- Jffi __ , M 
ADJUST E T OF 
BSE ATIONS WIT 
SIMULTANEOUS 
PUT TI NOF 
SI UALS D UN 
Dr. Nevio Rožic, ProfDr. Božidar Kanajet 
University of Zagreb, Faculty of Geodesy, Faculty of Mining, 
Geology and Petroleium, Zagreb, Croatia 
Received December 18, 1995 
Revised April 18, 1996 
Abstract 
The paper presents the possibility of modifying standard 
adjustment algorithms: direct obsen,ations, indirect 
observations, combined direct and indirect observations, and 
indirect obse1vations with constraints; residuals and 
unknowns are estimated simultaneously by solving 
appropriate systems of linear equations. Such a modification 
of standard adjustment algorithms corresponds to the use of 
modem personal computers and pocket calculators, since 
they support direct matrix algebra operations. 
Keywords: adjustment algorithms, systems of linear 
equation, personal computer 
1 INTRODUCTION 
UDC 528.181 
o s 
Modem personal computers equipped with program systems for table computations (Ingalsbe, 1988, Božic, 1994, Husnjak, 1994, Crnko et aL, 1995) 
and pocket calculators (Sharp Corporation, 1986) enable direct matrL'i: algebra 
operations. As a result of this, the use of adjustment algorithms in geodetic practice 
is nowadays more efficient than it used to be, since aH modem adjustment algorithms 
are theoretically defined with the use of matrix algebra, while computers enable their 
practical use. Computational procedures are thus accelerated and simplified and the 
possibility of errors occurenceis reduced. Professionals therefore do not need to 
know more difficu!t procedures for the solving of geodetic tasks. Direct matrix 
algebra operations also needs certain modifications to adjustment algorithms, which 
can be further adopted to possibilities for the computers. The standard form used in 
severa! publications (Wolf, 1968, Bjerhammat, 1973, Mikhail, Ackerman, 1976) is due 
to tradition not best suited to these possibilities. 
owadays the use of adjustment of indirect observations has advantages in the 
solving of various geodetic tasks (Caspary, 1988), while one of the possiblc 
Geodetski vestnik 40 (1996) l 
modifications is an algorithm which is used for simultaneous computation of 
residuals and unknowns. In the standard algorithm for indirect observations these 
quantities are determined gradually, beginning with unknowns (by solving normal 
equations) and followed by residua!s (by including unknowns into corresponding 
observation equations, Feil, 1989). A modified algorithm for indirect observations is 
presented in Hoepcke, 1980, but, if used in an appropriate way, it can also be used 
for direct observations, simultaneous adjustment of direct and indirect observations 
and the adjustment of indirect observations with constraints. 
2 DIRECT OBSERVATIONS 
functional model of direct observations is determined by a system of 
observation equations (Feil, 1989): 
v= e x- 1, P, 
mrl nxl 1xl nxl nxn 
where 
n - number of observations 
x - approximate value of unknown 
e - unit vector 
1 - vector of reduced observations 
v - vector of residuals 
P - weight matrix. 
(1) 
Unique solution to this system is obtained with the use of the least squares principle: 
/p v = minimum 
which also determines normal equations: 
From the use of the least squares principle comes out the basic control for the 
checking of correctness of residuals: 
By multiplying equation (1) with the weight matrix P from the left side and by 
rearranging the equation, we obtain: 
p V - p e X + P 1 = 0. 
Equations ( 4) and (5) determine the following system of linear equations 
(2) 
(3) 
(4) 
(5) 
(6) 
in which the matrix of coefficients is a symmetrical matrix of formats (n + 1) x (n -1). 
It should be emphasised that this matrix has the property of positive definiteness, but 
Geodetski vestnik 40 (1996) 1 
it is regu!ar and can be inverted through classical inversion. By inverting this matrix, 
the solution to the system of equations given in equation (6) is also determined: 
[v] = [ ~ Pe] 
1 
[-PIJ = [Q11 q12J[-PI] 
-x e P O O Q21 q22 O . 
In this manner, the residuals and the unknowns are estimated simultaneously. In 
equation (7), submatrices and a subvector are defined by inversion: 
q22 = -(elPe) -! = -qxx' 
lxl 
(7) 
(8) 
(9) 
-1 t 
Q11 = P - eq„ e , (10) 
nxn 
where qxx is a cofactor of the unknown. 
Since a theoretical form of the system of equations (6) is important for the practical 
use of this algorithm, it must therefore be understood in an analogy with the 
standard adjustment algorithm for direct observations, along with the system of 
observation equations. A division of system (6) into submatrices and subvectors has 
no effect on the efficiency of practical computation, since on a personal computer or 
pocket cakulator appropriate commands are used for inversion and multiplication. A 
comparison of this algorithm with a standard adjustment algorithm for direct 
observations shows that instead of solving one normal equation, equation (3), a 
system of ( n + 1) line ar equations is solved, equation ( 6), for the determination of 
unknown quantities (the corrections and the unknown). The task is not difficult, 
since matrix algebra computational operations can be used directly. 
he system of equations (6) and (7) also comprises the most general exampie of 
adjustment of direct observations, i.e. the adjustment of direct correlation 
observations. If adjustment of independent observations of different accuracy and 
direct independent observations of equal accuracy is performed, the algorithm is 
simplified, which depends on the form of the appertaining weight matrix P. 
3 INDIRECT OBSERVATIONS 
The functional model of indirect observations is also determined by a system of observation equations, but in contrast to direct observations it contains a greater 
number of unknowns: , 
v=Ax-1,P, 
nxl nxu uxl nxl nxn 
(11) 
where 
u - number of unknowns 
A - matrix of the coefficient of observation equations. 
Geodetski vestnik 40 (1996) 1 
The procedure for defining the adjustment algorithm with a simultaneous estimation 
of residuals and unknowns matches direct observations. From the use of the least 
squares principle, the control for checking the correctness of residuals is: 
By multiplying equation (11) with the weight matrix P from the left side and 
rearranging the equation, we obtain: 
P v - P A x + P l = O. 
Equations (12) and (13) determine a systern of linear equations: 
the solution of which is: 
[ v ] = [ ~ PA] - i [-PIJ = [Qu Q12J[-Pl] -x A P O O Q21 Q22 O . 
In this equation, the following submatrices are defined by inversion: 
Q22 = -(AtPA) -i = -Qxx' 
urn 
Ql2 = AQx,,' 
nxu 
Q = p -t - AQ i( 11 XX ' 
nxn 
where Qxx is a matrix of the cofactor of unknowns. 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
Equations (10) and (18) also include the matrix of cofactors of adjusted observations 
Q, i.e. in direct observations: 
Q = eqxxet 
nxn 
and in indirect observations: 
Q = AQ At. 
XX 
mm 
(19) 
(20) 
A comparison of this algorithm with the standard algorithm for adjustment of 
indirect observations shows that instead of (u) normal equations, a system of (n +u) 
linear equations is solved to estimate unknown quantities (residuals and unknowns). 
Geodetski vestnik 40 (1996) 1 
4 COMBINED DIRECT AND INDIRECT OBSERVATIONS 
In the combined form of adjustment of direct and indirect observations, a functional model is determined with error equations of direct observations: 
and observation equations of indirect observations: 
(21) 
(22) 
Unique solutions of the functional model is obtained with the use of the least squares 
principle in accordance with equation (2): 
such that the basic control for checking the correctness of residuals is: 
Taking into account (21) and (22) and multiplying them with appertaining weight 
matrices from the left side, and equation (24), a systern of linear equations is 
determined: 
(23) 
(24) 
(25) 
By the solution of the coefficient of this system, a vector of unknown quantities is 
determined, i.e. the vector of residuals of direct observations v1, vector of residuals 
of indirect observations v2 and the vector of approximate values of unknowns x at the 
same tirne: 
In this equation, the following matrices are defined by inversion: 
Q33 = -(A,' P1A1 + Ai p 2A2r 1 = -N -l = -Q,u 
uxu 
Q23 = A2Qxx ' n2XU 
Q22 = p-1 .2 - A2Q,,Ai , 
n2xn2 
Geodetski vestnik 40 (1996) 1 
(26) 
(27) 
(28) 
(29) 
Q13 = A1Qxx, 
nlX U 
Q12 = -A1QxxAi, 
n1xn2 
Qll = P; i - A1Q,xAJ , 
n1Xl11 
where Qxx is the matrix of the cofactor of unknowns. 
In this case, in contrast to the standard algorithm, a system of linear equations 
(n1 + n2 + u) must be solved. 
5 INDIRECT OBSERVATIONS WITH CONSTRAINTS 
In the combined form of indirect observations with constraints, the functional model is determined with observation equations of indirect observations: 
v=Ax-1,P 
rud nxu ux1 nxl mm 
and with constraint equations: 
B' x+ m = O , 
rxu uxl rxl rxl 
where: 
r - number of constraints 
m - vector of misclosure 
B - matrix of coefficients of constraint equations. 
(30) 
(31) 
(32) 
(33) 
(34) 
The basic check for the computation of residuals is determined with the use of the 
least squares principle: 
A'Pv + Bk = O, (35) 
where k is the vector of the correlate. 
Equations (33) and (34) and equation (32) determine the system of linear equations 
after their multiplication with the weight matrix P on the left side: 
(36) 
By the solution of this system, the vector of unknown quantities is determined, i.e. 
the vector of corrections of indirect observations v, the vector of approximate values 
of unknowns x and the vector of the correlate k, are determined simultaneously: 
Geodetski vestnik 40 (1996) 1 
[:x] = [:p p: :i-i 1-:ll = [~:: ~:: ~::11-:•i 
k O Bt O w Q 31 Q 12 Q J t:u . 
In this equation, matrices are defined by inversion: 
Q33 = (BtN-lB)-1 = Q •• , 
rxr 
Q23 = N-]BQk•' 
uxr 
Q22 = N-IBQkk BtN I_ N-1 = -Qxx, 
uxu 
Q
13 
= -AN 
nxr 
= p-l_ AQ At 
XX , 
where N is the matrix of coefficients of normal equations of indirect obse1vations: 
(37) 
(38) 
(39) 
(40) 
(41) 
(42) 
(43) 
(44) 
and Qxx is the matrix of the cofactor of unknowns. The number of linear equations 
which must be solved increases from (u + r) to (n + u + r). 
6 CONCLUSION 
Modem pocket calculators and personal computers considerably increase the efficiency of the use of adjustment algorithms in solving different geodetic 
tasks. Their efficiency is shown at the elementary level of use, above all in the 
possibility of direct performance of matrix computational operations, so that there is 
no discrepancy between the theoretical presentation of adjustment algorithms on one 
hand and practical computations on the other. In comparison with classical algebra, 
each matrix computational operation is more difficult and generally consists of a 
series of elementary operations. By direct use of matrix operations integrated in the 
computer/calculator, the computational procedure is accelerated and, simultaneously, 
the possibility of appearance of computational errors is reduced. 
The use of inversion, i.e. a command incorporated in the computer or calculator, also solves one of the problems which had an influence on the development and 
use of adjustment procedures, that is the solving of normal equations. In practice, 
dassical methods ( or their partially modernised versions) are still present due to 
tradition (Burmistrov, 1963, Čubranic, 1980, Klak 1982). As a result of direct 
inversion of the matrix of the coefficient and the use of indeterminate methods in 
thcir solution, these methods have become inefficient. The solving of a system of 
linear equations, the extent of which exceeds the number of normal equations in 
solving of the same geodetk task, is not difficult at all. This is also the conclusion of 
Geodetski vestnik 40 (1996) 1 
this paper, in which the adjustment algorithms for simultaneous cakulation of 
residuals and unknowns are based on the solving of a system of linear equations 
which are increased with regard to the corresponding normal equations in the 
standard algorithrn by the number of observations. It must be emphasised that the 
direct use of computational operations of matrix algebra in modern computers and 
calculators also has an influence of increasing the efficiency of practical 
computations, while the functional model or its coefficients are defined with regard 
to the geodetic task. The functional model can be determined in the classical way or 
by writing a special program in one of higher program languages (Rožič, 1992). 
In comparison with standard algorithms, the presented adjustment algorithms with simultaneous determination of residuals and unknowns mainly use the advantages 
of direct matrix algebra operations. They are therefore appropriate for practical use 
in solving all standard geodetic tasks in daily practice which are based on the use of 
adjustment algorithms. Tl}e use of modern computers and cakulators for the 
previously described algorithms was very rare, but expert programs should be written 
in higher program languages. On the other band, the production, standardisation, 
verification and licensing of such programs present a special problem which is not 
discussed in this paper. 
It can be established on the basis of the above that the most basic computational accessories of a modem geodetic professional in the use of adjustment algorithms 
are pocket calculators with the possibility of performing matrix algebra 
computational operations. Their use makes practical computations simpler and more 
efficient without the knowledge of programming, with the use of the modification of 
standard algorithms presented in this paper. Adjustment algorithms for conditional 
observations (Hoepcke, 1980) can be modified in a similar way, as well adjustment 
algorithms for conditional observations with unknowns. 
Example: 
Comparison of practical computation in the use of a standard adjustment algorithm and an adjustment algorithm with simultaneous cakulation of 
corrections and unknowns in indirect observations ( computation was performed on a 
SHARP PC-1403 pocket calculator). The coordinates of points T 1, T2, T3 and T4 are 
given and the approximate coordinates of point T (x
0
= 117.00 m, y
0
= 145.00 m), the 
position of which is unknown. On the basis of measured lengths si (i = 1, 2, ... , 4 ), the 
adjusted coordinates of point T must be determined (section of an are). This example 
is from Rožic, 1993, exercise 3.1.12. 
Measured lengths Coordinates of points 
TT1 = S1 = 105.60 m, T1, Y1 = 54.80 m, X1 = 172.94 m 
TT2 = Sz = 107.60 m, T2, y2 = 233.65 m, X2 = 177.55 m 
TT3 = S3 = 109.30 m, T3, y3 = 237.50 m, X3 = 59.76 m 
TT4 = S4 = 103.10 m, T4, Y4 = 57.38 m, X4 = 65.33 m 
Geodetski vestnik 40 (1996) 1 
Observation equations: v = A x - l, P = E 
A Ae -l s 
1
-0.53 0.851 l 0.321 l 0.541 l 0.861 -0.56 -0.83 -1.39 -0.24 -1.63 
0.53 -0.85 -0.32 -0.52 -0.85 
0.51 0.86 1.37 -1.38 -0.01 
A) Standard algorithm 
Normal equations: N x - n = O 
N Ne -n A1s 
[
1.1308 0.0079] [1.1387] [-1.1210] [0.0178] 
0.0079 2.8692 2.8771 -0.0849 2.7922 
Solving of normal equations with Choleski's algorithm: 
C -(C)..: 10 (C 1t 1 (ctt 1s cr 
[
1.06340 0.00745] [-1.05413] [ 0.94038 ] [0.95710] [0.95710] 
1.69385 -0.04547 -0.00414 0.59037 2.23462 2.23462 
Q Q e X 
[ 
0.88434 xx-0.00244] [0.88189] [0.991] 
-0.00244 0.34854 0.34610 0.027 
Calculation of residuals: v = A x - 1 
Ax -1 v 
l=~:~~~1 r-~:~:1 r-~:~~~1 0.499 -0.52 -0.023 0.527 -1.38 -0.853 
B) Simultaneous calculation of residuals and unknowns 
Defining the coefficients of systems of linear equations according to equation (14): 
1.00 -0.53 0.85 0.54 
1.00 -0.56 -0.83 -0.24 
[: ~] = 
1.00 0.53 -085 
[-~] = 
-0.52 
1.00 0.51 0.86 -1.38 
-0.53 -0.56 0.53 0.51 0.00 0.00 0.00 
0.85 -0.83 -0.85 0.86 0.00 0.00 0.00 
Geodetski vestnik 40 (1996) 1 
Solving of the system of equations according to equation (15): 
[:t :] l = [Q,, Q2l Q12] Q22 [ :x] 
0.50044 -0.01840 0.49932 -0.01844 -0.46816 0.29749 0.039 
-0.01840 0.48329 0.01783 0.49906 -0.49676 -0.28643 -0.826 
0.49932 0.01783 0.50092 0.01897 0.46742 -0.29767 -0.023 
-0.01844 0.49906 0.01897 0.51535 0.44710 0.29898 -0.853 
-0.46816 -0.49676 0.46742 0.44710 -0.88434 0.00244 -0.991 
0.29749 -0.28643 -0.29767 0.29898 0.00244 -0.34854 -0.027 
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Review: Tomaž Ambrožič 
Doc.D1: Bojan Stopar 
Translation from Croat language and professional review: Tomaž Ambrožič 
Doc.Dr. Bojan Stopar 
Geodetski vestnik 40 (1996) 1 
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