A fraction free Matrix Berlekamp/Massey algorithm

Kaltofen, Erich; Yuhasz, George

doi:10.1016/j.laa.2013.06.016

Cited by 12 publications

(9 citation statements)

References 21 publications

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“…Here, we also use the fraction free Berlekamp/Massey algorithm Kaltofen and Yuhasz 2013] to compute the determinants of H ℓ 1 −B and H ℓ 2 −B .…”

Section: Correcting 2 Errorsmentioning

confidence: 99%

Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding

Kaltofen,

Yang

2019

Preprint

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We present sparse interpolation algorithms for recovering a polynomial with ≤ B terms from N evaluations at distinct values for the variable when ≤ E of the evaluations can be erroneous. Our algorithms perform exact arithmetic in the field of scalars K and the terms can be standard powers of the variable or Chebyshev polynomials, in which case the characteristic of K is = 2. Our algorithms return a list of valid sparse interpolants for the N support points and run in polynomial-time. For standard power basis our algorithms sample at N = ⌊ 4 3 E + 2⌋B points, which are fewer points than N = 2(E + 1)B − 1 given by Kaltofen and Pernet in 2014. For Chebyshev basis our algorithms sample at N = ⌊ 3 2 E + 2⌋B points, which are also fewer than the number of points required by the algorithm given by Arnold and Kaltofen in 2015, which has N = 74⌊ E 13 + 1⌋ for B = 3 and E ≥ 222. Our method shows how to correct 2 errors in a block of 4B points for standard basis and how to correct 1 error in a block of 3B points for Chebyshev Basis.

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“…Here, we also use the fraction free Berlekamp/Massey algorithm Kaltofen and Yuhasz 2013] to compute the determinants of H ℓ 1 −B and H ℓ 2 −B .…”

Section: Correcting 2 Errorsmentioning

confidence: 99%

Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding

Kaltofen,

Yang

2019

Preprint

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“…We refer to [9,10] for a very nice classification of the BM algorithms for solving this problem and generalization to matrix sequences, see also [8]. Of particular importance for us is the solution of the underlying linear system in Toeplitz/Hankel form.…”

Section: Related Workmentioning

confidence: 99%

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Berthomieu

Boyer

Faugère

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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Sakata generalized the Berlekamp -Massey algorithm to n dimensions in 1988. The Berlekamp -Massey -Sakata (BMS) algorithm can be used for finding a Gröbner basis of a 0-dimensional ideal of relations verified by a table. We investigate this problem using linear algebra techniques, with motivations such as accelerating change of basis algorithms (FGLM) or improving their complexity.We first define and characterize multidimensional linear recursive sequences for 0-dimensional ideals. Under genericity assumptions, we propose a randomized preprocessing of the table that corresponds to performing a linear change of coordinates on the polynomials associated with the linear recurrences. This technique then essentially reduces our problem to using the efficient 1-dimensional Berlekamp -Massey (BM) algorithm. However, the number of probes to the table in this scheme may be elevated. We thus consider the table in the black-box model: we assume probing the table is expensive and we minimize the number of probes to the table in our complexity model. We produce an FGLM-like algorithm for finding the relations in the table, which lets us use linear algebra techniques. Under some additional assumptions, we make this algorithm adaptive and reduce further the number of table probes. This number can be estimated by counting the number of distinct elements in a multi-Hankel matrix (a multivariate generalization of Hankel matrices); we can relate this quantity with the geometry of the final staircase. Hence, in favorable cases such as convex ones, the complexity is essentially linear in the size of the output. Finally, when using the LEX ordering, we can make use of fast structured linear algebra similarly to the Hankel interpretation of Berlekamp -Massey.

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“…Further extensions have addressed fraction-free full-rank factoring of non-invertible matrices and fraction-free computation of the Moore-Penrose inverse [15]. More generally, applications exist in areas such as the Euclidean algorithm, and the Berlekamp-Massey algorithm [16].…”

Section: Introductionmentioning

confidence: 99%

Common Factors in Fraction-Free Matrix Decompositions

2020

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We consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauß–Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith–Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms that create them and give estimates of the frequency of their occurrence. Similar observations apply to the common factors in a fraction-free QR decomposition. Our conclusions are tested experimentally.

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A fraction free Matrix Berlekamp/Massey algorithm

Cited by 12 publications

References 21 publications

Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding

Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Common Factors in Fraction-Free Matrix Decompositions

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