only used for analysis and discussions about gross error 
location rather than to calculation of value of Qvv.p 
matrix. 
Equation(7) and (8) satisfy the condition tr(6)=r, where 
r is the redundant number because the main diagonal 
element of equation(8,b) is 
E m 
AB SIEH au Sich Dir Eu (Sho. (exi) 
m -— 
x m = . 
and tr (AG) 7, (gi 7D gu Si E aa gy "Si +. (Kat) 
Matrix Qvv.P is a singular idempotent matrix from which 
we get 
S 22 AKXi) 
gii -E sa "Ski ‚Bi "Bi 7yzBik Exi o ^ 
m - 
i @i- Dei -§ =-8 X a $us (Kéj3 
= 3 
so tr( G) = 0 and tr(G) =r 
The expanded expressions of matrix Qvv.P are very useful 
tool for the discussion of localizing gross errors. The 
present author gave the results of capability of 
localizing gross errors about iterated weight least 
squares method ( Wang Renxiang, 1988b ) and gave the 
conclusion about the power value of negative power 
function that taken 2.5--4.0 is better then 1.0--2.0 
in ( Wang Renxiang, 1989c ) by using the first order 
expression of matrix Qvv.P. In this paper, we will 
further extend the applications about the expanded 
Qvv.P matrix. 
2. CALCULATING Qvv.P MATRIX BY USING 
FAST RECURSIVE ALGORITHM 
As iterated weight leaset squares is performed, Qvv.P 
matrix will be changed with changing weight matrix P. 
It is time consuming for caculating  Qvv.P matrix 
according to the egation(2), when the normal equation 
with large dimension. If matrix Qvv.P is computed in 
first iteration, then the sequential iterated Qvv.P 
matrix can be calculated by using equation(6) and get 
o” = firt -œ (9) 
where 
m,k = up foot-not, denoting the number of repeated 
computation 
i = the number of row and column of changing 
diagonal element of weight matrix P 
G® =1-A (APA) 1 A-P , where P is the initial 
weight matrix P 
G = computed Qvv.P according to the fast recursive 
algorithm with changed weight matrix 
^ Aere Sig 
Jon 0 
rfi 4t ee (10) 
0. 0-Sigki 0-1 
x 
If all the elements of matrix P have got their own 
imcrement, calculate with the fast recursive algorithm 
according to k=i, otherwisse, if some elements of matrix 
P nothing changed,then the calculating will have to jump 
over the order. 
Qvv.P matrix calculated from the normal equation with 
large dimension by equation(9) is more fast then by 
equation(2), because after the first iteration, there 
is no inverb matrix in the computation. 
3. THE PROPERTIES OF WEIGHT ZERO RESIDUAL 
For localizing gross errors, in some robust estimate or 
iterated least squares method, at least the last 
iteration is always weighted zero value ( or near zero ) 
to the observation of which residual is rather large 
value in absolute. In this paper,we defined the residual 
computed with weighted zero to the observation as 
so-called ' weighted zero residual ' and symbolized Vi 
( Stefanovic, 1985 called ' swep residual 3. Tt is 
necessary to investigate the properties of weighted zero 
residual for further discussion about gross errors 
localization. 
Asumming that P = I. Firstly, we assign zero weight to 
observation i, ie. spi=-1 and Si--l/ gj. According 
to equation(9) we have 
1 
G! - Ti -G 
Using equatio(9) and (10), we get 
$= g. =1 j=1,m 
Qq, 
From above, we know that all the elements of i th 
column are enlarged by a factor of 1/g; If only one 
observation i is assigned zero weight then its weighted 
zero residual y; can be calculated from vi directly. 
ie. 
Vi=V;/% (11) 
On the otherhand, the elements of k th column will be 
BH Spada faut k=1, m, Ki 
Using 
Oik = dik/ y didkk 7 du / / qi Gre 
1 2 
then Bix = kk - Pj, k* Aki =(1-P; kx 
Let P =0.7 9 = 940,5 
In this cse, 
we get 20,0. 1 g. f. 25 
if gross error V, is included in 
observation k then to compare the magnitude of 
residuals, the observation i will be larger then the 
observation k. So, when correlation coefficient of 
residuals is big. The gross errors localizing is not 
reliable.. If the absolute value of residual is used as 
statistical equantity. 
Secondly, we further assign zero weight to observation k 
, le. 8p,--1, Sx »-1/8k , then we get 
G6 -G!- T .Ti-G? - T]. Gg 
According to equation(9) and (10), we have 
Bü =(Qij + Quk — Qk; * Aix)/(Qii | di - dii * dik) 
x; =(Qk; + Di - ij: yfgig "Ok - gk da) 
and gi-1 ,B«-1 > Bk 20 , 8 =0 
Now there is nolonger correlation between weighted zero 
residules Vi and y, In the same way, we obtain 
: zx PRR 2 
Bij z (Qj * di - 9k; :Pi.k/di.dkk y (17 83*x) Bi *-Qkk (12) 
&ij = (4; > di = 9, -Obik/du- di, )/ (1- Pi*x) Bii * dy 
In the following, 
we take two conditions for further 
discussions. 
3.1 Let Mi,k #0 
If Pisx=0 , then 
dij = 4i;/ di, 8kj = àj/ dk