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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
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Contents
• Introduction
• Computation of Polynomial Roots
• Bounds for Complex Roots
• Bounds for Real Roots
• Conclusions
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 2 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 3 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 4 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 5 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 6 / 39
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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)
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 7 / 39
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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
.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 8 / 39
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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
.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 9 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 10 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 11 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 12 / 39
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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 .
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 13 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 14 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 15 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 16 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 17 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 18 / 39
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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 )}
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 19 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 20 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 21 / 39
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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 .
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 22 / 39
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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 .
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 23 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 24 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 25 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 26 / 39
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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)
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 27 / 39
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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 }
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 28 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 29 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 30 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 31 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 32 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 33 / 39
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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) .
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 34 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 35 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 36 / 39
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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
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 37 / 39
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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.
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 38 / 39
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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).
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 38 / 39
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[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).
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 39 / 39
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THANK YOU VERY MUCH FOR YOUR ATTENTION !
Ştefănescu, Gerdt, Evlakov (U. Buch, JINR) () Estimates for Positive Roots of Polynomials October 29, 2009 39 / 39
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