Structured matrix methods for the computation of multiple roots of a polynomial

Abstract

This paper considers the application of structured matrix methods for the computation of multiple roots of a polynomial. In particular, the given polynomial f (y) is formed by the addition of noise to the coefficients of its exact formf (y), and the noise causes multiple roots off (y) to break up into simple roots. It is shown that structured matrix methods enable the simple roots of f (y) that originate from the same multiple root off (y) to be 'sewn' together, which therefore allows the multiple roots off (y) to be computed. The algorithm that achieves these results involves several greatest common divisor computations and polynomial deconvolutions, and special care is required for the implementation of these operations because they are ill-posed. Computational examples that demonstrate the theory are included, and the results are compared with the results from MultRoot, which is a suite of Matlab programs for the computation of multiple roots of a polynomial.

A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

“…The matrix A and vector b are structured because they are derived from the modified Sylvester subresultant matrix S t (f , α 0ḡ )Q t , which is defined in (17). This approximate equation is transformed to an exact equation by the addition of a matrix E, which has the same structure as A, to the left hand side, and a vector e, which has the same structure as b, to the right hand side, such that the approximation Ax ≈ b is transformed to the exact equation,…”

“…Resultant matrices are frequently used for this computation, and these matrices and other polynomial computations also occur in robotics [5], computer vision [6], computational geometry, for example, the implicitization of parametric curves and surfaces [9] and the construction of surfaces [10,11], control theory [13] and the computation of multiple roots of a polynomial [17,22]. There are several resultant matrices, including the Sylvester, Bézout and companion resultant matrices, of which the Sylvester matrix is the most popular, presumably because its entries are linear, even though it is larger than the Bézout and companion matrices.…”

“…. , allow the polynomialp(y) and its multiple roots to be computed, and this has been considered for power basis polynomials [17,22]. The work in this paper is therefore a necessary requirement for the extension of this polynomial root solver to the Bernstein basis.…”

Polynomial computations for blind image deconvolution

“…This error must be compared with the relative error of the blurred image, which is equal to, from (4), 4.86 × 10 −2 1 2 = 0.22. The AGCD computations and polynomial deconvolutions used in this paper are implemented by the method of SNTLN, and its effectiveness for the numerically robust implementation of these operations is shown in [31,36], where it is used for the computation of multiple roots of a polynomial. The results in Table 1 Table 1 The results of four deblurring methods applied to the image in Fig.…”

“…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, .…”