Abstract
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.
Highlights
Let deg f x be the degree of f x and C x be the set of univariate polynomials
We will illustrate for any given Sylvester matrix, as long as all the elements are allowed to be perturbed, we can always find k -Sylvester structure matrices hk Ek satisfy bk hk Range Ak Ek, where bk is the first column of Sk and Ak are the remainder column of Sk
As shown in the above table, we show that our method based on structured total least norm (STLN) algorithm converges quickly to the minimal approximate solutions, needing no more than 2 iterations whereas the method in [14] requires more iteration steps
Summary
Let deg f x be the degree of f x and C x be the set of univariate polynomials. 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. 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]. STLN is a problem formulation for obtaining an approximate solution
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