CONVERGENCE OF NEWTON S METHOD AND INVERSE FUNCTION THEOREM IN BANACH SPACE

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1 MATHEMATICS OF COMPUTATION Volume 68, Number 225, January 999, Pages S (99)999- CONVERGENCE OF NEWTON S METHOD AND INVERSE FUNCTION THEOREM IN BANACH SPACE WANG XINGHUA Abstrat. Under the hypothesis that the derivative satisfies some kind of weak Lipshitz ondition, a proper ondition whih makes Newton s method onverge, and an exat estimate for the radius of the ball of the inverse funtion theorem are given in a Banah spae. Also, the relevant results on premises of Kantorovih and Smale types are improved in this paper. We ontinue to disuss the problem of onvergene in the Newton method (.) x n+ = x n f (x n ) f(x n ), n =,,, to solve an operator equation f whih maps from some domain D in a real or omplex Banah spae X to another Banah spae Y, (.2) f(x) =. Now we ome bak to the problem whih we bypassed in []. We always assume that f (x ) exists and f (x ) f satisfies some kind of Lipshitz ondition similar to that of [] in some open ball B(x,r) D with enter x and radius r (or some losed ball B(x,r) D) in order to study the onvergene of Newton s method and the domain of the loal inverse funtion of f at x.. The domain of the inverse funtion The inverse funtion theorem asserts that there is an inverse funtion fx defined on some open ball B(f(x ),ε) Y with the property that fx (f(x )) = x, f(fx (y)) = y, y B(f(x ),ε), and fx is differentiable. Now we study the exat lower bound estimate of the radius of this ball. For this reason, we assume that f has a ontinuous derivative in the ball B(x,r), f (x ) exists and f (x ) f satisfies the enter Lipshitz ondition with the L average, f (x ) f (x) I ρ(x) (.) L(u)du, x B(x,r), Reeived by the editor Marh 2, 997 and, in revised form, June 6, Mathematis Subjet Classifiation. Primary 65H. Supported by the China State Major Key Projet for Basi Researh and the Zhejiang Provinial Natural Siene Foundation Amerian Mathematial Soiety

2 7 WANG XINGHUA where ρ(x) = x x and L is a positive integrable funtion in the interval (,r). By Banah s theorem, when r r, for all x B(x,r ), f (x) exists and f (x) f (x ) (.2) ρ(x), L(u)du where r satisfies (.3) r L(u)du =. Theorem.. Suppose that r r and b = r L(u)udu. Then under the hypothesis of ondition (.), we have (.4) B(f(x ),b/ f (x ) ) f(b(x,r )), and in the open ball on the left, f x exists and is differentiable. Moreover, the radius of the ball is the best possible. Lemma.2. Let (.5) where R satisfies h(t) =β t+ R t L(u)(t u)du, t R, (.6) L(u)(R u)du =. R Then when <β<b,his dereasing monotonially in [,r ], while it is inreasing monotonially in [r,r] and h(β) >, h(r )=β b<, h(r)=β>. Moreover, h has a unique zero in eah interval, denoted by r and r 2. They satisfy β<r < r (.7) b β<r <r 2 <R. Proof. It is obvious by the sign of h (t) = + t L(u)du that h(t) is pieewise monotone. By the positivity of L,weseethatϕ(t):= t t L(u)(t u)du is inreasing monotonially with respet to t. Infat,for<t <t 2, t2 ( t2 ( ϕ(t 2 ) ϕ(t )= L(u)du + ) t ) L(u)udu t t 2 t t 2 t t2 t2 ( L(u)du L(u)du ) t L(u)udu t t t 2 t ( = ) t L(u)udu >. t t 2 Thus we have β<r =h(r )+r =β+ϕ(r )r <β+ϕ(r )r =β+ r r L(u)(r u)du r = β + r b r r.

3 CONVERGENCE OF NEWTON S METHOD 7 By this lemma, Theorem. implies a more preise proposition, as follows. For this purpose, we assume the inequality (.) an be extended to the boundary, i.e. (. ) f (x ) f (x) I ρ(x) L(u)du, x B(x,r). Proposition.3. Suppose that r r and <β<b= r L(u)udu, wherer is determined by Lemma.2. Then, under the hypothesis of the ondition (. ), (.8) B(f(x ),β/ f (x ) ) f(b(x,r )), and in the losed ball on the left, fx exists, is differentiable, and its derivative (fx ) (y) =f (x) at y = f(x) satisfies (.2). Moreover, as a losed ball of the image, the radius r is as small as possible. Proof. Arbitrarily hoosing (.9) y B(f(x ),β/ f (x ) ), we onsider two sequenes {x n } Xand {t n } R, respetively given by (.) x n+ = x n f (x ) (f(x n ) y), n =,,, and (.) t n+ = t n + h(t n ), t =, n =,,. First, by the fat that h(t) +t inreases monotonially with respet to t and t =<t =β<r, we indutively find that {t n } inreases monotonially and is less than r.thus{t n }onverges to r. Then, by indution, for all n we will prove that (.2) x n+ x n t n+ t n. By (.9) and (.7), x x f (x ) f(x ) y β=t t. This means (.2) is true for n =. Suppose that (.2) is valid until some n. For τ, let (.3) x n +τ = x n + τ(x n x n ), t n +τ = t n + τ(t n t n ). We have x n +τ x x x + + x n x n 2 +τ x n x n (t t )+ +(t n t n 2 )+τ(t n t n ) = t n +τ <r r. Thus, by virtue of the equality x n+ x n = f (x ) (f(x n ) f(x n ) f (x )(x n x n )) ( = f (x ) f (x n +τ ) I ) (x n x n )dτ

4 72 WANG XINGHUA and (.), we obtain x n+ x n f (x ) f (x n +τ ) I xn x n dτ ρ(xn +τ ) tn +τ tn = L(u)(t n u)du = t n+ t n. L(u)du x n x n dτ L(u)du(t n t n )dτ tn L(u)(t n u)du This indiates that (.2) is valid for all n. The inequality (.2) above shows that the sequene {x n } is self-onvergent and so is onvergent. Taking the limit on both sides in (.), we see that x = lim x n satisfies (.4) f(x) =y. Also, sine x n x r,wehave (.5) x=fx (y) B(x,r ). For this reason we have to prove x satifying (.4) is unique in the losed ball. This will be given togather with the proof of the next proposition. Finally, the differentiablity of the inverse funtion follows by (.2). Remark. Exept for the differentiablity of the inverse funtion, the proposition is also true for β = b. Besides Proposition.3, we have the following proprosition, whih is alled the branh separation theorem Proposition.4. Suppose that r r<r 2 and <β<b,wherer,r 2 and b are determined by Lemma.2 and Theorem.. Then, under the ondition (. ), (.6) B(f(x ),β/ f (x ) ) f(b(x,r)\b(x,r )) =. Proof. Arbitrarily hoose (.7) Let y B(f(x ),β/ f (x ) ), x B(x,r). (.8) x n+ = x n f (x ) (f(x n ) y), n =,,, (.9) Sine (.2) x n+ x n+ = t n+ = t n + h(t n), t = x x, n =,,. ( f (x ) f (x n + τ(x n x n )) I ) (x n x n )dτ, we an prove that (.2) x n x n t n t n, n =,,. Hene, {x n} is also onvergent to x = lim x n. Therefore, there is only one x B(x,r ) in the open ball B(x,r) that satisfies (.4).

5 CONVERGENCE OF NEWTON S METHOD 73 Remark. The proof of Propositions.3 and.4 may be viewed as the proof of the existene and uniqueness theorems about the solution of the equation f(x) =y, and the premise (.9) and (.7) an be replaed by (.22) f (x ) (f(x ) y) β. Hene, setting y =andβ= f (x ) f(x ),wehave Theorem.5. Let β = f (x ) f (x ) b. Assume that r r<r 2 if β<b, or r = r if β = b, wherer,r 2 and b are determined by Lemma.2 and Theorem.. Then, under the onditions (. ), the equation (.2) has a unique solution (.23) x B(x f (x ) f(x ),r β) B(x,r ) in the losed ball B(x,r). 2. Further disussion of Lipshitz onditions In the ball B(x,r), the Lipshitz ondition with the onstant L is (2.) f(x) f(x ) L x x, where x, x B(x,r). If (2.) is only true for all x B(x,r)andx =x,then it is alled the enter Lipshitz ondition in []; if (2.) is valid for all x B(x,r) and for all x = x + τ(x x )(τ), then it is alled the radius Lipshitz ondition. Now, if (2.) is valid for all x B(x,r)andforallx B(x, r ρ(x)), then we all it the enter Lipshitz ondition in the insribed sphere. For a onstant or positive integrable funtion L, among all Lipshitz onditions with the onstant L or the average of L, there is an impliation relation as follows: Lipshitz ondition in a ball the enter Lipshitz ondition in the insribed sphere the radius Lipshitz ondition the enter Lipshitz ondition. Custom is the only reason why we give the names of different Lipshitz onditions. It is not neessary in essene; see Theorems 6.3 and 6.4 in []. Sometimes, however, we have to pay attention to suh austomed thinking beause it determines the development of the literature. 3. Convergene of Newton s method Suppose that f has a ontinuous derivative in the losed ball B(x,r), f (x ) exists and f (x ) f satisfies the enter Lipshitz ondition in the insribed sphere with the average of L, f (x ) (f (x) f (x )) ρ(xx ) L(u)du, (3.) ρ(x) x B(x,r), x ρmb(x, r ρ(x)), where ρ(x) = x x,ρ(xx )=ρ(x)+ x x r,andlis a positive nondereasing funtion in [,r]. Under this hypothesis, the onditions (.) and (. ) are of ourse satisfied, and thus Theorems. and.5, Propositions.3 and.4 hold.

6 74 WANG XINGHUA Theorem 3.. Assume that β = f (x ) f(x ) b and r r,whereband r are determined by Theorem. and Lemma.2. Then, under the ondition (3.), Newton s method (.) is defined for all n and onverges to a solution x of equation (.2), (3.2) x B(x,r β) B(x,r ). Moreover, for all n n, the best possible error bounds ( x x ) 2 n n x n (3.3) x n (r t n ) r t n and (3.4) ( ) 2 n n 2 x n+ x n + +4 r x xn+ x n x n (r t n ) t n+ (r t n ) 2 (t t n+ t n n+ t n ) are valid with t n+ = t n h(t n) (3.5) h (t n ), t =, n =,,. In order to prove Theorem 3., we need Proposition 3.2. Under the assumptions of Theorem 3., for any natural number n, we have (3.6) x n x n t n t n, (3.7) ( ) 2 f (x ) xn x n f(x n ) h(t n ), t n t n (3.8) f (x ) f(x n ) f (x ) f(x n ) h(t n) h(t n ) x n x n, t n t n and ( ) 2 xn x n (3.9) x n+ x n (t n+ t n ). t n t n Proof. By the hypotheses, (3.6) is true for n =. Now assume that it holds for some n. Then (3.) x n B(x,t n ) B(x,r ). Sine f(x n ) = f(x n ) f(x n ) f (x n )(x n x n ) = (f (x n +τ ) f (x n ))(x n x n )dτ, where we obtain f (x ) f(x n ) x n +τ = x n + τ(x n x n ), τ, f (x ) (f (x n +τ ) f (x n )) (x n x n ) dτ.

7 CONVERGENCE OF NEWTON S METHOD 75 By the hypothesis (3.), we have f (x ) f(x n ) = xn x n ρ(xn x n +τ ) ρ(x n ) L(u)du x n x n dτ L( x n x + u)( x n x n u)du. Sine L is a nondereasing funtion, ϕ(t) := t t L(ρ+u)(t u)du is nondereasing 2 with respet to t in [,r ρ]. In fat, when <t <t 2 r ρ,wehave ( ϕ(t 2 ) ϕ(t )= t 2 ) t 2 ( ( t 2 L ρ+ t ) ( 2 2 +u L ρ+ t )) 2 u udu + (t 2 t ) 2 t 2t 2 2 t (L(ρ + t ) L(ρ + u))du + t 2 2 Hene f (x ) f(x n ) x n x n 2 t2 t (L(ρ + u) L(ρ + t ))(t 2 u)du. xn x n tn t n L( x n x + u)( x n x n u)du x n x n 2 (t n t n ) 2 L(t n +u)(t n t n u)du x n x n 2 ( ) 2 xn x n = h(t n ), t n t n where we have used the indutive hypothesis (3.6). Therefore, (3.7) holds for all n, whih makes (3.6) hold. Sine (3.) x n x n f (x n ) f (x ) f (x ) f(x n ), we obtain (3.2) f (x ) f(x n ) f (x ) f(x n ) h(t n) x n x n (t n t n ) 2 f (x n ) f (x ). By (.2) and (.5) we have (3.3) f (x n ) f (x ) xn x h (t n ). L(u)du Combining (3.2) and (3.3) and using (3.5), we get that (3.8) is also true if (3.6) is true for some n. Inreasing n to n+ in (3.) and (3.3) and applying (3.7) and (3.3) to (3.), we get (3.9). So (3.6) an be ontinued, and (3.6)-(3.9) hold for all n.

8 76 WANG XINGHUA Proof of Theorem 3.. Obviously, {t n } is onvergent to r monotonially. Therefore, the sequene {x n } B(x,r ) onverges. Also by (.), f (x n ) is bounded uniformly. So from (3.4) f(x n )+f (x n )(x n+ x n )= we get lim x n = x. Finally, by (3.) and we obtain e n := f (x ) (f(x ) f(x n ) f (x n )(x x n )) = f (x ) (f (z τ ) f (x n ))(x x n )dτ, z τ := x n + τ(x x n ) e n = ρ(xnz τ) ρ(x n) x x n L(u)du x x n dτ L( x n x + u)( x x n u)du. Sine t t L(ρ + u)(t u)du is nondereasing with respet to t, wehave 2 Therefore, e n x x n 2 t n x x n r t n L( x n x +u)( x x n u)du x x n 2 (r t n ) 2 L(t n +u)(r t n u)du x x n 2 r ( x ) 2 x n = L(u)(r u)du. r t n x x n+ f (x n ) f (x ) e n r L(u)(r u)du ( x ) 2 x n tn. r t n L(u)du t n By the indution method, (3.3) follows. By (3.9), for all i andn n, we have ( ) 2 n n xn+ x n x n+i+ x n+i (t n+i+ t n+i ). t n+ t n Summing for all i results in the upper bound (3.4). It follows from (3.3) that x n+ x n x x n + x x n+ x x n + r t n+ (r t n ) 2 x x n 2. Then, using Gragg and Tapia [3], we obtain the proof of the lower bound (3.4).

9 CONVERGENCE OF NEWTON S METHOD Under the premise of a Kantorovih type About the onvergene of Newton s method, the main point of Kantorovih [2] type premise is to make (4.) h(t) =β t+ 2 Lt2, t R, beome a majorizing funtion. For this reason, as x x + x x r, itis suffiient to assume that (4.2) f (x ) (f (x) f (x )) L x x, for a positive onstant L. As (4.3) λ=lβ 2, orresponding to (.7), the zeros of h } r = 2λ (4.4) r 2 L satisfy (4.5) β r 2β L r 2 2 L, beause r =/L, R =2/L, b =/(2L) in this ase. So Theorems. and.5, Propositions.3 and.4 all have onrete forms. The onretization of Theorem 3. requires that the solution of the sequene (3.4) has a losed form t n = (4.6) q2n r, n =,,, q 2n where q = (4.7) 2λ + 2λ. (4.6) is independently given by [3] [5]. For instane, the onrete forms of Theorems.,.5 and 3. are, respetively, Theorem 4.. Let L be a positive onstant. Assume that f satisfies the ondition (4.8) f (x ) f (x) I L x x, x B(x, /L). Then f x (4.9) exists and is differentiable in the open ball B(f(x ), /(2L f (x ) )) f(b(x, /L)). Moreover, the radius of this ball (the left in (4.9)) is the best possible. Theorem 4.2. Let L be a positive onstant, β = f (x ) f(x ) and λ = Lβ. Assume that f satisfies the ondition 2 (4.) f (x ) f (x) I L x x, x B(x,r), where r r<r 2 if λ< 2,orr=r if λ = 2, while r and r 2 are determined by (4.4). Then the equation (.2) has a unique solution x satisfying (.23) in the losed ball B(x,r).

10 78 WANG XINGHUA Theorem 4.3. Let L be a positive onstant, β = f (x ) f(x ) and λ = Lβ 2. Assume that f satisfies the ondition (4.2). Then Newton s method (.) is well defined for all n and onverges to the solution x satisfying (3.2) of the equation (.2). Moreover, for all n, the best possible error bounds (4.) and (4.2) x x n q q2n 2n 2 n i= q i x x 2 n i= q x 2i x 2 x n+ x n + +4q 2n /( + q 2n ) 2 x x n q 2 n q 2i x n x n 2 β are valid with (4.7). i= q 2n x n x n Remark. It is a posterior estimation to use x n x n to estimate x x n. The posterior estimation in (4.2) an be obtained by setting n = n in (3.4). In the hypothesis of Kantorovih s type, more preise posterior estimations were studied by Potra [6] and Potra & Ptak [7]. 5. Under a premise of Smale type Under the hypotheses that f is analyti and satisfies f (x ) f (n) (5.) (x ) n!γ n, n 2, Smale [8] studied the onvergene and error estimation of Newton s iteration. Wang and Han [9] (also see [], []) ompletely improved Smale s results by introduing a majorizing funtion (5.2) h(t) =β t+ γt2 γt, t R. When γ x x <, it is easy to derive from (5.) that f (x ) f (x) h 2γ ( x x ) = ( γ x x ) 3 (see Lemma 3 in [2] or Lemma 3.5 in [3]). Hene, onditions (.) and (3.) are satisfied for the funtion L defined by (5.3) 2γ L(u) = ( γu) 3. Furthermore, for this L, the funtion h given in (.5) oinides with the one in (5.2). As α = γβ 3 2 2, orresponding to (.7), the zeros of h } r = +α ( + α) 2 8α (5.4) r 2 4γ satisfy (5.5) β r ( + )β ( ) 2 2 γ r 2 2γ,

11 CONVERGENCE OF NEWTON S METHOD 79 beause r =( 2 ) γ,r = 2γ,b =(3 2 2) γ in this ase. So Theorems. and.5, Propositions.3 and.4 all have onrete forms. The onretization of Theorem 3. requires that the solution of the sequene (3.4) has a losed form t n = (5.6) q2n q 2n η r, n =,,, where q = α ( + α) 2 8α α + ( + α) 2 8α, η = +α ( + (5.7) α)2 8α +α+ ( + α) 2 8α. For instane, the onrete forms of Theorems.,.5 and 3. are, respetively, Theorem 5.. Let γ be a positive onstant. Assume that f satisfies the ondition f (x ) f (x) I (5.8) ( γ x x ) 2, x B(x, ( 2 )/γ). Then f x (5.9) exists and is differentiable in the open ball B(f(x ), (3 2 2)/(γ f (x ) )) f(b(x, ( 2 )/γ)). Moreover, the radius of this ball (the left in (5.9)) is the best possible. Theorem 5.2. Let γ be a positive onstant, β = f (x ) f(x ) and α = βγ Assume that f satisfies the ondition (5.) f (x ) f (x) I ( γ x x ) 2, x B(x,r), where r r<r 2 if α<3 2 2,orr=r if α =3 2 2, while r and r 2 are determined by (5.4). Then the equation (.2) has a unique solution x satisfying (.23) in the losed ball B(x,r). Theorem 5.3. Let γ be a positive onstant, β = f (x ) f(x ) and α = βγ Assume that f satisfies the ondition (5.) f (x ) (f (x) f (x )) ( γ x x γ x x ) 2 ( γ x x ) 2, x x + x x r. Then Newton s method (.) is well defined for all n and onverges to the solution x satisfying (3.2) of the equation (.2). Moreover, for all n, the best possible error bounds (5.2) x x n η q 2n η q2n x x η 2 ( + α)( q 2n η) q2n x x

12 8 WANG XINGHUA and (5.3) + are valid with (5.7). 2 x n+ x n ( q 2n )( q 2n η) +4q 2n ( q 2n+ η) 2 ( ) 2 x x n q( q2n η) q 2n η x n x n 2 r ( η) q 2n q 2n q2n η q 2n x n x n The results above an be made more general by replaing (5.3). Now we take 2γ (5.3 ) L(u) = ( γu) 3, where is a positive number. In this ase the majorizing funtion is (5.2 ) and its zeros are (5.4 ) } r r 2 h(t) =β t+ γt2 γt, = +α ( + α) 2 4( + )α. 2( + )γ They satisfy (5.5 ) β r ( + + )β( + ) γ r 2 (+)γ, beause r =( + ) γ,r= (+)γ,b=(+2 2 (+)) γ.hene, we have Theorem 5.3. Let γ and be positive onstants, β = f (x ) f(x ) and α = βγ +2 2 (+). Assume that f satisfies the ondition (5. ) f (x ) (f (x) f (x )) ( γ x x γ x x ) 2 ( γ x x ) 2, x x + x x r. Then Newton s method (.) is well defined for all n and onverges to the solution x satisfying (3.2) of the equation (.2). Moreover, for all n, the best possible error bounds (5.2) and (5.3) are valid with (5.7 ) q = α ( + α) 2 4( + )α α + ( + α) 2 4( + )α, η = +α ( + α)2 4( + )α +α+ ( + α) 2 4( + )α.

13 CONVERGENCE OF NEWTON S METHOD 8 6. Under the premise of analytiity We ome bak to the analyti premise about f, to see what stronger onlusion an be obtained. When f is assumed to be analyti in the ball B(x,r), f an be expanded to a onvergent power series (6.) f(x) = n! f (n) (x )(x x ) n. n= If we suppose f (x ) f (n) (6.2) (x ) γ n, n 2, and write (6.3) where the sequene γ n satisfies g(t) = n=2 γ n n! tn, (6.4) γn lim sup n n! r, then f (x ) f satisfies the Lipshitz ondition about g in B(x,r). Thus, Theorem. asserts that fx exists in B(f(x ),b/ f (x ) ) and is analyti, where (6.5) and r satisfies (6.6) So, we have b = r r g (u)udu = r g(r ), g (u)du = g (r )=. Theorem 6.. Assume that f is analyti in the ball B(x,r) and r r.if (6.7) then the Euler series (6.8) x = x + n= y f(x ) < b f (x ), ( ) n d fx n! dy (y) y=f(x)(y f(x )) n onverges, and the onstant b in the right of (6.7), whih is determined by (6.5), is the best possible. When X = Y = C, wehave Theorem 6.2. Assume that f and F are analyti in the open ball B(x,r) C and r r. Then the onvergene radius, R(F fx ), of the Lagrange series (6.9) F (fx (y)) = F (x )+ n= ( ) n ( ( ) n ) d F x x (x) (y f(x )) n n! dx f(x) f(x ) x=x

14 82 WANG XINGHUA has an exat lower bound (6.) R(F fx ) b f (x ). It is a very tehnial thing to hoose the sequene {γ n } or the funtion g suh that it an give a bound of the different Taylor oeffiients of f and be onvenient to give the values of the parameters b and r in Theorems 6. and 6.2. In this paper we propose to hoose a different generating funtion G of the unit sequene {,, } with the positive onstants γ and, and then the funtion g an be obtained by g(t) = G ()γ {G(γt) G ()γt G()}. Example (Exponential type). Taking G(t)=e t as the exponential generating funtion of the unit sequene, we have Under the ondition g(t) = γ (eγt γt ). f (x ) f (n) (x ) γ n, n 2, we obtain that γr =ln +, γb =(+)ln +. Espeially, as =,wehave γr =ln2=.6934, γb =ln4 = Theorem 6. with the values above has been obtained in [4] by the method of taking the exponential generating funtion of the number of Shröder system as the majorizing sequene. Example 2 (Binomial type). Taking G(t) =+sign(m){( t) m } as the binomial generating funtion of the unit sequene, where m> andm is a real number, we have g(t) = { ( γt) m mγt }. mγ Under the ondition f (x ) f (n) (x ) (m +)(m+2) (m+n )γ n, n 2, we obtain that ( ) m+ γr =, +( γb =+ m+ ( ) m ) + m+. m Espeially, as m =and=,wehave γr = 2 =.29289, γb =3 2 2=.757.

15 CONVERGENCE OF NEWTON S METHOD 83 Theorem 6. with the values above has been obtained in [3] by the method of taking the normal generating funtion of the number of blankets added to n letters as the majorizing sequene. Also, as m = 2,=,wehave γr = 3 =.373, 4 γb = = As m = 2,=,wehave γr = 3 4, γb = 2. The required ondition of these simple numbers is not ompliated, i.e. f (x ) f (n) (2n 3)!! (x ) 2 n γ n, n 2, Example 3 (The first logarithmi type). Taking G(t)= ln ( t) as the first logarithmi generating funtion of the unit sequene, we have g(t) = γ ln γt t. Under the ondition f (x ) f (n) (x ) (n )!γ n, n 2, we obtain that Espeially, as =,wehave γr = +, γb = ln +. γr = 2, γb = ln 2 = Example 4 (The seond logarithmi type). Taking G(t) = +2t+( t)ln( t) as the seond logarithmi generating funtion of the unit sequene, we have g(t) = ( γt)ln( γt)+t. γ Under the ondition f (x ) f (n) (x ) (n 2)!γ n, n 2, we obtain that γr = e, γb = +e. Espeially, as =,wehave γr = e =.6322, γb = e =

16 84 WANG XINGHUA 7. Appliations to Smale s α-theory We ontinue the disussion of Chapter 7 in []. It is well known that Smale [8] first used the riterion (7.) α(f,x )=γ f (x ) f(x ) to judge x is an approximate zero of Newton s iteration of f, where γ=sup n 2 n! f (x ) f (n) n (7.2) (x ). Definition 7.. Suppose x D is suh that Newton s iteration (.) is well defined for f : D X Y and satisfies ( ) 2 n e(x n ) e(x n ), 2 for all positive integers n, wheree(x n ) denotes some measurement of the approximation degree between x n and x.thenx is said to be an approximate zero of f in the sense of e(x n ). The approximate zero defined in [8] was introdued in the sense of x n+ x n, while the seond kind of approximate zero is defined in the sense of x x n. Now a more reasonable definition for the seond kind was introdued in [5]. We find that it is not neessary to introdue the definition of an approximate zero in the sense of f (x ) f(x n ). In fat, similarly to Theorem 7.2 in [], by Theorem 5.3 we have Theorem 7.2. Let γ, and q be positive numbers, <q<. Assume that f satisfies the ondition f (x ) (f (x) f (x )) ( γ x x γ x x ) 2 (7.3) ( γ x x ) 2, γ x x + γ x x +. Then, as α 2q + ( + q)2 ( + q) 2 ( + q) 2 +4q (7.4) 2q for all natural numbers n, it follows that (7.5) x n x q 2n x n x, (7.6) x n+ x n q 2n x n x n, and (7.7) f (x ) f(x n ) q 2n f (x ) f(x n ), where x satisfies f(x )=. Espeially, as α(f,x ) (9 +8) (7.4a), 4

17 CONVERGENCE OF NEWTON S METHOD 85 x is an approximate zero of f in any sense of x x n, x n+ x n or f (x ) f(x n ). Proof. The representation in the inequality (7.4) at the right side an be obtained from (5.7 ) by representing q by α. Hene, under the hyposethis of (7.4), by Theorem 5.3 and Proposition 3.2, we have x n x r t n x n x, r t n x n+ x n t n+ t n x n x n t n t n and f (x ) f(x n ) h(t n) h(t n ) f (x ) f(x n ). Thus, Theorem 7.2 follows from the following lemma. Lemma 7.3. For (5.2 ), (5.4 ) and (5.7 ),ifα=βγ =2 2 (+),then (7.8) n η r t n = q2 r t n q 2n η q2 n q 2n, (7.9) and (7.) t n+ t n t n t n = h(t n ) h(t n ) = q2 n q2 q 2n n η n q 2n+ η q2 q 2n ( ) ( γt n) ( 2 ) tn+ t n q 2n. ( γt n ) t n t n 2 Proof. As (5.2) beomes (5.2 ), the representation (5.6) about r t n remains true provided that r and r 2, q and η are determined by (5.4 )and(5.7 ). Hene, (7.8)- (7.) follow immediately. Finally, similarly to Colloray 7.3 in [], we have Corollary 7.4. Let γ be a posiitve number. Assume that f (x ) exists, f is analyti in B(x, /γ), and for some q (, ) (7.) ( ) n! 2 (x ) f (n) (x ) q γ n, n 2. +q Then, as (7.2) α(f,x ) q, (7.5) holds. Espeially, as n! (x ) f (n) (x ) γn 9, n 2, and (7.2a) α(f,x ) 2 x is an approximate zero of Newton s iteration of f.

18 86 WANG XINGHUA Referenes [] Wang Xinghua, Convergene of Newton s method and uniqueness of the solution of equations in Banah spae, Hangzhou University, preprint. [2] L.V. Kantorovih and G.P. Akilov, Funtional Analysis, Pergamon Press, 982. MR 83h:462 [3] W.B. Gragg and R.A. Tapia, Optimal error bounds for the Newton-Kantorovih theorem, SIAM J. Numer. Anal., (974), -3. MR 49:8334 [4] A.M. Ostrowski, Solutions of Equations in Eulidean and Banah Spaes, Aademi Press, New York, 973. MR 5:76 [5] Wang Xinghua, Convergene of an iterative proedure, KeXue TongBao, 2(975), ; J. of Hangzhou University, 977, 2: 6-42; 978, 3: [6] F. A. Potra, On the a posteriori error estimates for Newton s method, Beitraege Numer. Math., 2(984), MR 85h:6528 [7] F. A. Potra and V. Ptak, Sharp error bounds for Newton s proess, Numer. Math., 34(98), MR 8:6527 [8] S. Smale, Newton s method estimates from data at one point, in The Merging of Disiplines: New Diretions in Pure, Applied and Computational Mathematis, R. Ewing, K.Gross and C. Martin eds, New York, Springer-Varlag, 986, MR 88e:6576 [9] Wang Xinghua and Han Danfu, On the dominating sequene method in the point estimates and Smale s theorem, Siene in China(Ser. A.), 33(99), MR 9h:658 [] Wang Xinghua, Some results relevant to Smale s reports, in From Topology to Computation: Proeedings of the Smalefest, M.W. Hirsh, J. E. Marsden and M. Shub eds., New York, Springer-Verlag, 993, MR 94f:26 [] Wang Xinghua, A summary on omplexity theorey, Contemporary Mathematis, 63(994), MR 94m:657 [2] Wang Xinghua, Han Danfu and Sun Fangyu, Point estimations on deformated Newton s iteration, Math. Num. Sin., 2(99), 45-56; Chinese J. Num. Math. Appl., 2(99), -3. MR 9d:584 [3] Wang Xinghua, Zheng Shiming and Han Danfu, Convergene on Euler s series, the iterations of Euler s and Halley s families, Ata Mathematia Sinia, 33(99), MR 92b:654 [4] Wang Xinghua and Han Danfu, The onvergene of Euler s series and ombinatorial skills, preprint, Hangzhou University, 996. [5] L. Blum, F. Cuker, M. Shub and S. Smale, Complexity and Real Computation, Part II: Some Geometry of Numerial Algorithms, City University of Hong Kong, preprint, 996. [6] Chen Pengyuan, Approximate zeros of quadratially onvergent algorithms, Mathematis of Computation, 63(994), MR 94j:6567 Department of Mathematis, Hangzhou University, Hangzhou 328 China