are 
zing 
alue 
APPENDIX 
MATHERMATICAL 
ABSTRICT: 
The increment of Qvv.P matrix due to the variation of 
weight matrix P can be exponded by using Neumann’s 
series and obtained both approximat and regirous 
expressions, which can be applicated in discussing the 
problams about the capability and the relibility of 
gross errors location. 
The appendix emphasizes in discussing the proparties of 
co-called ' weighted zero residual ' and the fast 
recursive algorithm for calculating Qvv.P matrix and its 
limitations. Serval examples with simulated data have 
been computed for the discussions. 
KEYWORDS: Qvv.P matrix, Standardized residual, Gross 
error location, Weighted zero residual. 
INTRODUCTION 
up to now, there are many weight functions used for 
localizing gross error by iterated weight least squares 
method and robust estimation. There is no unitized 
theory in use. However the characteristics of matrix 
Qvv.P and the variability of the relationship between 
Qvv.p and weight matrix p can be used for discussing the 
problems of localizing gross errors in a general way. 
The appendix is based on the papers(Wang Renxiang,1986a, 
1988b ) and makes further development. The results would 
be benefilted for the investigation of gross errors 
localization. 
1. THE RELATIONSHIP BETWEEN Qvv.P MATRIX 
AND THE INCREMENT OF MATRIX P 
1.1 The Expanded Qvv.p Matrix 
  
According to least squares method, the residuals of 
observation will be 
Vz-G-E (1) 
G= Quv-p=I1-A-@Par" Alp (2) 
where A = the design matrix 
Qvv = the cofactor matrix of residuals 
P = the weight matrix 
E = the vector of observational errors 
let  N-APA R= ANA, u=R-P 
then G = I-U (3) 
to simply, we take matrix P is a diagonal one, ie. 
P = diaglpi ^ Pi Ps 
if AP is the increment of P, ie. P =P + AP 
where 
AP = diag (Sp,--- Sp;--- ôpm /= 
Pi 
"M3 
APi -diag (0-6 p. 0) 
6pi- the increment of Pi 
According to the least squares method we get 
G-1-ÜU, U-R-P- AN 1A. P, N-APA-N-* AN , AN- AAPA 
ANALYSIS ABOUT Qvv.P MATRIX 
N" can be expanded by using Neumann’s series and 
obtained 
TE (cif (RAP -G (4) 
G = G+AG (5) 
1.2 The Increment of Qvv.P as Weight Matrix P only p. 
changs with 8pi 
As Pi gets an increment sp; , the increment of Qvv.P 
matrix will be 
5Pi SP © Sno 
1 
= —— (Bii —-1)———z— 
agi {14 185-1555 
1 
RD; n = - 
+((Bii.— 1) (3E ) cec )oRCOC 
1 
the equation above can be compressed as 
AG=Si -ARi:G 
where 
6p, Sp; 
S,7—— (1-7 @u-1)-—) 7% 
Pi Pi 
Bii 
AR; = ( 9 (gii- 1) 9 ) 
B mi 
then 6 = T;- G (6) 
1 0--Si - Bi 0 0 
where Tus | 0-41 9s, 085 42-9 
Equation(6) is a regorous expression for the changed 
Qvv.P matrix and provides that the denominator of Si is 
not equal to zero, ie. 
GPr 
Pi (j-(8ii-1 75 = 0 
1.3 The Approximate Expressions of Qvv.P Matrix 
  
Excluding the height order of equation(4), we get 
; ópi. q 
G=6-R( 89; )-6 (7) 
o “6Pm 
using Si instead of Spi( i= 1, m) in equation(7), we 
have 
y G 
(8,a) 
Equation(8,a) can also be expressed as 
: Si. P 
G -G-R ( Sn 
9 Sm 
(en ZU SS Bm "Sm 
a =G + Ei «S, ...... Cg 17 Si Pm . Sn . (8 b) 
gmi Si "mi Si-- {2 mm — 1)-Sm 
where 
Si £Si-Pi 
Equation(8,a) or (8,b) is of more precision than 
equation(7). However all the approximate equations are