Computing Approximation GCD of Several Polynomials by Structured Total Least Norm *

The task of determining the greatest common divisors (GCD) for several polynomials which arises in image compression, computer algebra and speech encoding can be formulated as a low rank approximation problem with Sylvester matrix. This paper demonstrates a method based on structured total least norm (STLN) algorithm for matrices with Sylvester structure. We demonstrate the algorithm to compute an approximate GCD. Both the theoretical analysis and the computational results show that the method is feasible.


Introduction
p .We wish to compute The problem of computing approximate GCD of several polynomials is widely applied in speech encoding and filter design [1], computer algebra [2] and signal processing [3] and has been studied in [4][5][6][7] in recent years.
Several methods to the problem have been presented.The generally-used computational method is based on the truncated singular decomposition(TSVD) [8] which may not be appropriate when a matrix has a special structure since they do not preserve the special structure (for example, Sylvester matrix).Another common method based on QR decomposition [9,10] may suffer from loss of accuracy when it is applied to ill-conditioned problems and the algorithm derived in [11] can produce a more accurate result for ill-conditioned problems.Cadzow algorithm [12] is also a popular method to solve this problem which has been rediscovered in the literature [13].

X. F. DUAN ET AL. 40
Somehow it only finds a structured low rank matrix that is nearby a given target matrix but certainly is not the closet even in the local sense.Another method is based on alternating projection algorithm [14].Although the algorithm can be applied to any low rank and any linear structure, the speed may be very slow.Some other methods have been proposed such as the ERES method [15], STLS method [16] and the matrix pencil method [17].An approach to be described is called Structured Total Least Norm (STLN) which has been described for Hankel structure low rank approximation [18,19] and Sylvester structure low rank approximation with two polynomials [20].STLN is a problem formulation for obtaining an approximate solution In this paper, we apply the algorithm to compute the structured preserving rank reduction of Sylvester matrix.We introduce some notations and discuss the relationship between the GCD problems and low rank approximation of Sylvester matrices in Section 2. Based on STLN method, we describe the algorithm to solve Problem 1.1 in Section 3. In Section 4, we use some examples to illustrate the method is feasible.

Main Results
First of all, we shall prove that Problem 1.1 always has a solution.
Theorem 2.1.Suppose that 1 , and are defined as those in Problem 1.1.There exist We have ˆˆˆ.
. For the real and imaginary parts of the coefficients of and of .We are considered with the continuous objective function , , , , .
We will prove that the function has a value on a closed and bounded set of its real argument vector which is smaller than elsewhere.Consider Clearly, any and with , , , , > can be discarded.So from above,we know that the coefficients of 1 2 can be bounded and so can the coefficients of ,0,0, ,0 In conclusion, the theorem is true.Now we begin to solve Problem 1.1, we first define a and an An extended Sylvester matrix or a generalized resultant is then defined by only if has rank deficiency at least 1.We have where k S is the -th Sylvester matrix generated by k From above, we know that (2.1) can be transformed to the low rank approximation of a Sylvester matrix.
If we use STLN [16] to solve the following overdetermined system , , where k is the first column of k and k b S A are the remainder columns of k , then we get a minimal perturbation We will give the following example and theorem to explain why we choose the first column to form the overdetermined system.
Example 2.1.Suppose three polynomials are given The matrix is the Sylvester matrix generated by The matrix is partitioned as Multiplying the vector   to the matrix , then we obtain Open Access ALAMT X. F. DUAN ET AL. 42 Next we will prove that k k A X b  has a solution.If we multiply the vector to two sides of the equation .
The solution X of equation ( 2.3) is equal to the coefficients of polynomials such that 1 , .
We can get and , we obtain a quotient and a remainder satisfy q .
Next, we will illustrate for any given Sylvester matrix, as long as all the elements are allowed to be perturbed, we can always find -Sylvester structure matrices k where k is the first column of and

S
, , n p t We can always find polynomials , it is always possible to find a -th Sylvester structure perturbation

STLN for Overdetermined System with Sylvester Structure
In this section, we will use STLN method to solve the overdetermined system According to theorem 2.3 and corollary 2.1, we can always find Sylvester structure   . Next we will use STLN method to find the minimum solution.
First, we define the Sylvester structure preserving perturbation of ] [ We can define a matrix such that .
1 We will solve the equality-constrained least squares problem where the structured residual is r By using the penalty method, the formulation (3.1) can be transformed into where is a large penalty value.Let Introducing a matrix of Sylvester structure and where satisfies that ( e following, we presen btain the matrix .Suppose to th e ve.Multiplying the vector 1 , where 1 ĝ ed is the polynomial with degree which is generat by the subvector of de which is generated by the subvector of 1 0 , is the polynomial with de wh h is gened by the subvector of gree n k  ic X : Here we will present simp , , ,

Approximate GCD Algorithm and Experiments
Problem The following algorithm is designed to solve 1.1.
to -th Sylvester matrix as the ab rst column of and the as a minimum.1) Form the k ove section, set the f remainder column . Compute k P and k Y as th above section., , , t f f p using Algorith and the CPU time   As shown in the above table, we show that our method based on STLN algorithm converges quickly to the minimal approximate solutions, needing no more than 2 iterations whereas the method in [14] requires more iteration steps.We also note that our algorithm still converges very quickly when the degrees of polynomials become large while the algorithm in [14] needs more iteration steps.Besides, our algorithm needs less CPU time than the AFMP algorithm.So the convergence speed of our method is faster.From the errors, we demonstrate that our method has smaller magnitudes compared with the method in [21].So our algorithm can generate much more accurate solutions.In this paper, we present that approximation GCD of several polynomials can be solved by a practical and reliable way based on STLN method and transformed to the approximation of Sylvester structure problem.For the matrices related to the minimization problems are all structured matrix with low displacement rank, applying the algorithm to solve these minimization problems would be possible.The complexity of the algorithm is reduced with respect to the degrees of the given polynomials.Although the problem of structured low rank ap-

A
are the remainder columns of . k E .Then the first o


, we can use the Al 1, 4.2, 4.3, 4.4 and 4.5 show that Alle to solve Problem 1.1.
are the remainder column of .
, t then we can com- 

Table 1 ,
[9,21]sent the performance of Algorithm 4.1 and compare the accuracy of the new fast algorithm with the algorithms in[9,21].