On Computation of Positive Roots of Polynomials and Applications to Orthogonal Polynomials. University of Bucharest CADE 2007.

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1 On Computation of Positive Roots of Polynomials and Applications to Orthogonal Polynomials Doru Ştefănescu University of Bucharest CADE February 2007

2 Contents Approximation of the Real Roots Bounds for Positive Roots On the Bound of Lagrange Another Bound Bounds for Orthogonal Polynomials Applications of the Hessian of Laguerre

3 Abstract We consider univariate nonconstant polynomials P with real coefficients. For computing the real roots of such polynomials it is convenient to isolate the roots. A key step in real isolation of roots its the computation of lower bounds for the real roots. This can be realized as soon as we obtained accurate upper bounds for positive roots. We present some bounds for positive roots and compare them with other methods. We also compute bounds for the roots of orthogonal polynomials. The computation of the roots of orthogonal polynomials is based on their isolation. Note that the orthogonal polynomials have real coefficients and all their zeros are real distinct simple and located in the interval of orthogonality. Therefore the methods of real root isolation using continued fractions or those based on Descartes rule of signs can be applied. It is sufficient to estimate the smallest positive root. And this can be done if we are able to compute LPR (the largest positive root). We obtain estimates for the largest positive roots using three approaches. We use results of Lagrange Kioustelidis and Ştefănescu on real roots of polynomials with real coefficients. Another approach is to study the ordinary differential equations satisfied by orthogonal polynomials through the Hessian of Laguerre. The results are compared with other bounds. 3

4 1 Approximation of the Real Roots Computation of the real roots i.e. desired degree of accuracy approximation of the real roots to any Isolation of the real roots i.e. the process of findingreal disjoint intervals such that each contains exactly one real root and every real root is contained in some interval If the polynomial P has no multiple roots (if not apply first square free factorization) it is associated through transformations x α x x α x x α x... with all α j integers α 1 0 α i > 0 for all i 1 a polynomial P which has one sign variation or has no positive roots. lb = α α is the smallest positive root of some polynomial f. The exponential behaviour of the CF (continued fractions) method can be eliminated by setting α i lb if lb 1 lb = 1 ub where ub is an upper bound for the postive roots (through the reciprocal polynomial X d f( 1 ) d = deg(f). X 4

5 2 On the Bound of Lagrange We first recall some results on upper bounds for positive roots of univariate polynomials with real coefficients. The first such bounds were obtained by Lagrange [11] and Cauchy [3]. Other limits were obtained by J. B. Kioustelidis [8] and D. Ştefănescu [16]. These bounds are expressed as functions of the degree and of the coefficients. We compare the bounds of Lagrange [11] Koustelidis [8] and a bound from our paper [16]. Other comparisons can be found in Akritas Strzebo nski Vigklas [2] and Ştefanescu [17]. In his treatise on the numerical solution of algebraic equations (1769) Lagrange obtained several bounds for the real roots of univariate polynomials with real coefficients. We remind two of them. Theorem 2.1 (Lagrange) Let F be a nonconstant monic polynomial of degree n over R and let {a j ; j J} be the set of its negative coefficients. Then an upper bound for the positive real roots of F is given by the sum of the largest and the second largest numbers in the set { } j a j ; j J. Apparently Theorem 2.1 result was completely forgoten. We found two modern references to the lectures [11] of Lagrange in the book [4] of L. Derwidué and the historical study [13] of R. Laubenbacher G. McGrath and D. Pengelley. The bound R + ρ is not mentioned in any of them. Note that the corresponding result for polynomials with complex coefficient is also true: Theorem 2.2 ( Lagrange over C ) If F (X) = X n +a 1 X n 1 + +a n 1 X+a n C[X] \ C an upper bound for the absolute values of the roots of F is given by R + ρ where R ρ are the largest numbers in the set { } a k 1/k ; 1 k n. In particular this gives: Corollary 2.3 (M. Fujiwara) An upper bound for the absolute values of the roots of F is n 2 max { a s 1/s }. s=1 Theorem 2.4 (Lagrange) Let P (X) = a 0 X d + + a m X d m a m+1 X d m 1 ± ± a d R[X] with all a i 0 a 0 a m+1 > 0. Let The number A = max { a i ; coeff (X d i ) < 0 }. 1 + ( A a 0 ) 1/(m+1) is an upper bound for the positive roots of P. 5

6 We give here two results that extends the bound given by Theorem 2.4 of Lagrange. Theorem 2.5 Let P (X) = a 0 X d + + a m X d m a m+1 X d m 1 ± ± a d R[X] with all a i 0 a 0 a m+1 > 0. We put A = max { a i ; coeff (X d i ) < 0 }. The number ( ( ) ) 1/(m s+1) 1/(m s+2) pa qa 1 + max a 0 + a a s sa a s 2 + a s 1 is an upper bound for the positive roots of P for any s { m} and p 0 q 0 such that p + q = 1. Proof: We consider x R x > 1. We have P (x) a 0 x d + + a m x d m a m+1 x d m a d a 0 x d + + a s x d s A(x d m ) (a 0 x s + + a s )x d s A xd m 1 x 1 (1) = (a 0 x s + + a s )(x 1)x d s A x 1 x d m + A x 1. The last right hand side of (1) is strictly positive provided (a 0 x s + + a s )(x 1)x m s A. (2) Now we put x = 1 + y and note that x j 1 + jy for all j N. We observe that (a 0 x s + + a s )(x 1)x m s (a 0 (1 + sy) + + a s 1 (1 + y) + a 0 ) y m s+1 = (a a s ) y m s+1 + (sa a s 2 + a s 1 ) y m s+2. 6

7 It follows that (2) is satisfied if with p q 0 p + q = 1. (a a s 1 + a s ) y m s+1 pa (sa a s 2 + a s 1 ) y m s+2 qa These inequalities are satisfied if { ( ) 1/(m s+1) ( pa qa y max a 0 + a a s sa a s 2 + a s 1 ) 1/(m s+2 and it follows that { ( ) 1/(m s+1) ( ) } 1/(m s+2) pa qa 1 + max a 0 + a a s sa a s 2 + a s 1 is an upper bound for the positive roots of the polynomial P. We also obtain Theorem 2.6 Let P (X) = a 0 X d + + a m X d m a m+1 X d m 1 ± ± a d R[X] with all a i 0 a 0 a m+1 > 0. We put The number 1 + max A = max { a i ; coeff (X d i ) < 0 }. { ( pa a a s ) 1/(m s+1) ( qa ) 1/(m s+2) sa a s 2 + a s 1 ( ) } 1/(m s+3) 2rA s(s 1)a 0 + (s 1)(s 2)a a s 2. is an upper bound for the positive roots of P for any s { m} and p 0 q 0 r 0 such that p + q + r = 1. 7

8 Proof: We consider x R x > 1. As in the proof of Theorem 2.5 P (x) 0 if (a 0 x s + + a s )(x 1)x m s A. (3) We consider again x = 1 + y and note that x j 1 + jy + Because x > 1 we have j(j 1) y 2 for all j 2. 2 a 0 x s + + a s 2 x 2 + a s 1 x + a s (a 0 + a a s ) + (sa a s 2 + a s 1 )y + ( s(s 1) 2 a 0 + a s 2 )y 2. It follows that the inequality (3) is satisfied if the following three conditions are fulfilled: (a a s )y m s+1 pa (sa a s 2 + a s 1 )y m s+2 qa (s(s 1)a 0 + (s 1)(s 2)a a s 2 )y m s+3 ra for p q r positive with p + q + r = 1. They give the upper bound { ( ) 1/(m s+1) pa 1 + max a a s ( qa ) 1/(m s+2) sa a s 2 + a s 1 ( ) } 1/(m s+3) 2rA. s(s 1)a 0 + (s 1)(s 2)a a s 2 (4) 8

9 2.1 Particular cases 1. In Theorem 2.5: For p = 1 we obtain the bound ( ) 1/(m s+2) A 1 +. sa a s 2 + a s 1 For p = 0 we recover the bound ( ) 1/(m s+1) A 1 + a a s 1 + a s which is our Theorem from [16]. 2. For p = q = 1 2 in Theorem 2.5 we obtain the bound 1 + max { ( ) 1/(m s+1) A 2(a a s ) ( ) 1/(m s+2) A. 2(sa a s 2 + a s 1 ) (5) 3. In Theorem 2.6 we consider and we obtain the bound p = 1 4 q = 1 4 r = max { ( ) 1/(m s+1) A 4(a a s ) ( ) 1/(m s+2) A 4(sa a s 2 + a s 1 ) ( A s(s 1)a 0 + (s 1)(s 2)a a s 2 ) 1/(m s+3) }. (6) 9

10 Example Let P 1 (X) = X 17 +X 13 +X 12 +X 9 +3X 8 +2X 7 +x 6 5X 4 +X 3 4X 2 6 P 2 (X) = X 13 + X 12 + X 9 + 3X 8 + 2X 7 + x 6 6X 4 + X 3 4X 2 7 and denote by B = B(m s p q) respectively T = T (m s p q r) the bounds given by (5) respectively by (6). For P 1 we have A = 6 m = 11 and for P 2 we have A = 7 and m = 6. P s p q B LPR P P P P P P P P P P P s p q r T LPR P P P P P P P

11 3 Another Bound J. B. Kioustelidis [8] gives the following upper bound for the positive real roots: Theorem 3.1 (Kioustelidis) Let P (X) = X d b 1 X m 1 b kx m k + g(x) with g(x) having positive coefficients and b 1 > 0... b k > 0. The number K(P ) = 2 max{b 1/m b 1/m k k }. is an upper bound for the positive roots of P. For polynomials with an even number of variations of sign we have the following bound: Theorem 3.2 (Ştefănescu) Let P (X) R[X] be such that the number of variations of signs of its coefficients is even. If P (X) = c 1 X d 1 b 1 X m 1 +c 2 X d 2 b 2 X m 2 + +c k X d k b k X m k +g(x) with g(x) R+[X] c i > 0 b i > 0 d i > m i > d i+1 for all i the number { (b1 ) 1/(d1 m 1 ) ( ) } 1/(dk m k ) bk S(P ) = max... c 1 is an upper bound for the positive roots of the polynomial P. Remark: Note that the bound of Lagrange returns only bounds surpassing unity so it cannot be used for some classes of orthogonal polynomials. For example the roots of Legendre polynomials are subunitary. c k 11

12 4 Applications to Orthogonal Polynomials Let us consider the polynomials of Laguerre and Chebyshev of first and second kind. Using Theorem 3.2 we obtain: Proposition 4.1 Let L n T n and U n be the orthogonal polynomials of degree n of Laguerre respectively Chebyshev of first and second kind. We have i. The number S(L n ) = n 2 is an upper bound for the roots of L n. n ii. The number S(T n ) = 2 is an upper bound for the roots of T n. n 1 iii. The number S(U n ) = is an upper bound for the roots 2 of U n. Proof: We use the representations ( ) n ( 1) k L n (X) = X k n k k! k=0 T n (X) = n 2 n/2 k=0 ( 1) k 2 n 2k n k ( n k k ) X n 2k U n (X) = and Theorem 3.2. n/2 k=0 ( ) n k ( 1) k 2 n 2k X n 2k k In the following tables we denote by LPR the largest positive root of the polynomial P. We used the gp-pari package. I. Bounds for Zeros of Laguerre Polynomials n L(P) K(P) S(P) LPR

13 II. Bounds for Zeros of Chebyshev Polynomials of First Kind n L(P) K(P) S(P) LPR III. Bounds for Zeros of Chebyshev Second Kind n L(P) K(P) S(P) LPR Note that for Chebyshev polynomials we have K(P ) = 2 S(P ). Other comparisons on roots of orthogonal polynomials were obtained by A. Akritas and P. Vigklas. 13

14 5 Applications of the Hessian of Laguerre Another approach for estimating the largest positive root of an orthogonal polynomial is the consideration of properties of its associated differential equation. This can be realized by considering the positivity of the Hessian of Laguerre for convenient polynomials. Definition of the Hessian of Laguerre Let us suppose that f(x) = a j X j is a univariate polynomial with real coefficients. We associate to f the corresponding bivariate polynomial and consider its Hessian We have We denote F Y Y = F (X Y ) = H (F ) = det F XX = F XY = j=1 a j X j Y n j j=1 F XX F XY F XY F Y Y. j(j 1)a j X j 2 Y n j j(n j)a j X j 1 Y n j 1 (n j)(n j 1)a j X j Y n j 2. f xx = F XX (x 1) f xy = F XY (x 1) 14

15 and compute f yy = F Y Y (x 1) H (F )(x 1) := det This determinant can be expressed in function of f f and f. In fact we have f xx f xy f xy f yy. 15

16 f xx = n j(j 1)a j x j 2 = f f xy = j(n j)a j x j 1 = n ja j x j 1 j 2 a j x j 1 = nf x j(j 1)a j x j 2 ja j x j 1 = (n 1)f xf f yy = (n j)(n j 1)a j x j = (n 2 n) a j x j 2n j a j x j + j(j + 1)a j x j 0 = (n 2 n) a j x j 2(n 1) j a j x j + j(j 1)a j x j 0 = (n 2 n) a j x j 2(n 1)x j a j x j 1 + x 2 j(j 1)a j x j 2 0 = (n 2 n)f 2(n 1)xf + x 2 f. 16

17 It follows that H (F )(x 1) = f (n 1)f xf (n 1)f xf (n 2 n)f 2 2(n 1)xf + x 2 f = n(n 1)ff (n 1) 2 f 2. Note that this differs only by a sign from the Hessian considered by Laguerre [12]: H (f) = (n 1) 2 f 2 n(n 1) ff 0. Theorem 5.1 (Laguerre) If the polynomial f has real simple roots then the Hessian H (f) is positive. Proof: We suppose that α 1... α n are the roots of f. We observe that f (x) f(x) = x α 1 x α n which gives (x α 1 ) (x α n ) 2 n f (x) 2 f(x)f (x) f (x) 2 ( ) f 2 n f 2 ff. f f 2 It follows that (n 1)f 2 nff 0 therefore H (f) = (n 1) 2 f 2 n(n 1) ff 0. 17

18 Applications to Bounds of Roots of Orthogonal Polynomials Let f R[X] be a polynomial of degree n 2 that satisfies the second order differential equation p(x) y + q(x) y + r(x) y = 0 (7) with p q and r univariate polynomials with real coefficients p(x) 0. Let us assume that all the roots of f are simple and real and let α be a root of f. Laguerre established We have 4(n 1)(p(α)r(α) + p(α)q (α) p (α)q(α)) (n + 2)q(α) 2 0. (8) The inequality (7) can be applied succesfully for finding upper bounds for the roots of orthogonal polynomials. Example 1. Consider the Legendre polynomial P n which satisfies the differential equation (1 x 2 )y 2xy + n(n + 1)y = 0. n + 2 From (7) it follows that La(n) = (n 1) n(n 2 + 2) for the roots of P n. We have is a bound Bounds for Zeros of Legendre Polynomials n La(P) LPR Example 2. Consider the Hermite polynomial H n which satisfies the differential equation y 2xy + 2ny = 0. 18

19 From (7) it follows that He(n) = (n 1) for the roots of H n. We have 2 n + 2 is a bound Bounds for Zeros of Hermite Polynomials n He(P) LPR Remark: The Hessian can give accurate bounds also for other orthogonal polynomials if convenient differential equations are examined. 19

20 References [1] A. G. Akritas A. W. Strzeboński: A comparative study of two real root isolation methods Nonlin. Anal: Modell. Control (2005). [2] A. Akritas A. Strzeboński P. Vigklas: Implementations of a new theorem for computing bounds for positive roots of polynomials Computing (2006). [3] A. L. Cauchy: Exercises de mathématiques Paris (1829). [4] L. Derwidué: Introduction à l algèbre supérieure et au calcul numérique algébrique Masson Paris (1957). [5] I. Z. Emiris E. P. Tsigaridas: Univariate polynomial real root isolation: Continued fractions revisited (2006) [6] M. Fujiwara: Über die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung Tôhoku Math. J (1916). [7] J. Herzberger: Construction of bounds for the positive root of a general class of polynomials with applications in Inclusion Methods for nonlinear problems with applications in engineering economics and Physiscs (Munich 2000) Comput. Suppl. 16 Springer Vienna (2003). [8] J. B. Kioustelidis: Bounds for positive roots of polynomials J. Comput Appl. Math (1986). [9] N. Kjurkchiev: Note on the estimation of the order of convergence of some iterative methods BIT (1992). [10] V. Kostov: On the hyperbolicity domain of the polynomial x n + a 1 x n a n Serdica Math. J (1999). [11] J. L. Lagrange: Traité de la résolution des équations numériques Paris (1798). (Reprinted in Œuvres t. VIII Gauthier Villars Paris (1879).) 20

21 [12] E. Laguerre: Mémoire pour obtenir par approximation les racines d une équation algébrique qui a toutes les racines réelles Nouv. Ann. Math. 2ème série (1880). [13] R. Laubenbacher G. McGrath D. Pengelley: Lagrange and the solution of numerical equations Hist. Math (201). [14] M. Mignotte D. Ştefănescu: Polynomials An algorithmic approach Springer Verlag (1999). [15] F. Rouillier P. Zimmermann: Efficient isolation of polynomial s real roots J. Comput. Appl. Math (2004). [16] D. Ştefănescu: New bounds for the positive roots of polynomials J. Univ. Comp. Sc (2005). [17] D. Ştefănescu: Inequalities on Upper Bounds for Real Polynomial Roots in Computer Algebra in Scientific Computing LNCS 4194 (2006). [18] W. Nuij: A note on hyperbolic polynomials Math. Scand (1968). [19] C. K. Yap: Fundamental problems of algorithmic algebra Oxford University Press (2000). 21