On some absolute positiveness bounds

Transcription

1 Bull. Math. Soc. Sci. Math. Roumanie Tome 53(101) No. 3, 2010, On some absolute positiveness bouns by Doru Ştefănescu Deicate to the memory of Laurenţiu Panaitopol ( ) on the occasion of his 70th anniversary Abstract We prove that some bouns for positive roots of univariate polynomials with real coefficients are absolute positiveness bouns. It is also prove that there exist positiveness bouns which are not absolute. Key Wors: Boun for absolute positiveness, Polynomial zeros Mathematics Subject Classification: Primary: 12D10; Seconary: 68W30. 1 Introuction The computation of bouns for positive roots of univariate polynomials with real coefficients is connecte to many other problems in mathematics an relate fiels. For example it is important for some algorithms for real root isolation an for the problem of testing positiveness of polynomials. This last problem is eciable (see A. Tarsi [14]). H. Hoon an D. Jaus introuce [5] the boun for absolute positiveness of a polynomial with real coefficients. They prove that some bouns for positive roots of univariate polynomials are also bouns for absolute positiveness. However, absolute positiveness nees to be chece for any particular boun. In this paper we observe that there exist upper bouns for positive roots that are not absolute positiveness bouns. Than we prove that some nown bouns for positive roots of univariate polynomials are also bouns for absolute positiveness. We also iscuss some cases where extensions of these results can be applie. The author gratefully acnowleges partial financial support from the grant PN II - IDEI 443, coe 1190/2008, aware by the CNCSIS - UEFISCSU, Romania.

2 270 Doru Ştefănescu 2 Bouns for Positive Roots Several bouns are nown for the absolute values of the roots of univariate polynomials with complex coefficients (see M. Maren [7] an M. Mignotte D. Ştefănescu [8]). They are expresse as functions of the egree an of he coefficients, an naturally they can be use also for the roots (real or complex) of polynomials with real coefficients. In the case of univariate polynomials with real coefficients these bouns are also upper bouns for the real roots, if any. On the other han, some specific bouns for real positive roots also exist (see. Aritas [2], D. Ştefănescu [13]). They are more accurate than those obtaine for the absolute values of complex roots, but espite the case of complex roots there is no general result that assure the absolute positiveness. It is esirable to prove that a boun for positive roots is also a boun for the absolute positiveness. 3 Bouns for Absolute Positiveness We remin that a number B > 0 is an absolute positiveness boun 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 boun for the positive roots of P an for the positive roots of all its erivatives. Remar: As H. Hoon notice [6], the bouns for complex roots of univariate polynomials over the reals are also bouns for absolute positiveness, ue to the theorem of Guaß Lucas (s. M. Maren [7]). In fact, if P is univariate with real coefficients, the convex hull K(P) of its zeros contains also the zeros of its erivative P. Its trace on the real line contains the real zeros of P an also all zeros of P. Remar: A natural question is whether an arbitrary positive boun of a univariate polynomial over the real numbers is also a positive absoluteness boun. As notice by H. Hoon [6], almost all nown upper bouns for positive roots are bouns for absolute positiveness. The next example shows that a boun for the positive roots is not necessarily an absolute positiveness boun. Example 1. Let P(X) = X 5 10X X 3 80X X 31. We observe that 1 is the largest (in fact the unique) positive root of P, so it is a boun for the positive roots. However, the erivative P (X) = 5(X 4 8X X 2 32X + 16) = 5(X 2) 4 has the positive root 2. So B(P) = 1 is an upper boun for the positive roots which is not a boun for the roots of the erivative P, hence it is not an absolute positiveness boun.

3 On Some Absolute Positiveness Bouns 271 We remin that a hyperbolic polynomial is a polynomial from R[X] which has only real roots. The following result is a corollary to nown facts. Proposition 1. Let P R[X] be a hyperbolic polynomial an let (α,β) be an interval incluing its roots. The roots of the erivative are also inclue in (α, β). This can be prove in at least two ways, using as previously the Gauß Lucas theorem or through the interlacing of roots of the polynomial an of its erivative. Notation: Let P R[X]\R be such that it has an even number of sign variations an can be represente as P(X) = c 1 X 1 b 1 X m1 + c 2 X 2 b 2 X m2 + + c X b X m + g(x), with g R +, 1 > 2,...,, c i > 0, b i > 0 an i > m i for all i. Denote S 1 (P) = max { (b1 c 1 ) 1/(1 m 1),..., ( b c ) 1/( m ) }. Proposition 2. Let P R[X] \ R be such that it has an even number of sign variations an can be represente as P(X) = c 1 X 1 b 1 X m1 + c 2 X 2 b 2 X m2 + + c X b X m + g(x), with g R +, 1 > 2,...,, c i > 0, b i > 0 an i > m i for all i. We have S 1 (P) > S 1 (P ). Proof: By Theorem 2 in [11] we now that S 1 (P) is a boun for the positive roots of P, so we have P(X) > 0 for all x B 1. Let s loo to the corresponing boun for the erivative of P. We have P (X) = 1 c 1 X 1 1 m 1 b 1 X m c X 1 m b X m 1 + g(x) We have = c 1X 1 b 1 X m a X b X m. a i = 1 a i, b i = m ib i, i = i 1, e i = e i 1. Therefore i e i = i e i an b i e i = m ib i i a i < b i a i,

4 272 Doru Ştefănescu since m i < i. We obtain that { (b ) S 1 (P 1/(1 m 1) ( ) } 1/( m ) ) max 1 b,,..., < max = S 1 (P). c 1 { (b1 c 1 ) 1/(1 m 1),..., c ( b c ) 1/( m ) } Corollary 3. The following inequalities hol S 1 (P) > S 1 (P ) >... > S 1 (P 1 1 ) > 0. Theorem 4. The number S 1 (P) is an absolute positiveness boun for the polynomial P. Proposition 5. Let P(X) = a 0 X + a 1 X a m X m a m+1 X m 1 ± ± a R[X] with all a i 0, a 0 > 0,, a m+1 > 0. Let A be the largest absoulute value of the negative coefficients of P. The boun of Lagrange ( ) 1 A m+1 L(P) = 1 + is an absolute positiveness boun for P. Proof: We remin that L(P) is an upper boun for the positive roots by a nown theorem of Lagrange... We have P (X) = a 0 X 1 + +( m)a m X m ( m i)a m+1 X m 2 ± ±a 1. If m = 1 we have no negative coefficient in P, so P has no positive roots because all its coefficients are positive. So L(P) is an absolute positiveness boun. If m < 1 we have m 2 0, so the erivative P has also negative coefficients. We enote by A the absolute value of the largest negative coefficient of P. We observe that A = max 0 j m+1 {( m j 1)a j ; coeff(x m j ) < 0} a 0 A max {( m j 1)} 0 j m+1 = A( m 1).

5 On Some Absolute Positiveness Bouns 273 It follows that ( m 1 L(P ) = 1 + A ) 1/(m+1) 1 + ( ) 1/(m+1) A = L(P). From this we euce that L(P) is a boun for absolute positiveness. Theorem 6. Let P(X) = X b 1 X m1 b X m + g(x), where b 1,...,b > 0 an g R + [X]. The number S 2 (P) = max{(b 1 ) (1/m1),...,(b ) (1/m ) } is an aboslute positiveness boun for P. Proof: We put Q(X) = 1 P (X) an we have Q(X) = X 1 m 1 We observe that g R + [X] an b 1 X m1 1 m b X m 1 + g (X). m j b j < b j. We conclue, as in Proposition 2, that S 2 (P) > S 2 (P ). Remar: The boun in Theorem 4 can be extene to polynomials having at least one sign variation. Most of such results are summarize in A. Aritas [2] an D. Ştefănescu [13]. These extensions are base on the formula in Theorem 4 an the proof of Theorem 6. It can be prove that they also are absolute. Applications an Computational Aspects Because classical orthogonal polynomials have real coefficients an all their zeros are real, any upper boun for the positive roots is a boun for abosolute positiveness.,we consier the polynomials P n (X) = L n (X) = n/2 ( 1) (2n 2)!!(n )!(n 2)! Xn 2, =0 n =0 ( ) n ( 1) X, n! Legenre Laguerre

6 274 Doru Ştefănescu an the bouns Nw(P) = a 2 1 2a 2a 0 (Newton, see [9]) an S 1 (P) (Theorem 4). They give the followinf bouns for absolute positiveness: Proposition 7. Let P n, respectively L n be the orthogonal polynomials of egree n of Legenre, respectively Laguerre. i. The numbers n(n 1) S 1 (P n ) = 2(2n 1) are bouns for the absolute positiveness P n. ii. The numbers 2(2n 2)! an Nw(P n ) = (n 1)!(n 2)! S 1 (L n ) = n 2 an Nw(L n ) = n 4 n 2 (n 1) 2 are bouns for the absolute positiveness L n. Proof: We use Theorem 4 an the boun of Newton. For Legenre polynomials we have the boun { } (n 2 + 1)(n 2 + 2) max ;1 n/2 (2n 2 + 1) n(n 1) which gives S 1 (P n ) = 2(2n 1). We note that S 1 (L n ) < Nw(L n ). Much better bouns can be obtaine using specific properties of orthogonal polynomials, see W. H. Foster I. Krasiov [4] an D. Ştefănescu [12]. Refinements of the Bouns for Absolute Positiveness Upper bouns for positive roots can be use for computing refine absolute positiveness bouns. A fruitful metho is to express the bouns as quotients between absolute values of negative an positive coefficients, as in Theorem 4. A. Aritas, A. Strzebońsi an P. Viglas [1], [2], for example, consier a polynomial an represents it as P(X) = α n X n + α n 1 X n α 0 R[X], α n > 0 P(X) = q 1 (X) q 2 (X)+q 3 (X) q 4 (X)+ +q 2m 1 (X) q 2m (X)+g(X), (1) where all polynomials q i an g have positive coefficients.

7 On Some Absolute Positiveness Bouns 275 This extens the proof of Theorem 6 in [11]. There we represente the polynomial as X b 1 X m1 b X m 1 ( X b 1 X m1) ( X b X m ), (2) which allows using the same evice as in Theorem 4. In fact, A. Aritas a.o [1] use the same boun as in Theorem 6 for a polynomial q 2i 1 q 2i, as we i for X b i X mi. In his Theorem 5 from [2], A. Aritas consiers the concepts of matching (or pairing, introuce in [11]) of a positive coefficient with a convenient negative one, an that of breaing up a positive coefficient into several parts to be paire with negative coefficients of lower orer terms. In fact, breaing up prove to be useful for pairing positive an negative coeffcients. It was use also by D. Ştefănescu [11] for proving that the number S 2 (P) is an upper boun for positive roots. In (2) (cf. [11]), the leaing coefficient 1 is broen into equal parts. A. Aritas [2] calle this evice Cauchy s leaing coefficient implementation of his Theorem 5. Remar: In the Theorem of Aritas [2] it is assume that P(X) has the representation (1), where all polynomials q i an g have positive coefficients. This can be realise for any polynomial with real coefficients having at least one sign variation, as prove by D. Ştefănescu [13]: Lemma 8. Any polynomial P R[X] having at least one sign variation can be represente as P(X) = c 1 X 1 b 1 X m1 + c 2 X 2 b 2 X m2 + + c X b X m + g(x), (3) with g R +, 1 > 2,...,, c i > 0, b i > 0, i > m i for all i. Note that the representation (3) is not unique. A. Aritas [2] use in [2] several representations of P. His main implementations, first λ an local max are base on the breaing of a coefficient into equal parts, respectively by powers of 2. Extening the proof of Theorem 6 from [11], we have obtaine in [13] a more general oun that involves two families of parameters (γ ij ) an (beta i ). A relate proceure was use by P. Batra V. Sharma [3] for multivariate polynomials, starting with the matrix of pouns (δ ij ) while in [13] we use a matrix (γ ij ) an a vector (β j ).

8 276 Doru Ştefănescu References [1] A. Aritas, A. Strzebońsi, P. Viglas: Implementations of a new theorem for computing bouns for positiveroots of polynomials, Computing, 78, (2006). [2] A. Aritas, Linear an Quaratic Complexity Bouns on the Values of the Positive Roots of Polynomials, Univ. J. Comput. Sci., 15, (2009). [3] P. Batra, V. Sharma, Bouns of Absolute Positiveness of Multivariate Polynomials, J. Symb. Comput.,45, (2010). [4] W. H. Foster, I. Krasiov: Bouns for the extreme zeros of orthogonal polynomials, Int. J. Math. Algorithms, 2, (2000). [5] H. Hong, D. Jaus, Testing Positiveness of Polynomials, J. Automate Reasoning, 21, (1991). [6] H. Hong, Bouns for the Absolute Positiveness of Multivariate Polynomials, J. Symb. Comput., 25, (1998). [7] M. Maren, Geometry of Polynomials, A. M. S. Survey, Provience, RI (1949). [8] M. Mignotte, D. Ştefănescu: Polynomials An algorithmic approach, Springer Verlag (1999). [9] A. van er Sluis: Upper bouns for the roots of polynomials, Numer. Math., 15, (1970). [10] G. Szegö: Orthogonal Polynomials, Proc. Amer. Math. Soc. Colloq. Publ., vol. 23, Provience, RI (2003). [11] D. Ştefănescu, New Bouns for Positive Roots of Polynomials, Univ. J. Comput. Sci., 11, (2005). [12] D. Ştefănescu, Bouns for Real Roots an Applications to Orthogonal Polynomials, in Proc. 10th Worshop on Computer Algebra in Scientific Computing, CASC 2007, LNCS 4470, , Springer Verlag, Berlin, Heielberg (2007). [13] D. Ştefănescu, Computation of Dominant Real Roots of Polynomials, Prog. & Comp. Software, 34, (2008). [14] A. Tarsi, The Completeness of Elementary Algebra an Geometry, (1930), reprinte at Institut Blaise Pascal, Paris (1967). Receive: University of Bucharest, Romania stef@rms.unibuc.ro