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10
On The Complexity Of Computing Determinants
 COMPUTATIONAL COMPLEXITY
, 2001
"... We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bi ..."
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Cited by 47 (17 self)
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We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bit operations; here denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors for positive real constants C 1 , C 2 , C 3 . The bit complexity (n results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n and O(n ) ring additions, subtractions and multiplications.
Approximate Factorization of Multivariate Polynomials via Differential Equations
 Manuscript
, 2004
"... The input to our algorithm is a multivariate polynomial, whose complex rational coe#cients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coe#cients by a small quantitity such that the resulting polynomial factors ove ..."
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Cited by 37 (9 self)
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The input to our algorithm is a multivariate polynomial, whose complex rational coe#cients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coe#cients by a small quantitity such that the resulting polynomial factors over C. Ideally, one would like to minimize the perturbation in some selected distance measure, but no e#cient algorithm for that is known. We give a numerical multivariate greatest common divisor algorithm and use it on a numerical variant of algorithms by W. M. Ruppert and S. Gao. Our numerical factorizer makes repeated use of singular value decompositions. We demonstrate on a significant body of experimental data that our algorithm is practical and can find factorizable polynomials within a distance that is about the same in relative magnitude as the input error, even when the relative error in the input is substantial (10 3 ).
Exact Certification of Global Optimality of Approximate Factorizations Via Rationalizing SumsOfSquares with Floating Point Scalars
, 2008
"... We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several wellknown factorization problems in hybrid symbolicnumeric computation. The idea is to transform a numerical sumofsquares (SOS) representation of a positive polynomial into an exact ..."
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Cited by 15 (9 self)
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We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several wellknown factorization problems in hybrid symbolicnumeric computation. The idea is to transform a numerical sumofsquares (SOS) representation of a positive polynomial into an exact rational identity. Our algorithms successfully certify accurate rational lower bounds near the irrational global optima for benchmark approximate polynomial greatest common divisors and multivariate polynomial irreducibility radii from the literature, and factor coefficient bounds in the setting of a model problem by Rump (up to n = 14, factor degree = 13). The numeric SOSes produced by the current fixed precision semidefinite programming (SDP) packages (SeDuMi, SOSTOOLS, YALMIP) are usually too coarse to allow successful projection to exact SOSes via Maple 11’s exact linear algebra. Therefore, before projection we refine the SOSes by rankpreserving Newton iteration. For smaller problems the starting SOSes for Newton can be guessed without SDP (“SDPfree SOS”), but for larger inputs we additionally appeal to sparsity techniques in our SDP formulation.
Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials
 Manuscript
, 2006
"... We consider the problem of computing minimal real or complex deformations to the coefficients in a list of relatively prime real or complex multivariate polynomials such that the deformed polynomials have a greatest common divisor (GCD) of at least a given degree k. In addition, we restrict the defo ..."
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Cited by 14 (9 self)
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We consider the problem of computing minimal real or complex deformations to the coefficients in a list of relatively prime real or complex multivariate polynomials such that the deformed polynomials have a greatest common divisor (GCD) of at least a given degree k. In addition, we restrict the deformed coefficients by a given set of linear constraints, thus introducing the linearly constrained approximate GCD problem. We present an algorithm based on a version of the structured total least norm (STLN) method and demonstrate, on a diverse set of benchmark polynomials, that the algorithm in practice computes globally minimal approximations. As an application of the linearly constrained approximate GCD problem, we present an STLNbased method that computes for a real or complex polynomial the nearest real or complex polynomial that has a root of multiplicity at least k. We demonstrate that the algorithm in practice computes, on the benchmark polynomials given in the literature, the known globally optimal nearest singular polynomials. Our algorithms can handle, via randomized preconditioning, the difficult case when the nearest solution to a list of real input polynomials actually has nonreal complex coefficients.
Approximate Bivariate Factorization, a Geometric Viewpoint
, 2007
"... We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably squarefree, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its d ..."
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Cited by 9 (1 self)
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We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably squarefree, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its discriminant with respect to one variable, and the perturbation of the corresponding monodromy action on a smooth fiber. A novel geometric approach is presented, based on guided projection in the parameter space and continuation method above randomly chosen loops, to reconstruct from a perturbed polynomial a nearby composite polynomial and its irreducible factors. An algorithm and its ingredients are described.
Indecomposability of polynomials via Jacobian matrix
 Journal of Algebra
"... Abstract. Indecomposable polynomials are a special class of absolutely irreducible polynomials. Some improvements of important effective results on absolute irreducibility have recently appeared using Ruppert’s matrix. In a similar way, we show in this paper that the use of a Jacobian matrix gives s ..."
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Cited by 3 (3 self)
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Abstract. Indecomposable polynomials are a special class of absolutely irreducible polynomials. Some improvements of important effective results on absolute irreducibility have recently appeared using Ruppert’s matrix. In a similar way, we show in this paper that the use of a Jacobian matrix gives sharp bounds for the indecomposability problem. 1.
Approximate radical of ideals with clusters of roots
 in ISSAC ’06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation
, 2006
"... We present a method based on Dickson’s lemma to compute the “approximate radical ” of a zero dimensional ideal Ĩ in C[x1,..., xm] which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is “global ” in the sense that it ..."
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Cited by 2 (0 self)
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We present a method based on Dickson’s lemma to compute the “approximate radical ” of a zero dimensional ideal Ĩ in C[x1,..., xm] which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is “global ” in the sense that it does not require any local approximation of the zero clusters: it reduces the problem to the computation of the numerical nullspace of the so called “matrix of traces”, a matrix computable from the generating polynomials of Ĩ. To compute the numerical nullspace of the matrix of traces we propose to use Gauss elimination with pivoting, and we prove that if Ĩ has k distinct zero clusters each of radius at most ε in the ∞norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically equal to ε 2. We also prove that the computed approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically equal to ε 2. In the univariate case our method gives an alternative to known approximate squarefree factorization algorithms which is simpler and its accuracy is better understood.
1.1 Univariate Resultants
"... Consider the following homogeneous system of two linear equations in two unknowns: ..."
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Consider the following homogeneous system of two linear equations in two unknowns:
Article Submitted to Journal of Symbolic Computation From an approximate to an exact absolute polynomial factorization
"... We propose an algorithm to compute an exact absolute factorization of a bivariate polynomial from an approximate one. This algorithm is based on some properties of the algebraic integers over Z and is certified. It relies on a study of the perturbations in a Vandermonde system. We provide a sufficie ..."
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We propose an algorithm to compute an exact absolute factorization of a bivariate polynomial from an approximate one. This algorithm is based on some properties of the algebraic integers over Z and is certified. It relies on a study of the perturbations in a Vandermonde system. We provide a sufficient condition on the precision of the approximate factors, depending only on the height and the degree of the polynomial. 1.