Approximate factorization of multivariable polynomials

Abstract In this paper, some approaches are presented for the approximate factorization of unfactorable multivariable polynomials. An iterative algorithm which can be shown to be convergent is developed on the basis of coefficient matching in the least-squares sense. For the approximate separation of polynomials—a special case of approximate factorization—a noniterative algorithm is proposed by means of the singular value decomposition concept. Examples of the two-variable case are presented to illustrate the utility of the approaches. The techniques described can directly be utilized for the design of 2-D digital filters (see, e.g., Antoniou and Lu, 1987).

[1]  B. Moore,et al.  Singular value analysis of linear systems , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[2]  T. Marzetta.  Additive and multiplicative minimum-phase decompositions of 2-D rational power density spectra , 1982 .

[3]  Sanjit K. Mitra,et al.  Sum and product separabilities of multivariable functions and applications , 1975 .

[4]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[5]  S. Treitel,et al.  The Design of Multistage Separable Planar Filters , 1971 .

[6]  J. Le Roux.  2-D Bauer factorization , 1984 .

[7]  J. Justice,et al.  A Levinson-type algorithm for two-dimensional Wiener filtering using bivariate Szegö polynomials , 1977, Proceedings of the IEEE.

[8]  Alfred Fettweis,et al.  Digital circuits and systems , 1984 .

[9]  Anastasios N. Venetsanopoulos,et al.  A decomposition theorem and its implications to the design and realization of two-dimensional filters , 1985, IEEE Trans. Acoust. Speech Signal Process..

[10]  N.K. Bose,et al.  Problems and progress in multidimensional systems theory , 1977, Proceedings of the IEEE.

[11]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.

[12]  M. Ekstrom,et al.  Two-dimensional spectral factorization with applications in recursive digital filtering , 1976 .

[13]  B. Anderson,et al.  Polynomial Factorization via the Riccati Equation , 1976 .

[14]  Wu-Sheng Lu,et al.  Design of two-dimensional FIR digital filters by using the singular-value decomposition , 1987 .

[15]  Thomas S. Huang,et al.  Stability of two-dimensional recursive filters , 1972 .