and facilitates 
"fast" recursive 
ea of parallel 
> square root 
y, the ability to 
ositive definite 
r the update of 
iment of close- 
quare root free 
isformations 
is for standard 
Gauss-Markov 
rvation vector / 
> n, the goal is 
in such a way 
he elements of 
| by 
(1) 
servations, the 
(2) 
the Cholesky 
pper triangular 
A'A = U'DU, 
diagonal. The 
wo is that the 
S not. 
the right hand 
(3) 
(4) 
here Q is an m 
Risannxn 
ations may be 
R'Q'QR$ - R'Q'I. (5) 
Because H is 
Q is orthogonal hence OO = L 
nonsingular if AlAis, Eq. 5 reduces to 
Ré sl. (6) 
Equations 4 and 6 are then equivalent with U = R and d = 
ol. 
R and d are obtained by applying a series of orthogonal 
transformations to A and I. Thus the solution to this 
system may be determined without forming the normal 
equation matrix directly. 
Extending the system with an observational weight matrix 
P (assuming uncorrelated observations) the least- 
squares solution is given by 
X-(ATPA) ! A! PI. (7) 
Because P is diagonal, A can be simply premultiplied by 
P^ and the QR decomposition then applied to this 
modified design matrix. 
Now assume a sequential process where Eq. 4 
represents the reduced system at stage k - 1. The 
addition, deletion, or replacement of observations via 
orthogonal transformations is shown below and follows 
closely that of Gruen (1985). The addition of one 
observation equation to stage k - 1, including a set of new 
unknown parameters, results in the following form at 
stage k 
U|x| d 
a x: Um L (8) 
Here, a, represents the coefficient vector of the added 
observation, x’ is the new p x 1 parameter vector, and /, 
is the right hand side of the new observation equation. 
The total number of system parameters is n. Applying a 
series of n orthogonal transformations (in our case 
Givens Transformations) 
Q= 0,0, Q. (9) 
to Eq. 8 yields 
Q 50::9]Tp. oom 2 (10) 
and for the right hand side, 
Q 0 Ip = : (11) 
Jom Las 101 
Zeros in Eqs. 10 and 11 show that when new parameters 
are introduced, the rows and columns of the existing U 
matrix and the existing d vector must be extended with 
zeros. 
Finally, the solution vector for the updated system is 
obtained by back substitution into 
U| * |2d. (12) 
2.2 Givens Transformations 
To illustrate the use of Givens Transformations for the 
addition of one observation into an existing system, an 
expanded form of Eq. 8 is shown in Figure 1. 
  
Uy, Uy 4g c H4 À 
Uy Wa o Uy, d, 
Uzz + Un À 
Un d, 
Q 
a a, dy tis, d, l 
  
  
  
Figure 1: U matrix augmented by new coefficient 
vector 
Consider one row vector from the system Ux - d and a 
coefficient vector from the system Ax = /, 
Oe Ou ty +o Uy oo 
13 
0. Oa, a,, + a (13) 
One Givens Transformation replaces these two vectors 
with 
’ ’ , 
0 soe 0 u; Ui ..>. Uu. eo. 
(14) 
’ ’ 
0-00 al, aj + 
where 
u, = cu, + sa, 
a, =—su, +ca, (15) 
c+s°=1 
To annihilate a; to zero, the rotation parameters are 
computed from the diagonal elements of U and the 
135 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996