Send Orders for Reprints to Reprints@benthamscience.ae a Sort of New Improved Algorithm for Total Least Square

Aim to blemish of total least square algorithm based on error equation of virtual observation, this paper put forward and deduced a sort of new improved algorithm which selects essential unknown parameters among designing matrix , and then, doesn't consider condition equation of unknown parameters among designing matrix. So, this paper perfected and enriched algorithm, and sometimes, new method of this paper is better. Finally, the results of examples showed that new mothod is viable and valid.


INTRODUCTION
Literatures [1,2] made some reasonable improvement such as putting forward virtual observation method and listing error equations of virtual observation for total least square algorithm, but there existed some problem.Aim to blemish of total least square algorithm based on literatures [1,2], literatures [3] has put forward and deduced further improved algorithm which considered condition equation of unknown parameters among designing matrix.But some times, the improved algorithm isn't always better, so, this paper proposed and deduced a sort of new algorithm which selects essential unknown parameters among designing matrix, and then, doesn't consider condition equation of unknown parameters among designing matrix, perfects and enriches algorithm of total least square algorithm.Finally, imitative example showed that new method is viable and valid.

Error Equations of Actual Observations
We suppose there exists function model of adjustment of indirect observations based on actual observation [1][2][3]  We suppose any element of n,t B can be expressed by un- known parameters There, Y i ˆand Y j ˆ ( t j i , 1 and j i ) are independent.Then,based on Eq.1,we can obtain *Address correspondence to this author at the College of Engineering and Designing, Lishui University, Lishui, Zhejiang, 323000, P.R. China; Tel: +86 18969588403; +86 15925722009 E-mails: lsxydengyonghe@sina.com; a15925722009@163.com

R E T R A C T E D A R T I C L E
where: In Eq. 2, we don't consider n,t Bt, 1 x which is very small.Obviously, Eq. 2 of this paper is different from homologous equations of literatures [1][2][3].

Error Equations of Virtual Observations
Because we suppose any element of where: Based on Eq.3,We can obtain where: k---quantity of virtual observation( 0 k nt );  Obviously, Eq. 4 of this paper is still different from homologous equations of literatures [1][2][3].

Total Error Equations
Based on Eq. 2 and Eq.4,we can obtain total error equations where: . P--total weight matrix; Obviously, Eq. 5 of this paper is different from homologous equations of literatures [1][2][3].

Solving out Total Error Equations [4]
Based on least square method and Eq. 5, we can obtain estimated values of unknown parameters estimated value of mean square error of unit weight and estimated values of variances and covariance of unknown parameters , and results of the two method are not same, though the final alternate results are very small.In order to data consistence of estimated value of designing matrix, this paper selected that Obviously, Eq. 6, Eq. 7 and Eq. 8 of this paper are different from homologous equations of literatures [1][2][3].

FIRST IMITATIVE EXAMPLE
Suppose there existed a plane curve such as where: and a ˆ3 ---estimated values of unknown coefficient; X i and Y i ---measured data with error; and we obtained coordinations of 10 points such as Table 1 based on imitative example.

Error Equations of Actual Observations
Based on Eq. 9, we can obtain error equation where:

Error Equations of Virtual Observations
Because where:

Total Error Equations
Based on Eq. 11 and Eq. 12, we can obtain total error equations

Solving out Total Error Equations
Based on least square method and Eq.13, we can obtain estimated values of unknown parameters

Results of Comparison Analysis
From Table 2,Table 3, Table 4, Table 5, Table 6, and Table 7 of this imitative example, we can find out: (1) New method of this paper is better at algorithm being stable, mean square error of unit weight being small, and matrix of variances and covariance of a ˆ0 , a ˆ1, a ˆ2 and a ˆ3 being stable and small than the method of literatures [1,2].
(2) The results of new method of this paper is almost as same as the results of the method of literature [3].
In a word, this example showed that new mothod of this paper is viable and valid and still supported literature [3].

CONCLUSION
The new method of this paper is more strict in theory, because it thinks elements of designing matrix may have condition equations, and selects essential unknown parameters to express all virtual observation values among designing matrix.
The new method of this paper is another improved method which isn't the same as the method of literature [3], and has been deduced.So, this paper perfected and enriched total least square algorithm in theoretics.
Two examples showed that new mothod of this paper is viable and valid, and still supported and perfected literature [3] in practice.

1 L
quantity of actual observations; t--necessary quantity of unknown parameters; n,---estimated value of column vector of observations; B t n ˆ, ---estimated value of designing matrix( part of B ˆ may be constant and haven't error); X t ˆ1 , ---estimated value of column vector of unknown parameweight matrix of actual observations.
-weight matrix of virtual observations.
estimated value of mean square error of unit weight Sort of New Improved AlgorithmThe Open Construction and Building Technology Journal, 2015, Volume 9 243 3.6.Results of Iterative Computing Based on Method of Literatures[1,2]

. Results of Iterative Computing of This paper
If we select No. 1, No. 2, No. 3 and No. 4 of Table 1, we can obtain approximations of unknown coefficient

Table 3
[3][2]ing to new method of this paper,obtain Table4and Table5according to the method of Literatures[1][2], and obtain Table6and Table7according to Literatures[3].Because our aim is to solve out curve, so, in

Table 2 ,Table 3 ,Table 4 ,
Table 5, Table 6, and Table 7, this paper only listed mean square error of unit weight,coefficients and their variances and covariance.

Results of Iterative Computing Based on Method of Literatures [3]Table 6 .
Iterative Computing a ˆ0 , a ˆ1, a ˆ2 and a ˆ3 .

Table 7 . Iterative computing variances and covariance. Iterative times Matrix of variances and covariance of
a ˆ0 , a ˆ1, a ˆ2 and a ˆ3

Table 8 ,
we can obtain Table 9 and Table 10 according to new method of this paper,obtain Table 11 and Table 12 according to the method of Literatures [1-2], and obtain Table 13 and Table 14 according to Literatures [3].If we select No. 1, No. 2, No. 3 and No. 4 of Table 8, we can obtain approximations of unknown coefficient

Table 9 ,
Table 10, Table 11, Table 12, Table 13, and Table 14 of second imitative example, we can obtain same results as that of first imitative example.