Dedicated to Professor Apolodor Aristotel Răduţă’s 70th Anniversary
ON BOUNDS FOR REAL ROOTS OF POLYNOMIALS
DORU ŞTEFĂNESCU
Faculty of Physics, University of Bucharest, P.O.Box MG-11, Bucharest–Măgurele, România
E-mail: [email protected]
Received June 2, 2013
We give a device for computing bounds for positive roots of polynomials. Our
results allow the computation of absolute values for real and complex roots.
Keywords: polynomial roots, real roots, localization of roots
1. INTRODUCTION
The computation of the real roots of univariate polynomials with real coefficients is done using several algorithmic devices. Many of them are based on the isolation of the real roots, i.e. the computation of a finite number of intervals with the
property that each of them contains exactly one root. For that one of the steps is that
of computing bounds for the roots. This can be realized using classical bounds for
the absolute values of complex roots, see [9]. However there exist bounds specific to
real roots. We obtain a device for computing absolute positiveness bounds for the real
roots. The method is based on an improvement of the results of D. Ştefănescu [12]
on upper bounds for positive roots. It it useful for the isolation of these roots, i.e.
for the computation of a finite number of intervals such that each interval contains
exactly one root [1]. These methods allows us to compute an interval containing all
the positive roots of the derivatives of the polynomial. Analytical properties of polynomials and the algorithmic methods for the computation of their roots are relevant
for the study of many physical problems, see, for example [7], [10], [11].
2. BOUNDS FOR POSITIVE ROOTS
We construct new bounds for positive roots of polynomials and give a method
for obtaining bounds of absolute positiveness. We first compute an upper bound for
the for the unique positive root of a special polynomial.
Lemma 1 Let
P (X) = X d + X d−1 + · · · + X e+1 − γX e + be−1 X e−1 + · · · + b1 X + b0 ,
with the coefficients γ and bj strictly positive. Then γ 1/(d−e) is an upper bound for
the positive roots of the polynomials P (i) , for all i ∈ N.
RJP 58(Nos.
Rom.
Journ. Phys.,
9-10),
Vol. 1428–1435
58, Nos. 9-10,(2013)
P. 1428–1435,
(c) 2013-2013
Bucharest, 2013
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On bounds for real roots of polynomials
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Proof: We observe that if β > γ 1/(d−e) we have β d−e > γ, therefore β d−e − γ > 0.
Therefore
P (β) = β e (β d−e − γ) + β d−1 + · · · + β e+1 + be−1 β e−1 + · · · + bd > 0,
On the other hand, if we consider the polynomial
P 0 (X) = dX d−1 + · · · + (e + 1)X e − γeX e−1 + (e − 1)be−1 X e−2 + · · · + b1
we observe that
e
dβ d−1 − γeβ e−1 = dβ e−1 (β d−e − γ) > dβ e−1 (β d−e − γ) > 0.
d
It follows that P 0 (β) > 0.
Proposition 2 Let d > e ≥ 0 be integers and let’s consider the polynomial
d
X
k
ak X k−e − γ, where ad = 1, γ > 0 for all ai ≥ 0.
R(X) =
e
k=e+1
The unique positive root α of the polynomial R satisfies the inequality
1 + γ 1/d
α>
− 1,
M
with M = max{ad , ad−1 , . . . , ae+1 }.
Proof: Since the sequence of the coefficients of R has exactly one change of sign,
by the rule of Descartes R has a unique positive root. We put ae = 1 and we have
d d
X
X
k
k
k−e
ak X k−e − 1 − γ
ak X
−γ =
e
e
k=e
k=e+1
d−e X
e+j
aj+e X j − 1 − γ.
e
j=0
e+j
d
≤
for all j = 0, . . . , d − e,
j
j
=
On the other hand
so we obtain
d d−e d−e d−e X
X
X
X
k
e+j
d
d j
k−e
j
j
ak α
=
aj+e α ≤
aj+e α ≤ M
α .
e
e
j
j
k=e
j=0
j=0
Because
d−e d−1 X
d j X d j
α ≤
α = (1 + α)d − αd ,
j
j
j=0
j=0
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it follows that
d X
k
0 = R(α) = −1 − γ +
ak αk−e
e
k=e
≤ −1 − γ + M (1 + α)d − αd
< −1 − γ + M (1 + α)d ,
hence the conclusion.
Corollary 3 If d > e ≥ 0 are integers and γ > 0, ad = 1, ai ≥ 0, with e < i < d, the
unique positive root of the polynomial
d
X
k
R(X) =
ak X k−e − γ
e
k=e+1
1+γ 1/d
1/(d−e)
.
− 1, γ
lies in the interval
M
Proof: It is sufficient to observe that R = P (e) .
3. ABSOLUTE POSITIVENESS
We remind that a number B > 0 is an absolute positiveness bound of the univariate polynomial P ∈ R[X] if, for any t ∈ N, we have
P (t) (x) > 0 for all x ≥ B.
That means that B is an upper bound for the positive roots of P and for the positive
roots of all its derivatives. As H. Hoon have noticed [3], the bounds for complex
roots of univariate polynomials over the reals are also bounds for absolute positiveness, thanks to the theorem of Guaß–Lucas (s. M. Marden [8]). In fact, if P is
univariate with real coefficients, the convex hull of its zeros contains also the zeros
of its derivative P 0 . Its trace on the real line contains the real zeros of P and also all
zeros of P 0 . However, there exist bounds for positive roots which are not absolute,
see [13]. We use a new bound for positive roots of polynomials with real coefficients:
Theorem 4 Let
P (X) = a1 X d1 + a2 X d2 + · · · + as X ds − b1 X e1 − b2 X e2 − · · · − bt X 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
1
di −ej
bj
max
1≤i≤s,1≤j≤t
βj ai
d ≥e
i
j
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for any βj > 0 such that
β1 + · · · + βt ≤ 1.
Proof: We suppose x ∈ R, x > 0. We have
P (x) =
s
X
di
ai x −
i=1
≥
t
X
bj xej
j=1
s
X
di
(β1 + · · · + βt )ai x −
i=1
t
X
bj xej
j=1


s
t
t
X
X
X

βj ai xdi  −
bj xej
i=1
=
j=1
t
X
j=1
s
X
j=1
!
β j ai x
di
!
− bj x
ej
i=1



t
s
X
 X


di 
ej 


≥
β
a
x
−
b
x
j
i
j



j=1
i=1
di ≥ej


=
t
X
j=1



s
X
i=1
di ≥ej


 e
x j.
−
b
βj ai xdi −ej 
j


The last sum is positive if βj ai xdi −ej − bj > 0 for all i, j such that di ≥ ej , i.e. if
x>
bj
βj ai
1
di − ej ,
which proves the result.
Corollary 5 (D. Ştefănescu [12]) Let
P (X) = X d − b1 X d−m1 − · · · − bt X d−mt + g(X),
with b1 , . . . , bk > 0 and g(X) ∈ R+ [X].
The number
S1 (P ) = max{(tb1 )1/m1 , . . . , (tbt )1/mt }
is an upper bound for the positive roots of P .
Proof: Let Q(X) = X d − b1 X d−m1 − · · · − bt X d−mt . We observe that a bound for
the positive roots of g is also a bound for the positive roots of P . With the notation
RJP 58(Nos. 9-10), 1428–1435 (2013) (c) 2013-2013
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from Theorem 4 we have s = 1 and a1 = 1. We consider β1 = . . . βt =
follows that the number
1
t
and it
S1 (P ) = max{(tb1 )1/m1 , . . . , (tbt )1/mt }
is an upper bound for the positive roots of Q. But any upper bound for the positive
roots of the polynomial Q is also an upper bound for the positive roots of P .
Corollary 6 (J. B. Kioustelidis [4]) Let
P (X) = X d − b1 X d−m1 − · · · − bt X d−mt + g(X),
where b1 , . . . , bk > 0 and g ∈ R+ [X].
The number
1/m1
K(P ) = 2 · max{b1
1/mk
, . . . , bk
}
is an upper bound for the positive roots of P .
Proof: As in the proof of the previous result, it is sufficient to check that K is an
upper bound for the positive roots of the polynomial Q(X) = X d − b1 X d−m1 −
· · · − bk X d−mk and we consider
d−mi
1
βi =
for all i.
2
Without loss of generality we may suppose that
m1 < · · · < m t
and we have
d−m1
d−et
1
1
β1 + · · · + βt =
+···+
2
2
t
1 X
1
1
= m1
1 + m2 −m1 + · · · + mt −m1
2
2
2
m=1
t 1 X
1
1
1 + + · · · + t−1
≤ m1
2
2
2
m=1
≤
1
2m1
·
1 − 21t
1
1 < 2m1 −1 ≤ 1.
1− 2
Therefore K(P ) is an upper bound for the positive roots of Q, then also for those of
Q.
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Notation: Let P ∈ R[X] \ R be such that it has an even number of sign variations
and can be represented as
P (X) = c1 X d1 − b1 X m1 + c2 X d2 − b2 X m2 + · · · + ck X dk − bk X mk + g(X),
with g ∈ R+ , d1 > d2 , . . . , dk , ci > 0, bi > 0 and di > mi for all i.
We put
( 1/(dk −mk ) )
b1 1/(d1 −m1 )
bk
S2 (P ) = max
,...,
c1
ck
D. Ştefănescu proved in [13] that the bounds S1 (P ), S2 (P ) and K(P ) are absolute.
BOUNDS FOR ORTHOGONAL POLYNOMIALS
We consider now classical orthogonal polynomials. They are hyperbolic, i.e.
they have only real roots. Because of the interlacing of the roots with those of the
derivative a bound for the positive roots of an orthogonal polynomial is also a positiveness bound. We obtain newPbounds using the Hessian of Laguerre.
If we consider f (X) = nj=1 aj X j , a univariate polynomial with real coefficients, its Hessian is Hess(f ) = (n − 1)2 f 02 − n(n − 1)f f 0 . Laguerre [6] proved that
it is positive. This gives
Theorem 7 (E. Laguerre) Let f ∈ R[X] be a polynomial of degree n ≥ 2, that has
only real simple roots and that satisfies the second–order differential equation
p(x)y 00 + q(x)y 0 + r(x)y = 0,
(1)
with p, q and r univariate polynomials with real coefficients, p(x) 6= 0.
If α is a root of the polynomial f , then we have
4(n − 1) p(α)r(α) + p(α)q 0 (α) − p0 (α)q(α) − (n + 2)q(α)2 ≥ 0.
(2)
Using also the positivity of the Hessian we obtain another inequality:
Theorem 8 Let f ∈ R[X] be a polynomial of degree n ≥ 2 that satisfies the second–
order differential equation
p(x)y 00 + q(x)y 0 + r(x)y = 0,
(3)
with p, q and r real real polynomials, p(x) 6= 0.
Let us assume that all the roots of f are real and simple. If α is a root of f , then we
have (n − 3)q2 (α)2 − (n − 2)q(α)q3 (α) ≥ 0, where
q2 = q 2 + p0 q − pq 0 − pr,
q3 = 2p0 + q −q 2 − p0 q + pq 0 − pr − pq p00 + 2q 0 + r − p2 q 00 + 2r0 .
RJP 58(Nos. 9-10), 1428–1435 (2013) (c) 2013-2013
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Proof: We differentiate twice the relation
q(α)
g 0 (α) = −
· g(α).
2p(α)
7
(4)
and obtain
p(x)y 000 + p0 (x) + q(x) y 00 + q 0 (x) + r(x) y 0 + r0 (x)y = 0,
respectively
p(x)y (iv) + 2p0 (x) + q(x) y 000 + p00 (x) + 2q 0 (x) + r(x) y 00
+ q 00 (x) + 2r0 (x) y 0 + r00 (x)y = 0.
Therefore
g 00 (α) =
q2 (α)
· g(α).
3p(α)2
(5)
with q2 = q 2 + p0 q − pq 0 − pr and
q3 (α)
· g(α),
(6)
4p(α)3
where q3 = (2p0 + q) −q 2 − p0 q + pq 0 − pr − pq (p00 + 2q 0 + r) − p2 (q 00 + 2r0 ) . The
Hessian of g 0 is (n − 3)2 g 00 − (n − 2)(n − 3)g 0 g 000 and it is positive. So
g 000 (α) =
(n − 3)q2 (α)2 − (n − 2)q(α)q3 (α) ≥ 0.
(7)
Example 1. The Legendre polynomial of degree n satisfies the differential equation
(1 − x2 )y 00 − 2xy 0 + n(n + 1)y = 0.
Using the Hessian
s(1) of Laguerre we obtain the following upper bound for the roots
n+2
La(n) = (n − 1)
.
n(n2 + 2)
Example 2. The Hermite polynomial Hn , which satisfies the differential
equation
r
2
y 00 − 2xy 0 + 2ny = 0 and by (1) it follows that He(n) = (n − 1)
is a bound
n+2
for the roots of Hn .
On the other hand, by Theorem 8 we have the bound
s
p
2n2 + n + 6 + (2n2 + n + 6 + 32(n + 6)(n3 − 5n2 + 7n − 3)
.
4(n + 6)
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