A new semilocal convergence theorem for the Weierstrass method from data at one point
aa r X i v : . [ m a t h . NA ] S e p A NEW SEMILOCAL CONVERGENCETHEOREM FOR THE WEIERSTRASS METHODFROM DATA AT ONE POINT ∗ Petko D. Proinov
Abstract
In this paper we present a new semilocal convergence theorem from data atone point for the Weierstrass iterative method for the simultaneous computationof polynomial zeros. The main result generalizes and improves all previous onesin this area.
Key words: polynomial zeros, simultaneous methods, Weierstrass method,convergence theorems, point estimation
Let f be a monic polynomial of degree n ≥ f as a point in C n . Namely, a point ξ in C n with distinct coordinates issaid to be a root-vector of f if each of its coordinates is a zero of f . Starting froman initial point z in C n with distinct coordinates we build in C n the Weierstrass iterative sequence [1](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 ) . It is well-known that under some initial conditions the Weierstrass sequence (1) iswell-defined and tends to a root-vector of f . Iteration formula (1) defines the famous ∗ This paper is published in: C. R. Acad. Bulg. Sci 59 (2006), No 2, 131–136. f simultaneously.In 1962, Dochev [2, 3] proved the first local convergence theorem for the Weierstrassmethod. Since 1980 a number of authors [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] haveobtained semilocal convergence theorems for the Weierstrass method from data at onepoint (point estimation). In this note we present a new semilocal convergence theoremfor the Weierstrass method which improves and generalizes all these results.The main result (Theorem 1) of this note will be proved elsewhere.
Throughout the paper the norm k . k 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 with distinct coordinates we use the notations 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 , where d ( z ) = ( d ( z ) , . . . , d n ( z )) and d i ( z ) = min j = i | z i − z j | ( i = 1 , , . . . , n ) . Theorem 1.
Let f be a monic polynomial of degree n ≥ with simple zeros. Let ≤ p ≤ ∞ and /p + 1 /q = 1 . Define the real function (2) φ ( x ) = ( n − /q x (1 − x )(1 − /q x ) (cid:18) x ( n − /p (1 − /q x ) (cid:19) n − . Suppose that z is an initial point in C n with distinct coordinates satisfying (3) E ( z ) < / /q and φ ( E ( z )) ≤ . Then the following statements hold true. (i)
The Weierstrass iterative sequence (1) is well-defined and convergent to a root-vector ξ of f . Moreover, the convergence is quadratic if φ ( E ( z )) < . (ii) For each k ≥ we have the following a priori error estimate (4) (cid:13)(cid:13) z k − ξ (cid:13)(cid:13) p ≤ θ k λ k − − θλ k (cid:13)(cid:13) z − z (cid:13)(cid:13) p , where λ = φ ( E ( z )) and θ = 1 − /q E ( z ) . For all k ≥ we have the following a posteriori error estimate (5) (cid:13)(cid:13) z k +1 − ξ (cid:13)(cid:13) p ≤ θ k λ k − θ k λ k (cid:13)(cid:13) z k +1 − z k (cid:13)(cid:13) p , where λ k = φ ( E ( z k )) and θ k = 1 − /q E ( z k ) . Remark 1.
Let R ( n, p ) denote the unique solution of the equation φ ( x ) = 1 in theinterval (0 , / /q ), where φ is defined by (2). Then the assumption (3) of Theorem 1can also be written in the form E ( z ) ≤ R ( n, p ). In this section we compare Theorem 1 with all previous results of the same type. Notethat for z ∈ C n with distinct coordinates we have the obvious inequality(6) (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) p ≤ k W ( z ) k p δ ( z ) where δ ( z ) = min { d ( z ) , . . . , d n ( z ) } . Corollary 1.
Let f be a monic polynomial of degree n ≥ with simple zeros. Let ≤ p ≤ ∞ and /p + 1 /q = 1 . Suppose z is an initial point in C n satisfying (7) (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) p ≤ A ( n − /q + 2 /q + 1 , where A = 1 . . . . is the unique solution of the equation exp (1 /x ) = x . Then theWeierstrass sequence (1) is quadratically convergent to a root-vector ξ of f . Moreover,we have the error estimates (4) and (5) .Proof. The statement of Corollary 2 holds true even if we replace the right-hand side of(7) by R = 2 / ( m + √ m − b ) , where m = Aa + b + 1, a = ( n − /q and b = 2 /q .Define g ( x ) = ax (1 − x )(1 − bx ) . It is easy to show that g ( R ) = 1 /A and φ ( x ) < g ( x ) exp g ( x ) for 0 < x < /b . Thenby the definition of A we get φ ( R ) < Batra [13] has proved that the Weierstrass method is convergent under the condi-tion k W ( z ) k ∞ < δ ( z ) / (2 n ) . The following corollary improves Batra’s result as well asprevious results [4, 9, 10, 11, 12]. 3 orollary 2.
Let f be a monic polynomial of degree n ≥ with simple zeros. Let ≤ p ≤ ∞ and /p + 1 /q = 1 . Suppose z is an initial point in C n satisfying (8) (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 the Weierstrass sequence (1) is convergent to a root-vector ξ of f . Moreover, wehave the error estimates (4) and (5) .Proof. Denote by R the right-hand side of (8), i.e. R = 1 / (2 a + 2), where again a =( n − /q . By Theorem 1 it suffices to prove that φ ( R ) ≤
1. If n = 2, then φ ( R ) =(4 / − b ) / (4 − b ) ≤
1, where b = 2 /q . Further, suppose n ≥
3. It is easy to see that(9) φ ( R ) ≤ a ( a + 1)(2 a + 1)(2 a + 2 − b ) √ e. If a ≥
2, then (9) implies φ ( R ) ≤ a + 12 a + 1 √ e ≤ √ e < . If a ≤
2, then (9) implies φ ( R ) ≤ a ( a + 1)(2 a + 1)( a + 2) √ e ≤ √ e < Corollary 3.
Let f be a monic polynomial of degree n ≥ with simple zeros. Supposethat z is an initial 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 (10) (cid:18) x − x (cid:19) exp x − x = 1 . in the interval (0 , . Then the conclusion of Corollary 1 holds for p = 1 .Proof. It is easy to show that φ ( x ) < g ( x ) for 0 < x <
1, where g ( x ) denotes theleft-hand side of equation (10). Therefore, φ ( R ) < orollary 4 (Han [14]) . Let f be a monic polynomial of degree n ≥ with simple zeros.Assume ≤ p ≤ ∞ and /p + 1 /q = 1 . Suppose that (11) (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) p ≤ n (2 /n − n − /q + 2 /q (cid:18) − ( n − /q n (2 /n − n − /q + 2 /q (cid:19) . Then the Weierstrass iterative sequence (1) converges to a root-vector of f .Proof. Let R denote the right-hand side of (11). According to Corollary 2 it sufficesto prove the inequality R ≤ / (2 a + 2), where a and b are defined as in the proofof Corollary 1. Write R in the form R = g ( τ ), where g ( x ) = x (1 − ax ) and τ = n (2 /n − / ( a + b ). Since n (2 /n − < R ≤ g (1 / (2 a )) = 1 / (4 a ) ≤ / (2 a + 2)and the proof of the corollary is complete.Setting p = ∞ in Theorem 1 and taking into account (6) we obtain the followingcorollary. The first part of it is due to Zheng [5] and
Petkovi´c and
Herceg [15]. Thesecond part of the corollary is due to
Zheng [6].
Corollary 5 (Zheng [5, 6] and Petkovi´c and Herceg [15]) . Let f be a monic polynomialof degree n ≥ with simple zeros. Let < C < and λ := ( n − C (1 − C )(1 − C ) (cid:18) C − C (cid:19) n − ≤ where C = k W ( z ) k ∞ δ ( z ) . Then the Weierstrass method (1) is convergent to a root-vector ξ of f . Moreover, theerror estimate (4) holds with θ = 1 − C and p = ∞ . Petkovi´c and Herceg [15] have proved that the Weierstrass method is convergentunder the condition k W ( z ) k ∞ < δ ( z ) / ( an + b ) , where a = 1 . b = 0 . n ( n ≥ n ≥ Corollary 6.
Let f be a monic polynomial of degree n ≥ with simple zeros. Under theinitial condition (cid:13)(cid:13)(cid:13)(cid:13) W ( z ) d ( z ) (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ . n + 0 . the Weierstrass method is convergent to a root-vector ξ of f with the second order ofconvergence. Moreover, the estimates (4) and (5) hold for p = ∞ .Proof. The sequence φ n = φ (1 / (1 . n + 0 . ≤ n ≤
132 anddecreasing for n ≥ φ n ≤ φ < n ≥
2. Now the conclusion followsfrom Theorem 1. 5 orollary 7 (Wang and Zhao [8]) . Let f be a monic polynomial of degree n ≥ withsimple zeros. Let us assume that (12) k W ( z ) k ∞ ≤ C ( n ) δ ( z ) where C ( n ) = − min x> ( x (1 + x ) n − − x ) . Then the Weierstrass iterative sequence (1) converges to a root-vector of f .Proof. By Corollary 2 and (6) it suffices to prove that C ( n ) ≤ / (2 n ). This is obvious for n = 2 and n = 3 since C (2) = 0 .
25 and C (3) = 0 . . . . Define g ( x ) = 2 x − x (1 + x ) n − .It is easy to show that there exists a unique real number t such that 0 < t < / ( n − − g ′ ( t ) = 0. One can prove that t < / (2 n −
1) for n ≥
4. Then by the definitions of C ( n ) and t we get C ( n ) = max x ∈ [ 0 , / ( n − − g ( x ) = g ( t ) = 2( n − t / (1 + nt )which implies C ( n ) < / (3 n ) for n ≥
4. This completes the proof of Corollary 7.
Corollary 8 (Wang and Zhao [8]) . Let f be a monic polynomial of degree n ≥ withsimple zeros. Let us suppose that k W ( z ) k ≤ C ( n ) δ ( z ) where C ( n ) = − min x> (cid:18)X n − j =1 n − jj ! n x j +1 − x (cid:19) . Then the Weierstrass iterative sequence (1) converges to a root-vector of f .Proof. According to Corollary 3 and (6) it suffices to prove that C ( n ) ≤ .
3. For a given n ≥ f n ( x ) = P n − j =1 n − jj ! n x j +1 − x. It is easy to verify that for x > f n ( x ) ≤ X nj =1 x j +1 j ! − x = ( n + 1) f n +1 ( x ) − nf n ( x ) . This implies f n ( x ) ≤ f n +1 ( x ). Now by the definition of C ( n ) we get C ( n + 1) ≤ C ( n )for n ≥
2. Hence C ( n ) ≤ C (4) = 0 . . . . for n ≥ Remark 2.
Wang and
Zhao [7] have also proved that the Weierstrass method (1) isconvergent under the condition (cid:13)(cid:13) W ( z ) (cid:13)(cid:13) ≤ (3 − √ nn − δ ( z ) . Note that for n ≥ Remark on the SOR Weierstrass method
In this section we are concerned with the successive overrelaxation (SOR) Weierstrassmethod(13) z k +1 = z k − h k W ( z k ) , k = 0 , , , . . . , where h k = h k ( f ) ∈ (0 ,
1] is an acceleration parameter.
Petkovi´c and
Kjurkchiev [16] have noticed that usually the SOR method (13) is faster if h k is closer to 1. In 1995, Wang and
Zhao [8] considered the SOR method (13) with h k defined by(14) h k = min , . δ ( z k ) n P i =1 | W i ( z k ) | , According to Corollary 3 we can consider the SOR Weierstrass method (13) with h k defined by(15) h k = min , . n P i =1 (cid:12)(cid:12)(cid:12) W i ( z k ) d i ( z k ) (cid:12)(cid:12)(cid:12) . Note that the new h k is closer to 1. Moreover, if our acceleration parameter h k is lessthan 1, then it is greater than Wang-Zheng’s parameter h k by more than 50%. References [1]
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