Algorithms for computing the sparsest shifts of polynomials via the Berlekamp/Massey algorithm

As a sub-procedure our algorithm executes the Berlekamp/Massey algorithm on a sequence of large integers or polynomials. We give a fraction-free version of the Berlekamp/Massey algorithm, which does not require rational numbers or functions and GCD operations on the arising numerators and denominators. The relationship between the solution of Toeplitz systems, Padé approximations, and the Euclidean algorithm is classical. Fraction-free versions [3] can be obtained from the subresultant PRS algorithm [2]. Dornstetter [6] gives an interpretation of the Berlekamp/Massey algorithm as a partial extended Euclidean algorithm. We map the subresultant PRS algorithm onto Dornstetter's formulation. We note that the Berlekamp/Massey algorithm is more efficient than the classical extended Euclidean algorithm.

[1]  Stanley Cabay,et al.  Power Series Remainder Sequences and Pade Fractions Over an Integral Domain , 1990, J. Symb. Comput..

[2]  Erich Kaltofen,et al.  Early termination in Ben-Or/Tiwari sparse interpolation and a hybrid of Zippel's algorithm , 2000, ISSAC.

[3]  Richard J. Lipton,et al.  A Probabilistic Remark on Algebraic Program Testing , 1978, Inf. Process. Lett..

[4]  Erich Kaltofen,et al.  Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[5]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[6]  Marek Karpinski,et al.  Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields , 1988, SIAM J. Comput..

[7]  Michael Ben-Or,et al.  A deterministic algorithm for sparse multivariate polynomial interpolation , 1988, STOC '88.

[8]  Erich Kaltofen,et al.  Computing with Polynomials Given By Black Boxes for Their Evaluations: Greatest Common Divisors, Factorization, Separation of Numerators and Denominators , 1990, J. Symb. Comput..

[9]  Dima Grigoriev,et al.  Algorithms for Computing Sparse Shifts for Multivariate Polynomials , 1995, ISSAC '95.

[10]  James L. Massey,et al.  Shift-register synthesis and BCH decoding , 1969, IEEE Trans. Inf. Theory.

[11]  R. Haskins.  Mathematics for the analysis of algorithms (2nd ed.) , 1986, Proceedings of the IEEE.

[12]  D. Knuth,et al.  Mathematics for the Analysis of Algorithms , 1999 .

[13]  Marek Karpinski,et al.  A Zero-Test and an Interpolation Algorithm for the Shifted Sparse Polynominals , 1993, AAECC.

[14]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[15]  Erich Kaltofen,et al.  FOXBOX: a system for manipulating symbolic objects in black box representation , 1998, ISSAC '98.

[16]  J. Rosser,et al.  Approximate formulas for some functions of prime numbers , 1962 .

[17]  Erich Kaltofen,et al.  Early termination strategies in sparse interpolation algorithms , 2001 .

[18]  Joseph F. Traub,et al.  On Euclid's Algorithm and the Theory of Subresultants , 1971, JACM.

[19]  Marek Karpinski,et al.  Computational Complexity of Sparse Rational Interpolation , 1994, SIAM J. Comput..

[20]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[21]  Jean Louis Dornstetter.  On the equivalence between Berlekamp's and Euclid's algorithms , 1987, IEEE Trans. Inf. Theory.