Estimates for Positive Roots of Polynomials › digitalAssets › 120 › 120262_1Stefanescu.pdfIntroduction • The computation of accurate bounds for univariate complex polynomials

Estimates for Positive Roots of Polynomials

Doru Ştefănescu1 Vladimir Gerdt2 Simeon Evlakhov2

1University of Bucharest, Romania2JINR Dubna, Russia

29 October 2009, CADE 2009, PAMPLONA

Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 1 / 39

Contents

• Introduction

• Computation of Polynomial Roots

• Bounds for Complex Roots

• Bounds for Real Roots

• Conclusions


Introduction

• The computation of accurate bounds for univariate complexpolynomials has many appplications in various problems involvingpolynomials.

• We review some methods for computing polynomial bounds,emphasizing on the case of real zeros.

• For real roots we propose a general method that extends thetheorems of Kioustelidis, Ştefănescu a. o. These bounds areexpressed in function of the degree, the size of the coefficientsand families of parameters that can be properly chosen.

• We use these bounds for the evaluation of the sizes of polynomialdivisors.


Computation of Polynomial Roots

• One of the fundamental problems in Numerical Mathematics is thecomputation of polynomial zeros.

• The determination of the roots arises frequently in applications.

• Since the exact computation of the zeros in function of thecoeffients of the polynomial is not possible for generalpolynomials, for all practical purposes it is useful to handleefficient methods for estimation.

• Bounds for complex roots were obtained, among other, by Cauchy,Kuniyeda, Fujiwara, Landau and Montel.Upper and lower bounds can be derived using polynomial sizesdefined in function of the coefficients, such that the norm, thelength, the heigth and the measure.


Computation of Polynomial Roots (contd.)

• For computing real roots it is necessary to have efficient rootisolation methods.

• Root isolation means the computation of intervals containingexactly one real root.

• The CF (continued fraction) algorithms use estimates of thelowest bound for positive root.

• The computation of lowest bounds is equivalent to that of upperbounds.

• There are few methods that computes only bounds for real roots.

• We describe a new method for computing upper bounds forpositive roots.

• Finally we discuss bounds concerning roots of orthogonalpolynomials.


Bounds for Complex Roots

• We remind some of the most used bounds for complex roots ofunivariate polynomials.

• There exist many bounds for absolute values of complex roots ofpolynomials.

• All the bounds for complex roots can be used for computing upperbounds real roots.


Cauchy

Theorem (A.–L. Cauchy, 1829)

All the roots of the nonconstant complex polynomialP(X ) = a0 + a1X + · · · + adX d are contained in the disk |z| ≤ ξ ,where ξ is the unique positive solution of the equation

|ad |Xd = |a0| + |a1|X + · · · + |ad−1|X

d−1 . (1)


Kuniyeda

Theorem (M. Kuniyeda, 1916)

If and p, q > 0 are such that 1p +1q = 1 , then all the roots of the

polynomial P are contained in the disk |z| ≤ ξ , where

ξ =

1 +(

n−1∑

j=0

∣

∣

∣

∣

ajan

∣

∣

∣

∣

p) qp

1q

.


Fujiwara

Theorem (M. Fujiwara, 1926)

If λ1, . . . , λd ∈ (0,∞) and

1λ1

+ · · · +1λd

= 1,

then all the roots of the polynomial P are contained in the disk |z| ≤ ξ ,where

ξ = max1≤k≤d

(

λk∣

∣

an−dan

∣

∣

)

1k

.


Dominant Roots

• A root α ∈ C of the polynomial P ∈ C[X ] is called dominant if|α| > |β| for any other root β.

• The computation of dominant roots was considered by Newton(1707) in his Arithmetica Universalis, no. 133–137. His idea wasdeleloped by Daniel Bernoulli (1728), who used linear recurrentsequences for approaching the dominant roots.


Open Problems

• There exist still open problems concerning the estimation ofdominant roots.

• For example, let us suppose that a complex polynomial P hasexactly four dominant roots α1, . . . , α4 such that α1, α2 are realand α3, α4 are complex conjugate.

• For example, the powerfull methods of Bernoulli andLobachevskii–Graeffe give unconvenient results.


Bounds for Real Roots

• Bounds for Real Roots can be derived from bounds for ComplexRoots

• Lagrange obtained two upper bounds for positive real roots. Oneof them is surprisingly efficient.

• New bounds for positive roots were obtained by Kiostelidis (1986)and Ştefănescu (2005).

• We propose a general bound for positive roots.


Lagrange L1

Theorem (1)

[J.–L. Lagrange, 1769] LetP(X ) = a0X d + · · · + amX d−m − am+1X d−m−1 ± · · · ± ad ∈ R[X ] , withall ai ≥ 0, a0, am+1 > 0 . Let

A = max{

ai ; coeff (Xd−i) < 0

}

.

The number

1 +(

Aa0

)1/(m+1)

is an upper bound for the positive roots of P .


Lagrange L2

Note that in some special cases the following other bound of Lagrangecan be useful:

Theorem (2)

IfP(X ) = X d −

∑

j∈J

aj Xd−j + positive terms + . . . ,

andu√

|au| := R ≥ ρ := v√

|aj | ≥i

√

|ai | for all i 6= u, v .

the number R + ρ is an upper bound for the positive roots.


The Bound of Fujiwara

One of the most efficient is the following

Fw(P) = 2 · max

∣

∣

∣

∣

ad−iad

∣

∣

∣

∣

1/i

In fact, the previous bound is a general one for complex roots, wasobtained for the first time by Lagrange and rediscovered in 1916 byFujiwara [3]. It was recently used by V. Sharma [10]. For complex rootsit is one of the best.


Recent Bounds

Specific bounds for positive real roots were obtained by Kioustelidis(1986, [4]) and Ştefănescu (2005, [8]).

• They can be applied to polynomials having roots smaller than 1.

• They are more efficient than the classical bounds.

• They give methods for improving the bounds for particular classesof polynomials.


The Theorem of Kioustelidis (1986)

Theorem (3)

(J. B. Kioustelidis, 1986 [4]) Let

P(X ) = X d − b1Xd−m1 − · · · − bkX

d−mk +∑

j 6=m1,...,mk

ajXd−j ,

with b1, . . . , bk > 0 and aj ≥ 0 for all j 6∈ {m1, . . . , mk} .

The numberK (P) = 2 · {b1/m11 , . . . , b

1/mkk }

is an upper bound for the positive roots of P.


A result of Ştef ănescu (2005)

Theorem (4)

(D. Ştef ănescu, 2005 [8]) Let

P(X ) = X d − b1Xd−m1 − · · · − bkX

d−mk +∑

j 6=m1,...,mk

ajXd−j ,

with b1, . . . , bk > 0 and aj ≥ 0 for all j 6∈ {m1, . . . , mk} .

The number

B1(P) = max{(kb1)1/m1 , . . . , (kbk )

1/mk }

is an upper bound for the positive roots of P.


A Theorem of Ştef ănescu (2005)

We remind our main bound from [8]:

Theorem (5)

D. Ştef ănescu, 2005 [8] Let P(X ) ∈ R[X ] be such that the number ofsign variations of its coefficients is even. If

P(X ) = a1Xd1 − b1X

m1 + · · · + asX ds − bsX ms + g(X ) ,

where g(X ) ∈ R+[X ], aj > 0, bj > 0, dj > mj for all j , the number

St(P) = max1≤j≤s

{

(

bjaj

)1/(dj−mj )}


Improved Methods

Theorem (6)

Let

P(X ) = a1Xd1+a2X

d2 +· · ·+asX ds−b1Xe1−b2X

e2−· · ·−btX et ∈ R[X ] ,

where ai > 0 , bj > 0, d1 = deg(P) and d1 > d2 > · · · > ds .An upper bound for the positive roots of P is given by

max1≤i≤s1≤j≤tβj 6=0

(

γjibjβj ai

)

1di − ej


A General Bound (contd.)

for any βj ≥ 0 , γjk ≥ 0 such that

∑tj=1 βj ≤ 1 ,

∑si=1 γji ≥ 1 with γji = 0 if di < ej .

From the previous Theorem it is easy to obtain the bound K ofKioustelidis (Theorem 3) and the bound B1 of Ştefănescu (Theorem 5),that apply to polynomials of the formX d − b1X d−m1 − · · · − bkX d−mk + positive terms


A bound for s ≥ t

Proposition (7)

If s ≥ t and di > ej for all i and j, the number

B5(P) = max1≤i≤s1≤j≤t

(

bjai

)1/(di−ej )

is an upper bound for the positive roots of the polynomial

P(X ) = a1Xd1 + a2X

d2 + · · · + asX ds − b1Xe1 − b2X

e2 − · · · − btX et .


A New General Result

Theorem (8)

Let

P(X ) = a1Xd1+a2X

d2 +· · ·+asX ds−b1Xe1−b2X

e2−· · ·−btX et ∈ R[X ] ,

where ai > 0 , bj > 0, d1 = deg(P) and d1 > d2 > · · · > ds . An upperbound for the positive roots of P is given by

B6(P) = max1≤i≤s,1≤j≤t

di≥ej

(

bjβjai

)1

di−ej

for any βj > 0 such that

β1 + · · · + βt ≤ 1 .


A New General Result (contd.)

The following cases will be considered:

β1 = β2 = . . . = βt =1t

. (2)

and

β1 =12

, β2 = . . . = βt =1

2(t − 1). (3)

For (3) we obtained the following


B6 pseudocode

Input: P(X ) ∈ Q[X ]. Output: A real numberp := 0for i = 1 to s do

for j = 1 to t dor := di − ejif r > 0 then

q :=(

tbjai

)1

di−ej

fiif q > p then

p = qfi

ododreturn p


Complexity of the algorithm

Let coefficients of a polynomial and its degree be bounded by L bits,and let n be an integer such that

n ≫ max(L, log t)

Theorem (9)

The bit-complexity for the n − bit evaluation of a function which can beexponential function, or a trigonometric function, or an elementaryalgebraic function, or their superposition, or their inverse, or asuperposition of the inverses is given by

O(M(n) log2 n)

where M(n) is the complexity function for multiplication of n−bitintegers.


Complexity of the algorithm (contd.)Thus, the algorithm has the complexity of the bound estimation withaccuracy up to n digits is

O(

s · t · M(n) log2 n)

Here s and t are respectively the numbers of positive and negativecoefficients in the input polynomial.

The best (w.r.t. complexity) known algorithms for integer multiplicationare

• Karatsuba & Ofman (1962)

M(n) = O(

nlog2 3 ≈ n1,5849...)

• Schönhage and Strassen (1971)

M(n) = O(n · log n · log log n)


Numerical Results

We compare various results on upper bounds for positive polynomials.The following notation will be used:

L1(P) = 1 +(

Aa0

)1/(m+1)

L2(P) = R + ρ

Fw(P) = S(P) = 2 · max∣

∣

∣

ad−iad

∣

∣

∣

1/i

K (P) = 2 · max{b1/m11 , . . . , b1/mkk }


Numerical Results (contd.)

B1(P) = max{(kb1)1/m1 , . . . , (kbk )1/mk }

St(P) = max1≤j≤s

{

(

bjaj

)1/(dj−mj )}

B5(P) = max 1≤i≤s1≤j≤t

(

bjai

)1/(di−ej)

B6(P) = max 1≤i≤s,1≤j≤tdi≥ej

(

bjβjai

) 1di−ej

We denote by TUB the true upper bound for positive roots.


The case all di > ei

We consider the polynomials

P1(X ) = X 11 + 1.9 X 2 − 3

P2(X ) = X 11 + 3X 2 + 7X − 11.1

P3(X ) = X 11 + 32X 3 + 2X 2 − 37

P L1 L2 Fw K B1 B5 TUBP1 2.105 2.178 2.147 2.147 1.105 1.105 1.006P2 2.244 2.489 2.489 2.489 1.244 1.923 1.004P3 2.388 2.777 3.084 2.777 1.388 1.388 1.017


The case s = t

We consider the polynomials from [8] and Q5.

Q1(X ) = 3X 4 − X 3 + 7X 2 − 3X + 0.001

Q2(X ) = X 5 − 1.01X 4 + X 3 − 1.1X + 0.1

Q3(X ) = 3X 7 − X 6 + 7X 5 − 3X 2 + 0.001

Q4(X ) = 10X 9 − 17X 5 + 10X 4 − 13X + 1

Q5(X ) = 2X 13 − 3X 5 + 12X 4 + 3X − 15.7


The case s = t (contd.)

We obtain

P Fw K B1 St TUBQ1 3.055 2.0 0.857 0.428 0.421Q2 2.048 2.048 1.483 1.048 1.003Q3 3.055 2.0 0.949 0.753 0.725Q4 2.283 2.283 1.375 1.141 1.12Q5 2.471 2.343 1.303 1.069 1.025


The General CaseIn the general case we consider the bounds L1, L2, Fw , K , B1 and B6.For convenience, in B6, we consider

β1 = β2 = . . . = βt =1t

. (4)

We consider the following polynomials

R1 = X 21 + 7X 9 − 8X 8 − 9X 6 + 5X 5 − 11

R2 = X 21 − 5X 9 − 8X 8 − 9X 6 + 5X 5 − 11

R3 = X 7 + 2X 6 − 2X 4 − 3X 2 − X + 1

and we haveP L1 L2 Fw K B1 B6 TUBR1 12.000 2.331 2.352 2.346 1.279 3.428 1158R2 12.000 2.316 2.346 2.346 1.305 1.544 1.255R3 2.732 2.505 4.000 2.519 1.817 1.817 1.165


The General Case (contd.)

Note that for the choice (4) we always have B6(P) ≤ B1(P) . But forothere choices of can have B6(P) < B1(P). For example, if weconsider

β1 =12

, β2 = . . . = βt =1

2(t − 1). (5)

we obtain

P B1 B6 TUBR1 1.279 2.285 1.158R2 1.305 1.675 1.255R3 1.817 1.643 1.165

We observe that in this case B6 is smaller for R2 and R3. On the otherhand we have

B6(R3) < B1(R3) .


Chebychev Polynomials of First Kind Td(X )

Td(X ) =d2

⌊d/2⌋∑

k=0

(−1)k 2d−2k

d − k

(

d − kk

)

X d−2k

d L2 K B1 B2 B3 TUB3 0.7 1.732 0.866 0.866 1.225 0.86616 3.3 4.000 4.000 4.000 2.828 0.99563 6.8 7.937 15.875 15.875 11.941 0.999251 13.7 15.843 62.875 62.875 50.316 0.999356 16.4 18.868 89.000 89.000 71.648 0.999500 19.4 22.361 125.000 125.000 101.042 0.999


Chebychev Polynomials of First Kind Td(X ) (contd.)

5

10

15

20

25

Upp

er b

ound

Chebyshev I Kind Polynomials Td(X)

L2

Fw B

1B

3


Laguerre Polynomials Ld(X )

Ld(X ) =d

∑

k=0

(−1)k1k!

(

nk

)

X k .

d K B1 B6 Mth-1 Mth-2 TUB3 18.0 18.0 9.0 13.08 27.9 6.2816 512.0 2048.0 256.0 35 · 102 489.5 51.763 7938.0 13 · 104 3970.0 23 · 102 29 · 102 230.9300 18 · 104 13 · 106 9 · 104 3 · 104 7 · 105 972.4500 5 · 105 63 · 106 25 · 104 2 · 106 6 · 105 7050.4


Conclusions

• There exist still open problems for estimating the dominant roots.

• The general bounds for complex roots can give good estimates, inparticular the bound of Fujiwara.

• The bound L1 can be used only in particular cases, especiallywhen the largest positive root is greater than 1.

• The bound of Kioustelidis can be used for any polynomials havingpositive roots.

• The bound R + ρ of Lagrange is better than that of Fujiwara.

• For polynomials with an even number of sign variations the boundŞtefănescu ST gives the best result.

• For polynomials with an arbitrry number of sign variations suitableapplications of Theorem 6 can be used.

• Theorem 8 gives good estimates for all cases.


REFERENCES

[1] A. Akritas: Linear and Quadratic Complexity Bounds on theValues of the Positive Roots of Polynomials, Univ. J. Comput. Sci.,15, 523-537 (2009).

[2] M. Fujiwara: Über die obere Schranke des absoluten Betragesder Wurzeln einer algebraischen Gleichung, Tôhoku Math. J., 10,167–171 (1916).

[3] J. B. Kioustelidis: Bounds for positive roots of polynomials, J.Comput. Appl. Math., 16, 241-244 (1986).

[4] D. KNUTH: The Art of Computer Programming, Volume 2:Seminumerical Algorithms, Addison–Wesley, Reading Ma (1981).

[5] M. MARDEN: Geometry of Polynomials, AMS Math. Surv. andMonographs, Providence (1989). by Lagrange, IRMA Strasbourg025/2002, 1–17 (2002).

[6] M. Mignotte, D. Ştefănescu: Polynomials – An algorithmicapproach, Springer Verlag (1999).


[7] D. Ştefănescu: New Bounds for Positive Roots of Polynomials,Univ. J. Comput. Sci., 11, 2125-2131 (2005).

[8] D. Ştefănescu: Computation of Dominant Real Roots ofPolynomials, Prog. and Compt. Sotfware, 34, 69–74 (2008).

[9] V. Sharma: Complexity of real root isolation sing continuedfractions, Theor. Comp. Sci., 409, 292–310 (2008).

[10] E. Tsigaridas, I. Emiris: On the complexity of real root isolationusing continued fractions, Theor. Comp. Sci., 392, 158–173(2008).


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Estimates for Positive Roots of Polynomials › digitalAssets › 120 › 120262_1Stefanescu.pdfIntroduction • The computation of accurate bounds for univariate complex polynomials