Jérémy Berthomieu
ABSTRACT
Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp--Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence. Several algorithms solve this problem. The so-called Berlekamp--Massey--Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process. We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp--Massey--Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations. A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial. Finally, we give a partial solution to the transformation of this algorithm into an adaptive one.
- B. Beckermann and G. Labahn. 1994. A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants. SIAM J. Matrix Anal. Appl. 15, 3 (1994), 804--823. Google Scholar
Digital Library
- E. Berlekamp. 1968. Nonbinary BCH decoding. IEEE Trans. Inform. Theory 14, 2 (1968), 242--242. Google ScholarDigital Library
- J. Berthomieu, B. Boyer, and J.-Ch. Faugère. 2015. Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences. In Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation (ISSAC '15). ACM, New York, NY, USA, 61--68. Google ScholarDigital Library
- J. Berthomieu, B. Boyer, and J.-Ch. Faugère. 2017. Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences. Journal of Symbolic Computation 83, Supplement C (Nov. 2017), 36--67. Special issue on the conference ISSAC 2015: Symbolic computation and computer algebra. Google ScholarDigital Library
- J. Berthomieu and J.-Ch. Faugère. 2017. In-depth comparison of the Berlekamp -- Massey -- Sakata and the Scalar-FGLM algorithms: the non adaptive variants. (May 2017). https://hal.inria.fr/hal-01516708 preprint.Google Scholar
- J. Berthomieu and J.-Ch. Faugère. 2018. Experiments. (2018). http://www-polsys.lip6.fr/~berthomieu/ISSAC2018.htmlGoogle Scholar
- S. R. Blackburn. 1997. Fast rational interpolation, Reed-Solomon decoding, and the linear complexity profiles of sequences. IEEE Transactions on Information Theory 43, 2 (1997), 537--548. Google ScholarDigital Library
- R.C. Bose and D.K. Ray-Chaudhuri. 1960. On a class of error correcting binary group codes. Information and Control 3, 1 (1960), 68 -- 79.Google ScholarCross Ref
- M. Bras-Amorós and M. E. O'Sullivan. 2006. The Correction Capability of the Berlekamp--Massey--Sakata Algorithm with Majority Voting. Applicable Algebra in Engineering, Communication and Computing 17, 5 (2006), 315--335.Google ScholarCross Ref
- R. P. Brent, F. G. Gustavson, and D. Y.Y. Yun. 1980. Fast solution of Toeplitz systems of equations and computation of Padé approximants. Journal of Algorithms 1, 3 (1980), 259 -- 295.Google ScholarCross Ref
- D. G. Cantor and E. Kaltofen. 1991. On Fast Multiplication of Polynomials Over Arbitrary Algebras. Acta Informatica 28 (1991), 693--701. Google ScholarDigital Library
- J. W. Cooley and J. W. Tukey. 1965. An Algorithm for the Machine Calculation of Complex Fourier Series. Math. Comp. 19, 90 (1965), 297--301. http://www.jstor.org/stable/2003354Google ScholarCross Ref
- D. Cox, J. Little, and D. O'Shea. 2015. Ideals, Varieties, and Algorithms (fourth ed.). Springer, New York. xvi+646 pages. An introduction to computational algebraic geometry and commutative algebra. Google ScholarDigital Library
- J. Dornstetter. 1987. On the equivalence between Berlekamp's and Euclid's algorithms (Corresp.). IEEE Transactions on Information Theory 33, 3 (1987), 428--431. Google ScholarDigital Library
- J.-Ch. Faugère and Ch. Mou. 2011. Fast Algorithm for Change of Ordering of Zero-dimensional Gröbner Bases with Sparse Multiplication Matrices. In Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation (ISSAC '11). ACM, New York, NY, USA, 115--122. Google ScholarDigital Library
- J.-Ch. Faugère and Ch. Mou. 2017. Sparse FGLM algorithms. Journal of Symbolic Computation 80, 3 (2017), 538 -- 569.Google ScholarCross Ref
- P. Fitzpatrick and J. Flynn. 1992. A Gröbner basis technique for Padé approximation. J. Symbolic Comput. 13, 2 (1992), 133 -- 138. Google ScholarDigital Library
- P. Fitzpatrick and G.H. Norton. 1990. Finding a basis for the characteristic ideal of an n-dimensional linear recurring sequence. IEEE Trans. Inform. Theory 36, 6 (1990), 1480--1487. Google ScholarDigital Library
- A. Hocquenghem. 1959. Codes correcteurs d'erreurs. Chiffres 2 (1959), 147 -- 156.Google Scholar
- E. Jonckheere and Ch. Ma. 1989. A simple Hankel interpretation of the Berlekamp-Massey algorithm. Linear Algebra Appl. 125, 0 (1989), 65 -- 76.Google ScholarCross Ref
- N. Levinson. 1947. The Wiener RMS (Root-Mean-Square) error criterion in the filter design and prediction. J. Math. Phys. 25 (1947), 261--278.Google ScholarCross Ref
- J. L. Massey. 1969. Shift-register synthesis and BCH decoding. IEEE Trans. Inform. Theory it-15 (1969), 122--127. Google ScholarDigital Library
- Bernard Mourrain. 2017. Fast Algorithm for Border Bases of Artinian Gorenstein Algebras. In Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation (ISSAC '17). ACM, New York, NY, USA, 333--340. Google ScholarDigital Library
- Sh. Sakata. 1988. Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array. J. Symbolic Comput. 5, 3 (1988), 321--337. Google ScholarDigital Library
- Sh. Sakata. 1990. Extension of the Berlekamp-Massey Algorithm to N Dimensions. Inform. and Comput. 84, 2 (1990), 207--239. Google ScholarDigital Library
- Sh. Sakata. 2009. The BMS Algorithm. In Gröbner Bases, Coding, and Cryptography, Massimiliano Sala, Shojiro Sakata, Teo Mora, Carlo Traverso, and Ludovic Perret (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 143--163.Google Scholar
- N. Wiener. 1964. Extrapolation, Interpolation, and Smoothing of Stationary Time Series. The MIT Press, Cambridge, MA. Google ScholarDigital Library
Index Terms
- A Polynomial-Division-Based Algorithm for Computing Linear Recurrence Relations
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