Semilocal convergence of two iterative methods for simultaneous computation of polynomial zeros
aa r X i v : . [ m a t h . NA ] S e p SEMILOCAL CONVERGENCE OF TWOITERATIVE METHODS FOR SIMULTANEOUSCOMPUTATION OF POLYNOMIAL ZEROS ∗ Petko D. Proinov
Abstract
In this paper we study some iterative methods for simultaneous approxima-tion of polynomial zeros. We give new semilocal convergence theorems with errorbounds for Ehrlich’s and Nourein’s iterations. Our theorems generalize and im-prove recent results of
Zheng and
Huang [J. Comput. Math. 18 (2000), 113–122],
Petkovi´c and
Herceg [J. Comput. Appl. Math. 136 (2001), 283–307] and
Nedi´c [Novi Sad J. Math. 31 (2001), 103–111]. We also present a new sufficientcondition for simple zeros of a polynomial.
Key words: polynomial zeros, simultaneous methods, semilocal convergence,Ehrlich method, B¨orsch-Supan method, Nourein method
Let f be a monic complex polynomial of degree n ≥
2. A point ξ in C n is said to bea root-vector of f if its components are exactly the zeros of f with their multiplicities.There are a lot of iterations for simultaneous computation of all zeros of f (see themonographs of Sendov, Andreev, Kjurkchiev [1],
Petkovi´c, Herceg, Ilic [2]and
Kyurkchiev [3]). The famous one is
Weierstrass’ iteration [4](1) z k +1 = z k − W ( z k ) , k = 0 , , , . . . , where the operator W in C n is defined by W ( z ) = ( W ( z ) , . . . , W n ( z )) with W i ( z ) = f ( z i ) Q j = i ( z i − z j ) ( i = 1 , , · · · , n ) . ∗ This paper is published in: C. R. Acad. Bulg. Sci 59 (2006), No 7, 705–712. f is Ehrlich’ s iteration [5](2) z k +1 = F ( z k ) , k = 0 , , , . . . , where the operator F in C n is defined by F ( z ) = ( F ( z ) , . . . , F n ( z )) with(3) F i ( z ) = z i − f ( z i ) f ′ ( z i ) − f ( z i ) P j = i / ( z i − z j ) . Werner [6] has proved that the iteration function F can also be written in the form(4) F i ( z ) = z i − W i ( z )1 + P j = i W j ( z ) / ( z i − z j ) . Ehrlich’s method (2) with the iteration function F defined by (4) instead of (3) is knownas B¨orsch-Supan’s method since in such form it was proposed for the first time by B¨orsch-Supan [7]. The following iteration is due to
Nourein [8](5) z k +1 = G ( z k ) , k = 0 , , , . . . , where the operator G in C n is defined by G ( z ) = ( G ( z ) , . . . , G n ( z )) with(6) G i ( z ) = z i − W i ( z )1 + P j = i W j ( z ) / ( z i − z j − W i ( z )) . Nourein’s method is also known as B¨orsch-Supan method with Weierstrass’ correction.Since 1996, some authors [9, 10, 11, 12, 13, 14, 15, 16] have obtained semilocalconvergence theorems for Ehrlich’s and Nourein’s methods from data at one point. Thebest results on Ehrlich’ method are due to
Zheng and
Huang [14] and
Petkovi´c and
Herceg [15]. The best results on Nourein’s method are due to
Zheng and
Huang [14]and
Nedi´c [16].In this paper, we present new semilocal convergence theorems for Ehrlich’s andNourein’s iterations which generalize and improve all previous results in this area. Wealso present a new sufficient condition for simple zeros of a polynomial. The main resultsof the paper (Theorems 2.1, 3.1 and 4.1) will be proved elsewhere.Throughout the paper the norm k . k p (1 ≤ p ≤ ∞ ) in C n is defined as usual, i.e. k z k p = ( P ni =1 | z i | p ) /p . For a given point z in C n we define(7) d ( z ) = ( d ( z ) , . . . , d n ( z )) and δ ( z ) = min { d ( z ) , . . . , d n ( z ) } , where d i ( z ) = min {| z i − z j | : j = i } for i = 1 , , . . . , n . For a given point z in C n withdistinct components we use the notations(8) W ( z ) d ( z ) = (cid:18) W ( z ) d ( z ) , . . . , W n ( z ) d n ( z ) (cid:19) and E ( z ) = (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) p . Localization of simple polynomial zeros
The following theorem is an improvement of a result of
Zheng [17].
Theorem 2.1.
Let f be a monic polynomial of degree n ≥ . Suppose there exist ≤ p ≤ ∞ and a point z in C n with distinct components such that (9) E ( z ) < / /q and φ ( E ( z )) < . where q is defined by /p + 1 /q = 1 and φ is a real function defined on [0 , / /q ) by (10) φ ( x ) = ( n − /q x (1 − x )(1 − /q x ) (cid:18) x ( n − /p (1 − /q x ) (cid:19) n − . Then the following statements hold true. (i)
The polynomial f has only simple zeros. (ii) The closed disks D i = { z ∈ C : | z − ( z i − W i ( z )) | ≤ C | W i ( z ) |} , i = 1 , , . . . , n, where C = θλ/ (1 − θλ ) , λ = φ ( E ( z )) , θ = 1 − /q E ( z ) , are mutually disjointand each of them contains exactly one zero of f . Note that under the conditions (9) Weierstrass’ method (1) is convergent with thesecond order of convergence (see
Proinov [18]).
Theorem 3.1.
Let f be a monic polynomial of degree n ≥ , ≤ p ≤ ∞ and /p +1 /q =1 . Define the real function (11) φ ( x ) = ax (1 − ( a + 1) x )(1 − ( a + b ) x ) (cid:18) n − . ax − ( a + b ) x (cid:19) n − , where a = ( n − /q and b = 2 /q . Suppose that z is an initial point in C n with distinctcomponents satisfying E ( z ) < / ( a + b ) and φ ( E ( z )) ≤ . Then the following statements hold true.
Ehrlich’s iterative sequence (2) is well-defined and convergent to a root-vector ξ of f . Moreover, if φ ( E ( z )) < , then the order of convergence is three. (ii) For each k ≥ we have the following a priori error estimate (12) (cid:13)(cid:13) z k − ξ (cid:13)(cid:13) p ≤ A k θ k λ (3 k − / − θλ k (cid:13)(cid:13) W ( z ) (cid:13)(cid:13) p , where λ = φ ( E ( z )) , θ = ψ ( E ( z )) , A k = µ ( E ( z ) λ (3 k − / ) and the real functions ψ and µ are defined by ψ ( x ) = 1 − ( a + b ) x − ax and µ ( x ) = 11 − ax . (iii) For all k ≥ we have the following a posteriori error estimate (13) (cid:13)(cid:13) z k − ξ (cid:13)(cid:13) p ≤ µ k − θ k λ k (cid:13)(cid:13) W ( z k ) (cid:13)(cid:13) p , where λ k = φ ( E ( z k )) , θ k = ψ ( E ( z k )) and µ k = µ ( E ( z k ) . Setting p = ∞ in Theorem 3.1 we obtain the following corollary. Corollary 3.1.
Let f be a monic polynomial of degree n ≥ and let C be a real numbersatisfying (14) 0 ≤ C < n + 1 and ( n − C (1 − nC )(1 − ( n + 1) C ) (cid:18) − nC − ( n + 1) C (cid:19) n − < . Suppose that z is an initial point in C n satisfying (15) (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ C. Then Ehrlich’s method (2) is convergent to a root-vector ξ of f with the third order ofconvergence. Moreover, the error bounds (12) and (13) hold with p = ∞ . Corollary 3.1 improves Theorem 4.1 of
Petkovi´c and
Herceg [15]. Note thatPetkovic and Herceg have proved only linear convergence of Ehrlich’s method under thestronger condition k W ( z ) k ∞ ≤ C δ ( z ) with C satisfying (14) as well as(16) β := ( n − C (1 + ( n − C )(1 − nC )(1 − ( n − C ) (cid:18) − nC − ( n + 1) C (cid:19) n − < g ( β ) < − ( n − C C where g ( x ) = (cid:26) x for 0 < x ≤ / / (1 − x ) for 1 / < x < . Corollary 3.1 shows that Petkovic-Herceg’s assumptions (16) and (17) can be omitted.The following result improves and generalizes Theorem 1 of
Zheng and
Huang [14]as well as previous results [9, 10, 11, 12].
Corollary 3.2.
Let f be a monic polynomial of degree n ≥ , ≤ p ≤ ∞ and /p +1 /q =1 . Suppose z is an initial point in C n satisfying (18) (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) p ≤ n − /q + 2 . Then f has only simple zeros and Ehrlich’ method (2) converges to a root-vector ξ of f with the third order of convergence. Moreover, the error estimates (12) and (13) hold.Proof. From Theorem 2.1 and the proof of Corollary 2 of [18] we conclude that f hasonly simple zeros. It is easy to compute that φ ( R ) = a ( a + 1)( a + 2 − b ) (cid:18) n − . aa + 2 − b (cid:19) n − . for R = 1 / (2 a + 2), where a , b and φ are defined as in Theorem 3.1. By Theorem 3.1 itsuffices to prove that φ ( R ) ≤
1. If a > e −
1, then φ ( R ) < e/ ( a + 1) <
1. If a ≤ e − φ ( R ) < ( e − / ( a − b + 2) ≤ ( e − / < Remark.
In Corollary 3.2 we consider an initial condition of the type E ( z ) ≤ A ( n − /q + B .
Note that such initial conditions can be obtained for every
A >
1. For example, if
A > B = 2 /q + 14( A −
1) exp 1 A − . Corollary 3.3.
Let f be a monic polynomial of degree n ≥ . Suppose that z is aninitial point in C n satisfying (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ . n + 1 . . Then Ehrlich’s method (2) is convergent to a root-vector ξ of f with the third order ofconvergence. Moreover, the error estimates (12) and (13) hold with p = ∞ . roof. Define φ n = φ (1 / (1 . n + 1 . φ is defined by (11) with p = ∞ . Thesequence ( φ n ) is increasing for 2 ≤ n ≤
10 and decreasing for n ≥
10. Hence φ n ≤ φ < n ≥
2. Now the conclusion follows from Theorem 3.1.Corollary 3.3 improves Theorem 4.2 of
Petkovi´c and
Herceg [15]. Note that theauthors of this work have proved that Ehrlich’s method is convergent under the condition k W ( z ) k ∞ ≤ C ( n ) δ ( z ) , where C ( n ) = (cid:26) / ( n + 4 .
5) for n = 3 , / (1 . n + 5) for n ≥ . Corollary 3.4.
Let f be a monic polynomial of degree n ≥ . Suppose that z is aninitial point in C n satisfying (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ R = 0 . . . . . where R is the unique positive solution of the equation (19) (cid:18) x − x (cid:19) exp x − x = 1 . Then f has only simple zeros and Ehrlich’ method (2) converges to a root-vector ξ of f with the third order of convergence and the estimates (12) and (13) hold with p = 1 .Proof. It follows from Theorem 2.1 and the proof of Corollary 3 of [18] that f has onlysimple zeros. It is easy to show that φ ( x ) < g ( x ) for 0 < x < /
2, where φ ( x ) is definedby (11) with p = 1 and g ( x ) denotes the left-hand side of the equation (19). Therefore, φ ( R ) < Theorem 4.1.
Let f be a monic polynomial of degree n ≥ , ≤ p ≤ ∞ and /p +1 /q =1 . Define the real function (20) φ ( x ) = a x (1 − mx + bx )(1 − ( a + 2) x + x ) (cid:18) n − . a ( x − x )1 − mx + bx (cid:19) n − , where a = ( n − /q , b = 2 /q and m = a + b + 1 . Suppose that z is an initial point in C n with distinct components satisfying E ( z ) < / ( m + √ m − b ) and φ ( E ( z )) ≤ . Then the following statements hold true.
Nourein’s iterative sequence (5) is well-defined and convergent to a root-vector ξ of f . Moreover, if φ ( E ( z )) < , then the order of convergence is four. (ii) For each k ≥ we have the following a priori error estimate (21) (cid:13)(cid:13) z k − ξ (cid:13)(cid:13) p ≤ µ k θ k λ (4 k − / − θλ k (cid:13)(cid:13) W ( z ) (cid:13)(cid:13) p , where λ = φ ( E ( z )) , θ = ψ ( E ( z )) , µ k = µ ( E ( z ) λ (4 k − / ) and the real functions ψ and µ are defined by ψ ( x ) = 1 − mx + bx − ( a + 1) x and µ ( x ) = 1 − x − ( a + 1) x . (iii) For all k ≥ we have the following a posteriori error estimate (22) (cid:13)(cid:13) z k − ξ (cid:13)(cid:13) p ≤ µ k − θ k λ k (cid:13)(cid:13) W ( z k ) (cid:13)(cid:13) p , where λ k = φ ( E ( z k )) , θ k = ψ ( E ( z k )) and µ k = µ ( E ( z k ) . Setting p = ∞ in Theorem 4.1 we obtain the following corollary. Corollary 4.1.
Let f be a monic polynomial of degree n ≥ and let C be a real numbersuch that ≤ C < / ( n + 2 + √ n + 4 n − and (23) ( n − C (1 − ( n + 1) C + C )(1 − ( n + 2) C + 2 C ) (cid:18) − ( n + 1) C + C − ( n + 2) C + 2 C (cid:19) n − < . Suppose that z is an initial point in C n satisfying (15) . Then Nourein’s method (5) isconvergent to a root-vector ξ of f with the fourth order of convergence. Moreover, theerror bounds (21) and (22) hold with p = ∞ . Corollary 4.1 improves Theorem 2 of
Nedi´c [16]. Nedi´c has proved only linearconvergence of Nourein’s method under the condition k W ( z ) k ∞ ≤ C δ ( z ) with C satisfying 0 < C < n + 4 + √ n + 8 n and β < − nC n − C , where β denotes the left-hand side of (23).The following corollary improves and generalizes Theorem 2 of Zheng and
Huang [14] as well as previous results [10, 12, 13]. 7 orollary 4.2.
Under the assumptions of Corollary 3.2 f has only simple zeros andNourein’s method (5) converges to a root-vector ξ of f with the fourth order of conver-gence. Moreover, we have the error estimates (21) and (22) .Proof. Corollary 3.2 implies that f has only simple zeros. It it is easy to compute that φ ( R ) = 2 a ( a + 1)(2( a + 1) − (2 a + 1) b )(2 a + 2 a + 1)) (cid:18) n − . a (2 a + 1)2( a + 1) − (2 a + 1) b (cid:19) n − for R = 1 / (2 a + 2), where a , b and φ are defined as in Theorem 4.1. Taking into accountthat b ≤ φ ( R ) ≤ a ( a + 1)(2( a + 1) − (2 a + 1) b )(2 a + 2 a + 1)) (cid:18) n − a + 12 a (cid:19) n − and(25) φ ( R ) ≤ a + 12 a + 2 a + 1 (cid:18) n − . a + 12 a (cid:19) n − . By Theorem 4.1 it suffices to prove that φ ( R ) ≤
1. We shall consider two cases.
Case . Suppose a ≥ .
8. It follows from (25) that(26) φ ( R ) < g ( a ) = a + 12 a + 2 a + 1 exp (cid:18) a (cid:19) . The function g is decreasing on (0 , ∞ ). Therefore, φ ( R ) < g ( a ) < g (1 . < Case . Suppose a ≤ .
8. From (24) and the obvious inequality b ≤ a we obtain(27) φ ( R ) < h ( a ) = 2 a ( a + 1)(3 a + 2)(2 a + 2 a + 1) exp 32 . The function h is increasing on (0 , ∞ ). Therefore, φ ( R ) < h ( a ) < h (1 . < Corollary 4.3.
Let f be a monic polynomial of degree n ≥ . Suppose that z is aninitial point in C n satisfying (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ . n + 2 . . Then Nourein’s method (5) is convergent to a root-vector ξ of f with the fourth order ofconvergence. Moreover, the estimates (21) and (22) hold with p = ∞ . roof. Define φ n = φ (1 / (1 . n + 2 . φ is defined by (20) with p = ∞ . Thesequence ( φ n ) is increasing for 2 ≤ n ≤
19 and decreasing for n ≥
19. Hence φ n ≤ φ < n ≥
2. Now the conclusion follows from Theorem 4.1.Corollary 4.3 improves Theorem 3 of
Nedi´c [16]. Note that Nedi´c has proved thatNourein’s method is convergent under the condition k W ( z ) k ∞ ≤ C ( n ) δ ( z ) where C ( n ) = (cid:26) / (1 . n + 1 . ≤ n ≤ / (1 . n + 8 .
7) for n > . Corollary 4.4.
Let f be a monic polynomial of degree n ≥ . Suppose that z is aninitial point in C n satisfying (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ R = 0 . . . . , where R is the unique solution of the equation (28) x (1 − x + x ) exp x − x − x + x = 1 in the interval (0 , / (3 + √ . Then f has only simple zeros and Nourein’s sequence (5) converges to a root-vector ξ of f with the fourth order of convergence and error bounds (21) and (22) with p = 1 .Proof. By Theorem 3.4 f has only simple zeros. It is easy to show that φ ( x ) < g ( x ) for0 < x < / (3 + √
5) , where g ( x ) denotes the left-hand side of the equation (28) and φ ( x ) is defined by (20) with p = 1. Therefore, φ ( R ) < References [1]
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