Normalization of row reduced matrices

International audienceGiven a nonsingular $n \times n$ matrix of univariate polynomials over a field$\mathbb{K}$, we give fast and deterministic algorithms to compute itsdeterminant and its Hermite normal form. Our algorithms use$\widetilde{\mathcal{O}}(n^\omega \lceil s \rceil)$ operations in$\mathbb{K}$, where $s$ is bounded from above by both the average of thedegrees of the rows and that of the columns of the matrix and $\omega$ is theexponent of matrix multiplication. The soft-$O$ notation indicates thatlogarithmic factors in the big-$O$ are omitted while the ceiling functionindicates that the cost is $\widetilde{\mathcal{O}}(n^\omega)$ when $s =o(1)$. Our algorithms are based on a fast and deterministic triangularizationmethod for computing the diagonal entries of the Hermite form of a nonsingularmatrix

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“…where R is any − δ-column reduced form of A. To show this, we will rely on the following consequence of [28,Lemma 17]. Proof.…”

Section: Hermite Form Via Shifted Column Reductionmentioning

confidence: 99%

Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Labahn

Neiger

Zhou

2017

Journal of Complexity

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“…Applying the same transformation on M yields a reduced form of M with high probability, and from there the Popov form can be obtained. Lowering this cost further seems difficult, as indicated in the square case by the reduction from polynomial matrix multiplication to Popov form computation described in [23,Thm. 22].…”

Section: Introductionmentioning

confidence: 99%

“…In the nonsingular case, exploiting information on the pivots has led to algorithmic improvements for normal form algorithms [10,12,14,23]. Following this, we put our effort into two computational tasks: finding the location of the pivots in the normal form (the pivot support), and using this knowledge to compute this form.…”

Section: Introductionmentioning

confidence: 99%

Computing Popov and Hermite Forms of Rectangular Polynomial Matrices

Neiger

Rosenkilde

Solomatov

2018

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

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We consider the computation of two normal forms for matrices over the univariate polynomials: the Popov form and the Hermite form. For matrices which are square and nonsingular, deterministic algorithms with satisfactory cost bounds are known. Here, we present deterministic, fast algorithms for rectangular input matrices. The obtained cost bound for the Popov form matches the previous best known randomized algorithm, while the cost bound for the Hermite form improves on the previous best known ones by a factor which is at least the largest dimension of the input matrix.factors that are logarithmic in the input parameters, denoted by O˜(·). We let 2 ≤ ω ≤ 3 be an exponent for matrix multiplication: two matrices in K m×m can be multiplied in O(m ω ) operations. As shown in [5], the multiplication of two polynomials in K[x] of degree at most d can be done in O˜(d) operations, and more generally the multiplication of two polynomial matrices in K[x] m×m of degree at most d uses O˜(m ω d) operations.Consider a square, nonsingular M ∈ K[x] m×m of degree d. For the computation of a reduced form of M, the complexity O˜(m ω d) was first achieved by a Las Vegas algorithm of Giorgi et al. [7]. All the subsequent work mentioned in the next paragraph achieved the same cost bound, which was taken as a target: up to logarithmic factors, it is the same as the cost for multiplying two matrices with dimensions and degree similar to those of M.The approach of [7] was de-randomized by Gupta et al. [9], while Sarkar and Storjohann [23] showed how to compute the Popov form from a reduced form; combining these results gives a deterministic algorithm for the Popov form. Gupta and Storjohann [8, 10] gave a Las Vegas algorithm for the Hermite form; a Las Vegas method for computing the shifted Popov form for any shift was described in [18]. Then, a deterministic Hermite form algorithm was given by Labahn et al. [14], which was one ingredient in a deterministic algorithm due to Neiger and Vu [19] for the arbitrary shift case.The Popov form algorithms usually exploit the fact that, by definition, this form has degree at most d = deg(M). While no similarly strong degree bound holds for shifted Popov forms in general (including the Hermite form), these forms still share a remarkable property in the square, nonsingular case: each entry outside the diagonal has degree less than the entry on the diagonal in the same column. These diagonal entries are called pivots [13]. Furthermore, their degrees sum to deg(det(M)) ≤ md, so that these forms can be represented with O(m 2 d) field elements, just like M. This is especially helpful in the design of fast algorithms since this provides ways to control the degrees of the manipulated matrices.These degree constraints exist but become weaker in the case of rectangular shifted Popov forms, say m × n with m < n. Such a normal form does have m columns containing pivots, whose average degree is at most the degree d of the input matrix M. Yet it also contains n − m columns without pivots, which may all have lar...

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“…Column bases are produced by column reduced, Popov and Hermite forms and considerable work has been done on computing such forms, for example [1,8,9,12,13]. However most of these existing algorithms require that the input matrices be square nonsingular and so start with existing column bases.…”

Section: Introductionmentioning

confidence: 99%

“…However most of these existing algorithms require that the input matrices be square nonsingular and so start with existing column bases. It is however pointed out in [12,13] that randomization can be used to relax the square nonsingular requirement.…”

Section: Introductionmentioning

confidence: 99%

Computing column bases of polynomial matrices

Zhou

Labahn

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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Given a matrix of univariate polynomials over a field K, its columns generate a K [x]-module. We call any basis of this module a column basis of the given matrix. Matrix gcds and matrix normal forms are examples of such module bases. In this paper we present a deterministic algorithm for the computation of a column basis of an m × n input matrix with m ≤ n. If s is the average column degree of the input matrix, this algorithm computes a column basis with a cost of O ∼ nm ω−1 s field operations in K. Here the soft-O notation is Big-O with log factors removed while ω is the exponent of matrix multiplication. Note that the average column degree s is bounded by the commonly used matrix degree that is also the maximum column degree of the input matrix.

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Normalization of row reduced matrices

Cited by 12 publications

References 7 publications

Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Computing Popov and Hermite Forms of Rectangular Polynomial Matrices

Computing column bases of polynomial matrices

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