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

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Venue: | Manuscript |

Citations: | 36 - 9 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 ).

### Citations

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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... |

63 | Challenges of symbolic computation: my favorite open problems
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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... |

60 |
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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... |

50 | Factoring multivariate polynomials via partial differential equations
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Citation Context ...( � f) ≤ deg(f) limit the degrees in the individual variables, that is deg x( ˜ f) ≤ deg x(f) and deg y( ˜ f) ≤ deg y(f) (rectangular polynomials). Our algorithms are based on the exact algo=-=rithms in [7]-=-. All our methods are numerical and we execute our procedures with floating point scalars. We use the singular value decomposition (SVD) to determine the number of factors and approximate nullspace ve... |

46 | A method computing multiple roots of inexact polynomials
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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 ...: σm ≥ · · · ≥ σm−p−1 > ɛ ≥ σm−p ≥ · · · ≥ σ1. We do not wish to specify ɛ in advance, so we try to infer the “best” ɛ from the largest gap (i.e. the largest ratio σi=-=+1/σi) in the singular values. In [5], i-=-t is shown that when given a tolerance ɛ it is possible to certify the degree of the approximate GCD using gaps in the sequence τi = σ1(Si(g, h)) instead of the singular values of S1. The size of t... |

33 | Efficient algorithms for computing the nearest polynomial with constrained roots
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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 ... |

32 | Symbolic-numeric sparse interpolation of multivariate polynomials
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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 ... |

31 | Fast parallel absolute irreducibility testing
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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... |

26 |
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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 ...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 ... |

20 |
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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 ... |

19 |
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19 |
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19 | On approximate irreducibility of polynomials in several variables
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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... |

19 | The approximate gcd of inexact polynomials, Part I: A univariate algorithm. [http://www.neiu.edu/ zzeng/uvgcd.pdf
- Zeng
- 2004
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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... |

17 |
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Citation Context |

15 | Riemann surfaces, plane algebraic curves and their period matrices
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Citation Context ...G REMARKS The singular value decomposition was introduced to the area of hybrid symbolic/numeric algorithms in [3]. Since then, the SVD approach has been successfully applied to a variety of problems =-=[5, 9]-=-, most recently to the univariate squarefree factorization problem [32]. The past experience indicates that straightforward application of the SVD as a singular linear system solver may not yield usef... |

14 | Numerical factorization of multivariate complex polynomials
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Citation Context |

13 |
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12 | Displacement structure in computing the approximate GCD of univariate polynomials
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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 ... |

11 |
A geometric-numeric algorithm for absolute factorization of multivariate polynomials
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11 |
Reducibility of polynomials f(x,y) modulo p
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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, ... |

10 |
From an approximate to an exact absolute polynomial factorization
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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... |

6 |
Towards certified 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 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... |

5 |
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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]... |

2 | Transcendental and algebraic numbers. Translated from the 1st - Gelfond - 1960 |

1 |
The art of symbolic computation
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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, ... |

1 | 2003. Private email commun - Nagasaka |

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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(Show Context)
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... |