On the computation of the GCD (LCM) of 2-d polynomials N. P. Karampetakis Department of Mathematics Aristotle University of Thessaloniki Thessaloniki 54006, Greece P. Tzekis and H. K. Terzidis Technological Educational Institution of Thessaloniki School of Sciences Department of Mathematics P.O. Box Thessaloniki 54101, Greece

Contents Goals Computation of GCD-LCM of 1-d polynomials. Interpolation methods. 2-D DFT Computation of GCD-LCM of 2-d polynomials by using 2-D DFT techniques. Some special cases. Conclusions and further work.

Goals Computation of the Greatest Common Divisor (GCD) of 2-d polynomials. –It is linked with the computation of zeros of system representations, –Solution of polynomial (matrix) diophantine equations and applications to control design problems i.e. computation of stabilizing controllers. –Network theory, Communications, Computer Aided Design, Image restoration. Computation of the Least Common Multiple (LCM) of 2-d polynomials. –Integral part of algebraic synthesis methods in control theory (connected with the derivation of minimal fractional representations of rational models, which are essential for the study of a variety of algebraic design problems (Kucera)).

How to compute the GCD of 1-d polynomials; Euclidean algorithm. Numerical methods based on –Euclidean Algorithm (Fryer, Weinstock), –Generalized resultant test (Barnett, Vardulakis & Stoyle) –Matrix based methods (Blankinship, Barnett, Karcanias and Mitrouli) (see Pace and Barnett)). The computation of the GCD is a numerical ill-possed problem (If GCD(u(x),v(x))=v(x), where v(x) a divisor of u(x), then GCD(u(x)+d,v(x))=1). Approximate GCD (Karmarkar & Lakshman, Corless, Emiris, Karcanias, Mitrouli e.t.c.).

What about the GCD of 2-d polynomials; Needs modification of the Euclidean algorithm, as the ring R[z,w] is not Euclidean (Sebek 1994). Symbolic methods (Johnson, Pugh and Hayton 1995) Other methods –Euclidean Algorithm (problem of coefficient growth), –Polynomial remainder sequence (Sasaki et. al.1997) –Generalized subresultant method (Ochi et.al.1991) –Hensel lifting strategy (Zhi Li and Noda 2000 & 2001) –Sylvester-like resultant method (Phillai and Liang 1999). –Blackbox type algorithm (Zeng, Dayton 2004) –Modular algorithm (Corless et.al.1995) (the problem is reduced to univariate problem by using evaluation homomorphisms to eliminate variables, and reconstruct to GCD of the original inputs from these “images” using interpolations (dense and sparse method). Approximate GCD of multivariate polynomials (Zhi and Noda 2000, Zhi, Li and Noda 2001 e.t.c).

How to compute the LCM of 1-d polynomials; Existing procedures for LCM rely on: the standard factorisation of polynomials, computation of a minimal basis of a special polynomial matrix (Beelen & Van Dooren) and use of algebraic identities, GCD algorithms and numerical factorisation of polynomials (Karcanias and Mitrouli), Standard system theoretic concepts (Karcanias and Mitrouli)

Why we use interpolation methods ; Kurth (1967), Computation of the determinant of a polynomial matrix. Paccagnella & Pierobon (1976), use specific points for the interpolation (FFT technique). Schuster & Hippe (1992), Computation of the ordinary inverse of a polynomial matrix (Newton’s interpolation method). Karampetakis & Vologiannidis (2002), DFT calculation of the generalized and drazin inverse of a polynomial matrix (DFT technique). Petkovic & Stanimirovic (2006, 2007), Computation of the Drazin and generalized inverse of a polynomial matrix (Newton’s interpolation method). Direct approach using Vandermonde’s matrix, Newton’s interpolation method, Lagrange’s interpolation, DFT techniques (multipoint evaluation-interpolation method).

FFT algorithms make the difference in speed Very fast algorithms for computing the DFT (use properties of specific interpolation points) Advantages The speed of interpolation algorithms can be increased. Greatly benefited by the existence of a parallel environment (through symmetric multiprocessing or other techniques). Efficient algorithms available both in software and hardware.

What we call Discrete Fourier Transform;

Which is the complexity of the 2-D DFT/FFT ;

Computation of the GCD of 2-d polynomials

Example 1.

Computation of the LCM of 2-d polynomials

What happens when the interpolation points add extra divisors ;

Solution of this problem – Expand the circle where the interpolation points stands

What happens when we have univariate factors in the GCD ;

Solution of this problem

Algorithm (GCD of 2-d polynomials)

GCD *

Computation of Q 1 (x,y)

* GCD

Algorithm (GCD of 2-d polynomials)

Conclusions An algorithm for the computation of the GCD and LCM of two-variable polynomials has been developed, based on DFT techniques. Main advantages : a) speed and b) robustness, that all DFT techniques have. The proposed algorithm has been implemented in the Mathematica computer programming language. Further work Approximate GCD-LCM of 2-d polynomials. Use of other interpolation techniques and packages for the solution of the same problem and compare the computation complexity. Computational complexity and numerical accuracy. GCD-LCM of 2-d polynomial matrices by using either GCD Euclidean division algorithms of 1-D polynomial matrices (McDuffee, Wolovich), or algorithms based on real matrix operations (Emre and Silverman, Kung et. Al., Anderson et. Al.), or system theoretic approach (Silverman and Van Dooren)