Found Comput MathDOI 10.1007/s10208-015-9289-1

The Numerical Factorization of Polynomials

Wenyuan Wu1 · Zhonggang Zeng2

Received: 7 December 2014 / Revised: 4 August 2015 / Accepted: 18 August 2015© SFoCM 2015

Abstract Polynomial factorization in conventional sense is an ill-posed problem dueto its discontinuity with respect to coefficient perturbations, making it a challenge fornumerical computation using empirical data. As a regularization, this paper formulatesthe notion of numerical factorization based on the geometry of polynomial spaces andthe stratification of factorization manifolds. Furthermore, this paper establishes theexistence, uniqueness, Lipschitz continuity, condition number, and convergence of thenumerical factorization to the underlying exact factorization, leading to a robust andefficient algorithm with a MATLAB implementation capable of accurate polynomialfactorizations using floating point arithmetic even if the coefficients are perturbed.

Keywords Numerical polynomial factorization · Ill-posed problem ·Factorization manifold · Sensitivity

Mathematics Subject Classification 12Y05 · 13P05 · 65J20 · 65F22 · 65H04

Communicated by Felipe Cucker.

Zhonggang Zeng: Research supported in part by NSF under Grant DMS-0715127.

Wenyuan Wu: This work is partially supported by grant NSFC 11471307, 11171053.

B Zhonggang Zengz-zeng@neiu.edu

Wenyuan Wuwuwenyuan@cigit.ac.cn

1 Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute ofGreen and Intelligent Technology, CAS, Chongqing, China

2 Department of Mathematics, Northeastern Illinois University, Chicago, IL 60625, USA

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1 Introduction

Polynomial factorization is one of the fundamental algebraic operations in theory andin applications. It is also an enduring research subject in the field of computer algebra aswell as a significant success of symbolic computation (c.f. the survey [1]). Factorizationfunctionalities have been standard features of computer algebra systems such asMapleand Mathematica with a common assumption that the coefficients are representedexactly. Nonetheless, theoretical advancement and algorithmic development are stillin early stages in many cases. When a polynomial is approximately known with alimited accuracy in coefficients, the very meaning of its factorization as we know itbecomes a question, as illustrated in the following example. More precisely, a well-posed notion of numerical factorization has not been established, leaving a gap in thefoundation of its computation.

Example 1 We illustrate the central question of this paper: Assume the polynomial

f = x2y + 0.857143 xy2 + 0.833333 x2 + 1.38095xy

+ 0.571429 y2 + 0.555556 x + 0.476190 y (1)

is given as the empirical data of a factorable polynomial f . Knowing that the dataare imperfect with an error bound ‖ f − f ‖ ≤ 10−5, what is the factorization of theunderlying polynomial f ?

The factorization of f in conventional sense does not exist, while the underlyingpolynomial f is factorable but not known exactly. Intuitively, one can ask a moremodest question: Is there a factorable polynomial near f within the data errorbound 10−5? This latter question is similar to an open problem in [2], and the answeris ambiguous: The polynomial f is near many factorable polynomials, as shown inTable 1.

The numerical factorization of f within the error tolerance 10−5, as we shalldefine in Sect. 5, is the exact factorization of f in Table 1 and accurately approximatesthe factorization of f fromwhich f is constructed by rounding up digits. The nearestpolynomial to the data f , however, is not f but f3 whose factorization does notresemble that of f . In fact, the factorable polynomial with the smallest distance to thedata is almost certain to have an incorrect factorization structure by the factorizationmanifold embedding theorem in Sect. 4 whenever the underlying polynomial f hasmore than two factors.

As shown in this example, conventional factorization is a so-called ill-posed prob-lem for numerical computation since the factorization is discontinuous with respect todata perturbations. Consequently, fundamental questions arise such as if, under whatconditions, by computing which factorization and to what accuracy we can recoverthe factorization from empirical data. In this paper, we establish the geometry ofthe polynomial (topological) spaces in factorization manifold theorem and factoriza-tion manifold embedding theorem. Based on the geometry, we rigorously formulatethe notion of the numerical factorization. We prove the so-defined numerical factor-ization eliminates the ill-posedness of the conventional factorization and accuratelyapproximates the intended exact factorization (numerical factorization theorem) with

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Table 1 The polynomial f in (1) is near many factorable polynomials with various distances

a finite sensitivity measure that is conveniently attainable (numerical factorizationsensitivity theorem). As a result, the ill-posed factorization problem in numericalcomputation is completely regularized as a well-posed numerical factorization prob-lem that approximates the intended factorization with an accuracy in the same orderof the data precision.

Our results can be narrated as follows. The collection of polynomials possessinga nontrivial factorization structure is a complex analytic manifold of a positive codi-mension, and every such manifold is embedded in the closures of certain manifoldsof lower codimensions. This dimension deficit provides a singularity measurementof polynomials on the manifold and fully explains the ill-posedness their factoriza-tions: An infinitesimal perturbation reduces the singularity and pushes a polynomialaway from its native manifold into the open dense subset of polynomials with a trivialfactorization structure, making the exact factorization on the empirical data mean-ingless. Based on the geometric analysis, we formulate the notion of the numericalfactorization as the exact factorization of the polynomial on the nearby factorizationmanifold of the highest singularity having the smallest distance to the data. Underthe assumption that the data error is small, the original factorization can be recoveredaccurately by the numerical factorization of the data polynomial within a proper errorbound even if it is perturbed. From the tubular neighborhood theorem in differentialgeometry, the numerical factorization is a well-posed problem as it uniquely exists,is Lipschitz continuous and approximates the exact factorization of the underlyingpolynomial the data represent. The accuracy of the recovered factorization is in thesame order of the data accuracy since the factorization is Lipschitz continuous onthat manifold. Moreover, the conventional factorization becomes a special case of thenumerical factorization within a small error tolerance. Although a specific algorithmis not proposed here, the geometric analysis naturally leads to a two-staged computingstrategy for the numerical factorization: Identifying the factorization manifold by asquarefree factorization and a proper reducibility test, followed by the Gauss–Newtoniteration [3–6] for minimizing the distance to the factorization manifold.

This paper attempts to bridge differential geometry, computer algebra andnumericalanalysis. As an effective analytical tool that still appears to be underused, geometry hasled to many penetrating insights in numerical analysis (e.g. [7,8]) and effective algo-rithms such as homotopy methods based on Sard’s Theorem and Theorem of Bertini(e.g. [9,10]). Polynomial factorization problem has been studied from geometric

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perspective such as in [11–13]. This paper broadens the geometric analysis into anumerical computation of a basic problem in computer algebra by establishing thestratified complex analytic manifolds of factorization and their tubular neighborhood.In a seminal technical report [8], Kahan is the first to discover the hidden continuityon manifolds for generally discontinuous solutions of ill-posed algebraic problems.Recent works such as [6,14] made progress along this directions. This work pro-vides a complete regularization of a typical ill-posed algebraic problem in numericalpolynomial factorization by establishing its existence, uniqueness, Lipschitz continu-ity, convergence and condition number. Regularizations to this extent should now beexpected for other ill-posed algebraic problems that share a similar geometry.

For exact polynomial factorization, many effective methods have been developedover the past several decades. Those algorithms and complexity analyses have beenstudied extensively. The work of Sasaki et al. [15] introduces the techniques ofextended Hensel construction and the trace recombination that lead to factorizationalgorithms such as van Hoeij’s trace recombination [16] for univariate polynomial fac-torization of integer coefficients. The first polynomial-time factorization algorithms isgiven by Lenstra et al. [17] for univariate polynomial factorization, and by Kaltofenand Von zur Gathen [18,19] for multivariate polynomials. Rigorous proofs are alsoprovided in these works on the probabilities and the complexities. At present, thealgorithm having the lowest complexity for exact bivariate polynomial factorizationappears to be due to Lecerf [20].

Many authors made pioneer contributions to the numerical factorization problemof multivariate polynomials, such as pseudofactors by Huang et al. [21], the numericalreducibility tests by Galligo and Watt [22] and by Kaltofen and May [23], comput-ing zero sum relations by Sasaki [24], interpolating the irreducible factors as curvesby Corless et al. [11], and by Corless et al. [12]. Finding a nearby factorable poly-nomial as proposed in [1,2,11,13,22,23,25] has played an indispensable role in theadvancement of numerical polynomial factorization, even though such a backwardaccuracy alone is insufficient in numerical factorizations as illustrated in Example 1. In[26], Sommese, Verschelde andWampler developed a homotopy continuation methodalong with monodromy grouping, and Verschelde released and has maintained thefirst numerical factorization software as part of the PHC package [27] for solvingpolynomial systems. A breakthrough due to Ruppert’s differential forms [28] led to anovel hybrid factorization algorithm [29] by Gao, and the development of a numericalfactorization algorithm by Gao et al. [25,30]. Based on the formulation and analysisof this paper, we implemented a preliminary testing algorithm as a MATLAB mod-ule that shares a root similar to [25,29,30] along with several new developments fordemonstration of theories in this paper.

The results of this paper is not limited to multivariate polynomials. The numericalfactorization theory and computational strategy extend to the univariate polynomialfactorization, which is also known as polynomial root-finding where a recent majordevelopment enables accurate computation of multiple roots without extending thehardware precision even if the coefficients are perturbed [6]. This paper provides aunified framework for the numerical factorization including the univariate factorizationas a special case.

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2 Preliminaries

We consider polynomials in variables x1, . . . , x� with coefficients in the fieldC of complex numbers. The ring of these polynomials is commonly denoted byR = C[x1, . . . , x�]. The �-tuple degree of a polynomial f is defined as a vectordeg( f ) = (

degx1( f ), . . . , degx�( f )

)where degx j ( f ) is the degree of f in x j . For

any �-tuple degree n, denote

Rn := {p ∈ R ∣∣ deg(p) ≤ n

}

Rn := {p ∈ R ∣∣ deg(p) = n

}.

Here Rn is a vector space whose dimension is denoted by 〈n〉. Inequalities between�-tuple degrees are componentwise. With a monomial basis in lexicographical order,a polynomial f = f1xm1 + f2xm2 + · · · + f〈n〉xm〈n〉 in Rn corresponds toa unique coefficient vector denoted by � f � := ( f1, . . . , f〈n〉) ∈ C

〈n〉, suchas the polynomial f = 3 x21 x2 − 4 x1x2 + 5 x1 + 6 ∈ R(2,1) corresponding to� f � = (3, 0,−4, 5, 0, 6) ∈ C

6. A subset Ω ⊂ Rm corresponds to the subset�Ω� = {

�p� ∈ C〈m〉 ∣∣ p ∈ Ω

}in C

〈m〉. HereCn is the vector space of n-dimensional

vectors of complex numbers. All vectors in this paper are ordered arrays denoted byboldface lowercase letters or in the form of �·�. The Euclidean norm in C

〈n〉 inducesthe polynomial norm as ‖ f ‖ := ∥∥� f �

∥∥2, making Rn a topological metric space.

There are no differences between factoring a polynomial and factoring its nonzeroconstant multiple. We say p and q are equivalent, denoted by p ∼ q, if p = αqfor α ∈ C \ {0}. A metric is needed in the quotient space C[x1, . . . , x�]/ ∼ but notseen in the literature. We propose a scaling-invariant distance between polynomialsp and q as the sine of the principal angle between the subspaces span{�p�} andspan{�q�}, denoted by

sin(p, q) :=

⎧⎪⎨

⎪⎩

0 if p = q = 01 if p = 0, q = 0 or p = 0, q = 0∥∥∥ p

‖p‖ − �q�·�p�‖q‖ ‖p‖

q‖q‖

∥∥∥ if p = 0, q = 0.(2)

Here the “·” denotes the standard vector dot product. Let Pf be the projectionmappings to span{� f �} for any polynomial f . It is known that sin(p, q) ≡ ‖Pp−Pq‖2(c.f. [31]) and is thus a distance in the quotient space C[x1, . . . , xl ]/ ∼.

A polynomial f is factorable if there exist nonconstant polynomials g and h suchthat f = g h, otherwise it is irreducible. We say α f1 f2 . . . fk is a factorizationof f if α ∈ C is a constant scalar and the factors f1, . . . , fk ∈ C[x1, . . . , xl ] sothat the product α f1 . . . fk is equivalent to f . Here we abuse the notation α f1 . . . fkas it represents either the polynomial product or the factorization that consists of thearray of factors α, f1, . . . , fk , depending on the context.

We say two factorizations α f1 f2 . . . fk and β g1 g2 . . . gk of the same numberof factors are equivalent, denoted by α f1 f2 . . . fk ∼ β g1 g2 . . . gk , if there is apermutation {σ1, . . . , σk} of {1, . . . , k} such that f j ∼ gσ j for j = 1, . . . , k. For twofactorizations α f1 f2 . . . fk and β g1 g2 . . . gm with different number of factors, say

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k < m, they are equivalent if we can append constant factors fk+1 = . . . = fm = 1so that α f1 f2 . . . fm ∼ β g1 g2 . . . gm . If f1, f2, . . . , fk are all irreducible,then α f1 f2 . . . fk is an irreducible factorization. The irreducible factorization ofa polynomial is unique as an equivalence class.

A factorization γ g1 . . . gm is regarded as an approximate factorization of fif the backward error sin( f, γ g1 . . . gm) is small enough and acceptable in theunderlying application. The forward error of the factorization γ g1 . . . gm is thedifference between the factors g1, . . . , gm and their counterparts in f = α f1 . . . fkvia a proper metric that is needed but not properly established in the literature. Here weextend the distancemeasurement sin(·, ·) to the distance between two factorizationsas

dist(α f1 . . . fk, γ g1 . . . gk

) := min(σ1,...,σk )∈Σ

{max1≤ j≤k

{sin( f j , gσ j )

}}(3)

where Σ is the collection of all permutations (σ1, . . . , σk) of (1, . . . , k). Thisdistance extends to two factorizations with different numbers of factors by appendingdummy constant factors. Clearly, two factorizations are equivalent if and only if theirdistance is zero.

A polynomial is squarefree if its irreducible factorization consists of pairwisecoprime factors. A squarefree factorization α f k11 . . . f krr consists of squarefreepolynomials f1, . . . , fr as components that are pairwise coprime but may or maynot be irreducible. Again, we use the notation α f k11 f k22 . . . f krr to represent either thepolynomial that equals to the result of the polynomialmultiplicationor the factorizationconsists of the factors α, f1, . . . , f1, f2, . . . , f2, . . . , fr , . . . , fr where each f jrepeats k j times for j = 1, . . . , r .

If α f k11 . . . f krr is an irreducible squarefree factorization of a polynomial f with

degree m, we shall use M = mk11 . . .mkr

r to denote the factorization structure,or simply the structure of f , where m j = deg( f j ) = 0, k j ≥ 1 for j = 1, . . . , rand k1m1 + · · · + krmr = m = deg( f ). We shall also say such an M is one of thefactorization structures of the degree m and denote deg(M) = m. Any permutationof mk1

1 , . . . ,mkrr in M = mk1

1 . . .mkrr is considered the same structure.

There are two cases for a factorization structure M to be called trivial whenM is the factorization structure of either an irreducible polynomial or a univariatepolynomial with no multiple roots. A factorization structure is nontrivial if it is nottrivial.

3 Factorization Manifolds

The factorization of a polynomial f is an equivalence class in which a specificrepresentative α f k11 . . . f krr can be extracted using a set of auxiliary equations

b1 · � f1� = · · · = br · � fr � = 1

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where b1, . . . ,br are unit vectors of proper dimensions. We call such vectorsb1, . . . ,br the scaling vectors. Scaling vectors can be chosen randomly. A morenatural choice during computation is the normalized initial approximation of � fi � sothat bi · � fi � ≈ ‖ fi‖2 = 1 for i = 1, . . . , r . For any factorizations γ pk11 . . . pkrrand μqk11 . . . qkrr scaled by equations bi · �pi � = bi · �qi � = 1 for i = 1, . . . , r ,it is clear that ‖p1‖, . . . , ‖pr‖, ‖q1‖, . . . , ‖qr‖ ≥ 1 since the scaling vectors are ofunit norms, and the following lemma applies.

Lemma 1 Let γ pk11 . . . pkrr and μqk11 . . . qkrr be two factorizations with componentssatisfying ‖pi‖, ‖qi‖ ≥ 1 for i = 1, . . . , r . Then

dist(γ pk11 . . . pkrr , μqk11 . . . qkrr

) ≤ max1≤i≤r

∥∥pi − qi∥∥. (4)

Proof It is straightforward to verify that sin(pi , qi ) ≤ ‖pi − qi‖ whenever‖pi‖, ‖qi‖ ≥ 1 for i = 1, . . . , r . Thus (4) holds. �Suppose f possesses an irreducible squarefree factorization α f k11 . . . f krr and thedegrees deg( fi ) = mi for i = 1, . . . , r . The factorization structure M equals tomk1

1 . . .mkrr . All the polynomials sharing this factorization structure form a subset

FM := {f ∈ Rm

∣∣ f = αgk11 . . . gkrr where α ∈ C, g j ∈ Rm j , j = 1, . . . , r

are irreducible and pairwise coprime}

of Rm where m = deg(M). For almost all unit scaling vectors bi ∈ C〈mi 〉 for

i = 1, . . . , r , a polynomial f ∈ FM possesses irreducible factors α, f1, . . . , frsuch that α f k11 . . . f krr = f and b1 · � f1� = · · · = br · � fr � = 1 so that the array(γ, �p1�, . . . , �pr �

) = (α, � f1�, . . . , � fr �

)is a solution to the equation

φ(γ, �p1�, . . . , �pr �

) = (� f �, 1, . . . , 1

)(5)

for γ ∈ C, p j ∈ Rm j , j = 1, . . . , r , where the mapping φ is defined by

φ : C × C〈m1〉 × · · · × C

〈mr 〉 −→ C〈m〉 × C × · · · × C(

γ, �p1�, . . . , �pr �) �−→ (

�γ pk11 . . . pkrr �, b1 · �p1�, . . . , br · �pr �).

(6)Let qi = (

kipi

) αpk11 . . . pkrr . Then the Jacobian of φ can be written as

J (α, �p1�, . . . , �pr �)

=

⎡

⎢⎢⎢⎢⎢⎣

column(�pk11 . . . pkrr �) Cm1(q1) Cm2(q2) · · · Cmr (qr )column(b1)H

column(b2)H

. . .

column(br )H

⎤

⎥⎥⎥⎥⎥⎦

(7)

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where column(·) represents the column block generated by a vector (·), the notation(·)H denotes theHermitian transpose of thematrix (·), and Cmi (qi ) is the convolutionmatrix [6] associated with qi so that Cmi (qi ) ·�h� = �qi h� holds for any h ∈ Rmi ,i = 1, . . . , r . We need several lemmas for establishing the main theorems of thepaper.

Lemma 2 For α ∈ C \ {0}, b j ∈ C〈m j 〉 and p j ∈ Rm j with b j · �p j � = 0 for

j = 1, . . . , r , the Jacobian in (7) is injective if and only if p1, . . . , pr are pairwisecoprime.

Proof Assume p1, . . . , pr are pairwise coprime and thematrix-vector multiplication

J (α, �p1�, . . . , �pr �) · (−a, �v1�, . . . , �vr �) = 0. (8)

Then b1 · �v1� = · · · = br · �vr � = 0 as well as∑r

i=1 qivi = a∏r

i=1 pkii that lead

to∑r

i=1 kiαp1 . . . pi−1vi pi+1 . . . pr = ap1 p2 . . . pr . Thus

k1 α v1 p2 . . . pr = a p1 p2 . . . pr −r∑

i=2

ki α p1 . . . pi−1 vi pi+1 . . . pr

that contains the factor p1. Because gcd(p1, p j ) = 1 for j = 2, . . . , r , there isa polynomial s such that v1 = sp1. The degree deg(v1) ≤ deg(p1) leads to sbeing a constant. Since b1 · �p1� = 0, 0 = b1 · �v1� = s b1 · �p1�, hence s = 0.Consequently v1 = 0. Similarly we can prove that vi = 0 for i = 2, . . . , r .Substituting v1 = · · · = vr = 0 into (8), we have apk11 . . . pkrr = 0 and thus a = 0.Therefore, the Jacobian is injective.

Conversely, to prove that the injectivity of the Jacobian in (7) implies p1, . . . , prare pairwise coprime, assume there are some i = j such that gcd(pi , p j ) = 1. Thenwe shall prove that the Jacobian must be rank-deficient. Without loss of generality,we can assume p1 = e s and p2 = e t for some polynomials e, s and t wheree = gcd(p1, p2) is nonconstant. Then there are three possible cases. As case one, ifb1 ·�s� = 0 and b2 ·�t� = 0, then it is easy to show that

(0, 1

k1�s�, −1

k2�t�, 0, . . . , 0

)

is a nonzero solution to (8). As case two, if b1 · �s� = c = 0 and b2 · �t� = 0, thenwe can consider w = c

β1e s− s, where β1 = b1 · �p1�. It is straightforward to verify

that

b1 · �w� = c

β1b1 · �p1� − b1 · �s� = c − c = 0.

Since e = gcd(p1, p2) which is nontrivial, we have deg(ces) > deg(s) and conse-quently w = 0. Thus

(−cβ1

α, 1k1

�w�, 1k2

�t�, 0, . . . , 0)is a nonzero solution of (8).

For the third casewhere b1 ·�s� = c = 0 and b2 ·�t� = d = 0, let v1 = cβ1

e s−s and

v2 = − dβ2

e t + t where β2 = b2 · �p2�. Then( dα

β2− cα

β1, 1

k1�v1�,

1k2

�v2�, 0, . . . , 0)

is a nonzero solution of (8). Therefore, the Jacobian is a rank-deficient matrix. �

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Lemma 3 Let M = mk11 . . .mkr

r be a factorization structure of degree m andassumea sequence {p j }∞j=1 ⊂ FM converges to q ∈ Rm. Then there is a subsequence

of {p j }∞j=1 whose irreducible factorizations converge to a factorization αqk11 . . . qkrrof q with deg(qi ) = mi for i = 1, . . . , r . Further assume q ∈ FM. Then theirreducible factorizations of {p j }∞j=1 converge to the irreducible factorization of q.

Proof Let Si = {� f �

∣∣ f ∈ Rmi , ‖ f ‖ = 1}

and p j = α j pk1j1 . . . pkrjr be

an irreducible factorization of p j , where p ji ∈ Si for i ∈ {1, . . . , r} andj = 1, 2, . . .. Denote

P j = (α j , �p j1�, . . . , �p jr �) ∈ C × S1 × · · · × Sr .

There is a subsequence { j1, j2, . . .} of {1, 2, . . .} such that limσ→∞ ‖p jσ i − qi‖ = 0

for i ∈ {1, . . . , r} since Si ’s are compact. As a result, the subsequence{α jσ

}

converges to certain α ∈ C. Namely, the subsequence {P jσ }∞σ=1 converges to a point

(α, �q1�, . . . , �qr �) such that q = αqk11 . . . qkrr and deg(qi ) ≤ mi for i = 1, . . . , r .From deg(q) = m we have deg(qi ) = mi for i = 1, . . . , r . By Lemma 1, theirreducible factorizations of p jσ for σ = 1, 2, . . . converge to the factorization

αqk11 . . . qkrr since

dist(α jσ pk1jσ 1 . . . pkrjσ r , αqk11 . . . qkrr ) ≤ max

i‖p jσ i − qi‖ −→ 0

when σ → ∞. Moreover, if q ∈ FM, then αqk11 . . . qkrr is an irreduciblesquarefree factorization of q by the uniqueness of factorizations. Furthermore, theirreducible squarefree factorizations of the whole sequence {p j } must converge to

the factorization αqk11 . . . qkrr since otherwise there would be a δ > 0 and a sub-

sequence of {P j }∞j=1 converging to (α, q1, . . . , qr ), with q = αqk11 . . . qkrr and

dist(αqk11 . . . qkrr , αqk11 . . . qkrr

) ≥ δ, contradicting the uniqueness of the factorizationof q. �Lemma 3 directly leads to the following corollaries.

Corollary 1 Let f be a polynomial with a factorization structure M = mk11 . . .mkr

r

of degree m and an irreducible squarefree factorization α f k11 . . . f krr whose com-ponents satisfies ‖ f1‖ = · · · = ‖ fr‖ = 1. For any ε > 0, there is a neighborhoodΩ f of f in Rm such that every g ∈ Ω f ∩ FM corresponds to a unique

(β, g1, . . . , gr ) with g = βgk11 . . . gkrr , � f1� · �g1� = · · · = � fr � · �gr � = 1 and

√|α − β|2 + ‖ f1 − g1‖2 + · · · + ‖ fr − gr‖2 < ε.

Proof For any δ > 0, Lemma 3 implies that there is a neighborhood Ω f,δ of

f in Rm such that the irreducible squarefree factorization βgk11 . . . gkrr of every

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g ∈ Ω f,δ ∩ FM satisfies dist(βgk11 . . . gkrr , α f k11 . . . f krr

)< δ. We can assume

δ < 12 mini = j {sin( fi , f j )} and max j {sin( f j , g j )} < δ. Since i = j implies

sin( fi , g j ) ≥ sin( fi , f j ) − sin( f j , g j ) > δ,

no the other permutations of g1, . . . , gr satisfy max j {sin( f j , g j )} < δ. Furtherassume g1, . . . , gr are the unique representatives in their respective equivalenceclasses satisfying � f j � · �g j � = 1 for j = 1, . . . , r . Then

� f j � · � f j − g j � = 0, ‖ f j − g j‖ = ‖g j‖ sin( f j , g j ) and

‖g j‖2 = ‖ f j‖2 + ‖ f j − g j‖2 = 1 + ‖g j‖2 sin2( f j , g j ),

leading to

‖ f j − g j‖ = sin( f j , g j )√1 − sin2( f j , g j )

<δ√

1 − δ2

for j = 1, . . . , r . Therefore, for any ε > 0, the assertion holds when δ is small. �Corollary 2 Polynomials of degree m with a trivial factorization structure form anopen subset of Rm.

Proof For a univariate degree m, the assertion follows from the continuity of polyno-mial roots with respect to the coefficients. Assume m is multivariate and the assertiondoes not hold. Then there is an irreducible polynomial f of degree m and a sequenceof factorable polynomials {p j }∞j=1 approaching f . Because there are finitely manyfactorization structures in Rm, there exists a nontrivial factorization structure Mand a subsequence {p jσ }∞σ=1 in FM. By Lemma 3, the irreducible factorizationsof this subsequence converge to a nontrivial factorization of f , contradicting theirreducibility of f . �We can now establish the following factorization manifold theorem. A subset S in thetopological space Rm is a complex analytic manifold of dimension k in Rm if, forevery p ∈ S, there exists an open subset Ω of Rm containing p and a biholomorphicmapping from �S∩Ω� ⊂ C

〈m〉 onto an open subset of Ck . The codimension, namely

the dimension deficit, of S is denoted by codim(S) := dim(Rm)− dim(S) = 〈m〉− k.The factorization manifold theorem is at core of the geometry on the polynomialfactorization. This result and the proof are fundamental but not seen in the literature.

Theorem 3 (Factorization manifold theorem) Let M = mk11 . . .mkr

r be a factoriza-tion structure with deg

(M

) = m. Then FM is a complex analytic manifold in Rdand

codim(FM) = 〈m〉 −(〈m1〉 + · · · + 〈mr 〉 + 1 − r

). (9)

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Proof Let f ∈ FM with an irreducible squarefree factorization α f k11 . . . f krr alongwith deg( f j ) = m j , ‖ f j‖ = 1 and b j = � f j � for j = 1, . . . , r so that (6) definesa holomorphic mapping φ : C

k → C〈m〉+r with k = 1+ 〈m1〉 + · · · + 〈mr 〉 and

φ(α, � f1�, . . . , � fr �

) = (� f �, 1, . . . , 1

).

By Corollary 2, there is a neighborhood Δ of(α, � f1�, . . . , � fr �

)in the space

C × C〈m1〉 × · · · × C

〈mr 〉 and every(α, � f1�, . . . , � fr �

) ∈ Δ forms an irreducible

squarefree factorization α f k11 . . . f krr . By Lemma 2, the Jacobian of φ is of fullrank k at (α, � f1�, . . . , � fr �). As a result, the Inverse Mapping Theorem ensuresthat certain k components of φ form a biholomorphic mapping φ from an openneighborhood Σ of

(α, � f1�, . . . , � fr �

)in C

k to an open subset Π of Ck . We

can assume Σ ⊂ Δ.The mapping φ must contain the last r components of φ since φ would not

be injective without those scaling constraints. Without loss of generality, we assumeφ consists of the last k components of φ and we split φ(x) into φ(x) = u, andφ(x) = (

v, w)where u ∈ C

〈m〉+r−k , v ∈ Ck−r and w ∈ C

r . Let

Π ={v ∈ C

k−r∣∣ (v, 1, . . . , 1

) ∈ Π}

which is open in Ck−r . Then the holomorphic mapping

μ : Π −→ C〈m〉

v �−→ (φ ◦ φ−1(v, 1, . . . , 1), v

)

maps Π to μ(Π) ⊂ FM since Π × {(1, . . . , 1)} ⊂ Π and Σ ⊂ Δ. Define theprojection

ψ : C〈m〉+r−k × Π −→ Π

(u, v) �−→ v.

By Corollary 1, there is an open neighborhood Ω ⊂ C〈m〉−k+r × Π of � f � in C

〈m〉such that every �p� ∈ Ω ∩ �FM� corresponds to a unique (γ, �p1�, . . . , �pr �) ∈ Σ

with φ(γ, �p1�, . . . , �pr �) = (�p�, 1, . . . , 1

), namely μ ◦ ψ(�p�) = �p�. Define

Π = ψ(Ω ∩ �FM�).

We have μ(Π) ⊃ Ω ∩ �FM�). Then for every v ∈ Π , there is a u ∈ C〈m〉+r−k

such that (u, v) ∈ Ω ∩ �FM� corresponds to a unique (γ, �p1�, . . . , �pr �) ∈ Σ

with φ(γ, �p1�, . . . , �pr �) = (u, v, 1, . . . , 1), implying

u = φ ◦ φ−1(v, 1, . . . , 1) and μ(v) = (u, v).

Namely μ(Π) ⊂ Ω ∩�FM� and thus μ(Π) = Ω ∩�FM�. Since Ω is open and μ

is continuous, hence μ−1(Ω) = μ−1(Ω ∩ FM) = Π is open in Ck−r . Therefore,

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ψ is biholomorphic from Ω ∩ �FM� onto Π with the inverse μ. Namely FM isa complex analytic manifold of dimension k − r , and (9) follows. �We shall refer to FM as the factorizationmanifold associated with the factorizationstructure M. Its dimension deficit indicates how ill-posed the factorization is forpolynomials on themanifold. For a polynomial p of degree m, we say the singularityof p and its factorization structure M is k if FM is of codimension k in Rm.A polynomial is singular in terms of factorization if its singularity is positive, ornonsingular otherwise.

Corollary 4 A polynomial is singular if and only if its factorization structure is non-trivial, and nonsingular polynomials of degree m form an open dense subset ofRm.

Proof For both type of trivial factorization structures, the corresponding factorizationmanifold has a singularity zero from (9) by a straightforward verification. To provecodim(FM) > 0 for any nontrivial structure M, it suffices to show that for anydegrees n = 0 and n = 0 such that n + n is a non-univariate degree, we have

〈n + n〉 > 〈n〉 + 〈n〉 − 1. (10)

In fact, if is straightforward to verify (10) for � = 2, namely we have the inequality

(n1 + n1 + 1)(n2 + n2 + 1) > (n1 + 1)(n2 + 1) + (n1 + 1)(n2 + 1) − 1

if n1 + n1 > 0 and n2 + n2 > 0, and the inequality (10) for any positive integer� ≥ 2 follows an induction. Let M be trivial. FM is open in Rm by Corollary2. It is dense in Rm since it equals Rm minus finitely many singular factorizationmanifolds of lower dimensions. �Corollary 4 provides an ultimate explanation why polynomial factorization is an ill-posed problem: Any polynomial p having a nontrivial factorization is singular interms of factorization. Almost any perturbation Δp results in the data p = p + Δpthat is pushed off the native manifold into the open dense subset of nonsingularpolynomials, altering the factorization to a trivial one. This discontinuity makes theconventional factorization ill-posed and a challenge in numerical computation. Whenthe factorization structure is preserved, however, the irreducible factorization is Lip-schitz continuous as asserted in the following corollary. It is this continuity that makesnumerical factorization possible.

Corollary 5 (Factorization continuity theorem) The irreducible squarefree factoriza-tion is locally Lipschitz continuous on a factorization manifold: For any polynomialf ∈ FM with an irreducible squarefree factorization α f k11 . . . f krr , there are con-stants δ, η > 0 such that, for every polynomial g ∈ FM satisfying ‖ f − g‖ < δ,the irreducible squarefree factorization βgk11 . . . gkrr of g satisfies

dist(α f k11 . . . f krr , βgk11 . . . gkrr

) ≤ η ‖ f − g‖.

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Proof We can assume ‖ f1‖ = · · · = ‖ fr‖ = 1 and thus define the mapping φ in(6) with b j = � f j � for j = 1, . . . , r . Using the notations in the proof of Theorem 3,the mapping ζ(�g�) = φ−1

(ψ(�g�), 1, . . . , 1

)is holomorphic and thus Lipschitz

continuous for any g ∈ FM near f . Thus the assertion of this corollary followsfrom Lemma 1. �

4 Geometry of Factorization Manifolds

Factorization manifolds form a topologically stratified space Rm in which everysingular factorization manifold is embedded in manifolds of lower singularities aswe shall elaborate in detail. There are two embedding operations on a factorizationstructure:

– The degree combining operation is adding two �-tuple degrees of the same mul-tiplicity while keeping other components of the factorization structure unchanged:

. . . nkii . . . nk jj . . . −→ . . . (ni + n j )

k . . . where ki = k j = k. (11)

– Themultiplicity splitting operation decomposes a component of a factorizationstructure into two as follows:

. . . nkii . . . −→ . . . nkii nkii . . . where ki = ki + ki . (12)

A factorization structure N is embedded in M, denoted by N ≺ M, if N = Mor M can be obtained by applying a sequence of embedding operations on N. Forexample,

(4, 3)5(1, 6)2(3, 2) ≺ (4, 3)3(4, 3)2(1, 6)2(3, 2) (splitting (4, 3)5 to (4, 3)3(4, 3)2)

≺ (4, 3)3(5, 9)2(3, 2) (combining (4, 3)2(1, 6)2 to (5, 9)2)

The relation ≺ is a partial ordering among factorization structures.

Theorem 6 (Factorization manifold embedding theorem) Let N be a factorizationstructure and f ∈ FN. For any factorization structure M with deg(N) = deg(M) =m, we have f ∈ FM if and only if N ≺ M. Furthermore, codim(FN) > codim(FM)

in Rm if N ≺ M and N = M.

Proof Assume N ≺ M. To prove f ∈ FM, it suffices to show f ∈ FN if N

is obtained from N by one of the embedding operations (11) and (12). If N isobtained by degree combining (11), then we can write f = f ki f kj g with fi ∈ Rniand f j ∈ Rn j being irreducible and coprime. By Corollary 4, there is a polynomialsequence {hl}∞l=1 ⊂ Rni+n j converging to zero such that fi f j + hl is irreducible

for all l = 1, 2, . . .. Thus f ∈ FN since ( fi f j + hl)kg ∈ FN converges to f for

l → ∞. If N is obtained by multiplicity splitting (12), then we can write f = f kii gwith fi ∈ Rni . There is a sequence {hl}∞l=1 ⊂ Rni converging to zero such that

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Fig. 1 Stratification of factorization manifolds in R(3,2), where N≺ M indicates FN ⊂ FM

fi +hl is irreducible for all l = 1, 2, . . .. Thus f ∈ FN since f kii ( fi +hl)ki g ∈ FN

with ki + ki = ki converges to f for l → ∞.Conversely, assume f ∈ FM with an irreducible squarefree factorization

α f k11 . . . f krr . There is a sequence {gl}∞l=1 ⊂ FM converging to f . Write

M = mk′11 . . .m

k′ss and gl = βl g

k′1

l1 . . . gk′s

ls for l = 1, 2, . . .. Clearly deg(M) ≥deg(N). If deg(M) = deg(N), then Lemma 3 implies that we can further assumethe limit lim

l→∞ gl j = g j ∈ Rm j for j = 1, . . . , s and βl → β. Due to

α f k11 . . . f krr = β gk′1

1 . . . gk′s

s and the uniqueness of factorizations, we can factorpolynomials g1, . . . , gs and combine equivalent irreducible factors into higher mul-tiplicities to reproduce the squarefree irreducible factorization α f k11 . . . f krr . Namely,the structure M can be obtained by a sequence of embedding operations on N,leading to N ≺ M.

The inequality codim(FN) > codim(FM) follows from a straightforward verifi-cation using (9) on (11) and (12). �

The factorization manifold embedding theorem implies the geometry of polyno-mial factorization: The subset Rm of degree m polynomials is a disjoint union offactorization manifolds that are topologically stratified in such a way that every fac-torization manifold of positive singularity is embedded in the closure of a factorizationmanifold of lower singularity. As an example, Fig. 1 illustrates such a stratificationamong all the factorization manifolds through corresponding factorization structuresin R(3,2).

We define the distance between a polynomial and a factorization manifold

dist(f,FM

) = infg∈FM

(|| f − g||). (13)

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Let N be the factorization structure of f . The distance dist(f,FM

) = 0 if andonly if f ∈ FM, which is equivalent to N ≺ M by the factorization manifoldembedding theorem. As a consequence, the native manifold FN of f distinguishesitself as the unique factorizationmanifold that is of the highest singularity (i.e., highestcodimension) among all the factorization manifolds having a distance zero to f . Moreprecisely, a polynomial f belongs to a factorization manifold FN if and only if, inRm,

codim(FN)

= max{codim(FM)

∣∣∣ deg(M) = deg( f ) = m and dist

(f,FM

) = 0}.

On the other hand, a polynomial f ∈ FN with N ⊀ M implies that the distancedist

(f ,FM

)is positive. Since there are finitely many factorization manifolds, there

exists a minimum positive distance

θ f = minFM � f

dist(f ,FM

)> 0. (14)

The constant θ f is the critical gap of f from unembedded singularities, and it is the

very window of opportunity for numerical factorization. When the polynomial f ∈FN is represented by an empirical version f with a small perturbation ‖ f − f ‖ <12 θ f , the underlying factorization structure can still be identified by the followinglemma.

Lemma 4 Let f be a polynomial with a factorization structure N with θ f be given

in (14). For any empirical data f of f satisfying ‖ f − f ‖ < 12 θ f , the factorization

structure N of f is uniquely identifiable using the data f by

codim(FN)

= max{codim(FM) | deg(M) = deg( f ) = m and dist

(f,FM

)< ε

}(15)

in Rm for any ε satisfying dist(f,FN

)< ε < 1

2 θ f .

Proof A straightforward verification. �In summary, singular polynomials form factorization manifolds with positive codi-mensions and nonsingular polynomials form an open dense subset in Rm. Thosefactorization manifolds topologically stratify in such a way that every singular mani-fold belongs to the closures of some manifolds of lower singularities. Almost all tinyperturbations on a singular polynomial alter its factorization structure in such a waythat the singularity reduces and never increases. There is a gap from any singularpolynomial to higher singularity, and this gap ensures the lost factorization structurecan be recovered by finding the highest singularity manifold nearby if the perturba-tion is small. As a result, identifying the factorization structure is well posed as anoptimization problem.

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5 The Notion of Numerical Factorization

We shall rigorously formulate the concept of the numerical factorization to removethe ill-posedness of the conventional factorization and to achieve the main objectiveof recovering the exact factorization accurately using the imperfect empirical data.The numerical factorization should approximate the underlying factorization with anaccuracy the data deserve. The following problem statement gives a precise descriptionof the problem that numerical factorization is intended to solve.

Problem 1 (Numerical factorizationproblem) Let f be apolynomial as the empiricaldata of an underlying polynomial f whose irreducible factorization α f1 . . . fr is tobe computed. Assuming the data error

∥∥ f − f∥∥ is sufficiently small, find an irreducible

factorization α f1 . . . fk of a certain polynomial f such that both the backward errorand forward error are in the order of data error and the unit round-off:

∥∥ f − α f1 . . . fk∥∥ = O

(‖ f − f ‖ + u)

(16)

dist(α f1 . . . fk, α f1 . . . fr

) = O(‖ f − f ‖ + u

). (17)

where u is the unit round-off in the floating point arithmetic.

Problem 1 goes a step further from the Open Problem 1 in [2] in which only thebackward error is required to be small. Notice that the factorization α f1 . . . fk in(17) does not need to have the same number of factors as the underlying factorizationα f1 . . . fr . When r = k and the distance is small in (17), some of the factors areapproximately constants.

The assumption of the data error ‖ f − f ‖ to be sufficiently small should not beconfused with the local convergence of an iterative method. It is not a requirementof having a good approximation of the solution as local convergence implies. Rather,it assumes we are not trying to solve a problem that is too faraway to be relevant.From conventional factorization where data error is required to be zero, allowingsufficiently small data perturbation while still achieving a high accuracy is the signif-icant advancement that the numerical factorization is intended to achieve. Basic dataintegrity is required in all scientific studies, albeit almost always implicitly. It needs tobe explicitly stated here because we are in a frontier of solving discontinuous problemsfor which the difference is profound between exact data and perturbed ones, and thelatter may be all we have.

Let f be the polynomial in Problem 1with the factorization structure M and f beits empirical data representation, as illustrated in Fig. 2. By the factorization manifoldembedding theorem, the data polynomial f is away from the native factorizationmanifold FM with a reduced singularity. Note that the data polynomial f is alsonear all the factorization manifolds FN with M ≺ N and the native manifold FM

is not the nearest in distance but highest in singularity by Lemma 4. Upon identifyingthe factorization structure M, it is then natural to calculate the exact irreduciblefactorization α f k11 . . . f krr of the polynomial f ∈ FM that is the nearest to f anddesignate it as the numerical irreducible factorization of f since Corollary 5 suggests

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Fig. 2 Illustration of the numerical factorization of f

that the (exact) irreducible factorization of f approximates that of f . The followingdefinition is the detailed formulation.

Definition 7 (Numerical factorization) For a given polynomial f and a backwarderror tolerance ε > 0, we say α f1 f2 . . . fs is a numerical irreducible factorizationof f within ε if α f1 f2 . . . fs ∈ FM is an irreducible factorization and

minγ∈C

∥∥ f − γ f1 f2 . . . fs∥∥ = min

g∈FM

‖ f − g‖ = dist(f, FM

)< ε, (18)

where M is the factorization structure of the degree m = deg( f ) with the highestsingularity

codim(FM) = max{codim(FN)

∣∣ deg(N) = m and dist( f,FN) < ε}

(19)

in Rm. We call α f k11 . . . f krr a numerical irreducible squarefree factorization off within ε if it is squarefree and it is a numerical irreducible factorization of fwithin ε.

We shall use the abbreviated term numerical factorization for either the numericalirreducible factorization or the numerical irreducible squarefree factorizationwhen thedistinction is insignificant in the context. The formulation of the numerical factoriza-tion follows the same “three-strikes” principles that have been effectively applied tothe regularization of other ill-posed algebraic problems [14]: The numerical factor-ization of f is the exact factorization of a nearby polynomial f within a backwarderror tolerance ε (backward nearness principle). The nearby polynomial f is of thehighest singularity among all the polynomials in the ε-neighborhood of f (maxi-mum singularity principle). The nearby polynomial f is the nearest polynomial to

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the given f among all the polynomials with the same singularity as f (minimumdistance principle).

The error tolerance ε in Definition 7 depends on the particular application, thehardware precision, the underlying polynomial f and the data error ‖ f − f ‖. Theinterval for setting ε will be established in the numerical factorization theorem inSect. 6. Notice that a polynomial f can easily have different numerical factorizationswithin different error tolerances approximating different factorizations (c.f. Example 5in Sect. 9).

6 Regularity and Sensitivity of Numerical Factorization

As a concept attributed to Jacques S. Hadamard, a mathematical problem is well posedif its solution holds existence, uniqueness and continuity with respect to data. Further-more, Lipschitz continuity of the solution is crucial for numerical computation as itimplies a finite sensitivity with respect to data perturbations and round-off. With thegeometry established in Sects. 3 and 4, the well-posedness of numerical factorizationis a direct consequence of the tubular neighborhood theorem, which is one of thefundamental results in differential topology. The following elementary version of thetubular neighborhood theorem is adapted from its abstract form for complex analyticmanifolds in C

n .

Lemma 5 (Tubular neighborhood theorem) [32] Every complex analytic manifold iscontained in a tubular neighborhood.More precisely, for every complex analytic man-ifold Π in C

n, there is an open subset Ω of Cn containing Π and a projectionmap-

ping π : Ω −→ Π such that, for every z ∈ Ω , its projection π(z) ∈ Π is the uniquedistance-minimization point from z to Π , namely ‖π(z) − z‖2 = min

u∈Π

∥∥u − z∥∥2.

Furthermore, the mapping π is locally Lipschitz continuous.

We can now establish the main theorem, which asserts the properties that are desirablefrom the numerical factorization as formulated in Definition 7, provides a completeregularization and, in essence, achieves the objectives of numerical factorization inProblem 1.

Theorem 8 (Numerical factorization theorem) Let f ∈ Rm be a polynomial withits critical gap θ f as in (14) and an irreducible factorization α f1 f2 . . . fk . Thenθ f > 0 and the following properties of numerical factorization hold.

(i) Conventional factorization is a special case of numerical factorization: Thenumerical factorization of f within any ε ∈ (0, θ f ) is identical to the exact

irreducible factorization of f .(ii) Computing numerical factorization is a well-posed problem: There is a neigh-

borhood Ω f of f in Rm such that every f ∈ Ω f is associated with a

constant δ f ≤ ‖ f − f ‖ such that the numerical factorization α f1 f2 . . . fl of

f uniquely exists within ε for all ε ∈(δ f ,

12 θ f

)and is Lipschitz continuous

with respect to f .

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(iii) Numerical factorization is backward accurate: For every f ∈ Ω f and ε in(δ f ,

12 θ f

), the numerical factorization α f1 f2 . . . fl of f within ε satisfies

∥∥ f − α f1 f2 . . . fl∥∥ ≤ ‖ f − f ‖. (20)

(iv) The conventional factorization can be accurately recovered from empirical data:

For every f ∈ Ω f as empirical data of f and ε ∈(δ f ,

12 θ f

), the numer-

ical factorization α f1 f2 . . . fl of f within ε has the identical structure asα f1 f2 . . . fk and

dist(α f1 f2 . . . fl , α f1 f2 . . . fk

)< η f ‖ f − f ‖. (21)

where η f > 0 is a constant depends on f .

Proof Let M denote the factorization structure of f . Then FM is the manifold ofthe highest singularity within ε ∈ (0, θ f ) of f , and f itself is the polynomial of

minimum distance zero on FM from f , and thus (i) holds.Let Σ be the tubular neighborhood of FM described in Lemma5 and let Ω f ⊂ Σ

be a neighborhood of f such that every f ∈ Ω f satisfies ‖ f − f ‖ < 12 θ f . Set

δ f = dist(f,FM

). Then, for every ε ∈ (δ f ,

12θ f ) the equality (19) holds since

dist(f,FN

)> 1

2 θ f > ε for every N � M. By the tubular neighborhood theorem,

there exists a unique f = π( f ) ∈ FM with minimal distance to f . As a result, thenumerical factorization of f uniquely exists as the exact irreducible factorization off , and the numerical factorization is locally Lipschitz continuous since π is locallyLipschitz continuous along with Corollary 5, leading to part (ii).

Part (iii) is true since ‖ f − f ‖ ≤ ‖ f − f ‖. TheLipschitz continuity of the numericalfactorization also implies (21) and part (iv). �

In simpler terms, Numerical Factorization Theorem ensures that every factorablepolynomial f is allowed to be perturbed, while its factorization can still be recoveredas long as the empirical data f is still in the neighborhood Ω f . For each data

representation f of f , there is a window (δ f ,12 θ f ) for setting the error tolerance

ε for recovering the factorization of f . The fact that the lower bound δ f of theerror tolerance ε is no larger than the data error ‖ f − f ‖ is significant in practicalcomputation: If a data error bound η > 0 for ‖ f − f ‖ is known or can be estimatedin an application, the error tolerance can be set at ε = η or a moderate multiple ofthe unit round-off, whichever is larger. The upper bound 1

2 θ f appears to be difficultto estimate but not needed as long as it is not too small.

With a proper error tolerance ε, the numerical factorization of the data f withinε approximates the exact factorization of the underlying polynomial f with anaccuracy in the same order of the data accuracy. Namely, the numerical factorizationwe formulated in Definition 7 achieves the objective of the numerical factorizationproblemas specified inProblem1.Furthermore, computing the numerical factorization

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is a well-posed problemwith a finite sensitivity that can be established in the followingtheorem.

The assumption f ∈ Rm in Theorem 8 implies that the degree of the datapolynomial f is the same as the degree of the underlying polynomial f . If thedata error is sufficiently small, the degree of the data polynomial is at least equalto the degree of the underlying polynomial. When the perturbation is of a higherdegree but sufficiently small in magnitude in monomial basis, the coefficients of thoseextra monomials are below the error tolerance. As illustrated in Example 2 in Sect. 9,higher degree perturbations in monomial basis generally are negligible in practicalcomputation.

The condition f ∈ Ω f in Theorem 8 is to ensure the data error to be “sufficientlysmall” as one of main assumptions of this paper. As tubular neighborhood theoremdictates, the given polynomial needs to be close to manifold since the projectionto the manifold may not be unique otherwise. As a result, accurate recovery of thefactorization cannot be guaranteed from perturbed data with large errors. Theories andalgorithms for numerical factorizations with data perturbations of large magnitudes,higher degrees and in polynomial bases other than monomials are open problems forfurther studies.

Theorem 9 (Numerical factorization sensitivity theorem) Let α f k11 . . . f krr be thenumerical factorization of f within certain ε and g be sufficiently close to f sothat its numerical factorization within ε can be written as and γ gk11 . . . gkrr withdeg(g j ) = deg( f j ) = m j for j = 1, 2, . . . , r . Further assume J (·) is as defined in(7) where b j ∈ C

〈m j 〉 with ‖b j‖2 = 1 and b j · � f j � = 1 for j = 1, . . . , r . Then

lim supg→ f

dist(α f k11 . . . f krr , γ gk11 . . . gkrr

)

‖ f − g‖ ≤ η

∥∥∥J (α, � f1�, . . . , � fr �)+∥∥∥2

< ∞(22)

where η is a constant associated with f and b1, . . . ,br .

Proof A straightforward verification using Theorem 2 in [32] and Lemma 1. �The inequality (22) depends on the choices of the specific representative factoriza-tion α f k11 . . . f krr in the equivalent class of factorizations and the scaling vectorsb1, . . . ,br . Independent of those choices, we define the positive real number

κε( f ) := inf{ ∥

∥J (β, �h1�, . . . , �hr �)+∥∥2

∣∣∣ βhk11 . . . hkrr ∼ α f k11 . . . f krr ,

b j ∈ C〈m j 〉, ‖b j‖2 = 1, b j · �h j � = 1, j = 1, . . . , r

}(23)

as the condition number of the numerical factorization of f within ε whereα f k11 . . . f krr is a numerical squarefree irreducible factorization of f within ε. From

Lemma 2, this condition number is finite since the factorization α f k11 . . . f krr of fis squarefree, and κε( f ) becomes large when the Jacobian J (β, �h1�, . . . , �hr �)+is near rank-deficient when two of the factors h1, . . . , hr are a small perturbationaway from having nonconstant GCD. Consequently, the nature of the computation

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stability of numerical factorization become apparent: The numerical factorizationα f k11 . . . f krr of f is ill-conditioned if there exist two factors fi and f j that arenear non-coprime polynomials so that a small perturbation of f can increase thesingularity above that of f = α f k11 . . . f krr .

7 On the Numerical Squarefree Factorization

Every polynomial f has a unique squarefree factorization α f k11 . . . f krr wheref1, . . . , fk are pairwise coprime squarefree polynomials. Such squarefree factoriza-tions are important in its own right and usually easier to compute than irreduciblefactorizations. Our numerical factorization test algorithm and implementation startwith finding a numerical squarefree factorization followed by numerical irreduciblefactorizations of the squarefree components. Naturally, the notion of numerical square-free factorization and its properties are in question.

Similar to (irreducible) factorization structure, we can define a squarefree factor-ization structure N = nk11 . . . nkrr of polynomials having a squarefree factorization

f = α f k11 . . . f krr where k1 ≤ . . . ≤ kr where f j ∈ Rn j is squarefree forj = 1, . . . , r and pairwise coprime. Also let SN denote the collection of polynomi-als in Rm having a squarefree factorization structure N. Notice that Lemma 2 appliesto squarefree factorizations since the irreducibility of factors is not required. It is alsoa straightforward verification that Lemma 3 and Corollary 1 still hold for SN. As aresult, the subset SN is also a complex analytic manifold in Rd of codimension

codim(SN) = 〈d〉 −(〈n1〉 + · · · + 〈nr 〉 + 1 − r

).

The embedding properties of squarefree factorization manifolds hold as well.Similar to Definition 7, we can formulate the numerical squarefree factorization

of a polynomial f within an error tolerance ε as the exact factorization of apolynomial f ∈ SN where SN is the squarefree factorization manifold of thehighest codimension among all manifolds intersecting the ε-neighborhood of fand f is the nearest polynomial from f on SN. Such a numerical squarefreefactorization is a generalization of the conventional exact squarefree factorizationand accurate approximation to the exact squarefree factorization of the underlyingpolynomial f . Furthermore its computation is a well-posed problem with a finitesensitivity measure.

For a unit vector z = (z1, . . . , z�) ∈ C�, let ∂z p denote the directional derivative

of p ∈ C[x1, . . . , x�] along (the direction of) z, namely

∂z p = z1∂p

∂x1+ · · · + z�

∂p

∂x�

.

The following lemma is the basis for the numerical squarefree factorization.

Lemma 6 Every f ∈ C[x1, . . . , x�] has a squarefree factorization αhk11 . . . hkrrwith non-constant factors h1, . . . , hr and distinct multiplicities k1, . . . , kr ≥ 1.Furthermore, for almost all unit vectors z ∈ C

�,

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gcd(v, l · ∂zv − w) ={h j for l = k j with j = 1, . . . , r1 if l /∈ {k1, . . . , kr } (24)

where v and w are cofactors of u = gcd( f, ∂z f ) such that f = uv and ∂z f = uw.

Proof The existence of h1, . . . , hr is obvious. For almost all z ∈ C�, ∂zh j = 0

for j = 1, . . . , r . Thus v ∼ h1 . . . hr and w ∼r∑

j=1

k j (∂zh j )∏

i = j

hi , leading to

(24). �

The numerical squarefree factorization can be computed by a sequence of numericalgreatest common divisor replacing the exact GCD in (24).

By Lemma 2, the Jacobian of the mapping φ in (6) is injective at the least squaressolution of φ(β, �g1�, . . . , �gr �) = (� f �, 1, . . . , 1), implying the Gauss–Newtoniteration locally converges to this least squares solution if the data f and the initialiterate are sufficiently accurate.

Due to its similarity with the numerical irreducible squarefree factorization, weomit the detailed elaboration of the numerical squarefree factorization in this paper.

8 Computation of Numerical Factorizations

Overall, computing the numerical factorization consists of two stages. The first stageidentifies the factorization structure along with initial approximations of the numericalfactors. In the second stage, the numerical factors are refined to minimize the distancefrom the given polynomial to the manifold associated with the factorization structure.

In the first stage, the factorization structure can be computed by a sequence numer-ical squarefree factorizations, rank-revealing of the Ruppert matrices [23,28,29],generalized eigenvalue computation and numerical greatest common divisor calcu-lation. Initial approximations of the numerical factors are obtained as by-products.The computation requires standard numerical linear algebra operations with complex-ities of O(n3) where n is the number of relevant monomials of the polynomial andthe numerical factors.

In the second stage, the initial factor approximations p1, . . . , pr of degreesm1, . . . ,mr can be scaled to unit norms

∥∥p1∥∥ = · · · = ∥∥pr

∥∥ = 1. The map-ping φ in (6) becomes well defined by setting up the scaling vectors bi = �pi � fori = 1, . . . , r , The second stage of the numerical factorization algorithm is essentiallythe process of solving for the least squares solution to the overdetermined nonlinearsystem

φ(z) = (� f �, 1, . . . , 1), with z ∈ C × C〈m1〉 × · · · × C

〈mr 〉 (25)

using the Gauss–Newton iteration

z j+1 = z j − J (z j )+ φ(z j ), j = 0, 1, . . . . (26)

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where J (z)+ is the pseudo-inverse of the Jacobian J (z) of φ(z) given in (7) andz0 = (α, �p1�, . . . , �pr �).

Detailed discussion on the Gauss–Newton iteration and conditions for its conver-gence can be found in [3,5,14,32]. In a nutshell, the iteration (26) locally convergesto the least squares solution z∗ that is the point satisfying

∥∥φ(z∗) − (� f �, 1, . . . , 1)∥∥2 = min

z∈C×C〈m1〉×···×C〈mr 〉

∥∥φ(z) − (� f �, 1, . . . , 1)∥∥2

(27)if both the residual

∥∥φ(z∗) − (� f �, 1, . . . , 1)

∥∥2 and the initial error ‖z0 − z∗‖2 are

small.

Lemma 7 Let α f k11 . . . f krr be the numerical squarefree irreducible factorization off within ε with m j = deg( f j ) for j = 1, . . . , r . Then, for almost all unit vectors

bi ∈ C〈mi 〉, i = 1, . . . , r , there is a factorization α∗ f k1∗1 . . . f kr∗r ∼ α f k11 . . . f krr

such that z∗ = (α∗, � f∗1�, . . . , � f∗r �

)is the least squares solution to the equation

(25) with residual

∥∥φ(z∗) − (� f �, 1, . . . , 1)∥∥2 = ‖ f − α f k11 . . . f krr ‖ (28)

where φ is defined as in (6).

Proof Clearly∥∥φ(z) − (� f �, 1, . . . , 1)

∥∥2 ≥ ‖ f − α f k11 . . . f krr ‖ by Definition 7. On

the other hand, setting f∗i = 1bi ·� fi � fi for i = 1, . . . , r and an appropriate α∗ yields

(28). �Under the main condition that the Jacobian J (z∗) is of full rank, the Gauss–Newtoniteration converges locally [14]. The local convergence of the Gauss–Newton iterationrequires two conditions: The initial iterate z0 must be near the least squares solutionz∗ and the residual ‖φ(z∗)−(� f �, 1, . . . , 1)‖2 must be sufficiently small. From (28),the residual ‖φ(z∗) − (� f �, 1, . . . , 1)‖2 is bounded by the data error ‖ f − f ‖. As aresult, the residual requirement will be satisfied if the data error is sufficiently small.

The convergence of the two-staged method relies on the basic assumption of theproblem data being sufficiently close to the underlying polynomial. Accurate datalead to sufficiently small residual and an initial iterate being close to the least squaressolution so that the local convergence conditions of Gauss–Newton iteration are sat-isfied based on the alpha-theory [3,5]. Conversely, numerical factorization from poordata is not guaranteed to succeed partly because the Gauss–Newton iteration is locallyconvergent. The algorithmic and technical details of the numerical factorization arebeyond the scope of this paper and will be elaborated in a separate report.

9 Implementation, Software and Sample Results

Our numerical factorization test algorithm is implemented for both univariate andmul-tivariate polynomials as a function PolynomialFactor in the MATLAB packageNAClab for numerical algebraic computation as an upgrade and an expansion from its

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predecessor Apalab [33]. The entire NAClab package is freely available1, includingnumerical factorization, numerical rank-revealing, numerical computation of multi-plicity structure at zeros of nonlinear systems, numerical greatest common divisors,etc. We shall present several sample results highlighting the theories in this paper andthe major improvement areas of our test algorithm and the resulting software: Effi-ciency, accuracy, versatility anduser friendliness. The numerical factorization softwareis made available for other researchers to experiment with. All the tests are carriedout on a Samsung Series 7 XE700T1A tablet computer with 4GB memory and Inteli5-2467M CPU at 1.60 GHz running on Windows 7 64-bit operating system. The testlog and relevant MATLAB/Maple scripts can be downloaded online.2

The MATLAB package NAClab provides a user-friendly interface for numericalalgebraic computations. Polynomials can be entered and output as intuitive strings forcasual users. The function PolynomialFactor can be conveniently executed bysetting an error tolerance at or slightly above the magnitude of the data error.

Example 2 (Basic software application) Consider the polynomial

f = −30 + 20 x + 18 x2 − 12 x3 + 12.000007 x y − 8 x2 y + 3 × 10−7 x3 y − 5 y2

+3 x2 y2 + 2 x y3

≡ (3 x2 + 2 x y − 5) (y2 − 4 x + 6) + 0.0000003 x3 y + 0.000007 x y

as the data polynomial of (3 x2 + 2 x y − 5) (y2 − 4 x + 6) with a perturbation ofmagnitude about 7× 10−6. Setting the error tolerance slightly larger as, say 10−5, acall of the functionality PolynomialFactor in NAClab yields>> p = ’-30+20*x+18*xˆ2-12*xˆ3+12.000007*x*y-8*xˆ2*y+3e-07*xˆ3*y-5*yˆ2+3*xˆ2*yˆ2+2*x*yˆ3’>> PolynomialFactor(p,1e-5,’row’)ans =(-30.0000005908641) * (1-0.6*xˆ2-0.400000158490569*x*y) * (1-0.666666625730982*x+0.166666656432746*yˆ2)

The resulting numerical factorization has the same order of accuracy as the data.Notice that the degree of the perturbation is not limited to the degree of the underlyingpolynomial.

Example 3 (Univariate factorization) Accurate factorization of univariate polynomi-als with multiple roots has been a challenge in numerical computation. Conventionalsoftware functions for polynomial root-finding, such as MATLAB roots and Maplefsolve cannot factor such polynomial accurately and output scattered root clusters.For example, let

f (x) = x100 − 222.222222222222 x99 + · · · − 8.53544016536406 1030 x

+ 1.47799829703274 1029

≈ (x − 4.444444444444444)10(x − 3.333333333333333)20

(x − 2.222222222222222)30 × (x − 1.111111111111111)40 (29)

1 http://homepages.neiu.edu/~naclab.html.2 http://homepages.neiu.edu/~zzeng/NumFactorTests.zip.

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In contrast, our PolynomialFactor is an advanced polynomial root-finderthat is capable of accurate computation for multiple roots without extendingmachine precision even if the coefficients are perturbed. On this example, ourPolynomialFactor yields a factorization containing accurate roots and multi-plicities:

>> PolynomialFactor(f,1e-10,’row’)ans =(x-4.44444444445)ˆ10 * (x-3.33333333333)ˆ20 * (x-2.22222222222)ˆ30* (x-1.11111111111)ˆ40

This is a substantial improvement over its predecessor [6].

Since available software implementations for multivariate factorizations are builton different platforms, based on different notions of numerical factorizations, and withdifferent designing emphases, comprehensive comparisons are not feasible. Amongthem, Maple factor is built for the exact factorization. Developed by Vershelde, thepackage PHC[27] is a general-purpose polynomial system solver whose factorizationoption -f is perhaps the first implemented numerical factorization software. ThisPHC option initiates the implementation of a factorization algorithm [26] in numericalcomputation based on the homotopy continuationmethod. TheMaple codeappfac isdeveloped byKaltofen et al. [30], and the algorithm uses similar reducibility test basedon [28,29], which appears to be superior in factoring polynomial with highly perturbeddata.The computing examples in the remainder of this section are designed to showcasethe differences and improvement areas of our test algorithm and implementation.

Example 4 (Stewart-Gough platforms) In [26], the authors tested three polynomialsderived from the Stewart-Gough platform manipulator in mechanical engineering:

⎧⎨

⎩

g1 = F1(q0, q1, q2, q3) (q20 + q21 + q22 + q23 )3

g2 = F2(q0, q1, q2, q3) (q20 + q21 + q22 + q23 )3

g3 = ap33(q0 + bq3)(q0 + cq3)(q0 + i q3)5(q0 − i q3)5(30)

where F1, F2 ∈ C[q0, q1, q2, q3]. The polynomials g1 and g2 both have 910 terms,while g3 has 24. We test PHC in windows 7 command prompt using the compiledexecutable file phc.exe provided by its authors compared with our interpretive codePolynomialFactor in MATLAB. To level the base of accuracy comparison, wedisabled the Gauss–Newton iteration option in our PolynomialFactor in thistest since the iterative refinement was not developed for the computed factors whenPHC was released. Table 2 lists the elapsed execution times and the errors, where theforward errors are measured on the known factors only. Since the three implementa-tions are tested on different platforms, the comparisons should be considered indirect.Nonetheless, the results appear to show our algorithm are efficient and accurate onthose polynomials. It also appears that PHC has been improved substantially as it runsmuch faster and outputs more accurate factors than it is reported in 2004. It also needsto be pointed out that PHC is based on homotopy method that is naturally parallel andis generally much faster on parallel computers. The test is conducted on sequentialmachine since, at this point, parallelization of PolynomialFactor has not beenconsidered.

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Table 2 Factorization results on polynomials in (30) derived from Stewart-Gough platforms

g1 g2 g3

Maple factor Not designed for empirical data

appfac –

PHC Elapsed time 1382.8 1410.1 1.48

Backward error 9.6 × 10−10 0.9945 1.7 × 10−12

Forward error 1.3 × 10−12 1.1 × 10−12 3.4 × 10−13

PolynomialFactor(without refinement)

Elapsed time 376.5 480.3 0.79

Backward error 4.3 × 10−15 4.1 × 10−15 4.8 × 10−14

Forward error 2.0 × 10−15 8.5 × 10−16 5.0 × 10−16

Results of PolynomialFactor are in boldface

Example 5 (A polynomial with 5 numerical factorizations) An issue of significantimportance on the concept of numerical factorization is that a polynomial may havedifferent numerical factorizations within different error tolerances approximating dif-ferent conventional factorizations. A numerical factorization algorithm in this contextneeds mechanisms for targeting specific factorizations. For example, the polynomial

p1 = x7y + x5y3 + x6 − x y7 − x3y5 − 3 x2 y4 + 2(x3 y3− x2 y6 + x6 y2 − x y5 − y6 + x4)

+6 x2y2+7 x4y2+4 x5y+.999001 x y3+1.998002001 x y+4.999001 x3 y+2.999001 y2

−1.000999 x2 + .001(y3 + x3 + 3 x y2 + x3y2 + 3 x2y + x4 y + x y4 + x2y3 + 2 y + 2 x)

−2.001997998999 (31)

can be considered as empirical data of p1 itself and any one of the four factorablepolynomials

p2 = (xy + 1) (.001(x2y + xy2 + y3 + x3 + 2x) + 6x3y + 2x5y + 2x2y2 − 1.000999x2 + 2x4

+x6 − y6 + x4y2 + 4xy − x2y4 + 2.999001y2 − 2xy5 − 2.001997999 + .002x − 2xy3)

p3 = (−2xy3 + 2xy+2x3y+x4 + 2y2 − y4 − 1.000999+.001y + .001x)(x2 + y2+2)(xy + 1)

p4 = (x3 − xy2 + x + x2y − y3 + y + x2 − y2 + 1.001)(x + y − 1)(x2 + y2 + 2)(xy + 1)

p5 = (x + y + 1)(x2 − y2 + 1)(x + y − 1)(x2 + y2 + 2)(xy + 1) (32)

with data errors of variousmagnitudes listed in Table 3. In other words, the polynomialp1 has a numerical factorization 1 · p1 within an error tolerance between 0 and10−14, and numerical factorizations within error tolerances roughly in the intervals(10−13, 10−12), (10−9, 10−8), (10−6, 10−5) and (10−3, 10−2) approximating theexact factorizations of p2, p3, p4, p5 in (32) respectively. Our formulation ofthe numerical factorization includes the error tolerance ε and our implementationPolynomialFactor provides such an option. Based on the choices of those errortolerances, PolynomialFactor calculates all five numerical factorizations (32)with forward accuracies in the same orders of the data errors as shown in Table 3.Other algorithms such as Maple factor, PHC and appfac are designed to computeone numerical factorization from a given polynomial data.

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Table 3 Forward accuracies of Maple factor, PHC, appfac and PolynomialFactor (in boldface)from the data polynomial p1 in (31) calculating the factorization of either p1, p2, p3, p4 or p5 (32)

Underlying polynomial tobe factored

p1 p2 p3 p4 p5

Data error sin(p1, p j ) 0 7.4×10−14 1.1×10−10 2.6×10−7 4.9×10−4

Maple factor 0 – – – –

appfac+refinement – – – 2.4×10−7 –

PHC (no refinement) – – 1.4×10−9 – –

PolynomialFactor 7.0×10−16 8.9×10−14 9.2×10−11 2.1×10−7 5.6×10−4

10 Conclusions

Conventional factorization is an ill-posed problem in the sense that it is infinitely sen-sitive to data perturbations. The reason for such hypersensitivity is revealed by thegeometry of polynomial factorization, and the singularity can be quantified by thedimension deficit of the factorization manifold. The numerical factorization as formu-lated in this paper generalizes the concept of conventional factorization and eliminatesthe ill-posedness. By establishing the fundamental theorems for geometric structure ofmultivariate factorization, we proved that the numerical factorization uniquely existsand possesses Lipschitz continuity with respect to data under the overall assumptionthat the data error is small. Consequently, the numerical factorization achieves theobjective of recovering the factorization accurately even if the polynomial data areempirical and the accuracy is in the order of data precision. An algorithm is imple-mented as a MATLAB module, and numerical results support this conclusion.

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