## Approximate Factorization of Multivariate Polynomials via Differential Equations (2004)

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Venue: | Manuscript |

Citations: | 39 - 11 self |

### BibTeX

@INPROCEEDINGS{Gao04approximatefactorization,

author = {Shuhong Gao and Erich Kaltofen and John May},

title = {Approximate Factorization of Multivariate Polynomials via Differential Equations},

booktitle = {Manuscript},

year = {2004},

pages = {167--174},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 ).

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Citation Context ...˜ S) ≤ ɛ3. Note that M is not a Sylvester matrix and �M − S�2 ≤ 2ɛ3. Now, M has a dimension 1 nullspace, so let �w = [ũ, ˜v] be the vector which spans the nullspace of M with �ũ�=-=2 = 1. Theorem 6.4 in [31] (reformulated for o-=-ur purpose in [11, section 8]) bounds the distance between w and �w in terms of ɛ3 so that as ɛ3 → 0, ɛ4 = �w − �w�2 → 0. Thus for sufficiently small ɛ1 and ɛ2 we have tdeg(ũ) = td... |

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Citation Context ...then determine ℓ directly from m − p = β(ℓ − 1, n). For univariate polynomials, the coefficient matrix for the linear system (1) and (3) is nothing but the well-known Sylvester matrix for g a=-=nd h. In [3]-=- the Sylvester matrix is used to get an approximate algorithm for the univariate GCD. For multivariate polynomials, we still call the coefficient matrix corresponding to (1) and (3) the Sylvester matr... |

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Citation Context ...so provide insight in the quality of a computed approximate factorization. No polynomial-time algorithm is known for computing the nearest factorizable polynomial f [min] , which is open problem 1 in =-=[15]-=-. In [12] a polynomial-time algorithm is given for computing the nearest polynomial with a complex factor of constant degree. In practice, that algorithm is much slower than any of the numerical solut... |

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Citation Context ...by way of the Bertini/Hilbert irreducibility theorems. One may interpolate the remaining variables, that either with a dense or the new sparse numerical algorithms [10], together with the homotopy of =-=[18] w-=-hen needed. We have reported success with this approach in section 3.4 for the case that the irreducibility radius is small (10 −16 ). Since the generalized Sylvester and Ruppert matrices have a fas... |

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Citation Context ...h ways and use the one with smaller backwards error. In [33] a different method is proposed, which is based entirely on multivariate approximate GCDs and which generalizes the univariate algorithm in =-=[32]. -=-We experimentally compare our approach to that one in the next subsection. 3.4 Experiments Example 3.6. [22] We illustrate our algorithm by factoring the following polynomial: f := (x 2 + yx + 2y − ... |

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Citation Context ...e insight in the quality of a computed approximate factorization. No polynomial-time algorithm is known for computing the nearest factorizable polynomial f [min] , which is open problem 1 in [15]. In =-=[12] a-=- polynomial-time algorithm is given for computing the nearest polynomial with a complex factor of constant degree. In practice, that algorithm is much slower than any of the numerical solutions—and ... |

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Citation Context ...egree 12 have small absolute coefficient error, 10 −16 , and have approximate factors of multiplicities 1, 3 and 5.. We computed the tri-variate approximate factors via sparse numerical interpolatio=-=n [10]-=-, which is possible here because the 7 forward error in the approximate factor coefficients is small.. We carried out all computations in double precision. Our running times, no more than 200 seconds ... |

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Citation Context ...ake use of bivariate factor coefficient bounds, seem unrealistically large. There is an extensive literature on the problem of factoring multivariate polynomials over the real or complex numbers. In [=-=14]-=- one of the first polynomial-time algorithms is given for input polynomials with exact rational or algebraic number coefficients, and the problem of approximate factorization is already discussed ther... |

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Citation Context ...able counterpart, say within machine floating point precision, one can attempt to run exact methods with floating point arithmetic, such as Hensel lifting or curve interpolation. The work reported in =-=[28, 27, 6, 13, 26, 4, 2, 25, 30]-=- studies recovery of approximate factorization from the numerical intermediate results. A somewhat related topic are algorithms that obtain the exact factorization of an exact input polynomial by use ... |

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Citation Context ...[19]. Once we have our approximations of g1 and h1 we can compute an approximate GCD by doing least squares approximate division. A very similar multivariate approximate GCD algorithm was proposed in =-=[33] -=-but a tolerance ɛ is required there, and an additional Gauss-Newton iteration step is used to improve the GCD further. AMVGCD: Approximate Multivariate GCD. Input: g and h in C[x1, . . . , xn]. Outpu... |

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Citation Context ...able counterpart, say within machine floating point precision, one can attempt to run exact methods with floating point arithmetic, such as Hensel lifting or curve interpolation. The work reported in =-=[28, 27, 6, 13, 26, 4, 2, 25, 30]-=- studies recovery of approximate factorization from the numerical intermediate results. A somewhat related topic are algorithms that obtain the exact factorization of an exact input polynomial by use ... |

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Citation Context ...rical analysis of any of our algorithms are the notions of “near” and “small”. Our experiments show that our algorithms perform well even for polynomials with a relatively large irreducibility=-= radius [22, 17, 23]-=-. In section 2 we provide an initial analysis for the approximate GCD algorithm by proving that the approximate GCD converges to the exact GCD as the perturbation error goes to 0. However, our worst c... |

17 |
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Citation Context ...The matrix A can be formed in the following two steps: First, we reduce the polynomial ggi and gjfx with respect to f at y = α for 1 ≤ i, j ≤ r by using approximate division of univariate polynom=-=ials [34]; then we so-=-lve the least squares problem: min �rem(ggi − (ai,1g1fx + · · · + ai,rgrfx), f)�2 at y = α to find the value of unknown elements ai,j. Let Eg(x) = det(Ix−A), the characteristic polynomial ... |

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13 |
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Citation Context ...rical intermediate results. A somewhat related topic are algorithms that obtain the exact factorization of an exact input polynomial by use of floating point arithmetic in a practically efficient way =-=[1]-=-. A different line of methods bounds from below the distance from the input polynomial to the nearest factorizable polynomial, that is, the irreducibility radius [22, 17]. Not only do such bounds help... |

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Citation Context ... · · · fr, (4) where fi ∈ C[x, y] are distinct and irreducible over C. Define Then Ei = f fi ∂fi ∂x ∈ C[x, y], 1 ≤ i ≤ r. (5) fx = E1 + E2 + · · · + Er and EiEj ≡ 0 mod f for all i=-= �= j. Theorem 3.1. [24] Suppose f ∈ C[x-=-, y] with bi-degree (m, n), i.e., deg x f = m, deg y f = n. Then f is absolutely 4 irreducible if and only if the equation � � ∂ g = ∂y f ∂ � � h , (6) ∂x f has no nonzero solution g, ... |

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Citation Context ...rical analysis of any of our algorithms are the notions of “near” and “small”. Our experiments show that our algorithms perform well even for polynomials with a relatively large irreducibility=-= radius [22, 17, 23]-=-. In section 2 we provide an initial analysis for the approximate GCD algorithm by proving that the approximate GCD converges to the exact GCD as the perturbation error goes to 0. However, our worst c... |

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Citation Context ...nstant multiple of the cofactors g1 and h1. This null vector can be computed numerically without computing the full SVD by using an iterative method. For our implementation we use the method given in =-=[19]-=-. Once we have our approximations of g1 and h1 we can compute an approximate GCD by doing least squares approximate division. A very similar multivariate approximate GCD algorithm was proposed in [33]... |

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1 |
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Citation Context ...n the algorithms are nested, as is the case for our factorization method. The worst case analysis together with our many successful experiments thus make our methods “good heuristics” in the sense=-= of [16]-=-. Clearly, our work is not finished. Now that we have a universal bivariate approximate factorization algorithm, meaning that closeness to a reducible polynomial is no longer a necessary requirement, ... |

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1 |
From an approximate to an exact absolute polynomial factorization. Paper submitted
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Citation Context ... numerical intermediate results. A somewhat related topic are algorithms that obtain the exact factorization of an exact input polynomial by use ofsoating point arithmetic in a practically ecient way =-=[1-=-]. A dierent line of methods bounds from below the distance from the input polynomial to the nearest factorizable polynomial, that is, the irreducibility radius [22, 17]. Not only do such bounds help ... |

1 |
Towards certi irreducibility testing of bivariate approximate polynomials
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Citation Context ...rical analysis of any of our algorithms are the notions of \near" and \small". Our experiments show that our algorithms perform well even for polynomials with a relatively large irreducibili=-=ty radius [22, 17, 23]-=-. In section 2 we provide an initial analysis for the approximate GCD algorithm by proving that the approximate GCD converges to the exact GCD as the perturbation error goes to 0. However, our worst c... |