Fast real and complex root-finding methods for well-conditioned polynomials
aa r X i v : . [ c s . S C ] F e b Fast real and complex root-finding methods forwell-conditioned polynomials
Guillaume Moroz [email protected]รฉ de Lorraine, CNRS, Inria, LORIAVillers-lรจs-Nancy, France
ABSTRACT
Given a polynomial ๐ of degree ๐ and a bound ๐ on a conditionnumber of ๐ , we present the ๏ฌrst root-๏ฌnding algorithms that re-turn all its real and complex roots with a number of bit operationsquasi-linear in ๐ log ( ๐ ) . More precisely, several condition num-bers can be de๏ฌned depending on the norm chosen on the coe๏ฌ-cients of the polynomial. Let ๐ ( ๐ฅ ) = ร ๐๐ = ๐ ๐ ๐ฅ ๐ = ร ๐๐ = q(cid:0) ๐๐ (cid:1) ๐ ๐ ๐ฅ ๐ .We call the condition number associated with a perturbation ofthe ๐ ๐ the hyperbolic condition number ๐ โ , and the one associatedwith a perturbation of the ๐ ๐ the elliptic condition number ๐ ๐ . Foreach of these condition numbers, we present algorithms that ๏ฌndthe real and the complex roots of ๐ in ๐ (cid:16) ๐ log ( ๐๐ ) polylog ( log ( ๐๐ )) (cid:17) bit operations.Our algorithms are well suited for random polynomials since ๐ โ (resp. ๐ ๐ ) is bounded by a polynomial in ๐ with high probability ifthe ๐ ๐ (resp. the ๐ ๐ ) are independent, centered Gaussian variablesof variance 1. KEYWORDS
Polynomial equation, Root ๏ฌnding, Condition numbers, Real roots,Complex roots
The problem of ๏ฌnding all the real or complex solutions of a poly-nomial equation ๐ ( ๐ง ) = ๐ is a polynomial of degree ๐ withinteger coe๏ฌcients of bit size bounded by ๐ , the state-of-the-artmethods to ๏ฌnd the real or complex roots of ๐ require a number ofbit operations in ๐ ( ๐ ( ๐ + ๐ ) polylog ( ๐๐ )) [2, 28]. In the case wherethe polynomial is well-conditioned, the best methods in the stateof the art also require at least a quadratic number of bit operationsto ๏ฌnd its roots. By well-conditioned, we mean that the variationof the roots of ๐ with respect to the variation of its coe๏ฌcients issmall ([7, chapter 12], [8, chapter 14] and references therein).For ill-conditioned polynomials, the distance between two rootscan as small as 2 โ ๐๐ . Pan considered optimal an algorithm thatused ๐ ( ๐ ) arithmetic operations, where the number of bit opera-tion for each arithmetic operation is in ๐ ( ๐๐ ) , and in this sense,he provided a near-optimal algorithm. On the other hand, when apolynomial is well-conditioned, the distance between two roots isnot exponentially small in ๐ .Random polynomials are well-conditioned with a high proba-bility. More precisely, let ๐ ( ๐ฅ ) be a polynomial of degree ๐ whereeach of its coe๏ฌcients is a Gaussian random variable of variance (cid:0) ๐๐ (cid:1) . There exist constants ๐ด > ๐ต > elliptic condition number (see De๏ฌnition 1.1) is lower than ๐ ๐ด withprobability higher than 1 โ / ๐ ๐ต [10]. A similar result was provenfor the so-called hyperbolic condition number when the varianceis 1 [13]. Moreover, the distribution of the roots of polynomialswith random coe๏ฌcients is well understood ([14] and referencestherein). Thus it makes sense to provide algorithms that performsbetter than the general case for random polynomials and for well-conditioned polynomial.Provided that we know a bound ๐ on a condition number of ๐ ,we will show that it is indeed possible to ๏ฌnd all the roots of ๐ witha number of operations quasi-linear in ๐ and polynomial in log ( ๐ ) .Even though a condition number was not explicitly used, theanalysis of root-๏ฌnding methods for well-conditioned polynomi-als started with Smale [32] who studied the probability of failureof the Newton method. The Newton method is one of the most fa-mous iterative method, that converges quadratically toward a sin-gle root ๐ of ๐ provided that the initial point is close enough to ๐ ([7,chapter 8], [12, chapter 3], [8, chapter 15] and references therein).It was later shown that it is even possible to construct a set ๐ ๐ of ๐ log ( ๐ ) points such that for all polynomials ๐ and each root ๐ of ๐ , there exists a point in ๐ ๐ such that the Newton iterationeventually converges toward ๐ [19]. Explicit bounds polynomialin the condition number were derived and improved for multivari-ate polynomial system of equations, based notably on homotopymethods ([3, 9, 11, 25] among others). One drawback of those ap-proaches is that they require to evaluate ๐ on at least ๐ points,which leads to a number of arithmetic operations at least quadraticin ๐ . Some methods based on modi๏ฌed Newton operators, such asthe Weierstrass method ([4] and references therein) or the Aberth-Ehrlich method [16] were implemented with success, notably inthe software MPSolve [5, 6].For general polynomials, including ill-conditioned ones, fast nu-merical factorization is the ๏ฌrst approach to provide the state of theart bound in ๐ ( ๐ ( ๐ + ๐ ) polylog ( ๐๐ )) [28]. However this methodis di๏ฌcult to implement.Another family of methods that are e๏ฌcient in practice are thesubdivision methods. The idea is to subdivide recursively a domainthat contains the roots of ๐ in subdomains, and to reject or acceptthe subdomains according to criteria that guarantee that a subdo-main contains one or zero root. For real roots, the criteria that onemay use are notably the Descartesโ rule of signs ([29] and refer-ences therein), the Budanโs theorem [17, 31, 36], or the Sturmโstheorem [1] among others. For complex roots, one may use Pel-letโs test [2] or Cauchyโs integral theorem [20, 21] among others.Combining subdivision approaches with Newton iterations allowsto match the complexity bound of Panโs algorithm for real [30].Subdivision methods are more commonly implemented, notably uillaume Moroz in the software ANewDsc [24], SLV [35], the package RootFindingin Maple [26], the package real_roots in sage [34], Ccluster [22]among others.We can also mention approaches based on the computation ofthe eigenvalues of the companion matrix associated to ๐ [27]. Theseapproach has the advantage of being numerically stable in manycases [15]. These methods are implemented notably in Matlab [33]and numpy [18]. Focusing on univariate polynomial equations,we develop new algorithms that are for the ๏ฌrst time polynomial inthe logarithm of a condition number, and quasi-linear in the degree.Our approaches work for two classical condition numbers that wede๏ฌne here for ๐ฅ in the interval [ , ] and for ๐ง in the complex unitdisk ๐ท ( , ) .Following the theory of condition number associated to the root-๏ฌnding problem [8, chapter 14 and 16], we introduce the followingde๏ฌnitions. De๏ฌnition 1.1.
Given the polynomial ๐ ( ๐ฅ ) = ร ๐๐ = ๐ ๐ ๐ฅ ๐ = ร ๐๐ = q(cid:0) ๐๐ (cid:1) ๐ ๐ ๐ฅ ๐ , let ๐ ( ๐ก ) = cos ๐ ( ๐ก ) ๐ ( tan ( ๐ก )) . The real hyperboliccondition number associated to ๐ is: ๐ R โ ( ๐ ) = max ๐ฅ โ[ , ] min (cid:18) k ๐ k | ๐ ( ๐ฅ )| , ๐ k ๐ k | ๐ โฒ ( ๐ฅ )| (cid:19) The real elliptic condition number associated to ๐ is: ๐ R ๐ ( ๐ ) = max ๐ก โ[ , ๐ ] min k ๐ k | ๐ ( ๐ก )| , โ ๐ k ๐ k | ๐ โฒ ( ๐ก )| ! For ๐ ( ๐ง ) with ๐ง in the unit disk, letting ๐ ๐ ( ๐ฅ ) = ๐ ( ๐ฅ๐ ๐๐ ) ,we de๏ฌne the complex hyperbolic and the complex elliptic con-dition numbers as ๐ C โ ( ๐ ) = max ๐ โ[ , ๐ ] ๐ R โ ( ๐ ๐ ) and ๐ C ๐ ( ๐ ) = max ๐ โ[ , ๐ ] ๐ R ๐ ( ๐ ๐ ) respectively.The justi๏ฌcation for the name hyperbolic and elliptic comes fromthe fact that when the ๐ ๐ are independent, centered Gaussian vari-ables of variance 1, then the density of the root distribution in [ , ] converges to 1 /( ๐ ( โ ๐ก )) when ๐ converges to in๏ฌnity. Similarly,when the ๐ ๐ are independent, centered Gaussian variables of vari-ance 1, then the root distribution has density โ ๐ /( ๐ ( + ๐ก )) [14].Remark that by symmetry of the weights we consider in frontof the coe๏ฌcients, we can reduce the problem of ๏ฌnding all theroots in R or in C to the problem of all ๏ฌnding all the roots in [ , ] and C respectively, through the changes of variable ๐ฅ โฆโ โ ๐ฅ and ๐ฅ โฆโ / ๐ฅ .For our algorithms, we consider polynomials with bit-streamcoe๏ฌcients, where the ๏ฌrst ๐ bits can be accessed in ๐ ( ๐ ) bit op-erations. Our output is a list of approximate zero as introduced bySmale [32], in the sense that for any point ๐ง returned by our algo-rithm, the sequence ๐ง ๐ + = ๐ง ๐ โ ๐ ( ๐ง ๐ )/ ๐ โฒ ( ๐ง ๐ ) converges quadrat-ically toward its associated root of ๐ . We can now state our mainresult. Theorem 1.2.
Let ๐ ( ๐ฅ ) be a polynomial of degree ๐ , with bit-streams coe๏ฌcients.There exist two algorithms that ๏ฌnds all its real roots in the inter-val [ , ] in ๐ ( ๐ log ( ๐๐ ) polylog ( log ( ๐๐ ))) with ๐ = ๐ R โ ( ๐ ) and ๐ = ๐ R ๐ ( ๐ ) respectively. TypeDomain Hyperbolic Elliptic [ , ] ๐ท ( , ) [ , ] ๐ท ( , ) ๐ฃ ๐ ( ๐ฅ ) ๐ฅ ๐ (cid:0) ๐๐ (cid:1) sin ๐ ( ๐ฅ ) cos ๐ โ ๐ ( ๐ฅ ) โ ( ๐ฅ ) ๐ฅ tan ( ๐ฅ ) ๐ log ( ๐๐ ) ๐ ๐ ( log ( ๐ / ๐ )) ๐ ( p ๐ / ๐ ) ๐ ๐ ๐ + ๐ ( p ๐ / ๐ ) ๐พ Eq. (1) Eq. (2) Eq. (3) Eq. (4) ๐ Table 1: Values for the initialisation of the variables in Step ๐ด of Algorithm 1 There exist two algorithms that ๏ฌnds all its complex roots in theunit disk in ๐ ( ๐ log ( ๐๐ ) polylog ( log ( ๐๐ ))) with ๐ = ๐ C โ ( ๐ ) and ๐ = ๐ C ๐ ( ๐ ) respectively. The main idea of our algorithms is to approximate ๐ with apiecewise polynomial function, where each polynomial has a de-gree in ๐ ( log ( ๐๐ )) . This is achieved by partitioning the interval [ , ] and the unit disk following the distribution of the roots. Thenusing Kantorovichโs theory, we show that a good enough approx-imation the roots of the piecewise polynomial is a set of approxi-mated roots associated to all the roots of ๐ . Our method is summa-rized in Algorithm 1.For the correctness of Algorithm 1, we prove in key Lemma 2.1that if a polynomial ๐ of small degree is su๏ฌciently close to a se-ries ๐ , then the problem of ๏ฌnding the root of ๐ can be reduced tothe problem of ๏ฌnding the roots of ๐ . Then in Section 3.4 and 4.4,we show that the piecewise polynomials that we construct in Al-gorithm 1 satisfy the assumptions of Lemma 2.1.For the bound on the number of bit operations, the main stepsthat we need to analyse in Algorithm 1 are Step ๐ต and Step ๐ถ . InStep ๐ถ we need to solve ร ๐๐ = ๐ ๐ polynomials of degree ๐ . Using aclassical algorithm with the state-of-the-art complexity ([28, Theo-rem 2.1.1] and [2]), we can ๏ฌnd all the roots in the unit disk of eachpolynomial with an error bounded by 2 โ ๐ , and with a number ofbit operations in ๐ ( ๐ polylog ( ๐ )) . Then, since ๐ is in ๐ ( log ( ๐๐ )) and the sum of the ๐ ๐ is in ๐ ( ๐ / log ( ๐๐ )) in all cases (see Table 1),we conclude that the bound on the number of bit operations toperform Step ๐ถ is in ๐ ( ๐ log ( ๐๐ ) polylog ( log ( ๐๐ ))) .In Step ๐ต , if we perform the loop as written in Algorithm 1,this leads to a number of operations quadratic in ๐ . Instead, in Sec-tion 3.5 and 4.5, we show how we can modify Step ๐ต such that thenumber of bit operations for this step is in ๐ ( ๐ log ( ๐๐ ) polylog ( log ( ๐๐ ))) .First we will prove in Section 2 that we can reduce the root-๏ฌnding problem to the problem of ๏ฌnding the roots of a smallerdegree polynomial. Then in Section 3 and 4, we will prove the cor-rectness and bound the complexity of Algorithm 1 for polynomialswith small hyperbolic condition number and small elliptic condi-tion number respectively. Finally in Section 5, we will discuss openquestions related to our approach. ast real and complex root-finding methods for well-conditioned polynomials Algorithm 1
Root-๏ฌnding algorithm
Input: ๐ : list of ๐ + ๐ก : type of the monomial weight ( ๐๐๐๐๐๐ก๐๐ or โ๐ฆ๐๐๐๐๐๐๐๐ ) ๐ : bound on the condition number (see De๏ฌnition 1.1) Output: ๐๐๐ ๐ข๐๐ก : list of the approximate roots of the function (ร ๐๐ = ๐ [ ๐ ] ๐ฅ ๐ if ๐ก is โ๐ฆ๐๐๐๐๐๐๐๐ ร ๐๐ = ๐ [ ๐ ] q(cid:0) ๐๐ (cid:1) ๐ฅ ๐ if ๐ก is ๐๐๐๐๐๐ก๐๐ A. Initialization
Variables depending on ๐ก (see Table 1): ๐ฃ โ list of ๐ + โ โ a scale function ๐พ โ list of ๐ real numbers, centers of disks ๐ โ list of ๐ real number, radii of disks ๐ โ list of ๐ integers ๐ โ โ
10 log ( ๐๐ )โ ๐ค โ list of ๐ -th roots of unity for โค ๐ โค ๐ do ๐ง [ ๐ ] โ list of the ๐ ๐ -th roots of unity ๐๐๐ ๐ข๐๐ก โ empty list B. Evaluation for โค ๐ < ๐ and 0 โค ๐ < ๐ dofor โค ๐ < ๐ ๐ do ๐ [ ๐, ๐, ๐ ] โ ร ๐๐ = ๐ [ ๐ ] ๐ฃ [ ๐ ]( ๐พ [ ๐ ] + ๐ [ ๐ ] ๐ค [ ๐ ]) ๐ง [ ๐, ๐ ] ๐ up to precision k ๐ k โ ๐ C. Interpolation and root-๏ฌnding for โค ๐ < ๐ dofor โค ๐ < ๐ ๐ do ๐ โ polynomial such that ๐ ( ๐ค [ ๐ ]) = ๐ [ ๐, ๐, ๐ ] for all ๐ with coe๏ฌcients up to precision k ๐ k โ ๐ ๐ โ roots of ๐ up to precision k ๐ k โ ๐ for โค ๐ < size of ๐ do Append โ ( ๐พ [ ๐ ] + ๐ [ ๐ ] ๐ [ ๐ ]) ๐ง [ ๐, ๐ ] to ๐๐๐ ๐ข๐๐ก return ๐๐๐ ๐ข๐๐ก Given a polynomial or an analytic series ๐ , we will denote by ๐ โฒ and ๐ โฒโฒ the derivative and the second derivative of ๐ , and by ๐ ( ๐ ) the ๐ -th derivative of ๐ . Given a vector ๐ฃ , we will denote by k ๐ฃ k , k ๐ฃ k and k ๐ฃ k โ the classical norm 1, 2 and in๏ฌnity of ๐ฃ . The trans-pose of ๐ฃ is denoted by ๐ฃ ๐ and its conjugate transpose by ๐ฃ ๐ป and if ๐ค is another vector, ๐ฃ ๐ป ยท ๐ค denotes their scalar product. For a matrix ๐ด , we denote by k ๐ด k ๐ the induced norm sup ๐ฅ โ k ๐ด๐ฅ k ๐ /k ๐ฅ k ๐ .For a polynomial ๐ ( ๐ฅ ) = ร ๐๐ = ๐ ๐ ๐ฅ ๐ = ร ๐๐ = q(cid:0) ๐๐ (cid:1) ๐ ๐ ๐ฅ ๐ , we de-note by k ๐ k the norm 1 of the vector ( ๐ ๐ ) , and by k ๐ k ๐ the norm2 of the vector ( ๐ ๐ ) .Finally, we will denote by ๐ผ the interval [ , ] , by ๐ the unit disk,and by ๐ท ( ๐พ, ๐ ) the complex disk of radius ๐ centered at ๐พ . Based on Kantorovichโs theory, we show that if a polynomial anda series have coe๏ฌcients close enough, then the roots of the poly-nomial are in the basin of quadratic convergence of the roots ofthe series.We state the following theorem for complex roots in the unitdisk ๐ท ( , ) โ C . Remark that in the case where ๐ and ๐ have realcoe๏ฌcients, it holds for their real roots in the interval [ , ] โ R Lemma 2.1.
Let ๐ ( ๐ฅ ) = ร โ ๐ = ๐ ๐ ๐ฅ ๐ be an analytic series with ra-dius of convergence greater than . Assume that there exist ๐ > , ๐ > , ๐ > and an integer ๐ > ( ๐ ๐ ) such that for all point ๐ง in the unit disk: โข | ๐ ( ๐ง )| โค ๐ /( ๐ ๐ ) implies | ๐ โฒ ( ๐ง )| > ๐ / ๐ , โข | ๐ โฒโฒ ( ๐ง )| < ๐๐ โข for all ๐ > ๐ we have | ๐ ๐ | โค ๐ / ๐ .Let ๐ ( ๐ฅ ) = ร ๐๐ = ๐ ๐ ๐ฅ ๐ be a polynomial of degree ๐ such that for all โค ๐ โค ๐ we have | ๐ ๐ โ ๐ ๐ | โค ๐ / ๐ .Then, for each root ๐ of ๐ in the unit disk, ๐ has no other rootin ๐ท ( ๐ , /( ๐ ๐ )) and ๐ has a root in the disk ๐ท ( ๐ , /( ๐ ๐ )) . More-over, if ๐ has a root ๐ in the unit disk, then ๐ has a root in the disk ๐ท ( ๐, /( ๐ ๐ )) . Proof.
First, let ๐ be a root of ๐ in the unit disk. Then | ๐ ( ๐ )| = | ๐ ( ๐ ) โ ๐ ( ๐ )| โค ๐ ๐ + ๐ + ๐ ๐ โค ๐ ๐ + ๐ using the bounds on the di๏ฌer-ence of the coe๏ฌcients of ๐ and ๐ . In particular, with the lowerbound on ๐ , we have ๐ โฅ log ( ๐ ) + ๐ / ๐ โฅ
20 since ๐ ๐ โฅ . This implies that | ๐ ( ๐ ) โค ๐ ( ๐ + )/ ๐ โค ๐ /( ๐ ๐ ) ยท ( ๐ + )/ ( ๐ / ) โค ๐ /( ๐ ๐ ) . This implies that | ๐ โฒ ( ๐ )| โฅ ๐ / ๐ . In turn, wehave ๐ฝ = | ๐ ( ๐ )/ ๐ โฒ ( ๐ )| โค /( ๐ ๐ ) , and ๐พ = max ๐ง โ ๐ (| ๐ โฒโฒ ( ๐ง )/ ๐ โฒ ( ๐ )|) โค ๐ ๐ . Thus, 2 ๐ฝ๐พ โค / โค
1. Using Kantorovichโs theory [12, Theo-rem 88], this ensures that ๐ has a root in ๐ท ( ๐, ๐ฝ ) which impliesthat ๐ has a root in the disk ๐ท ( ๐, /( ๐ ๐ )) . Moreover, using Kan-torovichโs theory again [12, Theorem 88], since 2 ๐พ โค ๐ ๐ , thisimplies that ๐ is the only root of ๐ in the disk ๐ท ( ๐ , /( ๐ ๐ )) .Reciprocally, let ๐ be a root of ๐ in the unit disk. Then | ๐ ( ๐ )| = | ๐ ( ๐ ) โ ๐ ( ๐ )| โค ๐ ๐ + ๐ using the bounds on the di๏ฌerence of thecoe๏ฌcients of ๐ and ๐ . Similarly | ๐ โฒ ( ๐ ) โ ๐ โฒ ( ๐ )| โค ๐ ๐ ( ๐ + ) ๐ + + ๐ ๐ + ๐ โค ๐ ( ๐ + ) ๐ + . And for all ๐ง โ ๐ we have also | ๐ โฒโฒ ( ๐ง ) โ ๐ โฒโฒ ( ๐ง )| โค ๐ (cid:0) ๐ + (cid:1) / ๐ โ + ๐ ( ๐ + ๐ + )/ ๐ โค ๐ ( ๐ + ) ยท ๐ .This implies that: | ๐ ( ๐ )| โค ๐ ๐ + ๐ | ๐ โฒ ( ๐ )| โฅ ๐ / ๐ โ ๐ ( ๐ + ) / ๐ | ๐ โฒโฒ ( ๐ง )| โค ๐๐ + ๐ ( ๐ + ) / ๐ In particular, with the lower bound on ๐ , we have ๐ โฅ log ( ๐ ) + ๐ / ๐ โฅ
20, which implies | ๐ โฒ ( ๐ )| โฅ ๐ /( ๐ ) and | ๐ โฒโฒ ( ๐ง )| โค ๐ ( ๐ + / ) . Such that ๐ฝ : = | ๐ ( ๐ )/ ๐ โฒ ( ๐ )| โค ( ๐ + ) ๐ / ๐ โ and ๐พ : = max ๐ง โ ๐ (| ๐ โฒโฒ ( ๐ง )/ ๐ โฒ ( ๐ )|) โค ( ๐ + / ) ๐ โค . / ๐ ๐ .Let ๐ = /( . ๐ ๐ ) . Using Kantorovichโs theory [12, Theorem 85]this implies that ๐ is the unique root of ๐ in the disk ๐ท ( ๐ , ๐ ) .Moreover, ๐ฝ /( ๐ ) โค . ๐ ๐ ( ๐ + )/ ๐ โค . ( ๐ + )/ ๐ / โค / ๐ โฅ
19, which is the case since ๐ ๐ โฅ . In this case, using uillaume Moroz Kantorovichโs theory again [12, Theorem 88], 2 ๐ฝ๐พ โค ๐ฝ / ๐ โค ๐ has a root in ๐ท ( ๐ , ๐ฝ ) . Moreover, since ๐ฝ /( ๐ ) โค / ๐ฝ โค ๐ / ๐ has a root ๐ in the disk ๐ท ( ๐ , ๐ / ) .In particular, ๐ is the only root of ๐ in the disk ๐ท ( ๐, ๐ / ) , whichimplies that ๐ โ ๐ท ( ๐, /( ๐ ๐ )) and thus ๐ โ ๐ท ( ๐ , /( ๐ ๐ )) . (cid:3) In this section we consider the polynomial ๐ ( ๐ฅ ) = ร ๐๐ = ๐ ๐ ๐ฅ ๐ , overthe interval [ , ] and over the complex unit disk ๐ท ( , ) . For a complex number ๐พ and a real number ๐ , we de๏ฌne the poly-nomial ๐ ๐พ,๐ ( ๐ฅ ) = ๐ ( ๐พ + ๐๐ฅ ) . Lemma 3.1.
Let ๐ > be a real and ๐พ a complex number in ๐ท ( , ) such that either ๐ โค โ| ๐พ | , or ๐ โค ๐ /( ๐๐ ) . Let ๐ ๐ be the coe๏ฌcientsof ๐ฅ ๐ in ๐ ๐พ,๐ ( ๐ฅ ) . For all ๐ > ๐ : | ๐ ๐ | โค ๐ k ๐ k Proof.
We distinguish 2 cases. For the case where 2 ๐ โค โ| ๐พ | , the coe๏ฌcient of ๐ฅ ๐ in ๐ ๐พ,๐ is ๐ ๐ = ร ๐๐ = ๐ ๐ ๐ (cid:0) ๐๐ (cid:1) | ๐พ | ๐ โ ๐ ๐ ๐ โค ๐ ร ๐๐ = ๐ ๐ ๐ (cid:0) ๐๐ (cid:1) | ๐พ | ๐ โ ๐ ( โ | ๐พ |) ๐ โค k ๐ k / ๐ .Then, for the case ๐ โค ๐ /( ๐๐ ) we have ๐ ๐ = ร ๐๐ = ๐ ๐ ๐ (cid:0) ๐๐ (cid:1) | ๐พ | ๐ โ ๐ ๐ ๐ โค ๐ ๐ (cid:0) ๐๐ (cid:1) k ๐ k . Using the inequality ๐ ! โฅ โ ๐๐ ( ๐ / ๐ ) ๐ we get (cid:0) ๐๐ (cid:1) โค โ ๐๐ ( ๐๐ / ๐ ) ๐ . Which implies, forall ๐ > ๐ that ๐ ๐ โค k ๐ k ( ๐ /( ๐ )) ๐ โค k ๐ k / ๐ . (cid:3) [ , ] Let ( ๐พ ๐ ) ๐ โ ๐ = and ( ๐ ๐ ) ๐ โ ๐ = be the sequences: ๐พ ๐ = โ
23 13 ๐ ๐ ๐ = (
13 13 ๐ if 0 โค ๐ < ๐ โ โ ๐พ ๐ if ๐ = ๐ โ ๐ = โ log (cid:16) ๐๐๐ (cid:17) โ is chosen such that ๐ ๐ โ โค ๐ ๐๐ . Remarkthat the union of the intervals [ ๐พ ๐ โ ๐ ๐ ,๐พ ๐ + ๐ ๐ ] is the interval [ , ] .Finally, for 0 โค ๐ < ๐ , Lemma 3.1 implies that the coe๏ฌcients ๐ ๐ polynomial ๐ ( ๐พ ๐ + ๐ก๐ ๐ ) satisfy ๐ ๐ โค k ๐ k / ๐ for all ๐ > ๐ . ๐ท ( , ) In this section, we de๏ฌne a set of disks that cover the disks unit diskof radius 1 centered at 0, while their centers and radii still satisfythe conditions of Lemma 3.1.Let ( ๐ ๐ ) ๐๐ = be the sequence: ๐ ๐ = ( โ ๐ if 0 โค ๐ < ๐ ๐ = ๐ Then for 0 โค ๐ < ๐ , let ๐พ ๐ = ( ๐ ๐ + ๐ ๐ + ) and ๐ ๐ = ( ๐ ๐ + โ ๐ ๐ ) ,such that ( ๐พ ๐ ) ๐ โ ๐ = and ( ๐ ๐ ) ๐ โ ๐ = are the sequences: ๐พ ๐ = ( โ
34 12 ๐ if 0 โค ๐ โค ๐ โ โ
12 12 ๐ if ๐ = ๐ โ ๐ ๐ = (
38 12 ๐ if 0 โค ๐ โค ๐ โ
34 12 ๐ if ๐ = ๐ โ ๐ = โ log (cid:16) ๐๐๐ (cid:17) โ is chosen such that ๐พ ๐ โ โค ๐ ๐๐ .Let ๐ ๐ = ๐ min ( ๐ + ,๐ + ) . The following lemma shows that theunion of the disks ๐ท ( ๐พ ๐ ๐ ๐๐๐ ๐ , ๐ ๐ ) for 0 โค ๐ < ๐ and 0 โค ๐ < min ( ๐ + ,๐ + ) contains the disk ๐ท ( , ) . Lemma 3.2.
The disk of center ๐พ ๐ and radius ๐ ๐ covers a sector ofangle ๐ min ( ๐ + ,๐ + ) of the ring between the concentric circles centeredat of radii ๐ ๐ and ๐ ๐ + . Proof.
Consider the ring between the circles of radii ๐ ๐ and ๐ ๐ + and let ๐ผ ๐ be the angle of the sector covered by the disk ๐ท ( ๐พ ๐ , ๐ ๐ ) . Using classical trigonometric formula we have ๐ ๐ = ๐พ ๐ + ๐ ๐ + โ ๐พ ๐ ๐ ๐ + cos (cid:0) ๐ผ ๐ (cid:1) , and we also have ๐ ๐ = ( ๐ ๐ + โ ๐พ ๐ ) , whichimplies:sin (cid:16) ๐ผ ๐ (cid:17) = vt โ ( ๐พ ๐ + ๐ ๐ + โ ๐ ๐ ) ๐พ ๐ ๐ ๐ + = vt ๐ ๐ ( ๐พ ๐ + ๐ ๐ + ) โ ( ๐ ๐ + โ ๐พ ๐ ) ๐พ ๐ ๐ ๐ + = vuuuuuuuuut ( ๐ ๐ + โ ๐พ ๐ ) ๐ ๐ + ( + ๐ ๐ + ๐พ ๐ )โ ( ( ๐ ๐ + โ ๐พ ๐ )( ๐ ๐ + + ๐พ ๐ ) ๐ ๐ + ๐พ ๐ ) = ๐ ๐ + โ ๐พ ๐ ๐ ๐ + s ( + ๐ ๐ + ๐พ ๐ ) โ ( ๐ ๐ + ๐พ ๐ + ) A variation analysis shows that for 1 โค ๐ฅ โค
2, the expression ( + ๐ฅ ) โ ( + ๐ฅ ) is greater or equal to 5. Moreover, ๐ ๐ + โ ๐พ ๐ ๐ ๐ + isgreater than ๐ + if ๐ < ๐ โ ๐ + if ๐ = ๐ โ ๐ผ ๐ โฅ sin (cid:16) ๐ผ ๐ (cid:17) โฅ โ min ( ๐ + ,๐ + ) โฅ ๐ min ( ๐ + ,๐ + ) (cid:3) Remark that like for the real case, for 0 โค ๐ < ๐ , Lemma 3.1implies that the coe๏ฌcients ๐ ๐ polynomial ๐ ( ๐พ ๐ + ๐ก๐ ๐ ) satisfy ๐ ๐ โคk ๐ k / ๐ for all ๐ > ๐ . We show in this section that the polynomials computed in Algo-rithm 1 computes the correct approximate roots of ๐ . For that, weshow that with the parameters chosen in the algorithm, Lemma 2.1applies correctly and thus, the approximate truncated polynomialsthat we use return the correct roots. We focus on the complex case.The real case can be proven with similar arguments. ast real and complex root-finding methods for well-conditioned polynomials Let ๐ โฅ ๐ be a real number, let ๐พ โ ๐ท ( , ) and ๐ > ๐ โค โ | ๐พ | or ๐ โค ๐ /( ๐๐ ) . Moreover, assume that ๐ โฅ ๐ /( ๐๐ ) . Denote by ๐ ๐พ,๐ ( ๐ง ) the polynomial ๐ ( ๐พ + ๐๐ง ) and denoteby ๐ ๐ its coe๏ฌcients. For ๐ง โ ๐ , | ๐ โฒ ๐พ,๐ ( ๐ง )| = ๐ | ๐ โฒ ( ๐พ + ๐๐ง )| and ๐ โฒโฒ ๐พ,๐ ( ๐ง ) = ๐ ๐ โฒโฒ ( ๐พ + ๐๐ง ) . Lemma 3.3.
With ๐ = k ๐ k , ๐ = ๐๐ ๐ , ๐ = ๐ โ ( ๐ ) and ๐ = โ ( ๐ ๐ )โ , ๐ ๐พ,๐ satis๏ฌes all the assumptions of Lemma 2.1. Proof.
First, by de๏ฌnition of ๐ โ , if | ๐ ๐พ,๐ ( ๐ง ) = ๐ ( ๐พ + ๐๐ง )| โค ๐ / ๐ ,then | ๐ โฒ ( ๐พ + ๐๐ง )| โฅ ๐๐ / ๐ , which implies | ๐ โฒ ๐พ,๐ ( ๐ง )| โฅ ๐๐ /( ๐๐ ) โฅ ๐ / ๐ .For the second derivative of ๐ โฒโฒ ๐พ,๐ ( ๐ง ) , remark ๏ฌrst that | ๐พ + ๐๐ง | โค| ๐พ | + ๐ โค + ๐ /( ๐๐ ) . Thus, for all 0 โค ๐ โค ๐ , we have | ๐พ + ๐๐ง | โค( + ๐ /( ๐๐ )) ๐ โค ๐ ๐ /( ๐ ) โค ๐ . Thus, | ๐ โฒโฒ ๐พ,๐ ( ๐ง )| โค ๐ | ๐ โฒโฒ ( ๐พ + ๐๐ง )| โคk ๐ k ๐๐ ๐ .Finally, for ๐ > ๐, | ๐ ๐ | โค k ๐ k / ๐ using Lemma 3.1. (cid:3) ๐ We focus now on the complexity of Step ๐ต in Algorithm 1. Wemodify the algorithm to be able to bound correctly the number ofbit operations of this step.The following lemma ๏ฌrst shows how to evaluate quickly thepoints near the unit circle. Lemma 3.4.
Let ๐ > be a real number and ๐ > be an in-tegers such that ๐ โค ๐ / ๐ . Given a complex number ๐ง such that | ๐ง | โค + ๐ / ๐ and an integer ๐ > , it is possible to compute the ๐ values ๐ ( ๐ง๐ ๐ ๐๐ / ๐ ) for โค ๐ < ๐ with an absolute error lowerthan k ๐ k / ๐ and with a number of bit operations in ๐ ( ๐ / ๐ ( ๐ + ๐ + log ( ๐ )) ยท polylog ( ๐ + log ( ๐ ))) . Proof.
For any 0 โค ๐ โค ๐ , remark that | ๐ง ๐ | โค ( + log ( ) ๐ / ๐ ) ๐ โค ๐ . Using fast algorithms, we can compute in quasi-linear time the๏ฌrst ๐ digits of the result of arithmetic operations [37]. Thus, wecan evaluate the ๏ฌrst ๐ + ๐ + ( ๐๐ ) digits of ๐ง ๐ with a numberof bit operations in ๐ ( ๐,๐, ๐ ) = ๐ (( ๐ + ๐ + log ( ๐ )) polylog ( ๐ + log ( ๐ ))) . This allows us notably to evaluate ๐ ( ๐ง ) with an errorlower than k ๐ k / ๐ in ๐ ( ๐๐ ( ๐,๐,๐ )) bit operations.For ๐ in ๐ ( ๐ / ๐ ) , we want to evaluate ๐ on the ๐ -th roots ofunity. We start by computing the polynomial ๐ ( ๐ ) = ๐ ( ๐ ) mod ๐ ๐ โ ๐ โ ๐ ( ๐๐ ( ๐,๐,๐ )) .Then we can use the fast Fourier transform to evaluate ๐ ( ๐ ๐ ๐๐ / ๐ ) for 0 โค ๐ < ๐ in ๐ ( ๐ / ๐ log ( ๐ ) ๐ ( ๐,๐, ๐ )) bit operations. (cid:3) Then we show how the points ๐ง in the disks ๐ท ( ๐พ ๐ , ๐ ๐ ) that sat-isfy | ๐ง | โค โ / ๐ and can be evaluated more e๏ฌciently. Lemma 3.5.
Let ๐ be a positive integer and ๐ > be an in-tegers such that ๐ โค ๐ + . Given a complex number ๐ง such that | ๐ง | โค โ / ๐ and an integer ๐ > , it is possible to compute the ๐ values ๐ ( ๐ง๐ ๐ ๐๐ / ๐ ) for โค ๐ < ๐ with an absolute error lower than k ๐ k / ๐ and with a number of bit operations in ๐ ( ๐ ( ๐ + log ( ๐ )) ยท polylog ( ๐ )) . Proof.
First, remark that | ๐ง ๐ | โค ๐ โ ๐ / ๐ . In particular, for ๐ > log ( ) ๐ ๐ , we have | ๐ง ๐ | โค / ๐ . Let e ๐ be the polynomial ๐ trun-cated to the degree ๐ ๐ = โ log ( ) ๐ ๐ โ . Each ๐ง ๐ for ๐ โค ๐ ๐ canbe computed with an error less than 1 / ๐ and with a number of bit operations in ๐ ( ๐ ) = ๐ ( ๐ polylog ( ๐ )) . The polynomial ๐ ( ๐ ) = e ๐ ( ๐ ) mod ๐ ๐ โ ๐ ( ๐ ๐ ๐ ( ๐ )) . Then, using the fast Fourier trans-form approach, we can compute ๐ ( ๐ง๐ ๐ ๐๐ / ๐ ) with a number of bitoperations in ๐ ( ๐ log ( ๐ ) ๐ ( ๐ )) . (cid:3) Thus, combining Lemma 3.4 and 3.5, with ๐ and ๐ in ๐ ( log ( ๐๐ )) , ๐ in ๐ ( log ( ๐ / ๐ )) , ๐ in ๐ ( ๐ ) , ๐ ๐ = / ๐ + , we can compute ๐ (( ๐พ ๐ + ๐ ๐ ๐ ๐ ๐๐ / ๐ ) ๐ ๐ ๐โ / ๐ ๐ ) for all 0 โค ๐ โค ๐ , 0 โค โ < ๐ ๐ and 0 โค ๐ < ๐ with a number of bit operations in ๐ ( ๐ log ( ๐๐ ) polylog ( ๐๐ )) . In this section, we consider the polynomial ๐ ( ๐ฅ ) = ร ๐๐ = q(cid:0) ๐๐ (cid:1) ๐ ๐ ๐ฅ ๐ and we de๏ฌne the function ๐ ( ๐ฅ ) = cos ๐ ( ๐ฅ ) ๐ ( tan ( ๐ฅ )) . Remark thatfor ๐ฅ โ [ , ๐ / ] , the function tan ( ๐ฅ ) is a bijection between theroots of ๐ in [ , ๐ / ] and the roots of ๐ in [ , ] . Moreover, let ๐ฃ ๐ ( ๐ฅ ) = q(cid:0) ๐๐ (cid:1) sin ๐ ( ๐ฅ ) cos ๐ โ ๐ ( ๐ฅ ) and let ๐ฃ ( ๐ฅ ) be the vector map ( ๐ฃ ( ๐ฅ ) , . . . , ๐ฃ ๐ ( ๐ฅ )) ๐ . Using the notations of the introduction, thefunction ๐ can be rewritten: ๐ ( ๐ฅ ) = ๐ ๐ป ยท ๐ฃ ( ๐ฅ ) Letting ๐ผ ๐ = p ๐ ( ๐ + โ ๐ ) , Edelman and Kostlan [14] observedthat the derivative of ๐ฃ satis๏ฌes the equation ๐ฃ โฒ ( ๐ฅ ) = ๐ด ยท ๐ฃ ( ๐ฅ ) , where ๐ด is the anti-symmetric linear matrix: ๐ด = ยฉ ยซ โ ๐ผ ๐ผ โ ๐ผ ๐ผ โ ๐ผ . . . . . . . . .๐ผ ๐ โ โ ๐ผ ๐ ๐ผ ๐ ยชยฎยฎยฎยฎยฎยฎยฎยฎยฌ This leads to the following relation: ๐ฃ ( ๐ฅ ) = ๐ ๐ฅ๐ด ๐ฃ ( ) As a corollary, for any point ๐ง โ ๐ท ( , ) : ๐ ( ๐ ) ( ๐ง ) = ๐ ยท ๐ ๐ง๐ด ๐ด ๐ ๐ฃ ( ) ๐ For any real ๐ฅ , observe that the matrix ๐ ๐ฅ๐ด is orthogonal because ๐ด is antisymmetric. This allows to prove the following lemma. Lemma 4.1.
For any real ๐ฅ : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ๐ ( ๐ ) ( ๐ฅ ) ๐ ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) โค k ๐ k (cid:16) max ( , p ๐๐ / ๐ ) (cid:17) ๐ Proof.
First using norm inequality, we have: (cid:12)(cid:12)(cid:12) ๐ ( ๐ ) ( ๐ฅ ) (cid:12)(cid:12)(cid:12) โค k ๐ k k ๐ด ๐ ๐ฃ ( )k For a positive integer ๐ , let ๐ด ๐ be the matrix ๐ด where all the entriesof indices ( ๐, ๐ ) with ๐ โฅ ๐ + ๐ โฅ ๐ + ๐ด is a tridiagonal matrix, and since ๐ฃ ( ) = ( , , ยท ยท ยท , ) ๐ ,we can deduce by induction that: ๐ด ๐ ๐ฃ ( ) = ๐ด ๐ ยท ยท ยท ๐ด ๐ฃ ( ) uillaume Moroz Let โ = j ๐ + k . For ๐ โค โ , we can bound the norm of ๐ด ๐ by: k ๐ด ๐ k โค p k ๐ด k k ๐ด k โ โค p ( ๐ โ )( ๐ + โ ( ๐ โ )) + p ๐ ( ๐ + โ ๐ )โค p ๐ ( ๐ + โ ๐ ) For ๐ > โ , we have k ๐ด ๐ k โค ๐ +
1. This allows us to deduce that:1 ๐ ! k ๐ด ๐ ๐ฃ ( )k โค ๏ฃฑ๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃณ ๐ q(cid:0) ๐๐ (cid:1) if ๐ โค ๐ + โ q(cid:0) ๐โ (cid:1) ( ๐ + ) ๐ โ โ โ ! ๐ ! otherwiseUsing the inequality ๐ ! โฅ โ ๐๐ (cid:16) ๐๐ (cid:17) ๐ we get (cid:0) ๐๐ (cid:1) โค โ ๐๐ (cid:16) ๐๐๐ (cid:17) ๐ .Moreover for ๐ > ( ๐ + )/
2, observe that q(cid:0) ๐โ (cid:1) โค โ ๐ โค โ โค ๐ ,and ( ๐ + ) ๐ โ โ โ ! ๐ ! โค ๐ โ โ , such that:1 ๐ ! k ๐ด ๐ ๐ฃ ( )k โค ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃณ (cid:18) q ๐๐๐ (cid:19) ๐ if ๐ โค ๐ + ๐ otherwise (cid:3) [ , ] Using the bound on the derivative of ๐ shown in the previous sec-tion we de๏ฌned a sequence of disks ๐ท ( ๐พ ๐ , ๐ ๐ ) that covers the realsegment [ , ๐ / ] such that the series ๐ ( ๐พ ๐ + ๐ ๐ ๐ง ) has the abso-lute value of its coe๏ฌcients ๐ ๐ decreasing exponentially for ๐ largeenough.For a real ๐ , let ๐ = (cid:24) ๐ q ๐๐๐ (cid:25) , and for 0 โค ๐ < ๐ let ๐พ ๐ and ๐ ๐ de๏ฌned by: ๐พ ๐ = ( ๐ + ) r ๐๐๐๐ ๐ = r ๐๐๐ (3)It is easy to check that the union of the corresponding diskscover the segment [ , ๐ / ] . The properties of the series ๐ ( ๐พ ๐ , ๐ ๐ ) will be analysed in Section 4.4. ๐ท ( , ) For the complex case, we need to de๏ฌne a sequence of disks ๐ท ๐ = ๐ท ( ๐พ ๐ , ๐ ๐ ) such that the union of the sets tan ( ๐ท ๐ ) covers an angu-lar sector of ๐ท ( , ) big enough. For that, we prove the followinglemma. Lemma 4.2.
Let โค ๐ โค ๐ / . If a set of complex disks ๐ท , . . . , ๐ท ๐ โ C covers the band ๐ต ๐ of points ๐ง with | ๐ผ๐ ( ๐ง )| โค ๐ and โค ๐ ๐ ( ๐ง ) โค ๐ / , then tan ( ๐ท ) , . . . , tan ( ๐ท ๐ ) covers the angular sector ๐ด ๐ of theunit disk between the angle โ ๐ and ๐ . Proof.
Using the integral expression of the function atan, re-mark that atan ( ๐ + ๐๐ ) = atan ( ๐ ) + โซ ๐ + ๐๐๐ง = ๐ + ๐ง ๐๐ง . In particular, aslong as ๐ โค ๐ , we have ๐ ๐ ( ๐ง ) โฅ
0, such that | + ๐ง | โค
1, whichallows us to conclude that | atan ( ๐ + ๐๐ ) โ atan ( ๐ )| โค | ๐ | . More-over, if ๐ โฅ ๐ + ๐ โค โค ๐ ๐ ( atan ( ๐ + ๐๐ )) โค ๐ /
4. Thus, for any point ๐ง โ ๐ด ๐ , we have atan ( ๐ง ) โ ๐ต ๐ , such that ๐ด ๐ โ tan ( ๐ต ๐ ) . (cid:3) Thus, we can cover a band of width q ๐ ๐๐ with ๐ = (cid:24) ๐ q ๐๐๐ (cid:25) disks de๏ฌned for 0 โค ๐ < ๐ by: ๐พ ๐ = ๐ r ๐๐๐๐ ๐ = r ๐๐๐ (4)This allows us to cover the angular sector of radius ๐ โฅ q ๐ ๐๐ ,and the number of sectors needed to cover the unit disk is ๐ = โ ๐ / ๐ โ โค โ ๐ q ๐๐๐ โ . We focus in this section on the complex case. The real case can beproven with similar arguments.For a complex number ๐พ and a real number ๐ , denote by ๐ ๐พ,๐ ( ๐ง ) the series ๐ ( ๐พ + ๐๐ง ) and denote by ๐ ๐ its coe๏ฌcients. For ๐ง โ ๐ , | ๐ โฒ ๐พ,๐ ( ๐ง )| = ๐ | ๐ โฒ ( ๐พ + ๐๐ง )| and ๐ โฒโฒ ๐พ,๐ ( ๐ง ) = ๐ ๐ โฒโฒ ( ๐พ + ๐๐ง ) . Lemma 4.3.
Let โค ๐พ โค ๐ / and ๐ = p ๐ /( ๐๐ ) be two real num-bers. With ๐ = k ๐ k , ๐ = ๐๐ ๐ , ๐ = ๐ ๐ ( ๐ ) and ๐ = โ ( ๐ ๐ )โ , ๐ ๐พ,๐ satis๏ฌes all the assumptions of Lemma 2.1. Proof.
First, by de๏ฌnition of ๐ ๐ , if | ๐ ๐พ,๐ ( ๐ง ) = ๐ ( ๐พ + ๐๐ง )| โค ๐ / ๐ ,then | ๐ โฒ ( ๐พ + ๐๐ง )| โฅ ๐ โ ๐ / ๐ , which implies | ๐ โฒ ๐พ,๐ ( ๐ง )| โฅ ๐ โ ๐ / ๐ โฅ ๐ / ๐ .For the second derivative of ๐ โฒโฒ ๐พ,๐ ( ๐ง ) , remark that ๐ โฒโฒ ( ๐ง ) = ๐ ยท ๐ด ๐ ๐๐ด๐ง ยท ๐ฃ ( ) = ๐ ยท ๐ด ๐ฃ ( ๐ง ) . Remark that ๐ฃ ( ๐ + ๐๐ ) = ๐ ๐๐ด ๐ฃ ( ๐๐ ) and k ๐ฃ ( ๐๐ )k = ( cosh ( ๐ ) + sinh ( ๐ )) ๐ / = cosh ๐ / ( ๐ ) . Using the in-equality cosh ( ๐ฅ ) โค ๐ ๐ฅ this leads to k ๐ฃ ( ๐ + ๐๐ )k โค ๐ ๐๐ . With | ๐ | โค ๐ , this leads to | ๐ โฒโฒ ๐พ,๐ ( ๐ง )| โค ๐ k ๐ k k ๐ด k ๐ log ( ) ๐ . Moreover, El-deman and Kostlan showed that ๐๐ด is similar to the Kac matrix [14,ยง4.3], and the absolute value of its eigenvalues is lower or equal to ๐ , such that k ๐ด k โค ๐ and ๐ k ๐ด k โค ๐๐ .Finally, for ๐ > ๐, | ๐ ๐ | โค k ๐ k / ๐ using Lemma 4.1. (cid:3) Thus, the two sequences of disks de๏ฌned in Equations (3) and (4)cover the interval [ , ] and the unit disk ๐ท ( , ) respectively, andthey satisfy the conditions of Lemma 4.3. ๐ In the elliptic case, evaluating a sequence of points in Step ๐ต of Al-gorithm 1 naively would be done roughly in a ๐ ( ๐ ) , or in ๐ ( ๐ / ) operations if we use the fast Fourier transforms. In both cases, thiswould exceed our complexity bound. The main idea in this sec-tion is to remark that if we are interested in computing an approx-imate value of the function ๐ = ร ๐๐ = ๐ ๐ ๐ฃ ๐ ( ๐ง ) up to k ๐ k / ๐ for agiven integer ๐ , then we can truncate ๐ to use a support of size in ๐ (โ ๐๐ ) . Lemma 4.4.
Given a function ๐ ( ๐ง ) = ร ๐๐ = ๐ ๐ ๐ฃ ๐ ( ๐ง ) and an inte-ger ๐ > , there exists โค โ โค ๐ข โค ๐ such that | ๐ข โ โ | โค โ ๐๐ and (cid:12)(cid:12)(cid:12) ๐ ( ๐ง ) โ ร ๐ข๐ = ๐ ๐ ๐ฃ ๐ ( ๐ง ) (cid:12)(cid:12)(cid:12) โค k ๐ k โ ๐ โ . ast real and complex root-finding methods for well-conditioned polynomials Proof.
Let ๐ be an integer, ๐ฅ be a real between 0 and1 and โ = max ( , โ ๐ฅ๐ โ p ( ) ๐ ( ๐ + )โ) and ๐ข = min ( ๐, โ ๐ฅ๐ + p ( ) ๐ ( ๐ + )โ) . Let ๐ผ be the union of theindices 0 , . . . , ๐ and ๐ข, . . . , ๐ . Using the Hoe๏ฌding inequality,we have ร ๐ โ ๐ผ (cid:0) ๐๐ (cid:1) ๐ฅ ๐ ( โ ๐ฅ ) ๐ โ ๐ โค ยท โ ( ๐ + ) . In par-ticular, this implies that | ร ๐ โ ๐ผ ๐ ๐ q(cid:0) ๐๐ (cid:1) sin ( ๐ง ) ๐ cos ( ๐ง ) ๐ โ ๐ | โคk ๐ k ร โ๐ = (cid:0) ๐๐ (cid:1) | sin ๐ ( ๐ง ) cos ( ๐ โ ๐ ) ( ๐ง )| . If ๐ง = ๐ + ๐๐ , wehave | cos ( ๐ง )| + | sin ( ๐ง )| = cosh ( ๐ ) . Thus, letting ๐ฅ = | sin ( ๐ง )|/ cosh ( ๐ ) , we can use the Hoe๏ฌding inequality and de-duce: | ร ๐ โ ๐ผ ๐ ๐ s(cid:18) ๐๐ (cid:19) sin ( ๐ง ) ๐ cos ( ๐ง ) ๐ โ ๐ | โค k ๐ k cosh ๐ / ( ๐ ) โ ( ๐ + )+ Moreover, comparing the coe๏ฌcients of the Taylorexpansion at 0 of cosh ( ๐ฅ ) and exp ( ๐ฅ / ) , remark thatcosh ๐ / ( ๐ ) โค ๐ ๐๐ . In particular, if | ๐ | โค p log ( ) ๐ / ๐ thatimplies cosh ๐ / ( ๐ ) โค ๐ . This allows us to conclude that | ๐ ( ๐ง ) โ ร ๐ข๐ = ๐ ๐ ๐ sin ๐ ( ๐ง ) cos ๐ โ ๐ ( ๐ง )| โค k ๐ k โ ๐ โ . (cid:3) Truncating ๐ can also be used to evaluate it e๏ฌciently on a set ofroots of unity using fast Fourier transform, as required for Step ๐ต of Algorithm 1. Lemma 4.5.
Let ๐ > be a real number and ๐ > be an inte-gers such that ๐ โค ๐ p ๐๐ / ๐ . Given a complex number ๐ง such that | ๐ผ๐ ( ๐ง )| โค p log ( ) ๐ / ๐ , let ๐ ๐ง ( ๐ ) = ร ๐๐ = ๐ ๐ ๐ฃ ๐ ( ๐ง ) ๐ ๐ . Given an inte-ger ๐ > , it is possible to compute the ๐ values ๐ ๐ง ( ๐ ๐ ๐๐ / ๐ ) for โค ๐ < ๐ with an absolute error lower than k ๐ k / ๐ with a number ofbit operations in ๐ ( p ๐ / ๐ ( ๐ + ๐ + log ( ๐ )) ยท polylog ( ๐ + log ( ๐ ))) . Proof.
For any 0 โค ๐ โค ๐ , and ๐ง = ๐ + ๐๐ , remark that | ๐ฃ ๐ ( ๐ + ๐๐ )| โค | cos ( ๐ง )| + | sin ( ๐ง )| = cosh ๐ / ( ๐ ) โค ๐ . Using fast algo-rithms, we can compute in quasi-linear time the ๏ฌrst ๐ digits ofthe result of arithmetic operations [37]. Moreover, using methodssuch as the FEE method [23], we can also evaluate trigonometric,exponential and factorial functions in quasi-linear time. Thus, wecan evaluate the ๏ฌrst ๐ + ๐ + ( ๐๐ ) digits of ๐ฃ ๐ ( ๐ง ) with a num-ber of bit operations in ๐ ( ๐,๐, ๐ ) = ๐ (( ๐ + ๐ + log ( ๐ )) polylog ( ๐ + log ( ๐ ))) . Using Lemma 4.4, this allows us notably to evaluate ๐ ๐ง ( ) with an error lower than k ๐ k / ๐ in ๐ (โ ๐๐๐ ( ๐,๐, ๐ )) bit opera-tions after truncating ๐ ๐ง .Let ๐ ๐ง ( ๐ ) be the polynomial of degree 4 โ ๐๐ approximating ๐ ๐ง ( ๐ ) . For ๐ in ๐ ( p ๐ / ๐ ) , we want to evaluate ๐ ๐ง on the ๐ -throots of unity. We start by computing the polynomial โ ๐ง = ๐ ๐ง mod ๐ ๐ โ ๐ โ ๐ (โ ๐๐๐ ( ๐, ๐,๐ )) . Then we can use the fast Fourier transform toevaluate โ ๐ง ( ๐ ๐ ๐๐ / ๐ ) for 0 โค ๐ < ๐ in ๐ ( p ๐ / ๐ log ( ๐ ) ๐ ( ๐,๐, ๐ )) bit operations. Thus the total number of bit operations is in ๐ (โ ๐๐ (โ ๐๐ + log ( ๐ )) ๐ ( ๐,๐, ๐ )) and ๐ (โ ๐๐ ) = ๐ ( ๐ + ๐ ) . (cid:3) Finally, Lemma 4.5 with ๐ and ๐ in ๐ ( log ( ๐๐ )) , ๐ in ๐ ( p ๐ / ๐ ) , ๐ in ๐ ( ๐ ) , ๐ ๐ in ๐ ( p ๐ / ๐ ) for all ๐ , we can compute all the val-ues in Step ๐ต of Algorithm 1 with a number of bit operations in ๐ ( ๐ log ( ๐๐ ) polylog ( ๐๐ )) . A third natural family of polynomials is of the form ๐ ( ๐ฅ ) = ร ๐๐ = โ ๐ ! ๐ ๐ ๐ฅ ๐ . When ๐ converges to in๏ฌnity, the density of itsroots distribution converges to 1 / ๐ . Thus, we can de๏ฌne a so-called ๏ฌat condition number as follow. De๏ฌnition 5.1.
Given the polynomial ๐ ( ๐ฅ ) = ร ๐๐ = q ๐ ! ๐ ๐ ๐ฅ ๐ , let ๐ ( ๐ฅ ) = ๐ ( ๐ฅ ) ๐ โ ๐ฅ / . The real ๏ฌat condition number associated to ๐ is: ๐ R ๐ ( ๐ ) = max ๐ฅ โ R min (cid:18) k ๐ k | ๐ ( ๐ฅ )| , k ๐ k | ๐ โฒ ( ๐ฅ )| (cid:19) For ๐ ( ๐ง ) with ๐ง in the unit disk, letting ๐ ๐ ( ๐ฅ ) = ๐ ( ๐ฅ๐ ๐๐ ) , we let ๐ C ๐ ( ๐ ) = max ๐ โ[ , ๐ ] ๐ R ๐ ( ๐ ๐ ) be the complex ๏ฌat condition numberassociated to ๐ .Considering this new condition number, several natural ques-tions occur. First, remark that the density of the distribution of theroots of ๏ฌat polynomials is close to the density of the distributionof the eigenvalues of random matrices. Whereas it was shown thatthe expectation of the hyperbolic condition number of the charac-teristic polynomial of complex standard Gaussian matrices of size ๐ is in 2 ฮฉ ( ๐ ) , it would be interesting to analyse the ๏ฌat conditionnumber of such characteristic polynomials.From an algorithmic point of view, remark that inthe ๏ฌat case, considering the vector of function ๐ฃ ( ๐ฅ ) = ( ๐ โ ๐ฅ / , ๐ฅ๐ โ ๐ฅ / , . . . , ๐ฅ ๐ ๐ โ ๐ฅ / , . . . ) , the derivation of ๐ฃ is ananti-symmetric operator, as for the elliptic case. Thus, afterdealing with boundary conditions, we should be able to derive analgorithm that ๏ฌnd the roots of such polynomials with numberof bit operations linear in ๐ and polynomial in the logarithm of ๐ ๐ . Such an algorithm might be well suited to ๏ฌnd the roots ofcharacteristic polynomials. As mentioned in introduction, the current bound on the number ofoperation to ๏ฌnd the roots of a multivariate polynomial systems ofequations is currently polynomial in its degree and in its conditionnumber. It would be nice to generalize for the multivariate case thetools that we used and developed for the univariate case.In particular, Lemma 2.1 is based on Kantorovichโs theory,where all theorems are valid for multivariate systems. Moreover,the distribution of the roots is also well described for the multivari-ate case [14]. Combining those results as we did for the univariatecase could help to improve the state-of-the-art bounds on the prob-lem of ๏ฌnding the roots of multivariate polynomial systems.
Although the algorithm we present here is quasi-linear in the de-gree of the polynomials and polynomial in the logarithm of its con-dition number, it requires that a bound on the condition numberis given as input. If the bound given as input is to low, the resultsmight be wrong. uillaume Moroz
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