Two methods for the calculation of the degree of an approximate greatest common divisor of two inexact polynomials

Polynomial computations for blind image deconvolution

“…. , min(m, n), in order to avoid computational problems that may arise [33][34][35]. Two of these operations introduce the parameters α 0 and θ 0 , whose initial values are computed from the solution of a linear programming problem.…”

“…Their introduction implies that the Sylvester matrix has a non-linear structure for which the method of SNTLN must be used, and they can be considered as degrees of freedom that can be optimised to obtain improved results. They are refined in the iteration (21), and the examples in [35] compare the results from the method of STLN, which is equivalent to the specification α 0 = θ 0 = 1 for all values of j in (21), and the method of SNTLN. These examples show that the method of STLN may return an incorrect value for the degree of an AGCD, that is, the horizontal and vertical extents of the PSF may be in error, and that significantly better results are obtained when the method of SNTLN is used.…”

“…The application of Theorem 4.1 to the inexact polynomials f (y) and g(y) requires that they be preprocessed by three operations [33][34][35], where the second and third operations introduce the parameters α 0 > 0 and θ 0 > 0 respectively, such that y = θ 0 w where w is the new independent variable, that is, an AGCD of the polynomialsf (w) = f (θ 0 w) and α 0ḡ (w) = α 0 g(θ 0 w) is computed. The values α 0 and θ 0 are refined in the iterative procedure for the calculation of an AGCD off (w) and α 0ḡ (w), and if θ * is the value of θ 0 at the termination of this procedure, then the inverse transformation w = y/θ * is applied to the results of the AGCD computation in order that the results be expressed in the same independent variable, y, as the specified polynomials f (y) and g(y).…”

“…Theorem 4.1 shows that the computation of the degree t of an AGCD off (w) and α 0ḡ (w) requires the calculation of the rank of their Sylvester matrix S(f , α 0ḡ ) and each subresultant matrix S k (f , α 0ḡ ). Computational experiments in [35] show that the SVD of S(f , α 0ḡ ) does not yield a good estimate of t, but two other methods, which yield good results and are based on Theorem 4.1, are described in [31,35,36]. They are, however, expensive because they require the computation of the SVD of S k (f , α 0ḡ ) for each value of k = 1, .…”

The Sylvester Resultant Matrix and Image Deblurring

“…It is shown in [28] that p(y) and q(y) must be processed by three operations before their Sylvester matrix S(p, q) is used to compute an AGCD. The first preprocessing operation arises because S(p, q) has, as noted above, a partitioned structure, which may cause numerical problems if the coefficients of p(y) are much smaller or larger in magnitude than the coefficients of q(y) since S(p, q) is not balanced if this condition is satisfied.…”