- Email: [email protected]
More Results on Newton's Method
Chapter 6 More Results on Newton's
Method
We exploit the weaker conditions introduced in Section 5 to improve on recent results on the local and semilocal convergence analysis of Newton's method.
6.1
Convergence radii for the Newton-Kantorovich method
Motivated by the works in Wang [341], [343] and using the same type of weak Lipschitz conditions we show that a larger convergence radius can be obtained for Newton-Kantorovich method (4.1.3). Recently in [341], [343] a Lipschitz-type condition
lift (X*)--I [f' (X) -- f' (X0)] II ~
f~(x)
L(u)du
(6.1.1)
JOs(x)
for all x E D, where L is an integrable, nondecreasing, nonnegative function, s(x) = IIx - x* II and x ~ = x* + O(x - x*) 0 E [0,1], w a s used to obtain larger convergence radii than the ones given by the relevant works in [136], [a401, [312]. Condition (6.1.1) is weaker than the famous Lipschitz condition IlF' (x* ) - l [F' (x) -
r'(y)]ll ~ w(llx- yll),
for all x, y E D,
(6.1.2)
where w is also a nondecreasing, nonnegative function. Condition (6.1.2) has been used e.g. in [295] (when w is a constant function). It turns out that (6.1.1) instead of the stronger (6.1.2) is really needed to show the local convergence of method (6.1.1) to x* [341], [343]. Based on this observation it became possible to obtain convergence radii larger than the ones obtained in [136],
[295], [340], [312]. Here we show that all the above mentioned results can be further improved under weaker or the same hypotheses and under less or the same computational cost. A numerical example is also provided where our results compare favorably with all earlier ones. Several areas of application are also suggested. 187
188
6. More Results on Newton's Method
We provide the following local convergence theorems: T h e o r e m 6.1.1 Let F" D C_ X ---+Y be a Frdchet-differentiable operator. Assume: (a) there exists a simple zero x* E D of equation F(x) - 0 so that F'(x*) -1 E L(Y, X), positive integrable nondecreasing functions L, Lo such that the radius Lipschitz condition with L average and the center radius Lipschitz condition with Lo average s(x) I I f ' ( x * ) - l [ f ' ( x ) - f'(x~ <_ L(u)du (6.1.3) JOs(z)
f
and
l l r ' ( x * ) - : [ r ' ( x ) - r'(x*)]ll < fo s(x) Lo(u)du
(6.1.4)
are satisfied respectively for all x c D, 0 E [0, 1]; (b) equation
/o
L(u)u du + r
/o
Lo(u)du- r - 0
(6.1.5)
has positive zeros. Denote by r* the smallest such zero; and
(6.1.6)
U(x*, r*) c_ D.
Then, sequence {Xn} (n >_ O) generated by method (~.1.3) is well defined, remains in U(x*, r*) for all n > O, and converges to x* provided that xo E U(x*, r*). Moreover the following estimates hold for all n >_ 1"
Ilxn - x*ll _ a 2 n - l l l x o - x*ll
(6.1.7)
where, a-
fo (x~ L ( u ) u d u
(6.1.8)
,
~(x0) [~- fo (x~ L0(~)d~] and
a C [0, 1).
(6.1.9)
P r o o f . We first show a E [0, 1). Indeed we can have in turn for 0 < u: < u2 and the monotonicity of L"
(u-~122 jl~ ul~fo~:)uL(u) du 1
~2
1
1 1
1
1
u1~ ~ jfOul u du - O.
189
6.1. Convergence radii for the Newton-Kantorovich method
Therefore -~1 fou L(v)dv is a nondecreasing function with respect to u. Then, we can have
0-
fo (~~ L(~)~ d~ ~(xo) ~(xo) ~ [ ~ - fo (x~ Lo(~)d~] .
<
fo L(~)~d~
< IIx0
-
--
r*
x*ll
~(xo)
< 1.
Let x E U(x*, r*). Using (6.1.4) and (6.1.6)we get
ILF'(x*)-I[F'(x)- F'(x*)]I I < fo ~(x) Lo(u)du < / i ~* Co(u)du < 1.
(6.1.10)
It follows from (6.1.10) and the Banach Lemma on invertible operators 1.3.2 that F'(X) -1 e L(Y, X ) and
1
1
IlF'(x)-lF'(x*)[t < < . - 1 - fo (x) Lo(u)du - 1 - fo* Lo(u)du
(6.1.11)
In particular Xo c U(x*, r* ) by hypothesis. Let us assume xn e U(x*, r* ). We obtain the identity X n + l - X * - - Xn ~ F ' ( X n ) - l F ( x n ) = ft(Xn)-l[F(x
x*
*) - r ( X n ) -
= [F'(Xn)-IF'(x*)]
(6.1.12)
F t ( X n ) ( X * - Xn)]
F ' ( x * ) - I ( F ' ( X n ) - F'(x~
- x*)dO .
By (6.1.3), (6.1.4) and (6.1.11)we get
Ilxn+l- X* I1-< IIF'(x~)-'f'(x*)ll < -
=
/o 1 [ [ F ' ( x * ) - I ( F ' ( X n )
-
F'(x~
)]x~ - x*lldO
f~ rs(xn) 30s(xn) L(u)dus(xn)dO 1 - fo (xn) Lo(u)du fo (xn) L ( u ) u d u 1 - Jof~(xn)Lo(u)du
<_aLIx~-
x*
I1<11~-
x*
II,
(6.1.13)
which shows the convergence of xn to x* and Xn+l r U(x*,r*). Hence sequence {Xn} (n _> 0) remains in U(x*,r*) by the principle of mathematical induction.
6. More Results on Newton's Method
190
Moreover s(xn) decreases monotonically. Furthermore by (6.1.13)we obtain more precisely
IlXn+l-
X$
f:(Xn) n('~)ud~
II
S(Xn) 2 [1 - fo (x~) Lo(u)du]
fo (x~ L ( u ) u d u
<
8(Xn) 2
s(xo) 2 [ 1 - fo (x~ Lo(u)du] _
a
_ ~s(xo---711Xn_ x, I 2
(6.1.14)
from which estimate (6.1.7) follows. 9 R e m a r k 6.1.1 In general
Lo(u) < L(u)
(6.1.15)
holds and Lo(u) L(u) can be arbitrarily large (see Section 5.1). If equality holds in (6.1.1518) then our theorem coincides with the corresponding Theorem 3.1 in [3~1]. However if strict inequality holds in (6.1.15) (see Example 5.1.5), then our convergence radius r* is greater than the corresponding radius r in Theorem 3.1. Moreover our convergence rate a is smaller than the corresponding q in Theorem 3.1. Summarizing, if Lo(u) < L(u)
(6.1.16)
r < r*
(6.1.17)
a < q.
(6.1.18)
then
That is our theorem provides a wider range of initial guesses xo and more precise error estimates on the distances IlXn- x*ll (n >_ O) than Theorem 3.1 and under the same computational cost since in practice the evaluation of function L requires that of Lo. In case L0, L are constants see section 5.1. T h e o r e m 6.1.2 Let F" D c_ X ~ Y be a Frdchet-differentiable operator. Assume: (a) there exists a simple zero x* E D of equation F(x) - O, so that F'(x*) -1 c L(Y, X ) , and a positive integrable function Lo such that condition (6.1.~) holds; (b) equation
1 r
~0r Lo(u)(r
- u)du- 1 - 0
(6.1.19)
6.1. Convergence radii for the Newton-Kantorovich method
191
has a minimum positive zero denoted by r~ ; and
(c) u(~*, ~;~)c D.
(6.1.20)
Then sequence {Yn} (n ~ O) generated by Yn+l -- Yn -- F ' ( x * ) - I F ( y n )
(n ~ O)
(6.1.21)
remains in U(x*, r~) for all n >_ 0 and converges to a unique solution x* of equation (~. 1.1) so that
Pf*~+l- X*Pl<_aoll*n-- X*PP
(6.1.22)
where ao 6 [0, 1) and 1 fo s(x~ ao = S(Xo) Lo(u)(s(xo) - u)du.
(6.1.23)
Proof. Simply use function L0 instead of L in the proof of Theorem 4.1 in [341].
R e m a r k 6 . 1 . 2 If Lo
L then Theorem 6.1.2 reduces to Theorem ~.1 in [3~1]. Moreover if (19) holds then -
~0 < ~;
(6.1.24)
q0 < a0
(6.1.25)
and
where by to, qo we denote the radius and ratio found in Theorem ~.1.
In the case when L, Lo are constants as expected our results improve Corollaries 6.3 and 6.4 in [342] which in turn compare favorably with the results obtained by Smale [312] and Debieu [139] respectively when F is analytic and satisfies
IlF'(x*)-lF(k)(x*)ll ~ k!~ k-~
(k ~ 2)
(6.1.26)
provided that L(u) -
27 (1 - T u ) 3
(6.1.27)
and 270 Lo(u) - (1 - 70u) 3
(6.1.28)
192
6. More Results on Newton's Method
for some 70 >_ O, 7 >- 0 with 70 _< 7.
(6.1.29)
For example our result corresponding to Corollary 6.4 gives
9 E 1)
r o E 0,~o~
(6.1.30)
whereas the corresponding radius in [342] is given by
[1)
r0E
0,~
.
(6.1.31)
Therefore (6.1.24) and (6.1.25) hold (if strict inequality holds in (6.1.29)). We can provide the local convergence result for twice Fr6chet-differentiable operators as an alternative to Theorem 6.1.1. T h e o r e m 6.1.3 Let F" D c_ X ---, Y be a twice Frdchet-differentiable operator. Assume: (a) there exists a simple zero x* E D of equation F(x) F'(x*) -1 E L(Y, X ) and nonnegative constants ~, b, ~o such that IIF'(x*)-~[S(x
+ e(x* - x)) -
IIF'(x*)-~_P"(x*)ll
r"(x*)] II _< ~(1
< b,
-
e)llx - x*ll,
O, so that
(6.1.32) (6.1.33)
and
IlF' (x* )-l [F" (x ~ - F"(x*)]l
_< eoOllx- =*11
(6.1.34)
for all x E D, 0 E [0, 1]; (b) U(x*,R*) c_ D, where,
R* = d-
2
d + v/d 2 + 4c
3b
and
,
c--lg+ 3
(6.1.35) lg
~o.
(6 1.36)
for all n >_ 0 and converges to x* provided that Xo E U(x*, R*). Moreover the following error bounds hold for all n > 0 X*
X* 2
(6.1.37)
and IIXn -- X* II -----h ~ - ~
IX0 -- X* II"
(6.1.38)
193
6.1. Convergence radii for the Newton-Kantorovich method
Proof. Let x c U(x*,R*). Using (6.1.33), (6.1.34) and the definition of R* we obtain in turn ft(x*)-l[ft(x
*) - f t ( x ) ] -
ft(x*)
-ft(x*)-l[Ft(x)-
F"(x*)(x
-
- x*) + Y " ( ~ * ) ( x
- x*)]
F ' ( z * ) - ~ { F " [ x * + O(z - x*)]
- F"(x*)}(x-
(6.1.39)
x*) - F ' ( x * ) F " ( x * ) ( x - x*)
and 1
*) - F'(x)]ll _
]]F'(z*)-I[F'(z
L
eo011~- x*ll~d0 + Vllx- x*ll
1 _< ~fo(R*)2 + bR* < 1.
(6.1.40)
It follows from (6.1.40) and the Banach Lemma on invertible operators that F ' ( x ) - l F ' ( x *) exists and ]]F'(x)-lF'(x*)ll <_
1 1 - T l lf.o x1
< -
X* I I ~ - b l l x
~o R,)~ 1--~(
- Z* II
"
-bR*
(s.1.41)
In particular by hypothesis Xo E U(x*,R*). F(x*) - 0 we obtain the identity X n + l -- X* -- Xn -- Y l ( x n ) - l
y(xn)
Assume Xn C U(x*,R*).
X*
*) - F ( X n ) -
F'(Xn)(X*
= [F'(xn)-lF'(x*)]F'(x*) -1
/o 1F"[xn
-- F ' ( X n ) - I [ F ( x
Using
- Xn)]
--[- O(X* -- Xn)](1 -- O)dO(xn - x*) 2 ~--- [ f ' ( X n ) - l -
F"(x*)](1
f'(x*)]F'(x*) -
-1
{/o 1[f"(Xn
-]-O(x* - X n ) )
O)dO(xn - x * ) 2
+ f0 ~ F " (~* )(1 - O)dO(xn - x*) ~ } .
(6.1.42)
By (6.1.32), (6.1.33), (6.1.41), (6.1.42) and the triangle inequality we get
[IXn+l -
x*
I1_<
1
-
1 X* b 5ellxnII + ~ eo 2 IIx~ - x* II~ - b[lxn
--
< hl]xn - x* I]2 < [IXn -- x* ]].
x*
II
IIx~--
x* 2 II
(6.1.43) (6.1.44)
194
6. More Results on Newton's Method
Estimate (6.1.44) implies
Xn+l
and lim
E U(x*,/~*)
n---~cx:)
Xn -- X*.
9
The analog of Theorem 6.1.2 is given by T h e o r e m 6.1.4 Under the hypotheses of Theorem 6.1.3 (excluding (6.1.~6)) and replacing R*, h by R;, ho respectively given by 2
R; =
(6.1.45)
do + v/dg + 4co ho - coR~ + do,
(6.1.46)
where, f0 co- ~
b d o = -~
and
(6.1.47)
the conclusions of Theorem 6.1.3 hold, for iteration (6.1.21). P r o o f . Using iteration (6.1.21) we obtain the identity Yn+l
-- X * - -
~- - - [ i ~ ' ( X * ) - l [ F ~ ( y n )
= --F'(X*) -1
=
-- X*)]
-- [i~(X *) -- [ g / ( X * ) ( y n
[/o1F " [ X *
--I- O ( y n --
/o] F"(x*)(1 - O)dO(yn - F Z ( x * ) - 1 {/o 1[FZ'(x* -I-
x*)](1 -
x*) 2 +
O(yn
-
O)dO(yn
/o1F"(x*)(1 -
x*)
-
F"(x*)](1
+ ~1 F" (x*)(y~-x ,)2 }
-
x*) 2
O)dO(yn - x*) 2 - O)dO(yn
-
x*) 2
(6.1.48)
By (6.1.33), (6.1.34), (6.1.48) and the definition of R~, ho we obtain (as in (6.1.43))
IlYn+t -- X*ll ~ [~~ollyn -- X*ll -~- ~] ][yn -- X*ll2 _ h011Y~ -- x*ll ~ < llY~ -- x*il,
(6.1.49)
which completes the proof of Theorem 6.1.4. 9 R e m a r k 6.1.3 (a) Returning back to Example 5.1.5 and using (6.1.32)-(6.1.36), (6.1.~5)-(6.1.~7) we obtain ~ - ~o - e - 1, b - 1, d - 1.5, c - g5 ( e - 1) , c o - e--1 6 , 1 do - 3, R* - .462484887
6.1. Convergence radii for the Newton-Kantorovich method
195
and we can set R~ - 1. Comparing with the earlier results mentioned above (see Example 5.1.5) we see that the convergence radius is significantly enlarged under our new approach. Note also that the order of convergence of iteration (6.1.21) is quadratic whereas it is only linear in the corresponding Theorem ~.1 in [3~1]. (b) The results can be extended easily along the same lines as above to the case when ~, ~o are not constants but functions. (c) The results obtained here can also be extended to hold for m-Frdchetdifferentiable operators (m >_ 2) along the lines of our works in [79], [68]. However we leave the details of the works for (b) and (c) to the motivated reader.
We present a new convergence theorem where the hypotheses (see Theorem 6.1.1) that functions L, Lo are nondecreasing are not assumed: T h e o r e m 6.1.5 Let F, x*, L, Lo be as in Theorem 6.1.1 but the nondecreasiness of functions L, Lo is not assumed. Moreover assume: (a) equation
ffo
~(L(u) + L o ( u ) ) d u - 1 - 0
(6.1.50)
has positive solutions. Denote by R* the smallest one; (b)
(6.1.51)
U(x*,R*) c D.
Then sequence {Xn} (n ~ O) generated by method (~.1.3) is well defined, remains in U(x*,R*) for all n >_ 0 and converges to x* provided that xo C g(x*,R*), so that: IIx~ - x'L1 _ b~lixo - x*]i
(~ >_ o)
(6.1.52)
where, b
fo (x~ L(u)du
(6.1.53)
1 - fo (x~ Lo(~)d~ and
be [0,1). Furthermore, if function L~ given by
(6.1.54)
L~(t) = tl-~L(t) is nondecreasing for some c~ E [0, 1], and R* satisfies -~
L(u)u du +
/o
Lo(u)du <_ 1.
(6.1.55)
196
6. More Results on Newton's Method
Then method (4.1.3) converges to x* for all xo E U(x*,R*) and IIx~ - x* II ~ b h~-~ IIx0 - x*ll
(6.1.56)
(~ ~ o)
where, k
h~ - Z ( 1 + ~)~ (k > 1).
(6.1.57)
i=0
Proof. Exactly as in the proof of Theorem 6.1.1 we arrive at I I x ~ + l - X* I1_<
fo (xn) L(u)u du
(6.1.58)
1 - fo (x~) Lo(u)du
< fo (x~) L(u)dus(xn)
-- 1-- fo (xn) Lo(u)du which show that if Xn E
U(x*,/~*) then
bllx~ - x* II < IlXn
Xn+l C U(x*,
R*) and
-
x*
II,
(6.1.59)
lim xn - x*. Esti-
n---,c~
mate (6.1.13) follows from (6.1.20). Furthermore, if function L~ is given by (6.1.15) and R* satisfies (6.1.16) then by Lemma 2.2 in [343, p. 408] for nondecreasing function
~ Z , ~ ( t ) - t~+Z
1 fo~
ugL(u)du,
(6.1.60)
we get IiXn+l-
X*
II-<
~l,a(8(Xn)) S(Xn) 1+~, 1 - JofS(xn)Lo(u)du ~1,c~ (S(X0)) S(Xn)l+a 1 - fo (xn) Lo(u)du b ~(x0)~ ]1~ - x*LI 1+~ < IIx~ - ~*IL,
(6.1.61)
(6.1.62)
which completes the proof of Theorem 6.1.5. 9 T h e o r e m 6.1.6 Let F, x*, Lo be as in Theorem 6.1.2 but the nondecreasness of function Lo is not assumed. Moreover assume: (a) equation
3
jr0r Lo(u)du -
1 - O,
(6.1.63)
has positive solutions. Denote by R~ the smallest one; (b) U(x*, R~) c_ D. Then, sequence {Xn} (n > O) generated by method (~.1.3) is well defined, remains in
6.1. Convergence radii for the Newton-Kantorovich method
197
U(x*, R~ ) for all n >_ 0 and converges to x* provided that xo E U(x*, R~). Moreover error bound (6.1.13) hold for all n >_ O; where
2 fo (x~ Lo( )d
b-
.
(6.1.64)
1 - fo (~~ Lo(u)du P r o o f . Simply use Lo instead of L in the corresponding proof of Theorem 1.2 in [343, p. 4 0 9 ] . . . R e m a r k 6.1.4 (a) In general (6.1.16) holds. If equality holds in (6.1.16) then Theorems 6.1.~ and 6.1.5 reduce to Theorems 1.1 and 1.2 in [343] respectively. However if strict inequality holds in (6.1.16) then clearly: our convergence conditions (6.1.11), (6.1.2~) are weaker than the corresponding ones (9) and (15) in [3~3]; convergence radii R* and R~ are also larger than the corresponding ones in [3~3] denoted here by p and po; and finally our convergence rates are smaller. For example in case functions Lo, L are constants we obtain
1 P -~' LR* b - 1 - LoR* '
1 Lo+L' 1 Po - 3L '
q
Lp 1-Lp' 1 and R o* -- 3L0 .
(6.1.65)
Therefore we have: p < R* < 2p,
(6.1.66)
Po <_ R~),
(6.1.67)
and b < q,
(6.1.68)
where q is the corresponding ratio in Theorem 1.1 in [3~3]. Note that if strict inequality holds in (6.1.26) so does in (6.1.28)-(6.1.30). Moreover the computational cost for finding Lo, L in practice is the same as of that of finding just L. As an example, as in Section 5.1 consider real function F given by (5.1.~0). Then using (6.1.3) and (6.1.4) we obtain no = e - 1 < e = L, R*-.225399674,
and Po - .12262648.
Hence, we have" p
p-.183939721,
R~-.193992236,
198
6. More Results on Newton's Method
and po < R;. (c) As already noted in [3~3, p. ~11] and for the same reasons Theorem 6.1.5 is an essential improvement over Theorem 6.1.~ if the radius of convergence is ignored (see Example 3.1 in [3~3]).
6.2
Convergence under weak continuous derivative
We provide a semilocal convergence analysis of Newton's method (4.1.3) under weaker Lipschitz conditions for approximating a locally unique solution of equation (4.1.1) in a Banach space. For p C [0, 1] consider the following HSlder condition: I F'(xo)-l[F'(x)
-
F'(y)] II ~ ellx- yll p.
(6.2.1)
In Rokne [303], Argyros [11] the HSlder continuity condition replaced (6.2.1) or in Argyros [93] the more general I[F'(Xo)-l[F'(x)
-
F'(y)] II _< v(llx- YlI)
(6.2.2)
was used, where v is a nondecreasing function on [0, R] for some given R. In order to combine the Kantorovich condition with Smale a-criterion [296], [312] for the convergence of method (4.1.3) (when F is an analytic operator), Wang in [341], [343] used the weak Lipschitz condition
]]ft(x0)-l[f'(x) -F'(y)]II
~
f,lx-~,,+,,x-xo,I
L(u)du
(6.2.3)
JIIx-xoll
instead of (6.2.1). Huang in [195] noticed that (6.2.3) is a special case of the Generalized Kantorovich condition Ix--Y + I~-zl)
IlF'(xo)-l[F'(x) - F'(y)][[ ~ r1~(
L(u)du
(6.2.4)
Js(llx-yll) m
m
for all x C U1, y E U2 where U1 - U(xo, t*), U2 - U ( x o , t * - I l x t* C (0, R) is the minimum positive simple zero of
x011), and
s(t)
f(t) -
L
L(u)(t- s-l(u))du-
t + ~7,
(6.2.5)
with functions L, s' continuously increasing satisfying L(u) > 0, on [0, R], s(0) - 0 and s'(t) > 0 on [0, R), respectively. Under condition (6.2.4) a semilocal convergence analysis was provided and examples were given where the Generalized
6.2. Convergence under weak continuous derivative
199
Kantorovich condition (6.2.4) can be used to show convergence of method (4.3.1) but (6.2.1) cannot. Here, we provide a semilocal convergence analysis based on a combination of condition (6.2.6) (which is weaker than (6.2.4)) and on the center generalized Kantorovich condition (6.2.7). The advantages of such an approach have already been explained in Section 5.1. We need the definitions: D e f i n i t i o n 6.2.1 (Weak Generalized Kantorovich Condition). We say xo satisfies the weak generalized Kantorovich condition if for all 0 E [0, 1], x c U1, y E U2 IlF'(xo)-~[F'(x) - F ' ( x + O(y - x))]l [ _<
f
s~ (llx-zoll+Oll~-yll) Lx(u)du
(6.2.6)
.] Sl ( l l x - x o l l )
holds, where functions S l , L1 are as s, L respectively.
D e f i n i t i o n 6.2.2 (Center Generalized Kantorovich Condition). We say xo satisties the center generalized Kantorovich condition if for all x E U1 so(I x-xoll) I l F ' ( x o ) - l [ F ' ( x ) - F'(xo)]ll < Lo(u)du (6.2.7)
]i
holds, where functions so, Lo are as s and L respectively.
Note that in general
80(U) < 81(U) < S(U)
(6.2.8)
Lo(u) < L l ( u ) < L(u)
(6.2.9)
and
hold. In case strict inequality holds in (6.2.8) or (6.2.9) then the results obtained in [195] can be improved under (6.2.6) or even more under the weaker condition (6.2.7). Indeed we can show the following semilocal convergence theorem for method (4.1.3). We first define functions f0, fl, by so(t) fo(t) L o ( u ) ( t - s o l ( u ) ) d u - t -~- r/, (6.2.10)
fd0
f l (t)
- f./f181(t) Ll(U)(t-
811(u))du- t -4- ?7
(6.2.11)
and sequences an, tn by an+l
-- an
tn+l -- tn
fl(tn) f;(tn)
(n > 0),
(to
0),
(6.2.12)
f(tn)
(n _> 0),
(to -- 0).
(6.2.13)
f'(tn)
200
6. More Results on Newton's Method
T h e o r e m 6.2.3 Let F" D c_ X ---, Y be a Frdchet-differentiable operator. Assume: there exists xo E D such that F'(xo) -1 E L ( Y , X ) ; and condition (6.2.5) holds. Then sequence {xn} generated by method (~.1.3) is well defined, remains in V(xo, a*) for all n > 0 and converges to a unique solution x* of equation F(x) - 0 in V(xo, a*), where a* is the limit of the monotonically increasing sequence {an} given by (6.2.12). Moreover the following estimates hold: (6.2.14)
I xn+l - Xnl] <_ a n + l - a n <_ t n + l -- t n ,
(6.2.15)
IlXn -- X*ll <-- a* - a n <_ t* - t n , a n <_ tn ,
(6.2.16)
a* < t *
(6.2.17)
Ilxn - x* II < (r - a*)qo(n) 50 < -
1 -
qo(n)(~o
1 -
52 ~-1
a*, 1
-
(6.2.18)
-
and IIF'(x~
< 5~"~
(6.2.19)
where r E [a*, t**] satisfies f (r) <_ O, a*
t** -
max {t, f(t) < O}
a*<_t
~
($o- - - ,
(6.2.20)
7"
n-1
qo(n) - H (A~
1,
(6.2.21)
i=0
and
.o
(r - ai) fo Fs(ai+O(a*--ai))L(u)dudO Js(ai)
Ai
(a* - - a i ) [ f o
Lo(u)dudu JFs(ai+O(a*-ai)) s(ai)
-
fo(r)]
<
1.
(6.2.22)
Furthermore if U2 in weak generalized Kantorovich condition (6.2.6) is U(x, t** tlx-xoII)NO, then equation (~.1.1) has a unique solution in the larger ball U(xo, t**)A D.
Proof. We have noticed that in the proof of Theorem 1 in [195, p. 117] weaker condition (6.2.6)can be used instead of (6.2.4)to obtain estimates on IIF'(x0) -1F(Xn)II. Moreover center generalized Kantorovich condition can be used instead of (6.2.7) to obtain the more precise error bounds on IIF'(xn)-IF'(xo)II. The rest of the proof goes along the lines of Theorem 1 in [195, p. 117] by simply replacing L, s, tn, t*, q(n), (~, Ai, f ' by L1, 81, an, a*, qo(n), 50, )~o, f~ respectively and using (6.2.6), (6.2.7). 9
201
6.2. Convergence under weak continuous derivative
R e m a r k 6.2.1 If equality holds in (6.2.8), (6.2.9) and (6.2.6) is replaced by (6.2.4) our theorem reduces to Theorem 1 in [195, p. 117]. Otherwise our results are more precise under the same hypotheses/computational cost since in practice computing s, L requires the computation of s l, L1, So, Lo. We need the following result on majorizing sequences. For simplicity we set wo - f~
and
w-
f'.
(6.2.23)
L e m m a 6.2.4 Assume there ezist parameters rl >_ O, (~ E [0, 2) and functions wo, w such that:
[/oI~(Ov)dO - w(O) ] + 5wo(~) <_ ~,
h~ - 2
Wo 2 and
~
1-
(6.2.24)
(6.2.25)
<1,
2{folly[222(~(1--(~)n+l) (~)n+l] --W[22~](~(1--(~)n+l)l} n+l +0
~l dO
2r/
(6.2.26)
hold for all ~ > O. Then, iteration {b~} (n > O) given by
bo - O,
bl = r/,
fo { [w(b~ + +
b~+2 = b~+l O ( b ~ + t - b~)]dO - w ( b ~ ) l } ( b ~ + t
1 - wo(bn+l)
- b~)
(6.2.27)
is monotonically increasing, bounded above by t** --
2rl 2-5'
(6.2.28)
and converges to some b* such that
0 < b* < b**.
(6.2.29)
Moreover, the following error bounds hold for all n > O:
0 _< b~+2 - b~+x < ~(bn+~ - b~) <
(6.2.30)
202
6. More Results on Newton's Method
The proof is omitted as it follows along the lines of Lemma 5.1.1 We can show the main semilocal convergence result for method (4.1.3): T h e o r e m 6.2.5 Assume: hypotheses of Lemma 6.2.4; conditions (6.2.6), (6.2.7); and U(xo, b*)c_ D.
(6.2.31)
Then sequence {xn} (n > 0) generated by method (4.1.3) is well defined, remains in U(xo, b*) for all n > 0 and converges to a solution x* c U(xo, b*) of equation F(x) - O. Moreover, the following estimates hold for all n > O" IlXn+l -- Xn ]1 ~_ bn+l - bn
(6.2.32)
[[Xn
(6.2.33)
and -
x*
II ~ b* - b~,
where iteration {bn} (n _> O) is given by (6.2.27). U(xo,b*) if
The solution x* is unique in
f0
1[w((1 + O)b*) - w(b*)]dO + wo(b*) < 1.
(6.2.34)
Furthermore if there exists Ro such that b* < Ro,
(6.2.35)
U(xo, Ro) C_ D
and
ol[w(b + O(b* + Ro)) - w(b*)]dO + wo(b*) _< 1,
(6.2.36)
then the solution x* is unique in U(xo, Ro).
Proof. Using induction we show xk E U(xo, b*) and estimate (6.2.32) hold for all k _> 0. For n - 0, (6.2.32) holds, since Ilxl - Xo]l- IIF'(xo)-lF(xo)ll <_ ~7- bl - b0,
(by (33)).
Suppose (6.2.32) holds for all n - O, 1,..., k + 1; this implies in particular that k
k
IlXk+l -- XoII ~ E
[]Xi+l -- Xill ~ E ( b i + l i=0 i=0
- hi) -
bk+l -- bo - bk+l.
We show (6.2.32) holds for n - k + 2. By (6.2.7) and (6.2.25) we get
]]F'(xo)-i[F'(xk+a)- F'(zo)]]]
<
Wo([IXk+~-Xolf)<_wo(bk+l) <
1. (6.2.37)
203
6.2. C o n v e r g e n c e u n d e r weak c o n t i n u o u s d e r i v a t i v e
It follows from (6.2.37) and the Banach Lemma on invertible operators 1.3.2 that Ft(xk+l) -1 exists and
xoll)] -~ ~ [1 -
wo(bk+l)] - 1 .
- Ft(xk)(Xk+l
-- Xk) -- r(xk)}ll
< [1 - Wo(llXk+l -
IlF'(Xk+l)-lF'(xo)ll
(6.2.38)
By (4.1.3), (6.2.6), and (6.2.23) we obtain IIF'(xo)-lF(xk+l)l]llF'(xo)-l{r(xk+l)
~ fO 1 [[ft(xo)-l[Ft(xk -Jr-O(Xk+l -- Xk)) -- F'(xk)][[ IlXknt-I -- xklldO <
foo1[w([[xk -
<
~001[w(bk
xoll + OIIxk+a -- x k [ I ) - W(IIXk -- Xoll)]dOl[xk+l - xkll
(6.2.39)
+ O(bk+l -- bk)) -- w(bk)]dO(bk+l - bk).
Using (4.1.3), (6.2.38) and (6.2.39) we get
Ilxk+2- Xk+~ll ~ ]lU(Xk+l)-l f'(xo)ll 9 ]lF'(xo)-lF(xk+l)][ < f l [ w ( b k + 0(bk+l - bk)) - w(bk)]dO(bk+l - bk) -
1 - wo(bk+l)
(6.2.40)
= bk+2 - bk+l.
Note also that:
I Xk-t-2--Xo[I "( I[Xk-[-2--Xk+IlI-I-IlXk-I-I--Xo] ~ b k + 2 - b k + l + b k + l - b o - bk+2 <_ b*. That is xk+2 E U ( x o , b*). It follows from (6.2.40) that {xn} (n >_ 0) is a Cauchy sequence in a Banach space X, and as such it converges to some x* E U(x0, b*). By letting k -+ c~ in (6.2.39) we obtain F ( x * ) - O. Moreover, estimate (6.2.33) follows from (6.2.32) by using standard majorization techniques. Furthermore, to show uniqueness, let y* be a solution of equation F ( x ) - 0 in V ( x o , Ro). It follows from (4.1.3)
Ily*-x~+~ll ~
[[F'(xk)-lF'(xo)[[
/o 1 < fo[~(llxk -
IPy* - x~llde
- x011 + 011y* - xkll) - w(llxk - xoll)]dO 1 - wo(llxk
< ~01[w(b * + 0(b* + R 0) ) - ~ ( b * ) ] d 0 1 - wo(b*)
< IlY* - xkll
(by (46)).
lly* - x~]l
- xoll) liy* - x~ II
(6.2.41)
204
6. More Results on Newton's Method
Similarly if y* E U(xo, b*) we arrive at Ily* - ~k+,ll _
f~
+ Oh*)- w(b*)]dOIlY* -
~kll,
1 - wo(b*)
(6.2.43)
(by (44)).
< Ily*-xkl[
(6.2.42)
In both cases above lira xk -- y*. Hence, we conclude k---~oc X :~ ~
y:~.
We can now compare majorizing sequences {tn}, {an}, {bn} under the hypotheses of Theorem 1 in [195, p. 117], and Theorems 6.2.3, 6.2.5 above" L e m m a 6.2.6 Under the hypotheses stated above and if strict inequality holds in (6.2.7) or (6.2.8) then the following estimates
(6.2.44)
bn+l < an+l < tn+l bn+ l - bn < an+ l
-
-
(6.2.45)
an < tn+ l -- tn
(6.2.46)
b* - bn < a* - an < t* - tn and
b* _< a* _< t*
(6.2.47)
hold for all n > O.
Proof. It follows easily by using induction, definitions of sequences {tn}, {an}, {bn} given by (6.2.12), (6.2.11), (6.2.27)respectively and hypotheses (6.2.8), (6.2.9). 9 R e m a r k 6.2.2 The above lemma justifies the advantages of our approach claimed at the beginning of the section.
6.3
Convergence
and
universal
constants
We improve the elegant work by Wang, Li and Lai in [342], where they gave a unified convergence theory for several Newton-type methods by using different Lipschitz conditions and a finer "universal constant". It is convenient to define majorizing function f and majorizing sequence {t~} by f ( t ) - r] - t +
~et 2 + ~ L t 3
(6.3.1)
and
tn+l -- tn
(t,) ff(tn)'
to
-
o
respectively where r/, t~, L are nonnegative constants.
(6.3.2)
205
6.3. Convergence and universal constants
P r o p o s i t i o n 6.3.1 [342, p. 207] Let 2
(6.3.3)
g + v/g 2 + 2L and
b - 2(t~ + 2v/t~2 + 2L). 3(~ + v/~2 + 2L) 2
(6.3.4)
Then, (a) function f is monotonically decreasing in [0, r], while it is increasing monotonically in Jr, +oc). Moreover, if ~l <__b then < O, f ( + ~ ) > r / > 0 .
f (~l) > O, f (r) - r l - b
Therefore f has a unique zero in two intervals denoted by t* and t** respectively, satisfying r
~/< t* < ~7/< r < t**,
(6.3.5)
when rl < b and t* - t**, when rl - b; (b) Iteration {t~} is monotonically increasing and converges to t*.
From now on we set D - U(xo, r). We can define a more precise majorizing sequence {Sn} (n > 0) by L ( S n + l -- 8 n ) -~- L s n -~8n+2 ~ 8n+1 -~- 3 2(1 -- ~ 0 8 n + l )
(Sn+l - sn) 2,
(6.3.6)
80 - - 0 , 81 --?~, for s o m e constant g0 C [0, g].
Iteration {tn} (n >__0) can also be written as L(tn+l
-- t n ) + Ltn +
tO -- 0, tl -- r].
t n + 2 -- t n + l -+- 3 L t n2+ l ] 211 -- ~ t n + l -- -~
(6.3.7)
'
Then under the hypotheses of Proposition 6.3.1, using (6.3.6), (6.3.7) and induction on n we can easily show: P r o p o s i t i o n 6.3.2 Under the hypotheses of Proposition 6.3.1, sequence {sn} is monotonically increasing and converges to some s* E (7, t*]. Moreover the following estimates hold: s~ < t ,
(n > 1)
O
(6.3.8) (n>O)
(6.3.9) (6.3.10)
and
s* < t* provided that L > 0 if ~o - g.
(6.3.11)
206
6. More Results on Newton's Method
Let xo e D such that F'(xo) -1 E L(Y, X ) . We now need to introduce some Lipschitz conditions: I[F'(xo)-l[F'(x) -
F'(xo)]ll _ 61Ix- xol,
IlF'(Xo)-l[F'(y) - F'(x)]ll ~
[e + L I I x -
for all x c U(xo, r)
1
(6.3.12)
1
xoll + -~LllY- xll IlY- xll, (6.3.13)
for all x, y" [lY- xll + ttx - XoI[ <_ r,
-'(.o)1 .. _< (e + LII -
IlF'(xo)-l[F'(x)
(6.3.14)
for all x E U(xo, r), IlF'(Xo)-l[F'(x) - F'(xo)]ll <_ g l l x - xo]l, for all x C U(xo, r)
(6.3.15)
IlF'(xo)-lF"(xo)ll < g
(6.3.16)
[[F'(xo)-lF"(Xo)[[ < go
(6.3.17)
IIF'(Xo)-l[F"(v)--F"(m)]ll
<_ Llly-xlI, for all Y,m" l v-xll+llm-xoll <_ r, (6.3.18)
[[F'(xo)-l[F"(x) - F"(xo)]ll <_ LIIx- xoll, for all x E U(xo, r)
(6.3.19)
and [l[i"(xo)-l[F"(x)
- F"(xo)]ll ~ L o l l x - Xoll, for 311 x c g(xo, r).
(6.3.20)
Note that we will use (6.3.12), (6.3.17), (6.3.20)instead of (6.3.15), (6.3.16) and (6.3.19) respectively in our proofs. We define as in [342]
K(1)(xo, e) - {F K(1) c~nt(Xo, e) - {F K (1) (~o, e, L) - {F K(1) cent(xo,f,L)- {F
~ Cl(xo,r) 9 (6.3.13) hdds for L - 0}, ~ C 1(xo, r)" (6.3.15) holds}, e C 1(~o, ~)" (6.3.13) holds}, c c l ( x o , r ) 9 (6.3.14) holds},
K(2)(xo, e, L ) - { r C C2(xo,r)" (6.3.16) and (6.3.18) hold}, K(u) cent(Xo, e, L) - { F C C2(xo, r)" (6.3.16) and (6.3.20) hold}. Moreover for our study we need to define:
K(1)(Xo,~o,~,L) -- {F E C l(x0,p)" (6.3.12) and (6.3.13) hold}, K(1) cent(Xo, g.o) - { F E C 1(xo, r)" (6.3.12) holds},
K(U)(mo,6,g,L)-
{F e C2(xo,r) 9 (6.3.12), (6.3.17) and (6.3.18) hold or (6.3.12), (6.3.16)and (6.3.18)hdd}.
We can provide the main semilocal convergence result for method
(4.1.3).
207
6.3. Convergence and universal constants
T h e o r e m 6.3.3 Suppose that F C K(1)(xo,go, g,L). If ~ <_ b, then s e q u e n c e { X n } generated by method (~.1.3) is well defined, remains in U(xo, s*) for all n >_ O, and converges to a solution x* c U(xo, r) of equation F(x) - O. Proof. Simply use (6.3.12)instead of (6.3.15) in Lemma 2.3 in [342] to obtain F'(x) -1 exists for all x E U(x0, r) and satisfies 1
IlF'(x)-lF'(Xo) I1 < -
(6.3.21)
eollx- x0fl
The rest follows exactly as in Theorem 3.1 in [342] but by replacing {tn}, {t*}, (6.3.11) by {Sn}, {S*}, (6.3.21)respectively. tt
1
T h e o r e m 6.3.4 Suppose that F E Kcent(Xo,~o ). If ~l < b, then the equation * F(x) - 0 has a unique solution x* in U(xo So), where s o* -- 8" if ~ -- b, and s~) E [s*,t*) if ~] < b. Proof. As in Theorem 3.2 in [342] but used (6.3.12)instead of (6.3.14). tt By Proposition 6.3.2 we see that our results in Theorems 6.3.3 and 6.3.4 improve the corresponding ones in [342] as claimed in the introduction in case g0 < g. At this point we wonder if hypothesis ~/< b can be weakened by finding a larger "universal constant" b. It turns out that this is possible. We first need the following lemma on majorizing sequences: L e m m a 6.3.5 Assume there exist nonnegative constants go, ~, L, ~l and a parameter 5 c [0, 2) such that for all n > 0
L ( 25) 2nT]_~_2L[12_5-(~)n]()n 2(~ ?]2-Jv~(~)n ?7 + 2-5
1-
(6.3.22)
r/<5.
Then, iteration { S n } given by (6.3.7) is monotonically increasing, bounded above by s** - 22'7 - 5 and converges to some s* such that -
-
0 <__ s* <__ s**.
(6.3.23)
Moreover the following estimates hold for all n > O"
n+l 0 < Sn+ 2 -- S n + l <_ -~(8nA-1 -- 8n) <~
r/.
(6.3.24)
Proof. The proof follows along the lines of Lemma 5.1.1. However there are some differences. We must show using induction on the integer k > 0 L -- (sk+l -- sk) 2 + Lsk(sk+l -- sk) + ~(8k+1 -- 8k) -[- ~0(~8k+1 < (~, 3
(6.3.25)
208
6. More Results on Newton's Method 0 ~ 8k+l
-
-
(6.3.26)
8k~
and (6.3.27)
0 < 1 - ~osk+l.
Estimates (6.3.25)-(6.3.27) hold for k - 0 by (6.3.22). But then (6.3.6) gives (6.3.28)
0 ~ 82 -- 81 ~ ~(81 -- S0).
Let us assume (6.3.24)-(6.3.27) hold for all k <_ n. We can have in turn" L
"~'(8k+1 -- 8k) 2 nt- L S k ( S k + l -- 8k) nt- ~(8k+l -- 8k) + ~0(~8k+1
L
6
+L.
_
1--(~)
+ t~0(~
1
k
1-( 5 -
(6.3.29)
~
~ + ~
k+l
52
~ < ~ by (6.3.22) -
We must also show (6.3.30)
Sk ~ 8"*.
For k - 0, 1, 2 we have s o - 0 < s_* *
,
81
-
?]<8
5 **, a n d s 2 - ~ + ~ < s * *
_
.
Assume (6.3.30) holds for all k _< n. It follows from the induction hypothesis (6.3.31) 5 k+l
1-(7) 1 5 2
2~/ r/<- 2 - ~ -s**
Moreover (6.3.27) holds from (6.3.22) and (6.3.31). Inequality (6.3.26) follows from (6.3.6), (6.3.27) and (6.3.25). Hence, sequence {Sn} is monotonically increasing, bounded above by s**, and as such it converges to some s* satisfying (6.3.23). w R e m a r k 6.3.1 Condition (6.3.22) can be replaced by a stronger but easier to verify condition
___ bo,
(6.3,32)
6.3. Convergence and universal constants
209
where,
2
bo =
+ L
(6.3.33)
+
( 5 - v/5)
g /3- ~ +
(6.3.34)
2go ,,:. 3 v'b
(6.3.35)
1.
(6.3.36)
for - ~-
Indeed (6. 3. 22) certainly holds if _
3 2
~/2
r/2
+
2"yo~ ~ 7 + 2 - 5 ~/-<5
+
(6.3.37)
since,
[1--(~)n+l] (~)
n+l < [1--(~)n I (~)n
(6.3.38)
holds for all n > 0 if 5 e [0, v/5- 1]. However (6.3.37) holds by (6.3.32).
Hence, we showed" Lemma 6.3.6 /f (6.3.22) replaces condition (6.3.22) in Lemma 6.3.5 then the conclusions for sequence { Sn } hold.
Remark 6.3.2 In Lemma 6.3.5 we wanted to leave (3.4) as uncluttered as possible, since it can be shown to hold in some interesting cases. For example consider Newton's method, i.e. set L - O. See Application 5.1.~ in Section 5.1. Note also that bo can be larger than b, which means that (3.1~) may hold but not n < b. Indeed, here are two such cases: choose (a) L - 1, "~o = " y - .5 and r l - .59 m
then b - . 5 8 3 (b) L -
and
bo-.590001591
.8, " y - .4 and ~l = .7
then b - . 6 8 2 7 6 2 4 2 3
and
bo-.724927596.
Below is the main semilocal convergence theorem for Newton's method involving Fr~chet differentiable operators and using center-Lipschitz conditions.
6. More Results on Newton's Method
210
T h e o r e m 6.3.7 Assume: (a) F C K (1) (xo, Co, e, L), and (b) hypotheses of Lemmas 6.3.5 or 6.3.6 hold. Then, sequence {xn} (n >_ 0) generated by method (~.1.3) is well defined, remains in U(xo, s*) for all n > O, and converges to a solution x* e U(xo, s*) of equation F(x) - O. Moreover the following estimates hold for all n>O: m
1
L[611X~+l - Xnll + ~IlXn -- Xoil] + Z 2 ]Xn+l -- Xn 12 IlXn+U - X~+lll __ 1 - "ToIIXn+I--XoII
(6.3.39) < L[l(sn-t-1 - 8n)-~- ~(8n -1
-
~0(Sn_t_l
-
8 0 ) ] -~- 2 ( 8 n + 1
_
8n)2
80)
--- 8n+2 -- 8n+l~
and IlXn -- X*
(6.3.40)
II ~ S* -- ~ ,
~h~, ~q~~
{~} (~ >_ o) i~ 9~wn by (6.3.6).
The solution x* is unique in
U(xo, s* ) if ?s* < 1.
(6.3.41)
Furthermore, if there exists R > s* such that U(xo, R) c_ D, and (6.3.42)
?o(s* + R) _< 2,
the solution x* is unique in U(xo, R). P r o o f . Let us prove that IIx~+~ - xk II -- Sk+l
--
(6.3.43)
8k
and U(Xk+I
~ S :r
-- 8 k + l )
C_
(6.3.44)
U(xk 8 -- 8k) ~
hold for all k > 0. For every z E U(Xl II Z -- X011 _~ II z -- X l l l - t - I l X l
~ S :r
--
Sl)
-- X011 --~ 8" -- 81 -Jr- 81 - - S* -- S0
implies z E U i ( X l , 8 * - Sl). Since also
[[/1-/oil- [IF'(xo)-lF(xo)[ <_ ~ - so, (6.3.43) and (6.3.44) hold for k - 0. Given they hold for n - 0, 1 , . . . , k, then k+l
I l X k + i - XOII ~_ E i--1
k+l
Ilxi --Xi-lll ~_ E ( S i i--1
8i-1)-
8 k + l - S O - 8k+1
211
6.3. Convergence and universal constants
and IlXk -4- O(Xk4-1 -- X k ) -- XOII ~ 8k Jr- 0(8k+1 -- 8k) < 8",
0 E
[0, 1].
Using (4.1.3) we obtain the approximation F(Xk+l)
-- F ( X k + l )
/o
--
- f(Xk)
- ft(Xk)(Xk+l
(6.3.45)
-- X k )
[ f z ( X k -4- O ( X k + l -- X k ) ) -- F t ( X k ) ] ( X k - t - 1
-- X k ) d O .
Then, we get by (6.3.12), (6.3.13) and (6.3.45) I[ft(Xo)-l
< L
f(xk+l
{1o L
(6.3.46)
)ll
[llxk -- XoII + OIIXk+l -- Zkll](1 -- O)dO + L
--< --I[Xk+16 -- xkll3 + ~llxk - Xo[[ [[Xk+l - - X k . . 2- I-I~ <
{
L
[~
1
(8k+1 -- 8k) -Jr- ~ ( S k -- 80)
]
-~}
+ ~
IlXk+l -- xkll 2
_-~l]Xk+l ")/
-- Xk[[
2
2
(8k+1 -- 8k) 9
Using (6.3.12) we get IIF'(xo)-l[F'(xk+~) - r'(xo)]ll ~ ~ollmk+l - xoll
(6.3.47)
_< 7osk+l <_ 7s* < 1 (by (6.3.22)). It follows from the Banach Lemma on invertible operators 1.3.2 and (6.3.22) that F1(Xn+l) -1 exists so that" IfF'(xk+~)-~F'(xo)ll < [1 - ~ollx~+~ - xoll] -1 ~ (1 - ~/otk+l) -1.
(6.3.48)
Therefore by (6.3.2), (6.3.46), and (6.3.48) we obtain in turn Ilxk+2 - xk+lll = I I ( F ' ( x k + I ) - I F ' ( x o ) ) ( F ' ( x o ) - I F ( x k + I ) ) I I < I l r ' ( x k + ~ ) - l r ' ( x o ) ] ] . [[FZ(xo)-IF(xk+I)][
(6.3.49)
~_ 8k+2 -- 8k+1.
Thus for every z E U(xk+2
S ~
sk+2) we have
IIz -- Xk4-111 ~ IIZ -- Xk+211-Jr- IlXk+2 -- Xk+l]] _~ 8* -- 8k+2 -- 8k§
(6.3.50)
~ 8k-t-1 ~ 8" -- 8 k § 1.
That is m
Z C V(Xk§
, 8* -- 8k+1).
(6.3.51)
212
6. More Results on Newton's Method
Estimates (6.3.49), (6.3.50) imply that (6.3.43), (6.3.44) hold for n - k + 1. The proof of (6.3.43) and (6.3.44) is now complete by induction. Lemma 6.3.5 implies {tn} (n > 0)is a Cauchy sequence. From (6.3.43), (6.3.44) {Xn} (n > 0) becomes a Cauchy sequence too, and as such it converges to some U(xo, s) such that (6.3.40) holds. The combination of (6.3.40) and (6.3.46) yields F ( x * ) - O. Finally to show uniqueness let y* be a solution of equation F ( x ) - 0 in U(xo, R). It follows from (6.3.12), the estimate
Ft(Xo) -1
lEt(y* + O(X* -- y * ) ) - Ft(Xo) ] dO
fo
_< ~o
/o 1 ly* + O(x*
- y*) - xolldO
70f01 [01Ix* - ~oll + (1 - 0)lly* - xoll]d0 < -~0-(s* + R)_< 1
z
(by (6.3.42)),
and the Banach Lemma on invertible operators that linear operator
L- o 1r ' ( y *
+ O(x* - y*))dO
is invertible. Using the identity 0 - F ( x * ) - F(y*) - L(x* - y*),
we deduce x* - y*. Similarly we show uniqueness in U(xo, s*) using (6.3.41)... As in [342] we also study the semilocal convergence of King's iteration [208]
Xn+l -- Xn -- F'(xm[n/m])-lF(xn)
(rt ~ O) (xo E D)
(6.3.52)
where [~] denotes the integer part of ~ , and Werner's method [345]
Xn+l - - z n - F ' ( y n ) - l F ( z n )
(6.3.53)
Yn+l -- Zn+l -- 12 f ' ( y n ) - l f ( X n + l )
(6.3.54)
Yo - xo.
(6.3.55)
Both methods improve the computational efficiency with (6.3.53)-(6.3.55) being more preferable because it raises the convergence order to 1 + x/~, although the number of function evaluations doubles when compared with Newton's method. Using (6.3.12) instead of (6.3.14) or (6.3.19) to compute upper error bounds on I [ F ' ( x ) - l F ' ( x o ) l l (see e.g. (6.3.48)) we can show as in Theorems 4.1 and 4.2 in [342] respectively:
6.~. A Deformed Method
213
T h e o r e m 6.3.8 Let F E K(1)(xo,~/o,~/,L) and assume hypotheses of Lemmas 6.3.5 or 6.3.6 hold. Then sequence {x~} generated by King's method (6.3.52) converges to a unique solution x* of equation F ( x ) - 0 in U(x0, s*). T h e o r e m 6.3.9 Let F E K (2) (xo, ~o, ~, L) and assume hypotheses of Lemmas 6.3.5 or 6.3.6 hold. Then sequence {x~} generated by Werner's method (6.3.53)-(6.3.55) converges to a unique solution x* of equation F ( x ) - 0 in U(x0, s*). Note that finer error bounds and a more precise information on the location of the solution x* are obtained again in the cases of Theorems 6.3.8 and 6.3.9 since more precise majorizing sequences are used.
6.4
A Deformed Method
We assume that F is a twice continuously F%chet-differentiable operator in this section. We use the following modification of King's iteration [208] proposed by Werner [345]" --1
Xn+l - - x n - F I
( xnq-Zn)2
Zn+l ~ Xn+l -- F'
(
Xn +2 Zn
(x0 E D) ( z 0 - x0) (n > 0)
F(xn),
)_1
(n > 0).
F(Zn+l)
(6.4.1)
(6.4.2)
Although the number of function evaluations increases by one when compared to Newton's method, the order of convergence is raised from 2 to 1 + v ~ [345]. Here in particular using more precise majorizing sequences than before we provide finer error bounds on the distances IIX~+l- x~ll , IIx~- x*ll and a more precise information on the location of the solution x*. Finally, we suggest an approach for even weakening the sufficient convergent condition (6.4.8) for this method along the lines of the works in Section 5.1. Let fl > 0, "y > 07 and 3'0 > 0 be given parameters. It is convenient to define real function f by f (t ) - fl - t + 1 -7------~' t e
[1]
0, -~ '
(6.4.a)
and scalar sequences {tn}, {tn) (n >_ 0) by tn+l - tn - f ' ( S n ) - l f ( t n )
Pn+l -- tn+l - f ' ( s n ) - l f ( t n + l ) -
tn+2 -- ~n+l --
f n + l
70~n -- 1
(6.4.4)
to -- ro -- O,
-
Sn --
to -- TO -- O,
tn -]- rn
-
tl = fl,
(6.4.5) (6.4.6)
214
6. More Results on Newton's Method
Yn+l
-
-
9n+l
tn+l
--
_8 n
7o~-
--
tn + Yn
1
2
1 --
Y,)2(1 - O)
(6.4.7)
where, jfO1
~/~n -- 7 ( t n + l
fo 1 fo 1
1 +
27(tn+
[1 --
gn+l --
O)
672(yn --tn)3Op(1-
[1 --
4
dO
-- ~n)O] 3
")/tn -- ")/(~n -- t n ) ( ~
o
+ V(1
dpdO ,
and fn+l -- ( f t ( ~ n + l )
-- ft(-gn))(Yn+l -- t n + l ) .
We need the following lemma on majorizing sequences: L e m m a 6.4.1 Assume: ~ - 9.r <_ a -
(6.4.8)
2 v~,
and
(6.4.9)
70 - 27.
Then iterations {tn}, {tn} are well defined and the following estimates hold: O--to--ro
tn
<
<
rn
<
tn+l
<
rn+l
<
O - - t o - - Y O < tn < Yn < tn+l < Yn+l <
t*
(~_> o),
(6.4.10)
(n _> o),
(6.4.11)
L < tn
(n >_2)
(6.4.12)
~n < r ~
(n___2)
(6.4.13)
t n + l - t n < tn+l--tn
Yn+l
-- t n + l
<
rn+l - tn+l
t*-tn <_t*-tn
(6.4.14)
(n >_ 1),
(6.4.15)
(n >_ 1),
(6.4.16)
(n>_O),
(6.4.17)
~*
F
-
(6.4.18)
lim t~, n--+ oo
and
t*-
lim t n = n ----~(ND
1 + c~ - V/(1 + c~)2 - 8c~
47
_< 1+
1
7
(6.4.19)
215
6.4. A Deformed Method
Proof. Estimates (6.4.10) and (6.4.19) follow using only (6.4.8) (see Lemma 2 in [177, p. 69]. The rest of the proof follows immediately by using induction on n and noticing that
1
1
27t- 1 <
[ (
for all t e 01 1 -
ft(t )
.~)1]
.
(6.4.20)
Indeed (6.4.20) reduces to showing" O
3 27
which is true by the choice of t. 9 R e m a r k 6.4.1 It follows from Lemma 6.J.1 that if {tn} is shown to be a majorizing sequence for {x~} instead of {t~} (shown to be so see Theorem 1 [179]) then we can obtain finer estimates on the distances IlZn -- ~ n l ,
IIx~+~ -- Z~ l, IIx~ -- X*II,
~n -- X*II
and an at least as precise information on the location of the solution x*. We also note that as it can be seen in the main semilocal convergence theorem that follows all that is achieved under the same hypotheses/computational cost:
T h e o r e m 6.4.2 Let F" D c_ X ~ Y be a thrice continuously Frdchet differentiable operator. Assume there exist xo c D, b _> 0, 7 > 0, 70 > 0 such that Ft(xo) -1 E L(Y,X),
If'(xo)-lF(xo)l
~ 3,
(6.4.21)
1Ft(xo)-lFU (xo) <_~, -~
(6.4.22)
IIF'(xo)-I[U(x) - F'(xo)] ] ___ ~ollx - ~oll
1 F' (xo) - 1Fut
g
72
(x)
__ (1 -
~/llx
-
xoll) 4
(6.4.23) (6.4.24)
for all
_( ( _( (1)1) x~U
U xo,
xo,
1-~
(6.4.25)
I-~
CD
(6.4.26)
and hypotheses of Lemma 6.~.1 hold. Then iterations {Xn}, {zn} (n > 0) generated by method (6.~.2) are well defined, remain in U(xo, t*) for all n > O, where t* is
216
6. More Results on Newton's Method
given by (6.4.19) and converge to a unique solution x* of equation F(x) - 0 in m
1
[lZn -- Xn[I ~_ ~ n - - t n I I X n + l -- Zn[I ~ Ilxn -- x*
[1 <
(6.4.27)
< r n -- t n
tn+l
< t* - tn
(6.4.28) (6.4.29)
~_ t* -- r n.
(6.4.30)
-- ~ n < t n + l
F - ~
-- r n
and [[Zn --
X'It ~_ -{* ---~n
Proof. We only need to show that the left-hand side inequalities in (6.4.27) and (6.4.28) hold. The rest follow from Lemma 2.1, standard majorization techniques and Theorem 1 in [179]. Let x c U(xo,t*). It follows from (6.4.9), (6.4.19) and (6.4.23) that [[Ft(xo)-l[F'(x)-
Ft(xo)][[
~
2"/ilx- xo[f ~ 2~t* < 1.
(6.4.31)
Using the Banach Lemma on invertible operators [6] and (6.4.31) we conclude that F'(x) -1 exists and 1
[[F'(x)-lF'(xo)I[ < -
1 -
2~lix-
(6.4.32)
~01f
By (6.4.2), (6.4.3)-(6.4.7), (6.4.20)above and (21)-(23)in Theorein 1 in [179, p. 70] we arrive at IlZn+I -- Xn+l ]t <
gn+l --
--
2~-$n-
1
f(tn+l) : m-t-1 -- tn-k-1 ~ - - ~
:
rn+I
-- tn-}-I
ft(Sn)
(with g~, t~ replacing rn, tn respectively in the estimate above (23) in [179]). Note also that (6.4.27) holds for n - 0, and by the induction hypothesis
ll~+l--~01[
<_
<_
1--
--. -y
Therefore the left-hand side of (6.4.27) holds for n + 1. We also have [[Zn+i -- Xol[ ~_ r n + l
~_ t* ~ t* ~_
( 1)1 1- -~
Hence, the same is true for Yn+I. Moreover by (24), (25) in [179], and (6.4.20) and (6.4.32) above we have Ilx~+~ - z~+~I! < --
1 2~8n-
(h'(~+~)-
ht(~n))(~n+l - t n + l )
1
= t n + 2 -- ~ n + l .
Hence, the left-hand side of (6.4.28) holds for n + 1. The rest of the proof follows exactly as in Lemma 6.4.1 with g~, tn, F replacing r~, t~ and t* respectively. 9
6.5. The Inverse Function Theorem
217
R e m a r k 6.4.2 (a) Let us denote by {~n} (n > 0) scalar sequence generated as (6.~.6), (6.~.7) by with tn, an, Pn, 9n+1 replaced bytn, an, rn, f(tn+l), respectively. It follows from the proof of Theorem 6.~.2 that {/~} is also a majorizing sequence for {x~} so that
0 - to - f o tn < tn,
< t , < ~ < tn+l ( fn+l ( t rn < r~
and
(n 2 0)
t < t*
where, t
-
lim
t~.
n - - + (N2
(b) It is expected that since sequence {tn} is finer than {tn} condition (6.~.8) can be dropped as a sufficient convergence condition for {-tn}, and be replaced by a weaker one. We do not pursue this idea here. However we suggest that such a condition can be found if we work along the lines of the more general majorizing sequences studied in Section 5.1.
6.5
T h e Inverse F u n c t i o n T h e o r e m
We provide a semilocal convergence analysis for Newton's method in a Banach space setting using the inverse function theorem. Using a combination of a Lipschitz as well as a center Lipschitz type condition we provide: weaker sufficient convergence conditions; finer error bounds on the distances involved, and a more precise information on the location of the solution than before [340]. Moreover, our results are obtained under the same or less computational cost, as already indicated in Section 5.1. Let x0 9 D such that U(xo, r) c_ D for some r > 0, and F'(xo) -1 9 L(Y, X ) . The inverse function theorem under the above stated conditions guarantees the existence of c > 0, operator F ' ( x o ) - l : U(F(xo),C) C_ Y --, X such that F'(xo) -1 is differentiable, (6.5.1)
F'(xo)-I (F(xo))=xo, and
F(F'(xo)-l(z)) = z
(6.5.2)
for all z 9 U(F(xo),C).
In particular in [341] the center Lipschitz condition
IlF'(xo)-~[F'(x)- F'(xo)]ll ~
~os(x)L(u)du
for all x 9 U(xo,r)
(6.5.3)
6. More Results on Newton's Method
218
was used, where s(x) -I1~-~0ll, and L is a positive integrable function on (0, r] to first study the semilocal convergence of the modified Newton method. Then using the same function L and the Lipschitz condition s(x,y)
[[F'(xo)-I[F'(x) - F'(y)]I] <
a~(z)
(6.5.4)
L(u)du
m
for all x E U(xo,r), y E U ( x , r - s(x)), where s ( x , y ) - s ( x ) + I Y xl] < r, the semilocal convergence of method (6.5.2) was studied in a unified way with the corresponding modified Newton's method. We assume instead of (6.5.3)"
IlF'(xo)-l[U(x)-
U(xo)]ll_ ~os(x) Lo(u)du
for all x E U(xo,r).
(6.5.5)
By Banach's Lemma on invertible operators 1.3.2 and
IIF'(xo)-I[U(x)- f ' ( x o ) ] l l _
f
~(x)
Lo(u)du < 1,
(6.5.6)
dO
for all r ~ E (0, r], x E U(xo, r~
[/o
IIF'(x)-lF'(xo)ll <_ 1 -
F'(x) -1 exists, and -1
Lo(u)du
]
(6.5.7)
with r ~ satisfying
fo '~ Lo(u)du - 1.
(6.5.8)
By simply using (6.5.5) instead of (6.5.3) we can show the following improvements of the corresponding results in [340]" T h e o r e m 6.5.1 Set bo - fo ~ Lo(u). Assume condition (6.5.5) holds for all r > r ~
Then the following hold:
U F(xo),
bo I) C r(U(~o, ~o)). iIF,(Xo)_~l
Ft(Xo) -1 exists and is differentiable on
u
( f(xo), iiF,(xo)_ll bo I) ,
and the radius of the ball is the best possible (under condition (6.5.5)).
(6.5.9)
219
6.5. The Inverse Function Theorem
L e m m a 6.5.2 Let t
fo(t) - / 3 -
t+
j~
L o ( u ) ( t - u)du
t e [o, R],
(~.5.1o)
where Ro satisfies
l foR~Lo(u)(Ro -
Ro
(6.5.11)
u)du - 1.
If ~ e (0, bo), then fo is monotonically decreasing on [0, r~ monotonically on [r~ R] so that fo(/3) > O, fo(r ~ - / 3 -
while it is increasing
bo < O, fo(Ro) - / 3 > O.
(6.5.12)
Moreover, fo has a unique root in each interval, denoted by r ~ and r ~ satisfying:
ro
(6.5.13)
Z < ~1~ < Voz < ~o < ~o < Ro.
If (6.5.5) is extended to the boundary, i.e. /i s(x) llF'(xo)-~[F'(x) - F'(xo)]ll _<
no(u)du
x C U(xo, r),
(6.5.14)
according to Lemma 6.5.2, Theorem 6.5.1 implies a more precise result" Proposition 6.5.3 If ~o
0 < 3 < bo -
f
(6.5.15)
Lo(u)udu,
JO
and
> ~o
(6.5.16)
then
-v( F(xo), iiF,(xo)_ Z ~11) c_F(~(xo,~~ F'(xo) -1 exists and is differentiable on-U(F(Xo), iir,(x~o)_l ); [F'(xo)-l]t(y)
-
f ' ( x ) -1,
y -
f(x)
satisfies (6.5.7), and radius r~ is as small as possible.
(6.5.17)
220
6. More Results on Newton's Method
P r o p o s i t i o n 6.5.4 Assume"
~ [~o, ~o], /3 C (0, bo) and condition
U
(6.5.14)
hold. Then
n r(U(xo, ~) \ ~(xo, ~o)) _ o
F(xo), ilF,(xo)_l]]
(6.5.18)
In particular if y - 0 and/3 - I I F ' ( x o ) - ~ r ( x o ) l l , we get" Theorem
6.5.5 Assume:
8 <- bo, ~
[~o, ~o],
if 9 < bo o~ ~ - ~1~ i f 9 -
bo.
Then, under condition (6.5.1~), then equation F ( x ) - 0 has a unique solution
x* c U(xo - F ' ( x o ) - l F ( x o ) , r ~ - / 3 ) C_ U(xo, rl~ in the ball U(xo, r).
Remark
6.5.1 Denote by ro, b, f , R, rl, r2 (the quantities used in [3~0]) corresponding to r ~ bo, fo, Ro, r ~ and r ~ Then in case strict inequality holds in (6.1.1~) we have ro
b
~o < ~,
~nd
fo(u)
u e [0, r], R < R o ,
(6.5.19)
~ < ~o,
(6.5.20)
which improve the corresponding results in [340], and under less computational cost since in practice the computation of L requires the computation of Lo. In case Lo, L are constants we get
1 ro-~,
1 ro~
fo(u) - / 3 - u + rl =
1 b-~-~,
Lou 2,
2~ 1 + V/1 -- 2L/3' 2/3
rlo = 1 + v/1 - 2Lo/~ '
1 2 bo---,2Lo R - ~ ,
Ro-
e f ( u ) - / 3 - u + -~Lu 2, r2 -and
2~
(6.5.21) (6.5.22)
,
(6.5.23)
2/3 1 - v/1 - 2Lo/3 "
(6.5.24)
1 - v/1 - 2L/3 rO= ~
2 Lo
Let us give a numerical example where strict inequality holds in (6.1.14) and therefore favorable estimates (6.5.19) hold"
221
6.5. The Inverse Function Theorem
E x a m p l e 6.5.1 Using Example 5.1.3 we again conclude that (5.1.63) (which is the condition used in (~.2.51) in this case) does not hold However our corresponding condition reduces to hA - 2Lo/3 < 1
(6.5.25)
h A - - -~2(1 -- x)(3 -- x) _< 1,
(6.5.26)
or
which
[0, 4-2v/-i~] C 0.
holds on 0o-
In order for us to study the convergence of method (4.1.3) we define scalar iterations {tn }7 {s~ } by f(t~) t n + l -- t n
f'(tn)
to -- 0
'
(6.5.27)
and 8n+2 -- 8n+l -
f 8 n+l n
L(u)(Sn+l - u)du f;(Sn+l)
,
SO - - O ,
S l - - /3.
(6.5.28)
We can prove the main semilocal convergence theorem for method (4.1.3)" T h e o r e m 6.5.6 Assume: conditions (6.5.~), (6.5.5); /3 <<_b; and r>rl hold, where b, r l a r e defined in Remark 6.5.1. Then, sequence {xn} (n > 0) generated by method (~.1.3) is well defined, remains in U(Xl, r l - / 3 ) for all n > O, and converges to a solution x* e g ( x l , rl -/3) of equation F(x) - O. Moreover the following estimates hold for all n > O" [[Xn+l -- Xn[] ~_ 8 n + l -- 8n ~ t n + l -- t n ]]Xn -- X*][ <__ r l -- 8n <_ r l -- t n .
(6.5.29) (6.5.30)
Furthermore if strict inequality holds in (6.1.1~) then 8n+1 -- 8n ~ t n + l -- tn~
(6.5.31)
8n < t n
(6.5.32)
and
hold for all n > 1.
222
6. M o r e R e s u l t s on N e w t o n ' s M e t h o d
Proof. It can easily be verified by using induction that (6.5.31), (6.5.32) and the right hand sides of double inequalities (6.5.29), (6.5.30) hold. Let us prove" IXk+l --
xkll <~ S k + l -- Sk,
(6.5.33)
r l -- S k + l ) C_ U ( X k , r l -- 8k),
(6.5.34)
and U(Xk+l,
hold for all k _> 0. For every z E U ( X l , rl - Sl), IIz - xoll _ IIz -
Xl II-Jr-II
implies z E U ( x o , rl - S o ) .
xI -- xoll _ r~
-
81 +
81
-
rl
-
8o
Since
IlXl - x o l - I l f ' ( x o ) - l f ( x o ) l l < ~ - s~,
(6.5.33)
and
(6.5.34)
hold for k - 0. Given they hold for n - 0, 1 , . . . , k, then k+l
k+l
I[Xk+l -- X01[ --< Z Ilxi i=1
IIx~ Jr O ( X k + l - - X k ) -
Xi-lll
<_ ~ ( ~ i i=1
- si-1)
- sk-t-1 - So -- 8 k + l ,
<__ rl
Xoll <-- 8k -Jr-O(8k+l - - S k )
0 e [0,11.
Using (6.5.2) we obtain the approximation F(Xk+e)
-- f ( X k + l ) F(xk) - Ft(Xk)(Xk+I -- X k ) 1 -[Ft(Xk + 0(Xk+l -- X k ) ) -- F t ( X k ) ] ( X k + I
~0
-- X k ) d O .
(6.5.35)
By (6.5.4), (6.5.27), (6.5.28) and the induction hypotheses we obtain in turn: ]lF'(xo)-~F(xk+l)][ < ~Ifs(Xk,xk+O(xk+I--Xk)) ~_
L(u)du[lXk+l
- Xk dO
Js(xk) o X~+l-X~ J L ( l l x k
1 <- IlXk+~_xkll ~ 1
~
x011 + u)(I Xk+l -- x k l l - u ) d u
llxk+~--xkl
L(llxk - Xoll + u)(llxk+l - x k l l - u)dullxk+l - xkll 2 fsk+~--sk
--< (8k+1 -- 8k) 2 dO .
~Sk-t-1
. L(~)(~k+l . . J sk tk+i
<_
~)d~
L(sk + u)(sk+l
(
Ilxk+~-~ll S k + l -- 8k
L(u)(tk+l - u)du - f(tk+l), Jtk
-
-
If
sk -- u)dul
~ <_
Xk+l
~+1 L ( u ) ( s k + l
-- X k l l 2
-- u ) d u
J Sk
(6.5.36)
223
6.5. The Inverse Function Theorem
where in (6.5.36) we used the fact that function g(t) - -~
for t E [0, r and
L(s + u ) ( t - u)du is nondecreasing
s] [340, p. 175]. By (6.5.7) for x - xk+l we get F'(Xk+l) -1 exists
E/oXk+lX~ ]
-1
]]F'(Xk+l)-~F'(x0)l] ~
1-
__< ( 1 - ~
Lo(u)du
sk+l Lo(~.~,d~_~)
--1
(6.5.a7)
--~(8k+1) __~ - f ' ( t k + l ) .
Using (6.5.2), (6.5.27), (6.5.2S), (6.5.36) and
[[xk§
(6.5.a7)w
get
Xk§ _< [[f'(xk+l)-*f'(xo)[[. <_ s k + 2 -- S k + l <_ t k + 2 -- t k + l .
Thus for every z C U(xk+2, rl - sk+2), we have
That is, Z
E U(xk+l,
rl-
8k+1).
(6.5.a9)
Estimates (6.5.38) and (6.5.39)imply (6.5.33) and (6.5.34) hold for n - k + 1. By induction the proof of (6.5.a3) (6.5.34) is completed. Estimate (6.5.30) follows from (6.5.29) by using standard majorization techniques. It also follows that {s~} (and {tn})is a Cauchy sequence. From (6.5.3a) and (6.5.34) {xn} becomes a Cauchy sequence too, and as such it converges to some x* E U(Xl, rl - / 3 ) (since U(Xl, r, - / 3 ) is a closed set). By (6.5.5), lIF'(xn)ll is uniformly bounded. Hence, by F(xn)
+ F'(Xn)(Xn+l
we get lim X n - X * .
- X n ) - - O,
9
n----+ ( x )
R e m a r k 6.5.2 If equality holds in (6.1.1~) then sn = tn (n > O) and Theorem 6.5.6 reduces to Theorem 3.1 in [3~0]. Otherwise our error bounds on the distances IIXn+l -- Xnll, llXn -- X'It are finer (see (6.5.29)-(6.5.32)) for all n > 1 than the ones in [3~0] using majorizing sequence {tn} instead of the more precise {sn}. We also note that our results are obtained under the same computational cost with the ones in [3~0] since in practice finding L requires finding Lo. The rest of the error bounds in [3~0] can also be improved by simply using Lo in (6.5.7) instead of L see Section 5.1. Instead we wonder if the convergence of scalar sequence {sn} can be established under weaker conditions than the ones given in Theorem 6.5.6.
6. More Results on Newton's Method
224
We showed in Section 5.1 that { 8 n } is monotonically convergent to s* if: (A) If Lo, L are constants then sequence {Sn} given by (6.5.28) becomes
L(sn+l -- 8n) 2 s~+2 - Sn+l + 2(1 - LoSn+l) '
80 - O, 81 -/~.
The advantages in this case have been explained in Section 5.1. (B) There exists a minimal positive number /~1 E [#, R0] satisfying equation -
1- s
L0( )d
r = 0.
(6.5.40)
Moreover R1 satisfies
Z IF~ILo(u)du < 1.
(6.5.41)
Indeed if 8 n + 1 ~ 1~1, then f88n n+l
L(u)du
Sn+2 <_ sn+l +
(Sn+l--Sn)
1-- f ? l Lo(u)du <
asnfSn+lL(u)du utl~n--"
-
1-
fo
# + f:n-~lL(u)du sn+l 1 - fo Lo( )d
foR1 L(u)du
I2~1 __ iI~1 9
Hence, {sn} is monotonically increasing (by (6.5.41)) and bounded above by /~1, and as such it converges to some s* E (/3,-R1]. (C) Other sufficient convergence conditions for sequence {s~} can be found in Section 5.1 but for more general functions L0 and L. Following the proof of Theorem 6.5.6 but using hypotheses (A) or (B) or (C) above we immediately obtain: T h e o r e m 6.5.7 Under hypotheses (A) or (B) or (C) above the conclusions of Theorem 6.5.6 for method (#.1.3) hold in the ball U(Xl, s * - # ) and s* < rl. Note that s* replaces rl in the left-hand side of estimate (6.5.30). Next we examine the uniqueness of solution x* in the ball U(xo, r l ) or U(xo, s*). R e m a r k 6.5.3 Assume: rlLo(rl) < 1,
(6.5.42)
225
6.5. The Inverse Function Theorem
~01Lo[Orl§
( 1 - 0)/~][0rl + ( 1 - 0)/~]d0 < 1.
(6.5.43)
Then under the hypotheses of Theorem 6.5.6 the solution x* is unique in U(xo, rl) if (6.5.~2) holds and in U(xo, R) for R >_ rl if (6.5.~3) holds. Indeed, if y* is a solution of equation F(x) - 0 in ~(xo, R), then by (6.5.5) we get for L - f l F'(y* + -
y*))dO
] I F ' ( x o ) - I ( L - F'(xo))tl <_ <_
Lo(u)dudO
~001Lo[O]lx*-
Xo]] + (1 - O)lly* - Xo]]]
9 [OIIx* - Xol] + (1 - O)lly* - Xo]]]dO < 1 by (6.5.~2),
(6.5.44)
if y* c U(xo,rl) or (6.5.~3) if y* E U(xo, R). It follows by the Banach Lemma on invertible operators and (6.5.~) that linear operator L is invertible. Using the identity 0 - F(x*) - F(y*) - L(x* - y*) we obtain in both cases x* - y * , which establishes the uniqueness of the solution x* in the corresponding balls U(xo, rl) and U(xo, R). Replace r~ by s* in (6.5.~2) and (6.5.~3) to obtain uniqueness of x* in U(xo, s*) or U(xo, R) under the hypotheses of Theorem 6.5.7. The corresponding problem of uniqueness of the solution x* was not examined in Theorem 1.5 in [3~0].
R e m a r k 6.5.4 Assume operator F is analytic and such that IJF'(xo)-lF(~)(xo)ll _< n!"fn-1
(n ~ 2).
(6.5.45)
Smale in [312] provided a semilocal convergence analysis for method (~.1.3) using (6.5.~5). Later in [177] Smale's results were improved by introducing a majorizing function of the form f (t) - / J -
t+ ~
7t 2
1 -
7t
t e [0, R],
(6.5.46)
(see also Section 6.~). Define function L by
27
L(u) - (1 - 7 u ) 3"
(6.5.47)
Then conditions (6.5.3) and (6.5.~) are satisfied and function f defined in Remark 6.5.1 coincides with (6.5.~6).
226
6. More Results on Newton's Method
Concrete forms of Theorems 6.5.1 and 6.5.5 were given by Theorems 5.1 and 5.2 respectively in [340]. Here we show how to improve on Theorems 5.1 and 5.2 along the lines of our Theorems 6.5.1 and 6.5.5 above. Indeed simply define functions f0, L0 corresponding to f, L by fo(t) - / 3 -
~/ot2
t+
1 -
t E [0, R]
70t
(6.5.48)
and Lo(u) (1 -
270 7oU) 3"
(6.5.49)
Again for this choice of Lo, function fo given by (6.5.10) coincides with (6.5.48). Moreover condition (6.5.5) is satisfied. Clearly ~o ___~
(6.5.50)
holds in general. If strict inequality holds in (6.5.50) then our results enjoy the ones corresponding to (6.1.14) benefits (see Remark 6.5.1). In particular as (6.5.51) the zeros of fo are 1 + ct -
r l~
470
rO= l + a + v / ( l + a ) 470
,o _
8a
V/(1 + a) 2 -
(
2-8a
1 ) 1 1- G ?-/o' R o -
1 2~o
and b o - ( 3 - 2 v / 2 )
1
~o
Then the concretizations of e.g., Theorems 6.5.1 and 6.5.5 are given respectively by: T h e o r e m 6.5.8 Let 7o > 0 be a given constant. Assume operator F satisfies 1 ___ (1 -~ollx - xoll) ~ - 1
[[F'(xo)-l[F'(x) - F ' ( x o ) ] ] l for all xcU
[ ( 1)1] xo,
1--~
-U.
Then F t ( x o ) -1 E L ( Y , X ) , and is diyerentiable in the ball
1)
7ollF,(xo)-i 1[ 9
(6.5.52)
227
6. 6. Exercises Moreover
Uo C U
(6.5.54)
and the radius in Uo is the best possible.
Theorem
6 . 5 . 9 Let 7o > 0 be a given constant, and
a - ~'~o <_ 3 -
2v/-2.
A s s u m e operator F satisfies 1
I~t(Xo)-l[Ft(x) - zb--'t(X0)]ll ~ (1 - ~ollx - x01 )2 - 1
(6.5.55)
f o r all x E -U(xo5 r), where r ~ < r < r~ if a < 3 - 2v/2, or r - r ~ if a - 3 - 2v/-2. T h e n equation F ( x ) - 0 has a unique solution x* satisfying (6.5.19) in the ball
U(xo,r) The rest of the results in [340] can be improved along the same lines. However we leave the details as exercises to the m o t i v a t e d reader. Iterative m e t h o d s of higher order using hypotheses on the second (or higher) Fr~chet-derivative as well as results involving outer or generalized inverses can be found in the Exercises t h a t follow.
6.6
Exercises
6.1 Let F 9 S c_ X --. Y be a three times FrSchet-differentiable operator defined on an open convex domain S of B a n a c h space X with values in a Banach space Y. Assume F ' ( x 0 ) -1 exists for some x0 C S, IIF'(x0)-lll <__ /35 IlF'(xo)-lF(xo)l[ _< w, IIF"(x)ll ___ M, LIF'"(x)II _< N, IIr'"(x)- F'"(Y)II _< LI x - Y L for ~1] ~, y e S, ~ d U (Xo, r~) c_ S, wh~r~ A-
M~],
B-
N/3~] 2,
ao - l - co 5 b o - - - 52A -5
Cdo-
an an+ l a n + l -- 1 - A a n ( c n + d n ) 5 _
_
-
32 Cn+ l = 2187
L/~r]35 7A( 1 +
A) ,
2A "-5-an+IOn+l,
-
a~+18ABan4-17c
~4(1§ 3
d n + l - 3bn+l (1 + :3b n + l ) c n + l
an+ld45
a
(u >_ o),
1 B - [0, 1 ( P (A) - 1 7C)] &rid and r - limn E i =n o (ci + di) If A E [0, ~] P(A) 1 C E 0 5 17 j5 where P (A) = 27 (A - 1)(2A - 1) (A 2 + A + 2) (A 2 + 2A + 4). ---+ (:X)
"
5
6. More Results on Newton's Method
228
Then show [186]" Chebyshev-Halley method given by yn-xn-F'(xn)-lF(xn) 2(yn-Xn))-Ft H n -- r ' (Xn) -1 I f ' (Xn -5 "5
(Xn)]
3Hr~ [ I - -~ 3Hn] ( Y n - Xn) Xn+ 1 -- Yn - "~ is well defined, remains in U (x0, rr/) and converges to a solution x* E 5 (x0, rr/) of equation F (x) - 0. Moreover, the solution x* is unique in U x0, ~
- r~ .
Furthermore, the following estimates hold for all n _> 0 (X3
[[xn - x* [I -< ~
OO
A (1 -5 A)] ~bl E ~'4i-1 /3 (ci -5 di)r/ _< ~ [1 + -ff
i=n
i=1
b_x
where 7 - bl"
6.2. Consider the scalar equation [185] f (x) -
o.
Using the degree of logarithmic convexity of f
L f (x) -
f (x) f " (x) f' (x) 2
the convex acceleration of Newton's method is given by
f (Xn) f'(Xn)
Xn+l = F ( X n ) - - X n -
1-5
Lf(xn) 2(1-Lf(xn))
1 (n > O) -
2k--1 f(b) < for some xo C R. Let k > 1754877, the interval [a, b] satisfying a + 2(k-Y) f'(b i --- ff((bb))9 If ILf (x)[ _< 1 and b and x0 E [a, b] with f (x0) > 0, and x0 > a + 2~-})
,
- ~
in [a, b], then show" Newton's method converges
to a solution x* of equation f (x) - 0 and x2~ >_ x*, X2n+l <_ x* for all n >_ 0. 6.3 Consider the midpoint method [681, [158]" -
-
(.n),
rn -
F' (*n)-',
Zn -- Xn -5 "~ Xn+l -- Xn -- Pn F ( X n ) ,
Pn -- F ! (Zn) -1
(Tt > 0 ) ,
for approximating a solution x* of equation F (x) - 0. Let F 9 f~ C_ X + Y be a twice Pr6chet-differentiable operator defined on an open convex subset of a Banach space X with values in a Banach space Y. Assume"
229
6.6. Exercises
(1) Fo E L (Y, X) for some Xo E f~ and IlColl < 9;
(2) Ilrof (~o)11 ~ ,; (3) IIF" (x)ll < M (x E fl); (4) IIF" ( x ) - F" (Y)It < K IIx- Y[I (x, y e a). D~not~ ao - M g ~ , bo - K g ~ ~. D ~ f i ~ s ~ q ~ c ~ an+~ - a n f (~n)~g (~n, bn),
bn+l
-
b n f (an)3g (an, bn) 2, f (x) -
96(1-x)(1--2x)
7(2-x)~
R-
2
x ~-
i2-x)~ + ~ "
If
where
0
h (~) =
2-3x2-X, and g (x, y) -
1
2 - a o 1--A ~
, u (x0, R~) c ~, -
A - f (a0) -1
then show: midpoint method {xn} (n _> 0) is well defined, remains in U(xo, Rrl) and converges at least R-cubically to a solution x* of equation F (x) = 0. The solution x* is unique in U(xo, 2 _ Rr/) N f~ and IlX~+l
X* -
3~-1
2
II <- ~---~o7
~
An
l--Z~.
6.4 Consider the multipoint method in the form [68], [158]: yn - zn - r . F
(zn),
rn - F'
(Xn) -1 ,
z~=x~+O(y~-x~), Hn-
1 -grn [F' ( X n ) - F I (Zn)] , 0 C (0 , 1] , 1
Xn+l -- Yn 2v -~Hn (Yn - Xn)
(It > 0),
for approximating a solution x* of equation F (x) = 0. Let F be a twiceFr6chet-differentiable operator defined on some open convex subset f~ of a Banach space X with values in a Banach space Y. Assume: (1) Co e L (Y, X),
for some xo E X and Ilroll ___/~;
(2) IlVoF (~o)11 ~ ~; (3) lIF" (x)ll _< M, (x e a); (4) IIF" ( x ) - F" (Y)II - K lix- yll ~,
(x, y) e ~, K > o, p e [o, 11.
Denote ao - M/3rl, bo - K/3rl I+p and define sequence a~+x -- a,~f(a~) 2 90 ( a n , bn) , b~+l - b n f (a~) 2+p go (an,
bn) I+p ,
f (x) - 2--2x-x 2 2 and 90 (x y) = x~+4x~ [2+(p+2)O~ly , 8 + 2(p+l)(p+2) 9
230
6. M o r e R e s u l t s on N e w t o n ' s M e t h o d
Suppose ao E (0, 89 and bo < hp (ao, 0 ) , where (p+l)(p+2)
hp (x, 0) - 412+(p+2)op] (1 - 2x)(8 - 4x 2 - x3). Then ' if U-(Xo' R~/) c-- ~, where R - ( 1 + ~ ) 1 ' A -- f ( a o ) - i show" 1-'),A _ iteration {x~ } (n > 0) is well defined, remains in U (x0, R~) for all n _> 0 and converges with R-order at least 2 + p to a solution x* of equation F (x) - 0. The solution x* is unique in U(x0, 2 - Rr/) N ft. Moreover, the following estimates hold for all n > 0
[Xn __ x*ll
l+p ) ~/ ((2+p)n--1 l+p ) < [1 + ~'~")/( (2+P)n--1 --
An 2 1 - 7(+P)
n
A
r]
'
where ~ / - ~al. 6.5. Consider the multipoint iteration [191]" Y n - - X n -- F n y
Fn - F ' (Xn) - 1 ,
(Xn),
2
Zn--Xn-sFnF(xn
),
H~ - r n [F' (z~) - F ' (x~)], 3 -1 3 [ I + ~Hn] Hn(yn-x~) Xn+ 1 -- Yn -- -~
( n >_0 ) ,
for approximation equation F ( x ) - 0. Let F 9 ft c X ~ Y be a three times F%chet-differentiable operator defined on some convex subset f~ of a Banach space X with values in a Banach space Y. Assume F ~ (xo) -1 C L (Y, X) (xo E f~), IFoll _< c~, IlVoF (xo)ll _
ao-co-1, an+l
__
- - 1 - Oaan n d n ~
Cn+l =
do-
bn+l - -
8(2-3bn)4 [
2-o 2(~-o),
20an+len+l -3
a~
03
(4_3b~)4 (2_3bn)2
wan
4-3b~+1 4-6b~+1 C n + l
( n > 0) _
9
Moreover, assume: U (xo, R/3) c_ ~, where
n R-lim
n---+OQ
d~,
0E
0,7
]
4
+ I-~S5 + 3(2-3bn)0 a n + l d n
and
dn+l --
'
,
0 <--5 <
i--0 27(20-1)(03-809. + 160-8) 17(1_0)2
0 __~ ~ <
3(20-1)(03 -802 + 160-8) 40(1_0)2
175 360 "
6. 6. Exercises
231
Then show: iteration {xn} (n _> 0) is well defined, remains in U (xo, R/3) for all n _> 0 and converges to a solution x* of equation F (x) - 0. Furthermore, the solution x* is unique in U (xo, 5-~ 2 - R/3) and for all n _> 0 (2-0)
1
/3
Ilxn - x* II < ~ gift <_ ~ 2(1-0)3,1/3 Z ")/4i+1 ikn
j>_n
bl where ~ / - 7o"
6.6. Consider the multipoint iteration [157]" y ~ -- X n -- r '
(x.)
~ r (xn)
a . --IF' (Xn + p (Yn -- Xn)) - r ' ( ~ . ) ] (Yn -- ~ ) . x~+i-y~-~
1F' (Yn) - 1 a ~
p E (0, 1],
(n>O)
for approximating a solution x* of equation F (x) - 0. Let F 9 t2 c_ X ~ Y be a continuously Fr6chet-differentiable operator in an open convex domain gt which is a subset of a Banach space X with values in a Banach space Y. Let Xo c f t such that Fo - F ' (xo) -1 C L (Y,X); Ilroll < ~, Ilyo- xolt _< ,, 2 P - 5, and IIF' ( x ) - f t (v)ll <_ K II~- vii for all x , y C f~. For bo - K/3r], define bn - bn-~f (bn _ ~)2 g (b~-l), where 2(l--x) f (x) - x2_nx+ 2
and
g(x)
-
x(x~-~+~) 8(1--x) 2
If b0 < r - .2922..., where r is the smallest positive root of the polynomial q ( x ) - 2x 4 - 1 7 x a + 4 8 x 2 - 4 0 x + 8 , and U(xo, KA---3)C_ f~, then show: iteration 1 and converges to a solution {x~} (n >_ 0) is well defined, remains in/F(xo, ~--~) x* of equation F(x) - O, which is unique in U ( x o , - ~ ) . 6.7. Consider the biparametric family of multipoint iterations [159]"
w-x~-r~F(~).
~-~
+p(y~-x~).
p e [0.11.
1 n [[gt (Zn) -- F I (Xn) ] , H n -- -~r 1 X n + l -- Yn -- -~Hn ( I - ~ - o z H n ) (Yn -- X n ) ,
(?~ ~ O)
where Fn - F ' (Xn) -1 (7t ~ 0) and a - -2/3 c IR. Assume Fo - F ' (X0) -1 C L ( K X ) exists at some xo E fro _c X, F : fro c_ X --+ Y twice Fr6chetdifferentiable, X, Y Banach spaces, IIFoll <_/3, IIFoF (zo)[[ _< r/, liE" (z)tl _< M, z E fro and I I F " ( x ) - F " ( y ) l l -< K l l x - y l l for all x , y C t2o. Denote ao = M/3~, bo = k/3~ 2. Define sequences
a n + 1 - a n f (an)29(an, bn),
bn+l - bnf (an)ag(an, bn) 2,
6. More Results on Newton's Method
232 where f (x) - 2 [2 - 2x - x 2 - I ~ 1 x ~]
-1
and
g (X, y) --
X5 -t- L~--x4 +
1+4[a[ 8 X 3 _t_ II+c~[ 2 X2
-'t- ~ -P - x y -~ ~2 +131p ~
for some real parameters c~ and p. Assume:
1 a0C (0, g),
3(8--16ao-4a~ +Taa+2a4) 3ao+2 '
bo
[c~] < min {6, r}, p E (0, 1] and p < h (]ct[), where r is a positive root of h ( x ) - - 6 b o ( 11- a o )
[ (24 - 48ao - 12%2 + 21a 3 + 6a 4 - 2bo (3ao + 2))
+ 6ag (2ao~ + 3 4 - 6ao - 2) 9 + 3ao~ (2ao - 1)x~], C (xo, Rr/) c_ fro, R - [1 + ~ (1 + [c~[ao)] 1-vA,1
~/ __ ~oo'al A - -
f (ao) -1 .
Then show: iteration {xn} (n _> 0) is well defined, remains in U (Xo, Rr/) for all n _> 0 and converges to a unique solution x* of equation F (x) - 0 in g(xo, 2 - Rr/) N fro. The following estimates hold for all n >_ 0: A n
1_~,3n A 1]"
6.8. Let f be a real function, x* a simple root of f and G a function satisfying G ( 0 ) - 1, G' ( 0 ) - 71 and IG" (0)l < +oc Then show: iteration f (xn) f'(Xn)'
Xn+1 - x n - G ( L f ( x n ) )
(n > 0) -
where
L f (x) -
f ( x ) f " (x) f'(x) 2
is of third order for an appropriate choice of Xo. This result is due to Gander [167]. Note that function G can be chosen
a (x)-
1+
a(z)-
1~ 2-x
(Chebyshev method); x
x
G (x) - 1 + 2(1---x)
(); (Super-Halley method).
6.6. Exercises
233
6.9. Consider the super-Halley m e t h o d [68] for all n > 0 in the form:
r(Xn)+F'(~)(yn-~n)-O,
3r (x~) + 3F' (~) [x~ +
2 (Yn -- Xn)] (Yn -- X n ) -~- 4 F ! ( Y n ) ( X n + l
- Yn) -- O,
for a p p r o x i m a t i n g a solution x* of equation F (x) - 0. Let F " ~ c_ X ~ Y be a three-times Fr~chet-differentiable operator defined on an open convex subset ~ of a B a n a c h space X with values in a B a n a c h space Y. Assume" (1) Fo - F ' (xo) -1 e L (Y, X ) for some x0 e ~ with [tF0J} < fl;
(2) 11roF (xo)Jl _< ,; (3) lit" (x)II <_ M (~ e ~z); (4) [[F'" (x) - F ' " (Y)t] -< n [Ix - y[[ (x, y e ~ ) ,
(L > 0).
Denote by a0 - M/~r/, co = L/3~/3, and define sequences
an+l -- a n f ( a n ) 2 g(an, Cn) , Cn+l - - C n f ( a n ) 4 g ( a n ,
Cn) 3 ,
where f (x) = x2_4x+~
g
(i-x) ~ + ~TY 9
Suppose: ao E (o, 89 co < h (ao), where
h (x) = ~(~x-~)(~-~)(x-~+e~)(x-~-e~) ...........
_
(x0,R~/) c ~,
17{i,/:)~ . . . . . . . . . .
[1 +
R-
,
a o ] l t--A'
2('i-a0)
and A - f (ao) -1. then show: iteration {Xn} (n > 0) is well defined, remains i n / ] (Xo, R~/) for all n _> 0 and converges to a solution x* of equation F (x) = 0. the solution x* is unique in U(xo, ~ - R~/) A ~ and
[ IfXn_X,t[<_
a0~/
1+2(
4n--I a~) ] a
1-
4~ - 1
An
.y---r-l_.y4nAT/
(n>0),_
ao
where 7 - ~ -
6.10. Consider the multipoint iteration m e t h o d [161]:
yn --- Xn --12' (Xn)--t12 (Xn) Gn --- F ' (Xn) -1 [F t (Xn Jr- -~ 2 (Yn -- Xn)) -- 12' (Xn)] , Xn+ 1 -- Yn
--
~3 G n [ I -
3 ~Gn] (Yn- Xn) (Tt _> O)
for a p p r o x i m a t i n g a solution x* of equation 12 (x) - 0. Let 12 " ~ c_ X ~ Y be a three times Fr~chet-differentiable operator defined on an open convex subset ~ of a B a n a c h space X with values in a B a n a c h space Y. Assume"
6. More Results on Newton's Method
234
(1) F0 - F' (Xo) -1 e L (Y, X) exists for some Xo Eft and IIr011 _ 0). Denote by ao - M/3~/, bo - N3~/2 and co - L3~/3. Define the sequence
an+l =an/(an) 29 (an, bn, Cn) , bn+l =bnf (an)39(an, bn,cn) 2 Cn+l =Cnf (an) 4 g(an, b~,cn) 3
where 2
f (X) -- 2_2x_x2_x3 ~ and .q (X, y Z) --
1 [27X 3 (X 2 + 2X -4- 5) -4- 18xy -4- 17z]
1 If ao E (0, 7), 17Co + 18a0bo < p (ao), where
p (x) -- 27 ( 1 - x) ( 1 - 2x) (x 2 + x + 2 )
U(xo, RII) C_a, -
R-[1-4--~-(1-t-ao)]
(x 2 + 2 x + 4 ) ,
I_~A, 1
~
_ _
al ~oo'
/~ __ f ( a 0 ) - I
then show: iteration {zn} (n _> 0) is well defined, remains in U (zo, Rr/) for all n >_ 0 and converges to a solution x* of equation F (z) - 0, which is unique in U(zo, 2 - Rr/) Cqft. Moreover the following estimates hold for all n>0:
]lXn -- X* [I ~
1 + 5ao7 ~x=-! (1_ +_ao7
~_1)] -3-
4~_1
,-)/ 3
~
1_,,/4n A T]"
6.11. Consider the Halley method [68], [131]
Xn+l -- Xn - - [ I -
LF (xn)] -1
F ' (Xn) -1 f (Xn)
(Tt ~_ O)
where
Ls (x) -- F ' (x) -1 F " ( x ) F ' ( z ) - l f
(X),
for approximating a solution x* of equation F (x) - 0. Let F 9 ft _C X --~ be a twice Fr6chet-differentiable operator defined on an open convex subset of a Banach space X with values in a Banach space Y. Assume:
6. 6. Exercises
235
(1) F ' (x0) - 1 E L (Y, X ) exists for some Xo C ft;
(2) IIF' (X0) -1
/[7' (X0)II
~ 9;
(3) I[F' (X0) - 1 Ftt (xo) II ~ "1';
(4) IF' (X0) -1 [F tt ( x ) -
F tt
(y)] II <-- M IIx- Yll (x, y ~ ~ )
If
212V@2+2M+'Y] [_7/(Xo,rl) C_ f~, 3 ~ 3[V/3,2+2M+712, M 3, (rl ~ r2) where rl, r2 are the positive roots of h (t) = / 3 - t +_~t 2 + -g-t then show: iteration {Xn} (n _> 0 ) i s well defined, remains in U (x0, rl) for all n >_ 0 and converges to a unique solution x* of equation F (x) - 0 in U (x0, rl). Moreover, the following estimates hold for all n _> 0:
IIX* -- Xn+l II < ( r l
l~n+l) /~ IIx*-x'~]])\ 3
(~20)3n A10 ~2_(~20)an (r2 -- r l ) < r l -- tn ~ A1_()~I0)an (r2 -- r l ) , /
0-
~1
A1 - ~/(~~176
r2~
< 1
V (ro--rl)2+rorl --
A2-
~2
- t o is the negative root of h, and tn+l -- H (tn), where H (t) - t -
h (t)/h' (t)
Lh (t) --
1 - ~ 1Lh (t)'
h (t)/h" (t) h' (t) 2
6.12. Consider the iteration [68], [158]
Xn+l -- Xn -- [I d- T (xn)] F n F (Xn)
(n ~> 0),
w h e r e r n -- F ' (Xn)-X a n d T (x n) - ~Xr n A r n f
(Zn) (n _> 0) , for approximat-
ing a solution x* of equation F (x) - 0. Here A 9 X x X --, Y is a bilinear operator with IIAI - a, and F " f~ c_ X --, Y is a Fr~chet-differentiable operator defined on an open convex subset ft of a Banach space X with values in a Banach space Y. Assume: (1) F ' (xo) -1 - Fo C L (Y, X ) exists for some xo E ~t with llFol[ _< ~;
(2) IIr0r (~0)ll _< ~; (3) lit' (x) - r ' (y)ll <_ k IIx - yll (x, y e
~) .
Let a, b be real numbers satisfying a E [0, 31) , b C (0, or), where
~[2a~--aa--X+,/l+Sa--4a~] (7 --
a(1-2a)
"
6. More Results on Newton's Method
236
b Define sequence 1, c o - 1, b o - 7b and do - 1 + 3"
Set a o -
an
a n + l = 1-aandn bn+ l __
Cn+l -- - ~
b "~an+ l Cn+ l
dn+l -
[
hi2
a + (l+bn) 2 dn,
(1 + b n + l ) C n + l
N-~n+l
and rn+l z_.,k=0dk (n_>0). I f a -- ~ r / E [0,89 5 ( x o , rr/) C_ ft, r = limn__~ rn, ct E [0, ~ ) , then show: iteration {xn} (n >_ 0 ) i s well defined, remains in V(xo, rrl) and converges to a solution x* of equation F (x) = 0, which is unique in U(xo, ~-~ - rri). Moreover the following estimates hold for all n > 0
IlXn+l -- Xnll ~ d ~ and 0(3
IIX*
-- Xn+ll]
~ (r -- rn)7]
E dkr/. k=n+l
6.13. Consider the Halley-method [68], [175] in the form: y~ - x~ - F ~ F (x~), Fn -- / ; ' (Xn) -1 , 1 X~+l - Yn + 5CF (Xn) H~ (y~ - Xn) (~ >_ 0),
LF (x~) = F n F " (Xn) F n F (Xn) ,
H n = [I - L r (Xn)] -1
(n_>0),
for approximating a solution x* of equation F (x) - 0. Let F " ft C_ X ~ Y be a twice F%chet-differentiable operator defined on an open convex subset ft of a Banach space X with values in a Banach space Y. Assume: (1) F0 E L (Y, X) exists for some xo E f t with ]tF011 <_ ~;
(2) IIF" (x)lt <_ M (x Eft); (3) IIr" (~) - F" (Y)II -< N Itx - Yll (x, y e a); (4) IlroF (xo)ll _< ,; (5) the p o l y n o m i a l p ( t ) - 7k t 2 - ~ 1t + ~ , roots rl and r2 with (rl < r2).
where M 2 + ~N _< k2 has two positive
Let sequences {sn}, {t~), (n _> 0) be defined by
8n--tn
p (tn) p'(tn)'
tn+l
--
Sn
1 Lp (tn) § 21-Lp(tn)
(Sn-
tin)
(It > 0 ) . -
If, U (x0, rl) _cft, then show: iteration {Xn} (ft ~ 0) is well defined, remains in U (x0, r]) for all n _> 0 and converges to a solution x* of equation F (x) - 0.
6. 6. Exercises
237
Moreover if rl < r2 the solution x* is unique in U (x0, r2). Furthermore, the following estimates hold for all (n > 0) (r2 -- rl) 0 4~ 1_04 n ,
0 - - ~r2
6.14. Consider the two-point method [68], [153]" yn=Xn-ft(Xn)-lF(xn), Hn -- p1 Ft (Xn)-i [F t (Xn -]- p (Yn -- Xn)) -- F' (Xn)] , X n + l -- Yn -- ~1H n F -~- H~] -1 (Yn - Xn)
P C (0, 1],
(n > 0 ) ,
for approximating a solution x* of equation F (x) - 0. Let F " f~ C_ X --+ Y be a twice-F%chet-differentiable operator defined on an open convex subset f~ of a Banach space X with values in a Banach space Y. Assume:
(1) F0 - F' (x0) -1 E L (Y, X) exists for some x0 E f~ with liP011 _< #; (2) llroF (~o)ll _< n;
(3) [IF"(*)II< V (. e ~); (4)
IIF"(x)
-
F"
(v)ll _ K 11~- vlt,
x, y ~ ~.
Denote by ao - M/3r/, bo - K/3r] 2. Define sequences
a~+l =a~f (an) 2 gp (a~, b~) , bn+l =b~f (a~)3gp (a~, b~) 2 , where f (x)
-
2(l-x) x~--~-xg2 and
gp (X, y)
-
-
3xa+2y(1-x)[(1-6p)x+(2+3p)] 24(1_x)2
If a0 C .
(0, 2)' bo < hp (ho), where hp (X) -- 3(2x--1)(x--2)(x--3+v/'5)(x--3--V/-5) 2(1-x)[(1-6p)x+2+3p] then show: iteration {Xn} (n > 0)is well defined, remains in U(xo __v_.)for all n > 0 and converges to a solution x* of equation F (x) - 0, which is unique in U(xo, ~). Moreover, the following estimates hold for all n > 0: --
'
3 n --1
llx*-Xn]}_< 1 + 2 ( 1 _ ; o ) where 7 = ~a o and kx = f (ao)- 1
7~~r/'l-A
ao
238
6.15.
6. More Results on Newton's Method
Consider the two-step method: yn-xn-Ft(xn)-lF(xn) Xn+l -- Yn -- Ft (Xn) - 1 F ( y n )
(?% ~ 0);
for approximating a solution x* of equation F (x) - 0. Let F 9 f~ C_ X C_ Y be a F%chet-differentiable operator defined on an open convex subset f~ of a Banach space X with values in a Banach space Y. Assume: (1) F0 - f ' (Xo) -1 C L (Y, X) for some x0 E f~,
Ilrol ~ ~;
(2) IIC0F(x0)ll (3) IIF' (x) - F' (Y)I[ <- K
IIx -
yll (x, y
C a).
Denote a0 - k~r / and define the sequence an+l - f (an) 2 g ( a n ) a n (n >_ 0), where f ( x ) - 2-2x-x2 ~ and g (x) - x 2 (x + 4 ) / 8 . If ao E (0, 89 , O (xo, R~]) c_ ao ~'~' /~ -- ll-+--ff---~,A,")/ -- aoa'! and A - f (a0) -1, then show: i t e r a t i o n
{Xn} (Tt _>O)
is well defined, remains in ~r (x0, Rr/) for all n > 0 and converges to a solution x* of equation F (x) - 0 , which is unique in U(xo, ~2 - Rr/) A ft. Moreover, the following estimates hold for all n > 0
IIXn- X* tl ~ [1 § 9~/
3 n.2:--1 ]
3n --I
An 1 - 73~A rl (n > 0).
6.16 (a) Assume there exist non-negative parameters K, M, L, g, p, ~/, 6 E [0, 1] such that" L<_K,
(1)
g + 2# < 1,
(2)
and hs-
( K + L S + T _ 4M) ~ ~]+M+2#<_5.
(3)
Show" iteration {tn} (n _> 0) given by to -- O, tl -- ?7, tn+2 -- tn+l +
K(tn+l-tn)+2(Mtn-t-tt) (tn+l -- tn) (n > O) 2(1-g-Ltn+m)
is nondecreasing, bounded above by t**, and converges to some t* such that O <_ t* <_ 22~_5- t**.
Moreover, the following estimates hold for all n _> 0
0 < tn+2
tn+l < ~(tn+l
tn) < (25-) n+l
6. 6. Exercises
239
(b) Let F" D C_ X + Y be a F%chet-differentiable operator. Assume: there exist an approximation A(x) E L(X, Y) of F'(x), an open convex subset Do of D, x0 E Do, a bounded outer inverse A # of A(xo), and parameters > 0 , K > 0 , M>_0, L > 0 , p_>0, C _> 0 such that (1)-(3) hold
IIA#F(x0)ll ~ W, IIA#[F'(x)- F'(y)]II ~ KFIm- YlI, I[A#[F'(x) - A(x)]ll _< M l l x -
x011 + ~,
and [[A#[A(x) - A(xo)]ll _ L l l x - xoll + e for all x, y E Do, and U(xo, t* ) c_ Do
Show: sequence { X n } (n ~ 0) generated by Newton-like method with A (Xn) # - [ I + A # ( A ( x n ) - A(xo))]-lA # is well defined, remains in U(xo, s*) for all n _> 0 and converges to a unique solution x* of equation A#F(x) - O, (7 (xo, t*) N Do Moreover, the following estimates hold for all n > 0
II~+~ -
xnll ~ t n + l -- t n ,
and IlXn -- X* I] ~ t* -- tn.
(c) Assume: there exist an approximation A(x) E L(X, Y) of F'(x), a simple solution x* E D of equation (1), a bounded outer inverse A1# of A(x*) and non-negative parameters K, L, M, #, f, such that"
-
IIA~[F'(x) - F'(y)]11 ~ K l l x - Yll, [IA~[F'(x) - A(x)] I! ~ ~r[Ix - x* II + ~, and IIA#~[A(x)
for all x, y E D;
-
A(x*)]ll _< LIIx- x*ll +
6. More Results on Newton's Method
240 - equation
(K-7- + A I + L ) r + / 2 + t ~ -
1=0
has a minimal non-negative zero r* satisfying
L r + t ~ < 1, and
U(x*,r*) c_ D. Show: sequence {x~} (n > 0) generated by Newton-like method is well defined, remains in U(x*, r*) for all n _> 0 and converges to x* provided that Xo E U(x*, r*). Moreover, the following estimates hold for all n _> 0:
tlX* -- Xn+lt[ _< 1
[~X*
]
*
- Xnfl
< (++/~)~*+~ t-Lr* - [ IIx* -- X~ It 6.17 (a) Let F " D C_ X ~ (m _> 2 integer). Assume:
Y be an m-times Fr6chet-differentiable operator
(al) there exist an open convex subset Do of D, x0 c Do, a bounded outer inverse F' (x0) # of F ' (x0), and constants ~ > 0, ai > 0, i - 2, ..., m + t such that for all x, y C Do the following conditions hold:
[tF'(xo)#(F ('~) ( x ) - F (m) (xo))II _< e,
c > o,
(6.6.1)
vx E u (xo, ~0)and some ~0 > O.
t[F' (xo) # r (xo)t[ _< ,~, [fF' (xo) # F + (~o)II - ~ the positive zeros s of p' is such that
p(~) ___ o, where, a2t 2
p(t) -- 71-- t + --~. + ' ' ' +
am+~
m!
tm.
Show: polynomial p has only two positive zeros denoted by t*, t** (t* _< t**).
6.6. Exercises
241
(a2) (;(xo, (~) q Do,
(~ -- max{ (~o,t*, t** }.
(a3) (~o E [t*, t**] or (~o > t**. Moreover show" sequence {an} (n >_ 0) generated by Newton's method with F'(xn) # - [I + F ' ( x o ) # ( F ' ( x n ) - F ' ( x o ) ) ] - l F ' ( x o ) # (n _> 0) is well defined, remains in g(xo, t*) and converges to a solution x* E U(xo, t*) of equation F ' ( x o ) # F ( z ) - 0; - the following estimates hold for all n > 0 IlXn+l -- Xn]] ~ tn+l -- tn
and
where {tn} (n _> 0) is a monotonically increasing sequence converging to t* and generated by t o -- O,
t n + l -- tn -- pP~(tn) (t~) 9
(b) Let F" D C_ X ~ Y be an m-times Pr~chet-differentiable operator (m >_ 2 an integer). Assume: (bl) condition (1) holds; (b2) there exists an open convex subset Do of D, Xo E Do, and constants c~, ~, r/_> 0 such that for any z E Do there exists an outer inverse F' (z) # of F'(z) satisfying N ( F ' ( x ) #) = N ( F ' ( z o ) #) and
IIF'(zo)#f(zo)ll_ W, IlF'(y)#~lF"[z+t(y-x)](1-t)dt(y-z)21] < [ am,+e
-L
m, IIv
_
_
zll
m-2
<
I
for all x, y E Do, am+a?]m-2 m' +...+~]r/<
1,
and D
U(Xo, r) c_ Do
with r - m i n { - - ~l - t o ~5o}
where, ro __ [am-t-e m! Tim-2 + ' " + 7 ] ~c~2 /9
-zll 2,
6. More Results on Newton's Method
242
Show: sequence {x~} (n _> 0) generated by Newton's method is well defined, remains in U(xo, r) for all n > 0 and converges to a solution x* of F ' ( x o ) # F ( x ) - 0 with the iterates satisfying N ( F ' ( z ~ ) #) - N ( F ' ( z o ) #) (n _> 0). Moreover, the following estimates hold for all n _> 0 I1-~+, - *~ii ___ ~g ri*, - .011, -
1_-~o I1~1 - *0fl,
and 1--r~z 1--r~ IIz~ - zolf _< ~--:~-o ilz, - zoll _< 1--=~o ~ - r.
6.18 Let X and Y be Banach spaces, and let L be a bounded linear operator on X into Y. A linear operator M : Y --+ X is said to be an inner inverse of L if L M A = L. A linear operator M : Y --+ X is an outer inverse of L if M L M = M. Let L be an m • n matrix, with rn > n. Any outer inverse M of L will be an n • rn matrix. Show: (a) If r a n k ( L ) = n, then L can be written as
0
'
where I is the n x n identity matrix, and A is an rn x rn invertible matrix.The n x m matrix
M--[I
B]A
-1
is an outer inverse of L for any n • ( m - n) matrix B. (b) If rank (L) = r < n, then L can be written as
L_A[I
0
0 0
C,
where A is an rn x rn invertible matrix, I is the r x r identity matrix, and C is an n x n invertible matrix. If E is an outer (inner) inverse of the matrix [ I L 0
0 ] then t h e n x m m a t r i x 0 ]'
M = C - l E A -1 is an outer (inner) inverse of g. (c) E is both an inner and an outer inverse of [ I0 be written in the form
E-
C
CM
-
C
[I
M].
01ifandonlyifEcan
6. 6. Exercises
243
(d) For any ( n - r) x r matrix T, the matrix E inverse of [ I [ 0
0 0
I T
0 ] is an outer 0 J
"
6.19 Let F 9D C X -+ Y be a F%chet-differentiable operator between two Banach space X and Y, A (z) E L (X, Y) (x E D) be an approximation to F ' (z). Assume that there exist an open convex subset Do of D, xo E Do, a bounded outer inverse A # of A ( = A (zo)) and constants r/,k > O , M , L , # , I > 0 such that for all x, y E Do the following conditions hold:
IIA#F(xo)II
~ ~,
IIA# (F' (x) - f ' (y))II ~ k IIx - YlI,
IIA # ( f ' (x) - A (x))[[ _< M I I x - xo[[ + #, IIA# (A (x) - A)II - L I]x - xoll + z,
b . - ~ + 1 < 1.
Assume h - ~r] _< ~1 ( l - b ) 2 ~ a "- max ( k , M + L), and U -
U(xo t*) _c
Do t* -- l-b-v/(l-b)~-2h. Then show (7 (i) sequence {Xn} (n _> 0) generated by xn+l - x n - A (Xn) # F (xn) (n _> 0) with A ( x n ) # - [ I + A # ( A ( x n ) - A ) ] - 1 A # remains in U and converges to a solution z* E U of equation' A # F (x) - 0.
(ii) equation
A#F(x)
-
0
has a
unique solution in /)Cq
{R(A #) + xo},
where
(xo, t*) ClDo if h - 7 1 (1 _ b)2 U(xo,t* 9 )VIDo if h < 17 ( l - b )
-~_ U
2,
R ( A # ) + xo "- {x + Xo " X E R ( A # ) } , and t* =
(iii) [Izn+l
1-b-+-4(1-b)2-2h
- xn[I I(t~)
_< t n + l
-- t n , ~ 2
tn + g-UTi-Y~)' f (t) - gt
Ilx* - a n I I
<-- t* -
tn,
where
to -
O, t n + l
--
- (1 - b) t + rl, and g (t) - 1 - Lt - g.
6.20 Let F 9D c_ X ~ Y be a F%chet-differentiable operator between two Banach spaces X and Y and let A (z) E L (X, Y) be an approximation of F ' (z). Assume that there exist an open convex subset Do of D, a point z0 E Do and constants r/, k > 0 such that for any x E Do there exists an outer inverse A (x) # of A (x) satisfying N ( A (x) #) - N ( A # ) , where A = A (xo) and A # is a
244
6. More Results on Newton's Method bounded outer inverse of A, and for this outer inverse the following conditions hold:
IIA#F (Xo)11 ~ r], IIA
(y)~ (F' (x + t(y - x)) - F' (x)) II ~ kt IIx - yll
1 for all x , y c Do and t E [0,1], h - 5kr/ < 1 and U(x0, r) c_ Do with r = l-h" ~ Then show sequence {xn} (n _> 0) generated by xn+l - x n -
A (x~) # F (x~) (n > 0) with A (x~) # satisfying N ( A (x~) #) - N (A #) remains in C (x0, r) and converges to a solution x* of equation A # F (x) - 0. 6.21 Show that Newton's method with outer inverses Xn+l - x n - F ~(x~) # (n _> 0) converges quadratically to a solution x* E U Cq { R ( F ' ( x o ) #) + x0} of equation f ' ( x o ) # f (x) A (x) - F' (x) (x E Do).
0 under the conditions of Exercise 6.4 with
6.22 Let F " D C_ X ~ Y be Frhchet-differentiable and assume that F ' (x) satisfies a Lipschitz condition
IIF' (x) - r ' (y)II ~ L IIx - YlI, x, y c D. 1 Assume x* c D exists with F (x*) = 0. Let a > 0 such that U (x*, a) C_ D. Suppose there is an
F ' (x*) # C s (F' (x*)) - { B E L (Y, X ) " B F ' (x*) B - B, B 7/= 0} such that ]lF ' (x*)# ]i ~ a and for any x E U (x*, 3-~a)' the set ft (F' (x)) contains an element of minimum norm. Then show there exists a ball U (x*, r) G D with cr < ~ 1 such that for any Xo e V (x*, r) the sequence {Xn} (n _> O) Xn+l -- X~ -- F' (Xn) # F (X~) (n >_ 0) with F ' (xo) # E argmin {tIBIt IB EFt (F' (xo)) } and with F' (x~) # - (I + F' (xo) # (F' (Xn) - F t ( x o ) ) ) - l F t (Xo) ~ converges 1 quadratically to 2* E U (xo, Z-~) V~{ R ( F ' (xo) #) + xo}, which is a solution of equation F ' ( x o ) # F ( x ) R ( F ' (xo)e)}
- O. Here, R ( F ' ( x o ) # ) + x o
- {x+xo
"x C
Method
We exploit the weaker conditions introduced in Section 5 to improve on recent results on the local and semilocal convergence analysis of Newton's method.
6.1
Convergence radii for the Newton-Kantorovich method
Motivated by the works in Wang [341], [343] and using the same type of weak Lipschitz conditions we show that a larger convergence radius can be obtained for Newton-Kantorovich method (4.1.3). Recently in [341], [343] a Lipschitz-type condition
lift (X*)--I [f' (X) -- f' (X0)] II ~
f~(x)
L(u)du
(6.1.1)
JOs(x)
for all x E D, where L is an integrable, nondecreasing, nonnegative function, s(x) = IIx - x* II and x ~ = x* + O(x - x*) 0 E [0,1], w a s used to obtain larger convergence radii than the ones given by the relevant works in [136], [a401, [312]. Condition (6.1.1) is weaker than the famous Lipschitz condition IlF' (x* ) - l [F' (x) -
r'(y)]ll ~ w(llx- yll),
for all x, y E D,
(6.1.2)
where w is also a nondecreasing, nonnegative function. Condition (6.1.2) has been used e.g. in [295] (when w is a constant function). It turns out that (6.1.1) instead of the stronger (6.1.2) is really needed to show the local convergence of method (6.1.1) to x* [341], [343]. Based on this observation it became possible to obtain convergence radii larger than the ones obtained in [136],
[295], [340], [312]. Here we show that all the above mentioned results can be further improved under weaker or the same hypotheses and under less or the same computational cost. A numerical example is also provided where our results compare favorably with all earlier ones. Several areas of application are also suggested. 187
188
6. More Results on Newton's Method
We provide the following local convergence theorems: T h e o r e m 6.1.1 Let F" D C_ X ---+Y be a Frdchet-differentiable operator. Assume: (a) there exists a simple zero x* E D of equation F(x) - 0 so that F'(x*) -1 E L(Y, X), positive integrable nondecreasing functions L, Lo such that the radius Lipschitz condition with L average and the center radius Lipschitz condition with Lo average s(x) I I f ' ( x * ) - l [ f ' ( x ) - f'(x~ <_ L(u)du (6.1.3) JOs(z)
f
and
l l r ' ( x * ) - : [ r ' ( x ) - r'(x*)]ll < fo s(x) Lo(u)du
(6.1.4)
are satisfied respectively for all x c D, 0 E [0, 1]; (b) equation
/o
L(u)u du + r
/o
Lo(u)du- r - 0
(6.1.5)
has positive zeros. Denote by r* the smallest such zero; and
(6.1.6)
U(x*, r*) c_ D.
Then, sequence {Xn} (n >_ O) generated by method (~.1.3) is well defined, remains in U(x*, r*) for all n > O, and converges to x* provided that xo E U(x*, r*). Moreover the following estimates hold for all n >_ 1"
Ilxn - x*ll _ a 2 n - l l l x o - x*ll
(6.1.7)
where, a-
fo (x~ L ( u ) u d u
(6.1.8)
,
~(x0) [~- fo (x~ L0(~)d~] and
a C [0, 1).
(6.1.9)
P r o o f . We first show a E [0, 1). Indeed we can have in turn for 0 < u: < u2 and the monotonicity of L"
(u-~122 jl~ ul~fo~:)uL(u) du 1
~2
1
1 1
1
1
u1~ ~ jfOul u du - O.
189
6.1. Convergence radii for the Newton-Kantorovich method
Therefore -~1 fou L(v)dv is a nondecreasing function with respect to u. Then, we can have
0-
fo (~~ L(~)~ d~ ~(xo) ~(xo) ~ [ ~ - fo (x~ Lo(~)d~] .
<
fo L(~)~d~
< IIx0
-
--
r*
x*ll
~(xo)
< 1.
Let x E U(x*, r*). Using (6.1.4) and (6.1.6)we get
ILF'(x*)-I[F'(x)- F'(x*)]I I < fo ~(x) Lo(u)du < / i ~* Co(u)du < 1.
(6.1.10)
It follows from (6.1.10) and the Banach Lemma on invertible operators 1.3.2 that F'(X) -1 e L(Y, X ) and
1
1
IlF'(x)-lF'(x*)[t < < . - 1 - fo (x) Lo(u)du - 1 - fo* Lo(u)du
(6.1.11)
In particular Xo c U(x*, r* ) by hypothesis. Let us assume xn e U(x*, r* ). We obtain the identity X n + l - X * - - Xn ~ F ' ( X n ) - l F ( x n ) = ft(Xn)-l[F(x
x*
*) - r ( X n ) -
= [F'(Xn)-IF'(x*)]
(6.1.12)
F t ( X n ) ( X * - Xn)]
F ' ( x * ) - I ( F ' ( X n ) - F'(x~
- x*)dO .
By (6.1.3), (6.1.4) and (6.1.11)we get
Ilxn+l- X* I1-< IIF'(x~)-'f'(x*)ll < -
=
/o 1 [ [ F ' ( x * ) - I ( F ' ( X n )
-
F'(x~
)]x~ - x*lldO
f~ rs(xn) 30s(xn) L(u)dus(xn)dO 1 - fo (xn) Lo(u)du fo (xn) L ( u ) u d u 1 - Jof~(xn)Lo(u)du
<_aLIx~-
x*
I1<11~-
x*
II,
(6.1.13)
which shows the convergence of xn to x* and Xn+l r U(x*,r*). Hence sequence {Xn} (n _> 0) remains in U(x*,r*) by the principle of mathematical induction.
6. More Results on Newton's Method
190
Moreover s(xn) decreases monotonically. Furthermore by (6.1.13)we obtain more precisely
IlXn+l-
X$
f:(Xn) n('~)ud~
II
S(Xn) 2 [1 - fo (x~) Lo(u)du]
fo (x~ L ( u ) u d u
<
8(Xn) 2
s(xo) 2 [ 1 - fo (x~ Lo(u)du] _
a
_ ~s(xo---711Xn_ x, I 2
(6.1.14)
from which estimate (6.1.7) follows. 9 R e m a r k 6.1.1 In general
Lo(u) < L(u)
(6.1.15)
holds and Lo(u) L(u) can be arbitrarily large (see Section 5.1). If equality holds in (6.1.1518) then our theorem coincides with the corresponding Theorem 3.1 in [3~1]. However if strict inequality holds in (6.1.15) (see Example 5.1.5), then our convergence radius r* is greater than the corresponding radius r in Theorem 3.1. Moreover our convergence rate a is smaller than the corresponding q in Theorem 3.1. Summarizing, if Lo(u) < L(u)
(6.1.16)
r < r*
(6.1.17)
a < q.
(6.1.18)
then
That is our theorem provides a wider range of initial guesses xo and more precise error estimates on the distances IlXn- x*ll (n >_ O) than Theorem 3.1 and under the same computational cost since in practice the evaluation of function L requires that of Lo. In case L0, L are constants see section 5.1. T h e o r e m 6.1.2 Let F" D c_ X ~ Y be a Frdchet-differentiable operator. Assume: (a) there exists a simple zero x* E D of equation F(x) - O, so that F'(x*) -1 c L(Y, X ) , and a positive integrable function Lo such that condition (6.1.~) holds; (b) equation
1 r
~0r Lo(u)(r
- u)du- 1 - 0
(6.1.19)
6.1. Convergence radii for the Newton-Kantorovich method
191
has a minimum positive zero denoted by r~ ; and
(c) u(~*, ~;~)c D.
(6.1.20)
Then sequence {Yn} (n ~ O) generated by Yn+l -- Yn -- F ' ( x * ) - I F ( y n )
(n ~ O)
(6.1.21)
remains in U(x*, r~) for all n >_ 0 and converges to a unique solution x* of equation (~. 1.1) so that
Pf*~+l- X*Pl<_aoll*n-- X*PP
(6.1.22)
where ao 6 [0, 1) and 1 fo s(x~ ao = S(Xo) Lo(u)(s(xo) - u)du.
(6.1.23)
Proof. Simply use function L0 instead of L in the proof of Theorem 4.1 in [341].
R e m a r k 6 . 1 . 2 If Lo
L then Theorem 6.1.2 reduces to Theorem ~.1 in [3~1]. Moreover if (19) holds then -
~0 < ~;
(6.1.24)
q0 < a0
(6.1.25)
and
where by to, qo we denote the radius and ratio found in Theorem ~.1.
In the case when L, Lo are constants as expected our results improve Corollaries 6.3 and 6.4 in [342] which in turn compare favorably with the results obtained by Smale [312] and Debieu [139] respectively when F is analytic and satisfies
IlF'(x*)-lF(k)(x*)ll ~ k!~ k-~
(k ~ 2)
(6.1.26)
provided that L(u) -
27 (1 - T u ) 3
(6.1.27)
and 270 Lo(u) - (1 - 70u) 3
(6.1.28)
192
6. More Results on Newton's Method
for some 70 >_ O, 7 >- 0 with 70 _< 7.
(6.1.29)
For example our result corresponding to Corollary 6.4 gives
9 E 1)
r o E 0,~o~
(6.1.30)
whereas the corresponding radius in [342] is given by
[1)
r0E
0,~
.
(6.1.31)
Therefore (6.1.24) and (6.1.25) hold (if strict inequality holds in (6.1.29)). We can provide the local convergence result for twice Fr6chet-differentiable operators as an alternative to Theorem 6.1.1. T h e o r e m 6.1.3 Let F" D c_ X ---, Y be a twice Frdchet-differentiable operator. Assume: (a) there exists a simple zero x* E D of equation F(x) F'(x*) -1 E L(Y, X ) and nonnegative constants ~, b, ~o such that IIF'(x*)-~[S(x
+ e(x* - x)) -
IIF'(x*)-~_P"(x*)ll
r"(x*)] II _< ~(1
< b,
-
e)llx - x*ll,
O, so that
(6.1.32) (6.1.33)
and
IlF' (x* )-l [F" (x ~ - F"(x*)]l
_< eoOllx- =*11
(6.1.34)
for all x E D, 0 E [0, 1]; (b) U(x*,R*) c_ D, where,
R* = d-
2
d + v/d 2 + 4c
3b
and
,
c--lg+ 3
(6.1.35) lg
~o.
(6 1.36)
for all n >_ 0 and converges to x* provided that Xo E U(x*, R*). Moreover the following error bounds hold for all n > 0 X*
X* 2
(6.1.37)
and IIXn -- X* II -----h ~ - ~
IX0 -- X* II"
(6.1.38)
193
6.1. Convergence radii for the Newton-Kantorovich method
Proof. Let x c U(x*,R*). Using (6.1.33), (6.1.34) and the definition of R* we obtain in turn ft(x*)-l[ft(x
*) - f t ( x ) ] -
ft(x*)
-ft(x*)-l[Ft(x)-
F"(x*)(x
-
- x*) + Y " ( ~ * ) ( x
- x*)]
F ' ( z * ) - ~ { F " [ x * + O(z - x*)]
- F"(x*)}(x-
(6.1.39)
x*) - F ' ( x * ) F " ( x * ) ( x - x*)
and 1
*) - F'(x)]ll _
]]F'(z*)-I[F'(z
L
eo011~- x*ll~d0 + Vllx- x*ll
1 _< ~fo(R*)2 + bR* < 1.
(6.1.40)
It follows from (6.1.40) and the Banach Lemma on invertible operators that F ' ( x ) - l F ' ( x *) exists and ]]F'(x)-lF'(x*)ll <_
1 1 - T l lf.o x1
< -
X* I I ~ - b l l x
~o R,)~ 1--~(
- Z* II
"
-bR*
(s.1.41)
In particular by hypothesis Xo E U(x*,R*). F(x*) - 0 we obtain the identity X n + l -- X* -- Xn -- Y l ( x n ) - l
y(xn)
Assume Xn C U(x*,R*).
X*
*) - F ( X n ) -
F'(Xn)(X*
= [F'(xn)-lF'(x*)]F'(x*) -1
/o 1F"[xn
-- F ' ( X n ) - I [ F ( x
Using
- Xn)]
--[- O(X* -- Xn)](1 -- O)dO(xn - x*) 2 ~--- [ f ' ( X n ) - l -
F"(x*)](1
f'(x*)]F'(x*) -
-1
{/o 1[f"(Xn
-]-O(x* - X n ) )
O)dO(xn - x * ) 2
+ f0 ~ F " (~* )(1 - O)dO(xn - x*) ~ } .
(6.1.42)
By (6.1.32), (6.1.33), (6.1.41), (6.1.42) and the triangle inequality we get
[IXn+l -
x*
I1_<
1
-
1 X* b 5ellxnII + ~ eo 2 IIx~ - x* II~ - b[lxn
--
< hl]xn - x* I]2 < [IXn -- x* ]].
x*
II
IIx~--
x* 2 II
(6.1.43) (6.1.44)
194
6. More Results on Newton's Method
Estimate (6.1.44) implies
Xn+l
and lim
E U(x*,/~*)
n---~cx:)
Xn -- X*.
9
The analog of Theorem 6.1.2 is given by T h e o r e m 6.1.4 Under the hypotheses of Theorem 6.1.3 (excluding (6.1.~6)) and replacing R*, h by R;, ho respectively given by 2
R; =
(6.1.45)
do + v/dg + 4co ho - coR~ + do,
(6.1.46)
where, f0 co- ~
b d o = -~
and
(6.1.47)
the conclusions of Theorem 6.1.3 hold, for iteration (6.1.21). P r o o f . Using iteration (6.1.21) we obtain the identity Yn+l
-- X * - -
~- - - [ i ~ ' ( X * ) - l [ F ~ ( y n )
= --F'(X*) -1
=
-- X*)]
-- [i~(X *) -- [ g / ( X * ) ( y n
[/o1F " [ X *
--I- O ( y n --
/o] F"(x*)(1 - O)dO(yn - F Z ( x * ) - 1 {/o 1[FZ'(x* -I-
x*)](1 -
x*) 2 +
O(yn
-
O)dO(yn
/o1F"(x*)(1 -
x*)
-
F"(x*)](1
+ ~1 F" (x*)(y~-x ,)2 }
-
x*) 2
O)dO(yn - x*) 2 - O)dO(yn
-
x*) 2
(6.1.48)
By (6.1.33), (6.1.34), (6.1.48) and the definition of R~, ho we obtain (as in (6.1.43))
IlYn+t -- X*ll ~ [~~ollyn -- X*ll -~- ~] ][yn -- X*ll2 _ h011Y~ -- x*ll ~ < llY~ -- x*il,
(6.1.49)
which completes the proof of Theorem 6.1.4. 9 R e m a r k 6.1.3 (a) Returning back to Example 5.1.5 and using (6.1.32)-(6.1.36), (6.1.~5)-(6.1.~7) we obtain ~ - ~o - e - 1, b - 1, d - 1.5, c - g5 ( e - 1) , c o - e--1 6 , 1 do - 3, R* - .462484887
6.1. Convergence radii for the Newton-Kantorovich method
195
and we can set R~ - 1. Comparing with the earlier results mentioned above (see Example 5.1.5) we see that the convergence radius is significantly enlarged under our new approach. Note also that the order of convergence of iteration (6.1.21) is quadratic whereas it is only linear in the corresponding Theorem ~.1 in [3~1]. (b) The results can be extended easily along the same lines as above to the case when ~, ~o are not constants but functions. (c) The results obtained here can also be extended to hold for m-Frdchetdifferentiable operators (m >_ 2) along the lines of our works in [79], [68]. However we leave the details of the works for (b) and (c) to the motivated reader.
We present a new convergence theorem where the hypotheses (see Theorem 6.1.1) that functions L, Lo are nondecreasing are not assumed: T h e o r e m 6.1.5 Let F, x*, L, Lo be as in Theorem 6.1.1 but the nondecreasiness of functions L, Lo is not assumed. Moreover assume: (a) equation
ffo
~(L(u) + L o ( u ) ) d u - 1 - 0
(6.1.50)
has positive solutions. Denote by R* the smallest one; (b)
(6.1.51)
U(x*,R*) c D.
Then sequence {Xn} (n ~ O) generated by method (~.1.3) is well defined, remains in U(x*,R*) for all n >_ 0 and converges to x* provided that xo C g(x*,R*), so that: IIx~ - x'L1 _ b~lixo - x*]i
(~ >_ o)
(6.1.52)
where, b
fo (x~ L(u)du
(6.1.53)
1 - fo (x~ Lo(~)d~ and
be [0,1). Furthermore, if function L~ given by
(6.1.54)
L~(t) = tl-~L(t) is nondecreasing for some c~ E [0, 1], and R* satisfies -~
L(u)u du +
/o
Lo(u)du <_ 1.
(6.1.55)
196
6. More Results on Newton's Method
Then method (4.1.3) converges to x* for all xo E U(x*,R*) and IIx~ - x* II ~ b h~-~ IIx0 - x*ll
(6.1.56)
(~ ~ o)
where, k
h~ - Z ( 1 + ~)~ (k > 1).
(6.1.57)
i=0
Proof. Exactly as in the proof of Theorem 6.1.1 we arrive at I I x ~ + l - X* I1_<
fo (xn) L(u)u du
(6.1.58)
1 - fo (x~) Lo(u)du
< fo (x~) L(u)dus(xn)
-- 1-- fo (xn) Lo(u)du which show that if Xn E
U(x*,/~*) then
bllx~ - x* II < IlXn
Xn+l C U(x*,
R*) and
-
x*
II,
(6.1.59)
lim xn - x*. Esti-
n---,c~
mate (6.1.13) follows from (6.1.20). Furthermore, if function L~ is given by (6.1.15) and R* satisfies (6.1.16) then by Lemma 2.2 in [343, p. 408] for nondecreasing function
~ Z , ~ ( t ) - t~+Z
1 fo~
ugL(u)du,
(6.1.60)
we get IiXn+l-
X*
II-<
~l,a(8(Xn)) S(Xn) 1+~, 1 - JofS(xn)Lo(u)du ~1,c~ (S(X0)) S(Xn)l+a 1 - fo (xn) Lo(u)du b ~(x0)~ ]1~ - x*LI 1+~ < IIx~ - ~*IL,
(6.1.61)
(6.1.62)
which completes the proof of Theorem 6.1.5. 9 T h e o r e m 6.1.6 Let F, x*, Lo be as in Theorem 6.1.2 but the nondecreasness of function Lo is not assumed. Moreover assume: (a) equation
3
jr0r Lo(u)du -
1 - O,
(6.1.63)
has positive solutions. Denote by R~ the smallest one; (b) U(x*, R~) c_ D. Then, sequence {Xn} (n > O) generated by method (~.1.3) is well defined, remains in
6.1. Convergence radii for the Newton-Kantorovich method
197
U(x*, R~ ) for all n >_ 0 and converges to x* provided that xo E U(x*, R~). Moreover error bound (6.1.13) hold for all n >_ O; where
2 fo (x~ Lo( )d
b-
.
(6.1.64)
1 - fo (~~ Lo(u)du P r o o f . Simply use Lo instead of L in the corresponding proof of Theorem 1.2 in [343, p. 4 0 9 ] . . . R e m a r k 6.1.4 (a) In general (6.1.16) holds. If equality holds in (6.1.16) then Theorems 6.1.~ and 6.1.5 reduce to Theorems 1.1 and 1.2 in [343] respectively. However if strict inequality holds in (6.1.16) then clearly: our convergence conditions (6.1.11), (6.1.2~) are weaker than the corresponding ones (9) and (15) in [3~3]; convergence radii R* and R~ are also larger than the corresponding ones in [3~3] denoted here by p and po; and finally our convergence rates are smaller. For example in case functions Lo, L are constants we obtain
1 P -~' LR* b - 1 - LoR* '
1 Lo+L' 1 Po - 3L '
q
Lp 1-Lp' 1 and R o* -- 3L0 .
(6.1.65)
Therefore we have: p < R* < 2p,
(6.1.66)
Po <_ R~),
(6.1.67)
and b < q,
(6.1.68)
where q is the corresponding ratio in Theorem 1.1 in [3~3]. Note that if strict inequality holds in (6.1.26) so does in (6.1.28)-(6.1.30). Moreover the computational cost for finding Lo, L in practice is the same as of that of finding just L. As an example, as in Section 5.1 consider real function F given by (5.1.~0). Then using (6.1.3) and (6.1.4) we obtain no = e - 1 < e = L, R*-.225399674,
and Po - .12262648.
Hence, we have" p
p-.183939721,
R~-.193992236,
198
6. More Results on Newton's Method
and po < R;. (c) As already noted in [3~3, p. ~11] and for the same reasons Theorem 6.1.5 is an essential improvement over Theorem 6.1.~ if the radius of convergence is ignored (see Example 3.1 in [3~3]).
6.2
Convergence under weak continuous derivative
We provide a semilocal convergence analysis of Newton's method (4.1.3) under weaker Lipschitz conditions for approximating a locally unique solution of equation (4.1.1) in a Banach space. For p C [0, 1] consider the following HSlder condition: I F'(xo)-l[F'(x)
-
F'(y)] II ~ ellx- yll p.
(6.2.1)
In Rokne [303], Argyros [11] the HSlder continuity condition replaced (6.2.1) or in Argyros [93] the more general I[F'(Xo)-l[F'(x)
-
F'(y)] II _< v(llx- YlI)
(6.2.2)
was used, where v is a nondecreasing function on [0, R] for some given R. In order to combine the Kantorovich condition with Smale a-criterion [296], [312] for the convergence of method (4.1.3) (when F is an analytic operator), Wang in [341], [343] used the weak Lipschitz condition
]]ft(x0)-l[f'(x) -F'(y)]II
~
f,lx-~,,+,,x-xo,I
L(u)du
(6.2.3)
JIIx-xoll
instead of (6.2.1). Huang in [195] noticed that (6.2.3) is a special case of the Generalized Kantorovich condition Ix--Y + I~-zl)
IlF'(xo)-l[F'(x) - F'(y)][[ ~ r1~(
L(u)du
(6.2.4)
Js(llx-yll) m
m
for all x C U1, y E U2 where U1 - U(xo, t*), U2 - U ( x o , t * - I l x t* C (0, R) is the minimum positive simple zero of
x011), and
s(t)
f(t) -
L
L(u)(t- s-l(u))du-
t + ~7,
(6.2.5)
with functions L, s' continuously increasing satisfying L(u) > 0, on [0, R], s(0) - 0 and s'(t) > 0 on [0, R), respectively. Under condition (6.2.4) a semilocal convergence analysis was provided and examples were given where the Generalized
6.2. Convergence under weak continuous derivative
199
Kantorovich condition (6.2.4) can be used to show convergence of method (4.3.1) but (6.2.1) cannot. Here, we provide a semilocal convergence analysis based on a combination of condition (6.2.6) (which is weaker than (6.2.4)) and on the center generalized Kantorovich condition (6.2.7). The advantages of such an approach have already been explained in Section 5.1. We need the definitions: D e f i n i t i o n 6.2.1 (Weak Generalized Kantorovich Condition). We say xo satisfies the weak generalized Kantorovich condition if for all 0 E [0, 1], x c U1, y E U2 IlF'(xo)-~[F'(x) - F ' ( x + O(y - x))]l [ _<
f
s~ (llx-zoll+Oll~-yll) Lx(u)du
(6.2.6)
.] Sl ( l l x - x o l l )
holds, where functions S l , L1 are as s, L respectively.
D e f i n i t i o n 6.2.2 (Center Generalized Kantorovich Condition). We say xo satisties the center generalized Kantorovich condition if for all x E U1 so(I x-xoll) I l F ' ( x o ) - l [ F ' ( x ) - F'(xo)]ll < Lo(u)du (6.2.7)
]i
holds, where functions so, Lo are as s and L respectively.
Note that in general
80(U) < 81(U) < S(U)
(6.2.8)
Lo(u) < L l ( u ) < L(u)
(6.2.9)
and
hold. In case strict inequality holds in (6.2.8) or (6.2.9) then the results obtained in [195] can be improved under (6.2.6) or even more under the weaker condition (6.2.7). Indeed we can show the following semilocal convergence theorem for method (4.1.3). We first define functions f0, fl, by so(t) fo(t) L o ( u ) ( t - s o l ( u ) ) d u - t -~- r/, (6.2.10)
fd0
f l (t)
- f./f181(t) Ll(U)(t-
811(u))du- t -4- ?7
(6.2.11)
and sequences an, tn by an+l
-- an
tn+l -- tn
fl(tn) f;(tn)
(n > 0),
(to
0),
(6.2.12)
f(tn)
(n _> 0),
(to -- 0).
(6.2.13)
f'(tn)
200
6. More Results on Newton's Method
T h e o r e m 6.2.3 Let F" D c_ X ---, Y be a Frdchet-differentiable operator. Assume: there exists xo E D such that F'(xo) -1 E L ( Y , X ) ; and condition (6.2.5) holds. Then sequence {xn} generated by method (~.1.3) is well defined, remains in V(xo, a*) for all n > 0 and converges to a unique solution x* of equation F(x) - 0 in V(xo, a*), where a* is the limit of the monotonically increasing sequence {an} given by (6.2.12). Moreover the following estimates hold: (6.2.14)
I xn+l - Xnl] <_ a n + l - a n <_ t n + l -- t n ,
(6.2.15)
IlXn -- X*ll <-- a* - a n <_ t* - t n , a n <_ tn ,
(6.2.16)
a* < t *
(6.2.17)
Ilxn - x* II < (r - a*)qo(n) 50 < -
1 -
qo(n)(~o
1 -
52 ~-1
a*, 1
-
(6.2.18)
-
and IIF'(x~
< 5~"~
(6.2.19)
where r E [a*, t**] satisfies f (r) <_ O, a*
t** -
max {t, f(t) < O}
a*<_t
~
($o- - - ,
(6.2.20)
7"
n-1
qo(n) - H (A~
1,
(6.2.21)
i=0
and
.o
(r - ai) fo Fs(ai+O(a*--ai))L(u)dudO Js(ai)
Ai
(a* - - a i ) [ f o
Lo(u)dudu JFs(ai+O(a*-ai)) s(ai)
-
fo(r)]
<
1.
(6.2.22)
Furthermore if U2 in weak generalized Kantorovich condition (6.2.6) is U(x, t** tlx-xoII)NO, then equation (~.1.1) has a unique solution in the larger ball U(xo, t**)A D.
Proof. We have noticed that in the proof of Theorem 1 in [195, p. 117] weaker condition (6.2.6)can be used instead of (6.2.4)to obtain estimates on IIF'(x0) -1F(Xn)II. Moreover center generalized Kantorovich condition can be used instead of (6.2.7) to obtain the more precise error bounds on IIF'(xn)-IF'(xo)II. The rest of the proof goes along the lines of Theorem 1 in [195, p. 117] by simply replacing L, s, tn, t*, q(n), (~, Ai, f ' by L1, 81, an, a*, qo(n), 50, )~o, f~ respectively and using (6.2.6), (6.2.7). 9
201
6.2. Convergence under weak continuous derivative
R e m a r k 6.2.1 If equality holds in (6.2.8), (6.2.9) and (6.2.6) is replaced by (6.2.4) our theorem reduces to Theorem 1 in [195, p. 117]. Otherwise our results are more precise under the same hypotheses/computational cost since in practice computing s, L requires the computation of s l, L1, So, Lo. We need the following result on majorizing sequences. For simplicity we set wo - f~
and
w-
f'.
(6.2.23)
L e m m a 6.2.4 Assume there ezist parameters rl >_ O, (~ E [0, 2) and functions wo, w such that:
[/oI~(Ov)dO - w(O) ] + 5wo(~) <_ ~,
h~ - 2
Wo 2 and
~
1-
(6.2.24)
(6.2.25)
<1,
2{folly[222(~(1--(~)n+l) (~)n+l] --W[22~](~(1--(~)n+l)l} n+l +0
~l dO
2r/
(6.2.26)
hold for all ~ > O. Then, iteration {b~} (n > O) given by
bo - O,
bl = r/,
fo { [w(b~ + +
b~+2 = b~+l O ( b ~ + t - b~)]dO - w ( b ~ ) l } ( b ~ + t
1 - wo(bn+l)
- b~)
(6.2.27)
is monotonically increasing, bounded above by t** --
2rl 2-5'
(6.2.28)
and converges to some b* such that
0 < b* < b**.
(6.2.29)
Moreover, the following error bounds hold for all n > O:
0 _< b~+2 - b~+x < ~(bn+~ - b~) <
(6.2.30)
202
6. More Results on Newton's Method
The proof is omitted as it follows along the lines of Lemma 5.1.1 We can show the main semilocal convergence result for method (4.1.3): T h e o r e m 6.2.5 Assume: hypotheses of Lemma 6.2.4; conditions (6.2.6), (6.2.7); and U(xo, b*)c_ D.
(6.2.31)
Then sequence {xn} (n > 0) generated by method (4.1.3) is well defined, remains in U(xo, b*) for all n > 0 and converges to a solution x* c U(xo, b*) of equation F(x) - O. Moreover, the following estimates hold for all n > O" IlXn+l -- Xn ]1 ~_ bn+l - bn
(6.2.32)
[[Xn
(6.2.33)
and -
x*
II ~ b* - b~,
where iteration {bn} (n _> O) is given by (6.2.27). U(xo,b*) if
The solution x* is unique in
f0
1[w((1 + O)b*) - w(b*)]dO + wo(b*) < 1.
(6.2.34)
Furthermore if there exists Ro such that b* < Ro,
(6.2.35)
U(xo, Ro) C_ D
and
ol[w(b + O(b* + Ro)) - w(b*)]dO + wo(b*) _< 1,
(6.2.36)
then the solution x* is unique in U(xo, Ro).
Proof. Using induction we show xk E U(xo, b*) and estimate (6.2.32) hold for all k _> 0. For n - 0, (6.2.32) holds, since Ilxl - Xo]l- IIF'(xo)-lF(xo)ll <_ ~7- bl - b0,
(by (33)).
Suppose (6.2.32) holds for all n - O, 1,..., k + 1; this implies in particular that k
k
IlXk+l -- XoII ~ E
[]Xi+l -- Xill ~ E ( b i + l i=0 i=0
- hi) -
bk+l -- bo - bk+l.
We show (6.2.32) holds for n - k + 2. By (6.2.7) and (6.2.25) we get
]]F'(xo)-i[F'(xk+a)- F'(zo)]]]
<
Wo([IXk+~-Xolf)<_wo(bk+l) <
1. (6.2.37)
203
6.2. C o n v e r g e n c e u n d e r weak c o n t i n u o u s d e r i v a t i v e
It follows from (6.2.37) and the Banach Lemma on invertible operators 1.3.2 that Ft(xk+l) -1 exists and
xoll)] -~ ~ [1 -
wo(bk+l)] - 1 .
- Ft(xk)(Xk+l
-- Xk) -- r(xk)}ll
< [1 - Wo(llXk+l -
IlF'(Xk+l)-lF'(xo)ll
(6.2.38)
By (4.1.3), (6.2.6), and (6.2.23) we obtain IIF'(xo)-lF(xk+l)l]llF'(xo)-l{r(xk+l)
~ fO 1 [[ft(xo)-l[Ft(xk -Jr-O(Xk+l -- Xk)) -- F'(xk)][[ IlXknt-I -- xklldO <
foo1[w([[xk -
<
~001[w(bk
xoll + OIIxk+a -- x k [ I ) - W(IIXk -- Xoll)]dOl[xk+l - xkll
(6.2.39)
+ O(bk+l -- bk)) -- w(bk)]dO(bk+l - bk).
Using (4.1.3), (6.2.38) and (6.2.39) we get
Ilxk+2- Xk+~ll ~ ]lU(Xk+l)-l f'(xo)ll 9 ]lF'(xo)-lF(xk+l)][ < f l [ w ( b k + 0(bk+l - bk)) - w(bk)]dO(bk+l - bk) -
1 - wo(bk+l)
(6.2.40)
= bk+2 - bk+l.
Note also that:
I Xk-t-2--Xo[I "( I[Xk-[-2--Xk+IlI-I-IlXk-I-I--Xo] ~ b k + 2 - b k + l + b k + l - b o - bk+2 <_ b*. That is xk+2 E U ( x o , b*). It follows from (6.2.40) that {xn} (n >_ 0) is a Cauchy sequence in a Banach space X, and as such it converges to some x* E U(x0, b*). By letting k -+ c~ in (6.2.39) we obtain F ( x * ) - O. Moreover, estimate (6.2.33) follows from (6.2.32) by using standard majorization techniques. Furthermore, to show uniqueness, let y* be a solution of equation F ( x ) - 0 in V ( x o , Ro). It follows from (4.1.3)
Ily*-x~+~ll ~
[[F'(xk)-lF'(xo)[[
/o 1 < fo[~(llxk -
IPy* - x~llde
- x011 + 011y* - xkll) - w(llxk - xoll)]dO 1 - wo(llxk
< ~01[w(b * + 0(b* + R 0) ) - ~ ( b * ) ] d 0 1 - wo(b*)
< IlY* - xkll
(by (46)).
lly* - x~]l
- xoll) liy* - x~ II
(6.2.41)
204
6. More Results on Newton's Method
Similarly if y* E U(xo, b*) we arrive at Ily* - ~k+,ll _
f~
+ Oh*)- w(b*)]dOIlY* -
~kll,
1 - wo(b*)
(6.2.43)
(by (44)).
< Ily*-xkl[
(6.2.42)
In both cases above lira xk -- y*. Hence, we conclude k---~oc X :~ ~
y:~.
We can now compare majorizing sequences {tn}, {an}, {bn} under the hypotheses of Theorem 1 in [195, p. 117], and Theorems 6.2.3, 6.2.5 above" L e m m a 6.2.6 Under the hypotheses stated above and if strict inequality holds in (6.2.7) or (6.2.8) then the following estimates
(6.2.44)
bn+l < an+l < tn+l bn+ l - bn < an+ l
-
-
(6.2.45)
an < tn+ l -- tn
(6.2.46)
b* - bn < a* - an < t* - tn and
b* _< a* _< t*
(6.2.47)
hold for all n > O.
Proof. It follows easily by using induction, definitions of sequences {tn}, {an}, {bn} given by (6.2.12), (6.2.11), (6.2.27)respectively and hypotheses (6.2.8), (6.2.9). 9 R e m a r k 6.2.2 The above lemma justifies the advantages of our approach claimed at the beginning of the section.
6.3
Convergence
and
universal
constants
We improve the elegant work by Wang, Li and Lai in [342], where they gave a unified convergence theory for several Newton-type methods by using different Lipschitz conditions and a finer "universal constant". It is convenient to define majorizing function f and majorizing sequence {t~} by f ( t ) - r] - t +
~et 2 + ~ L t 3
(6.3.1)
and
tn+l -- tn
(t,) ff(tn)'
to
-
o
respectively where r/, t~, L are nonnegative constants.
(6.3.2)
205
6.3. Convergence and universal constants
P r o p o s i t i o n 6.3.1 [342, p. 207] Let 2
(6.3.3)
g + v/g 2 + 2L and
b - 2(t~ + 2v/t~2 + 2L). 3(~ + v/~2 + 2L) 2
(6.3.4)
Then, (a) function f is monotonically decreasing in [0, r], while it is increasing monotonically in Jr, +oc). Moreover, if ~l <__b then < O, f ( + ~ ) > r / > 0 .
f (~l) > O, f (r) - r l - b
Therefore f has a unique zero in two intervals denoted by t* and t** respectively, satisfying r
~/< t* < ~7/< r < t**,
(6.3.5)
when rl < b and t* - t**, when rl - b; (b) Iteration {t~} is monotonically increasing and converges to t*.
From now on we set D - U(xo, r). We can define a more precise majorizing sequence {Sn} (n > 0) by L ( S n + l -- 8 n ) -~- L s n -~8n+2 ~ 8n+1 -~- 3 2(1 -- ~ 0 8 n + l )
(Sn+l - sn) 2,
(6.3.6)
80 - - 0 , 81 --?~, for s o m e constant g0 C [0, g].
Iteration {tn} (n >__0) can also be written as L(tn+l
-- t n ) + Ltn +
tO -- 0, tl -- r].
t n + 2 -- t n + l -+- 3 L t n2+ l ] 211 -- ~ t n + l -- -~
(6.3.7)
'
Then under the hypotheses of Proposition 6.3.1, using (6.3.6), (6.3.7) and induction on n we can easily show: P r o p o s i t i o n 6.3.2 Under the hypotheses of Proposition 6.3.1, sequence {sn} is monotonically increasing and converges to some s* E (7, t*]. Moreover the following estimates hold: s~ < t ,
(n > 1)
O
(6.3.8) (n>O)
(6.3.9) (6.3.10)
and
s* < t* provided that L > 0 if ~o - g.
(6.3.11)
206
6. More Results on Newton's Method
Let xo e D such that F'(xo) -1 E L(Y, X ) . We now need to introduce some Lipschitz conditions: I[F'(xo)-l[F'(x) -
F'(xo)]ll _ 61Ix- xol,
IlF'(Xo)-l[F'(y) - F'(x)]ll ~
[e + L I I x -
for all x c U(xo, r)
1
(6.3.12)
1
xoll + -~LllY- xll IlY- xll, (6.3.13)
for all x, y" [lY- xll + ttx - XoI[ <_ r,
-'(.o)1 .. _< (e + LII -
IlF'(xo)-l[F'(x)
(6.3.14)
for all x E U(xo, r), IlF'(Xo)-l[F'(x) - F'(xo)]ll <_ g l l x - xo]l, for all x C U(xo, r)
(6.3.15)
IlF'(xo)-lF"(xo)ll < g
(6.3.16)
[[F'(xo)-lF"(Xo)[[ < go
(6.3.17)
IIF'(Xo)-l[F"(v)--F"(m)]ll
<_ Llly-xlI, for all Y,m" l v-xll+llm-xoll <_ r, (6.3.18)
[[F'(xo)-l[F"(x) - F"(xo)]ll <_ LIIx- xoll, for all x E U(xo, r)
(6.3.19)
and [l[i"(xo)-l[F"(x)
- F"(xo)]ll ~ L o l l x - Xoll, for 311 x c g(xo, r).
(6.3.20)
Note that we will use (6.3.12), (6.3.17), (6.3.20)instead of (6.3.15), (6.3.16) and (6.3.19) respectively in our proofs. We define as in [342]
K(1)(xo, e) - {F K(1) c~nt(Xo, e) - {F K (1) (~o, e, L) - {F K(1) cent(xo,f,L)- {F
~ Cl(xo,r) 9 (6.3.13) hdds for L - 0}, ~ C 1(xo, r)" (6.3.15) holds}, e C 1(~o, ~)" (6.3.13) holds}, c c l ( x o , r ) 9 (6.3.14) holds},
K(2)(xo, e, L ) - { r C C2(xo,r)" (6.3.16) and (6.3.18) hold}, K(u) cent(Xo, e, L) - { F C C2(xo, r)" (6.3.16) and (6.3.20) hold}. Moreover for our study we need to define:
K(1)(Xo,~o,~,L) -- {F E C l(x0,p)" (6.3.12) and (6.3.13) hold}, K(1) cent(Xo, g.o) - { F E C 1(xo, r)" (6.3.12) holds},
K(U)(mo,6,g,L)-
{F e C2(xo,r) 9 (6.3.12), (6.3.17) and (6.3.18) hold or (6.3.12), (6.3.16)and (6.3.18)hdd}.
We can provide the main semilocal convergence result for method
(4.1.3).
207
6.3. Convergence and universal constants
T h e o r e m 6.3.3 Suppose that F C K(1)(xo,go, g,L). If ~ <_ b, then s e q u e n c e { X n } generated by method (~.1.3) is well defined, remains in U(xo, s*) for all n >_ O, and converges to a solution x* c U(xo, r) of equation F(x) - O. Proof. Simply use (6.3.12)instead of (6.3.15) in Lemma 2.3 in [342] to obtain F'(x) -1 exists for all x E U(x0, r) and satisfies 1
IlF'(x)-lF'(Xo) I1 < -
(6.3.21)
eollx- x0fl
The rest follows exactly as in Theorem 3.1 in [342] but by replacing {tn}, {t*}, (6.3.11) by {Sn}, {S*}, (6.3.21)respectively. tt
1
T h e o r e m 6.3.4 Suppose that F E Kcent(Xo,~o ). If ~l < b, then the equation * F(x) - 0 has a unique solution x* in U(xo So), where s o* -- 8" if ~ -- b, and s~) E [s*,t*) if ~] < b. Proof. As in Theorem 3.2 in [342] but used (6.3.12)instead of (6.3.14). tt By Proposition 6.3.2 we see that our results in Theorems 6.3.3 and 6.3.4 improve the corresponding ones in [342] as claimed in the introduction in case g0 < g. At this point we wonder if hypothesis ~/< b can be weakened by finding a larger "universal constant" b. It turns out that this is possible. We first need the following lemma on majorizing sequences: L e m m a 6.3.5 Assume there exist nonnegative constants go, ~, L, ~l and a parameter 5 c [0, 2) such that for all n > 0
L ( 25) 2nT]_~_2L[12_5-(~)n]()n 2(~ ?]2-Jv~(~)n ?7 + 2-5
1-
(6.3.22)
r/<5.
Then, iteration { S n } given by (6.3.7) is monotonically increasing, bounded above by s** - 22'7 - 5 and converges to some s* such that -
-
0 <__ s* <__ s**.
(6.3.23)
Moreover the following estimates hold for all n > O"
n+l 0 < Sn+ 2 -- S n + l <_ -~(8nA-1 -- 8n) <~
r/.
(6.3.24)
Proof. The proof follows along the lines of Lemma 5.1.1. However there are some differences. We must show using induction on the integer k > 0 L -- (sk+l -- sk) 2 + Lsk(sk+l -- sk) + ~(8k+1 -- 8k) -[- ~0(~8k+1 < (~, 3
(6.3.25)
208
6. More Results on Newton's Method 0 ~ 8k+l
-
-
(6.3.26)
8k~
and (6.3.27)
0 < 1 - ~osk+l.
Estimates (6.3.25)-(6.3.27) hold for k - 0 by (6.3.22). But then (6.3.6) gives (6.3.28)
0 ~ 82 -- 81 ~ ~(81 -- S0).
Let us assume (6.3.24)-(6.3.27) hold for all k <_ n. We can have in turn" L
"~'(8k+1 -- 8k) 2 nt- L S k ( S k + l -- 8k) nt- ~(8k+l -- 8k) + ~0(~8k+1
L
6
+L.
_
1--(~)
+ t~0(~
1
k
1-( 5 -
(6.3.29)
~
~ + ~
k+l
52
~ < ~ by (6.3.22) -
We must also show (6.3.30)
Sk ~ 8"*.
For k - 0, 1, 2 we have s o - 0 < s_* *
,
81
-
?]<8
5 **, a n d s 2 - ~ + ~ < s * *
_
.
Assume (6.3.30) holds for all k _< n. It follows from the induction hypothesis (6.3.31) 5 k+l
1-(7) 1 5 2
2~/ r/<- 2 - ~ -s**
Moreover (6.3.27) holds from (6.3.22) and (6.3.31). Inequality (6.3.26) follows from (6.3.6), (6.3.27) and (6.3.25). Hence, sequence {Sn} is monotonically increasing, bounded above by s**, and as such it converges to some s* satisfying (6.3.23). w R e m a r k 6.3.1 Condition (6.3.22) can be replaced by a stronger but easier to verify condition
___ bo,
(6.3,32)
6.3. Convergence and universal constants
209
where,
2
bo =
+ L
(6.3.33)
+
( 5 - v/5)
g /3- ~ +
(6.3.34)
2go ,,:. 3 v'b
(6.3.35)
1.
(6.3.36)
for - ~-
Indeed (6. 3. 22) certainly holds if _
3 2
~/2
r/2
+
2"yo~ ~ 7 + 2 - 5 ~/-<5
+
(6.3.37)
since,
[1--(~)n+l] (~)
n+l < [1--(~)n I (~)n
(6.3.38)
holds for all n > 0 if 5 e [0, v/5- 1]. However (6.3.37) holds by (6.3.32).
Hence, we showed" Lemma 6.3.6 /f (6.3.22) replaces condition (6.3.22) in Lemma 6.3.5 then the conclusions for sequence { Sn } hold.
Remark 6.3.2 In Lemma 6.3.5 we wanted to leave (3.4) as uncluttered as possible, since it can be shown to hold in some interesting cases. For example consider Newton's method, i.e. set L - O. See Application 5.1.~ in Section 5.1. Note also that bo can be larger than b, which means that (3.1~) may hold but not n < b. Indeed, here are two such cases: choose (a) L - 1, "~o = " y - .5 and r l - .59 m
then b - . 5 8 3 (b) L -
and
bo-.590001591
.8, " y - .4 and ~l = .7
then b - . 6 8 2 7 6 2 4 2 3
and
bo-.724927596.
Below is the main semilocal convergence theorem for Newton's method involving Fr~chet differentiable operators and using center-Lipschitz conditions.
6. More Results on Newton's Method
210
T h e o r e m 6.3.7 Assume: (a) F C K (1) (xo, Co, e, L), and (b) hypotheses of Lemmas 6.3.5 or 6.3.6 hold. Then, sequence {xn} (n >_ 0) generated by method (~.1.3) is well defined, remains in U(xo, s*) for all n > O, and converges to a solution x* e U(xo, s*) of equation F(x) - O. Moreover the following estimates hold for all n>O: m
1
L[611X~+l - Xnll + ~IlXn -- Xoil] + Z 2 ]Xn+l -- Xn 12 IlXn+U - X~+lll __ 1 - "ToIIXn+I--XoII
(6.3.39) < L[l(sn-t-1 - 8n)-~- ~(8n -1
-
~0(Sn_t_l
-
8 0 ) ] -~- 2 ( 8 n + 1
_
8n)2
80)
--- 8n+2 -- 8n+l~
and IlXn -- X*
(6.3.40)
II ~ S* -- ~ ,
~h~, ~q~~
{~} (~ >_ o) i~ 9~wn by (6.3.6).
The solution x* is unique in
U(xo, s* ) if ?s* < 1.
(6.3.41)
Furthermore, if there exists R > s* such that U(xo, R) c_ D, and (6.3.42)
?o(s* + R) _< 2,
the solution x* is unique in U(xo, R). P r o o f . Let us prove that IIx~+~ - xk II -- Sk+l
--
(6.3.43)
8k
and U(Xk+I
~ S :r
-- 8 k + l )
C_
(6.3.44)
U(xk 8 -- 8k) ~
hold for all k > 0. For every z E U(Xl II Z -- X011 _~ II z -- X l l l - t - I l X l
~ S :r
--
Sl)
-- X011 --~ 8" -- 81 -Jr- 81 - - S* -- S0
implies z E U i ( X l , 8 * - Sl). Since also
[[/1-/oil- [IF'(xo)-lF(xo)[ <_ ~ - so, (6.3.43) and (6.3.44) hold for k - 0. Given they hold for n - 0, 1 , . . . , k, then k+l
I l X k + i - XOII ~_ E i--1
k+l
Ilxi --Xi-lll ~_ E ( S i i--1
8i-1)-
8 k + l - S O - 8k+1
211
6.3. Convergence and universal constants
and IlXk -4- O(Xk4-1 -- X k ) -- XOII ~ 8k Jr- 0(8k+1 -- 8k) < 8",
0 E
[0, 1].
Using (4.1.3) we obtain the approximation F(Xk+l)
-- F ( X k + l )
/o
--
- f(Xk)
- ft(Xk)(Xk+l
(6.3.45)
-- X k )
[ f z ( X k -4- O ( X k + l -- X k ) ) -- F t ( X k ) ] ( X k - t - 1
-- X k ) d O .
Then, we get by (6.3.12), (6.3.13) and (6.3.45) I[ft(Xo)-l
< L
f(xk+l
{1o L
(6.3.46)
)ll
[llxk -- XoII + OIIXk+l -- Zkll](1 -- O)dO + L
--< --I[Xk+16 -- xkll3 + ~llxk - Xo[[ [[Xk+l - - X k . . 2- I-I~ <
{
L
[~
1
(8k+1 -- 8k) -Jr- ~ ( S k -- 80)
]
-~}
+ ~
IlXk+l -- xkll 2
_-~l]Xk+l ")/
-- Xk[[
2
2
(8k+1 -- 8k) 9
Using (6.3.12) we get IIF'(xo)-l[F'(xk+~) - r'(xo)]ll ~ ~ollmk+l - xoll
(6.3.47)
_< 7osk+l <_ 7s* < 1 (by (6.3.22)). It follows from the Banach Lemma on invertible operators 1.3.2 and (6.3.22) that F1(Xn+l) -1 exists so that" IfF'(xk+~)-~F'(xo)ll < [1 - ~ollx~+~ - xoll] -1 ~ (1 - ~/otk+l) -1.
(6.3.48)
Therefore by (6.3.2), (6.3.46), and (6.3.48) we obtain in turn Ilxk+2 - xk+lll = I I ( F ' ( x k + I ) - I F ' ( x o ) ) ( F ' ( x o ) - I F ( x k + I ) ) I I < I l r ' ( x k + ~ ) - l r ' ( x o ) ] ] . [[FZ(xo)-IF(xk+I)][
(6.3.49)
~_ 8k+2 -- 8k+1.
Thus for every z E U(xk+2
S ~
sk+2) we have
IIz -- Xk4-111 ~ IIZ -- Xk+211-Jr- IlXk+2 -- Xk+l]] _~ 8* -- 8k+2 -- 8k§
(6.3.50)
~ 8k-t-1 ~ 8" -- 8 k § 1.
That is m
Z C V(Xk§
, 8* -- 8k+1).
(6.3.51)
212
6. More Results on Newton's Method
Estimates (6.3.49), (6.3.50) imply that (6.3.43), (6.3.44) hold for n - k + 1. The proof of (6.3.43) and (6.3.44) is now complete by induction. Lemma 6.3.5 implies {tn} (n > 0)is a Cauchy sequence. From (6.3.43), (6.3.44) {Xn} (n > 0) becomes a Cauchy sequence too, and as such it converges to some U(xo, s) such that (6.3.40) holds. The combination of (6.3.40) and (6.3.46) yields F ( x * ) - O. Finally to show uniqueness let y* be a solution of equation F ( x ) - 0 in U(xo, R). It follows from (6.3.12), the estimate
Ft(Xo) -1
lEt(y* + O(X* -- y * ) ) - Ft(Xo) ] dO
fo
_< ~o
/o 1 ly* + O(x*
- y*) - xolldO
70f01 [01Ix* - ~oll + (1 - 0)lly* - xoll]d0 < -~0-(s* + R)_< 1
z
(by (6.3.42)),
and the Banach Lemma on invertible operators that linear operator
L- o 1r ' ( y *
+ O(x* - y*))dO
is invertible. Using the identity 0 - F ( x * ) - F(y*) - L(x* - y*),
we deduce x* - y*. Similarly we show uniqueness in U(xo, s*) using (6.3.41)... As in [342] we also study the semilocal convergence of King's iteration [208]
Xn+l -- Xn -- F'(xm[n/m])-lF(xn)
(rt ~ O) (xo E D)
(6.3.52)
where [~] denotes the integer part of ~ , and Werner's method [345]
Xn+l - - z n - F ' ( y n ) - l F ( z n )
(6.3.53)
Yn+l -- Zn+l -- 12 f ' ( y n ) - l f ( X n + l )
(6.3.54)
Yo - xo.
(6.3.55)
Both methods improve the computational efficiency with (6.3.53)-(6.3.55) being more preferable because it raises the convergence order to 1 + x/~, although the number of function evaluations doubles when compared with Newton's method. Using (6.3.12) instead of (6.3.14) or (6.3.19) to compute upper error bounds on I [ F ' ( x ) - l F ' ( x o ) l l (see e.g. (6.3.48)) we can show as in Theorems 4.1 and 4.2 in [342] respectively:
6.~. A Deformed Method
213
T h e o r e m 6.3.8 Let F E K(1)(xo,~/o,~/,L) and assume hypotheses of Lemmas 6.3.5 or 6.3.6 hold. Then sequence {x~} generated by King's method (6.3.52) converges to a unique solution x* of equation F ( x ) - 0 in U(x0, s*). T h e o r e m 6.3.9 Let F E K (2) (xo, ~o, ~, L) and assume hypotheses of Lemmas 6.3.5 or 6.3.6 hold. Then sequence {x~} generated by Werner's method (6.3.53)-(6.3.55) converges to a unique solution x* of equation F ( x ) - 0 in U(x0, s*). Note that finer error bounds and a more precise information on the location of the solution x* are obtained again in the cases of Theorems 6.3.8 and 6.3.9 since more precise majorizing sequences are used.
6.4
A Deformed Method
We assume that F is a twice continuously F%chet-differentiable operator in this section. We use the following modification of King's iteration [208] proposed by Werner [345]" --1
Xn+l - - x n - F I
( xnq-Zn)2
Zn+l ~ Xn+l -- F'
(
Xn +2 Zn
(x0 E D) ( z 0 - x0) (n > 0)
F(xn),
)_1
(n > 0).
F(Zn+l)
(6.4.1)
(6.4.2)
Although the number of function evaluations increases by one when compared to Newton's method, the order of convergence is raised from 2 to 1 + v ~ [345]. Here in particular using more precise majorizing sequences than before we provide finer error bounds on the distances IIX~+l- x~ll , IIx~- x*ll and a more precise information on the location of the solution x*. Finally, we suggest an approach for even weakening the sufficient convergent condition (6.4.8) for this method along the lines of the works in Section 5.1. Let fl > 0, "y > 07 and 3'0 > 0 be given parameters. It is convenient to define real function f by f (t ) - fl - t + 1 -7------~' t e
[1]
0, -~ '
(6.4.a)
and scalar sequences {tn}, {tn) (n >_ 0) by tn+l - tn - f ' ( S n ) - l f ( t n )
Pn+l -- tn+l - f ' ( s n ) - l f ( t n + l ) -
tn+2 -- ~n+l --
f n + l
70~n -- 1
(6.4.4)
to -- ro -- O,
-
Sn --
to -- TO -- O,
tn -]- rn
-
tl = fl,
(6.4.5) (6.4.6)
214
6. More Results on Newton's Method
Yn+l
-
-
9n+l
tn+l
--
_8 n
7o~-
--
tn + Yn
1
2
1 --
Y,)2(1 - O)
(6.4.7)
where, jfO1
~/~n -- 7 ( t n + l
fo 1 fo 1
1 +
27(tn+
[1 --
gn+l --
O)
672(yn --tn)3Op(1-
[1 --
4
dO
-- ~n)O] 3
")/tn -- ")/(~n -- t n ) ( ~
o
+ V(1
dpdO ,
and fn+l -- ( f t ( ~ n + l )
-- ft(-gn))(Yn+l -- t n + l ) .
We need the following lemma on majorizing sequences: L e m m a 6.4.1 Assume: ~ - 9.r <_ a -
(6.4.8)
2 v~,
and
(6.4.9)
70 - 27.
Then iterations {tn}, {tn} are well defined and the following estimates hold: O--to--ro
tn
<
<
rn
<
tn+l
<
rn+l
<
O - - t o - - Y O < tn < Yn < tn+l < Yn+l <
t*
(~_> o),
(6.4.10)
(n _> o),
(6.4.11)
L < tn
(n >_2)
(6.4.12)
~n < r ~
(n___2)
(6.4.13)
t n + l - t n < tn+l--tn
Yn+l
-- t n + l
<
rn+l - tn+l
t*-tn <_t*-tn
(6.4.14)
(n >_ 1),
(6.4.15)
(n >_ 1),
(6.4.16)
(n>_O),
(6.4.17)
~*
F
-
(6.4.18)
lim t~, n--+ oo
and
t*-
lim t n = n ----~(ND
1 + c~ - V/(1 + c~)2 - 8c~
47
_< 1+
1
7
(6.4.19)
215
6.4. A Deformed Method
Proof. Estimates (6.4.10) and (6.4.19) follow using only (6.4.8) (see Lemma 2 in [177, p. 69]. The rest of the proof follows immediately by using induction on n and noticing that
1
1
27t- 1 <
[ (
for all t e 01 1 -
ft(t )
.~)1]
.
(6.4.20)
Indeed (6.4.20) reduces to showing" O
3 27
which is true by the choice of t. 9 R e m a r k 6.4.1 It follows from Lemma 6.J.1 that if {tn} is shown to be a majorizing sequence for {x~} instead of {t~} (shown to be so see Theorem 1 [179]) then we can obtain finer estimates on the distances IlZn -- ~ n l ,
IIx~+~ -- Z~ l, IIx~ -- X*II,
~n -- X*II
and an at least as precise information on the location of the solution x*. We also note that as it can be seen in the main semilocal convergence theorem that follows all that is achieved under the same hypotheses/computational cost:
T h e o r e m 6.4.2 Let F" D c_ X ~ Y be a thrice continuously Frdchet differentiable operator. Assume there exist xo c D, b _> 0, 7 > 0, 70 > 0 such that Ft(xo) -1 E L(Y,X),
If'(xo)-lF(xo)l
~ 3,
(6.4.21)
1Ft(xo)-lFU (xo) <_~, -~
(6.4.22)
IIF'(xo)-I[U(x) - F'(xo)] ] ___ ~ollx - ~oll
1 F' (xo) - 1Fut
g
72
(x)
__ (1 -
~/llx
-
xoll) 4
(6.4.23) (6.4.24)
for all
_( ( _( (1)1) x~U
U xo,
xo,
1-~
(6.4.25)
I-~
CD
(6.4.26)
and hypotheses of Lemma 6.~.1 hold. Then iterations {Xn}, {zn} (n > 0) generated by method (6.~.2) are well defined, remain in U(xo, t*) for all n > O, where t* is
216
6. More Results on Newton's Method
given by (6.4.19) and converge to a unique solution x* of equation F(x) - 0 in m
1
[lZn -- Xn[I ~_ ~ n - - t n I I X n + l -- Zn[I ~ Ilxn -- x*
[1 <
(6.4.27)
< r n -- t n
tn+l
< t* - tn
(6.4.28) (6.4.29)
~_ t* -- r n.
(6.4.30)
-- ~ n < t n + l
F - ~
-- r n
and [[Zn --
X'It ~_ -{* ---~n
Proof. We only need to show that the left-hand side inequalities in (6.4.27) and (6.4.28) hold. The rest follow from Lemma 2.1, standard majorization techniques and Theorem 1 in [179]. Let x c U(xo,t*). It follows from (6.4.9), (6.4.19) and (6.4.23) that [[Ft(xo)-l[F'(x)-
Ft(xo)][[
~
2"/ilx- xo[f ~ 2~t* < 1.
(6.4.31)
Using the Banach Lemma on invertible operators [6] and (6.4.31) we conclude that F'(x) -1 exists and 1
[[F'(x)-lF'(xo)I[ < -
1 -
2~lix-
(6.4.32)
~01f
By (6.4.2), (6.4.3)-(6.4.7), (6.4.20)above and (21)-(23)in Theorein 1 in [179, p. 70] we arrive at IlZn+I -- Xn+l ]t <
gn+l --
--
2~-$n-
1
f(tn+l) : m-t-1 -- tn-k-1 ~ - - ~
:
rn+I
-- tn-}-I
ft(Sn)
(with g~, t~ replacing rn, tn respectively in the estimate above (23) in [179]). Note also that (6.4.27) holds for n - 0, and by the induction hypothesis
ll~+l--~01[
<_
<_
1--
--. -y
Therefore the left-hand side of (6.4.27) holds for n + 1. We also have [[Zn+i -- Xol[ ~_ r n + l
~_ t* ~ t* ~_
( 1)1 1- -~
Hence, the same is true for Yn+I. Moreover by (24), (25) in [179], and (6.4.20) and (6.4.32) above we have Ilx~+~ - z~+~I! < --
1 2~8n-
(h'(~+~)-
ht(~n))(~n+l - t n + l )
1
= t n + 2 -- ~ n + l .
Hence, the left-hand side of (6.4.28) holds for n + 1. The rest of the proof follows exactly as in Lemma 6.4.1 with g~, tn, F replacing r~, t~ and t* respectively. 9
6.5. The Inverse Function Theorem
217
R e m a r k 6.4.2 (a) Let us denote by {~n} (n > 0) scalar sequence generated as (6.~.6), (6.~.7) by with tn, an, Pn, 9n+1 replaced bytn, an, rn, f(tn+l), respectively. It follows from the proof of Theorem 6.~.2 that {/~} is also a majorizing sequence for {x~} so that
0 - to - f o tn < tn,
< t , < ~ < tn+l ( fn+l ( t rn < r~
and
(n 2 0)
t < t*
where, t
-
lim
t~.
n - - + (N2
(b) It is expected that since sequence {tn} is finer than {tn} condition (6.~.8) can be dropped as a sufficient convergence condition for {-tn}, and be replaced by a weaker one. We do not pursue this idea here. However we suggest that such a condition can be found if we work along the lines of the more general majorizing sequences studied in Section 5.1.
6.5
T h e Inverse F u n c t i o n T h e o r e m
We provide a semilocal convergence analysis for Newton's method in a Banach space setting using the inverse function theorem. Using a combination of a Lipschitz as well as a center Lipschitz type condition we provide: weaker sufficient convergence conditions; finer error bounds on the distances involved, and a more precise information on the location of the solution than before [340]. Moreover, our results are obtained under the same or less computational cost, as already indicated in Section 5.1. Let x0 9 D such that U(xo, r) c_ D for some r > 0, and F'(xo) -1 9 L(Y, X ) . The inverse function theorem under the above stated conditions guarantees the existence of c > 0, operator F ' ( x o ) - l : U(F(xo),C) C_ Y --, X such that F'(xo) -1 is differentiable, (6.5.1)
F'(xo)-I (F(xo))=xo, and
F(F'(xo)-l(z)) = z
(6.5.2)
for all z 9 U(F(xo),C).
In particular in [341] the center Lipschitz condition
IlF'(xo)-~[F'(x)- F'(xo)]ll ~
~os(x)L(u)du
for all x 9 U(xo,r)
(6.5.3)
6. More Results on Newton's Method
218
was used, where s(x) -I1~-~0ll, and L is a positive integrable function on (0, r] to first study the semilocal convergence of the modified Newton method. Then using the same function L and the Lipschitz condition s(x,y)
[[F'(xo)-I[F'(x) - F'(y)]I] <
a~(z)
(6.5.4)
L(u)du
m
for all x E U(xo,r), y E U ( x , r - s(x)), where s ( x , y ) - s ( x ) + I Y xl] < r, the semilocal convergence of method (6.5.2) was studied in a unified way with the corresponding modified Newton's method. We assume instead of (6.5.3)"
IlF'(xo)-l[U(x)-
U(xo)]ll_ ~os(x) Lo(u)du
for all x E U(xo,r).
(6.5.5)
By Banach's Lemma on invertible operators 1.3.2 and
IIF'(xo)-I[U(x)- f ' ( x o ) ] l l _
f
~(x)
Lo(u)du < 1,
(6.5.6)
dO
for all r ~ E (0, r], x E U(xo, r~
[/o
IIF'(x)-lF'(xo)ll <_ 1 -
F'(x) -1 exists, and -1
Lo(u)du
]
(6.5.7)
with r ~ satisfying
fo '~ Lo(u)du - 1.
(6.5.8)
By simply using (6.5.5) instead of (6.5.3) we can show the following improvements of the corresponding results in [340]" T h e o r e m 6.5.1 Set bo - fo ~ Lo(u). Assume condition (6.5.5) holds for all r > r ~
Then the following hold:
U F(xo),
bo I) C r(U(~o, ~o)). iIF,(Xo)_~l
Ft(Xo) -1 exists and is differentiable on
u
( f(xo), iiF,(xo)_ll bo I) ,
and the radius of the ball is the best possible (under condition (6.5.5)).
(6.5.9)
219
6.5. The Inverse Function Theorem
L e m m a 6.5.2 Let t
fo(t) - / 3 -
t+
j~
L o ( u ) ( t - u)du
t e [o, R],
(~.5.1o)
where Ro satisfies
l foR~Lo(u)(Ro -
Ro
(6.5.11)
u)du - 1.
If ~ e (0, bo), then fo is monotonically decreasing on [0, r~ monotonically on [r~ R] so that fo(/3) > O, fo(r ~ - / 3 -
while it is increasing
bo < O, fo(Ro) - / 3 > O.
(6.5.12)
Moreover, fo has a unique root in each interval, denoted by r ~ and r ~ satisfying:
ro
(6.5.13)
Z < ~1~ < Voz < ~o < ~o < Ro.
If (6.5.5) is extended to the boundary, i.e. /i s(x) llF'(xo)-~[F'(x) - F'(xo)]ll _<
no(u)du
x C U(xo, r),
(6.5.14)
according to Lemma 6.5.2, Theorem 6.5.1 implies a more precise result" Proposition 6.5.3 If ~o
0 < 3 < bo -
f
(6.5.15)
Lo(u)udu,
JO
and
> ~o
(6.5.16)
then
-v( F(xo), iiF,(xo)_ Z ~11) c_F(~(xo,~~ F'(xo) -1 exists and is differentiable on-U(F(Xo), iir,(x~o)_l ); [F'(xo)-l]t(y)
-
f ' ( x ) -1,
y -
f(x)
satisfies (6.5.7), and radius r~ is as small as possible.
(6.5.17)
220
6. More Results on Newton's Method
P r o p o s i t i o n 6.5.4 Assume"
~ [~o, ~o], /3 C (0, bo) and condition
U
(6.5.14)
hold. Then
n r(U(xo, ~) \ ~(xo, ~o)) _ o
F(xo), ilF,(xo)_l]]
(6.5.18)
In particular if y - 0 and/3 - I I F ' ( x o ) - ~ r ( x o ) l l , we get" Theorem
6.5.5 Assume:
8 <- bo, ~
[~o, ~o],
if 9 < bo o~ ~ - ~1~ i f 9 -
bo.
Then, under condition (6.5.1~), then equation F ( x ) - 0 has a unique solution
x* c U(xo - F ' ( x o ) - l F ( x o ) , r ~ - / 3 ) C_ U(xo, rl~ in the ball U(xo, r).
Remark
6.5.1 Denote by ro, b, f , R, rl, r2 (the quantities used in [3~0]) corresponding to r ~ bo, fo, Ro, r ~ and r ~ Then in case strict inequality holds in (6.1.1~) we have ro
b
~o < ~,
~nd
fo(u)
u e [0, r], R < R o ,
(6.5.19)
~ < ~o,
(6.5.20)
which improve the corresponding results in [340], and under less computational cost since in practice the computation of L requires the computation of Lo. In case Lo, L are constants we get
1 ro-~,
1 ro~
fo(u) - / 3 - u + rl =
1 b-~-~,
Lou 2,
2~ 1 + V/1 -- 2L/3' 2/3
rlo = 1 + v/1 - 2Lo/~ '
1 2 bo---,2Lo R - ~ ,
Ro-
e f ( u ) - / 3 - u + -~Lu 2, r2 -and
2~
(6.5.21) (6.5.22)
,
(6.5.23)
2/3 1 - v/1 - 2Lo/3 "
(6.5.24)
1 - v/1 - 2L/3 rO= ~
2 Lo
Let us give a numerical example where strict inequality holds in (6.1.14) and therefore favorable estimates (6.5.19) hold"
221
6.5. The Inverse Function Theorem
E x a m p l e 6.5.1 Using Example 5.1.3 we again conclude that (5.1.63) (which is the condition used in (~.2.51) in this case) does not hold However our corresponding condition reduces to hA - 2Lo/3 < 1
(6.5.25)
h A - - -~2(1 -- x)(3 -- x) _< 1,
(6.5.26)
or
which
[0, 4-2v/-i~] C 0.
holds on 0o-
In order for us to study the convergence of method (4.1.3) we define scalar iterations {tn }7 {s~ } by f(t~) t n + l -- t n
f'(tn)
to -- 0
'
(6.5.27)
and 8n+2 -- 8n+l -
f 8 n+l n
L(u)(Sn+l - u)du f;(Sn+l)
,
SO - - O ,
S l - - /3.
(6.5.28)
We can prove the main semilocal convergence theorem for method (4.1.3)" T h e o r e m 6.5.6 Assume: conditions (6.5.~), (6.5.5); /3 <<_b; and r>rl hold, where b, r l a r e defined in Remark 6.5.1. Then, sequence {xn} (n > 0) generated by method (~.1.3) is well defined, remains in U(Xl, r l - / 3 ) for all n > O, and converges to a solution x* e g ( x l , rl -/3) of equation F(x) - O. Moreover the following estimates hold for all n > O" [[Xn+l -- Xn[] ~_ 8 n + l -- 8n ~ t n + l -- t n ]]Xn -- X*][ <__ r l -- 8n <_ r l -- t n .
(6.5.29) (6.5.30)
Furthermore if strict inequality holds in (6.1.1~) then 8n+1 -- 8n ~ t n + l -- tn~
(6.5.31)
8n < t n
(6.5.32)
and
hold for all n > 1.
222
6. M o r e R e s u l t s on N e w t o n ' s M e t h o d
Proof. It can easily be verified by using induction that (6.5.31), (6.5.32) and the right hand sides of double inequalities (6.5.29), (6.5.30) hold. Let us prove" IXk+l --
xkll <~ S k + l -- Sk,
(6.5.33)
r l -- S k + l ) C_ U ( X k , r l -- 8k),
(6.5.34)
and U(Xk+l,
hold for all k _> 0. For every z E U ( X l , rl - Sl), IIz - xoll _ IIz -
Xl II-Jr-II
implies z E U ( x o , rl - S o ) .
xI -- xoll _ r~
-
81 +
81
-
rl
-
8o
Since
IlXl - x o l - I l f ' ( x o ) - l f ( x o ) l l < ~ - s~,
(6.5.33)
and
(6.5.34)
hold for k - 0. Given they hold for n - 0, 1 , . . . , k, then k+l
k+l
I[Xk+l -- X01[ --< Z Ilxi i=1
IIx~ Jr O ( X k + l - - X k ) -
Xi-lll
<_ ~ ( ~ i i=1
- si-1)
- sk-t-1 - So -- 8 k + l ,
<__ rl
Xoll <-- 8k -Jr-O(8k+l - - S k )
0 e [0,11.
Using (6.5.2) we obtain the approximation F(Xk+e)
-- f ( X k + l ) F(xk) - Ft(Xk)(Xk+I -- X k ) 1 -[Ft(Xk + 0(Xk+l -- X k ) ) -- F t ( X k ) ] ( X k + I
~0
-- X k ) d O .
(6.5.35)
By (6.5.4), (6.5.27), (6.5.28) and the induction hypotheses we obtain in turn: ]lF'(xo)-~F(xk+l)][ < ~Ifs(Xk,xk+O(xk+I--Xk)) ~_
L(u)du[lXk+l
- Xk dO
Js(xk) o X~+l-X~ J L ( l l x k
1 <- IlXk+~_xkll ~ 1
~
x011 + u)(I Xk+l -- x k l l - u ) d u
llxk+~--xkl
L(llxk - Xoll + u)(llxk+l - x k l l - u)dullxk+l - xkll 2 fsk+~--sk
--< (8k+1 -- 8k) 2 dO .
~Sk-t-1
. L(~)(~k+l . . J sk tk+i
<_
~)d~
L(sk + u)(sk+l
(
Ilxk+~-~ll S k + l -- 8k
L(u)(tk+l - u)du - f(tk+l), Jtk
-
-
If
sk -- u)dul
~ <_
Xk+l
~+1 L ( u ) ( s k + l
-- X k l l 2
-- u ) d u
J Sk
(6.5.36)
223
6.5. The Inverse Function Theorem
where in (6.5.36) we used the fact that function g(t) - -~
for t E [0, r and
L(s + u ) ( t - u)du is nondecreasing
s] [340, p. 175]. By (6.5.7) for x - xk+l we get F'(Xk+l) -1 exists
E/oXk+lX~ ]
-1
]]F'(Xk+l)-~F'(x0)l] ~
1-
__< ( 1 - ~
Lo(u)du
sk+l Lo(~.~,d~_~)
--1
(6.5.a7)
--~(8k+1) __~ - f ' ( t k + l ) .
Using (6.5.2), (6.5.27), (6.5.2S), (6.5.36) and
[[xk§
(6.5.a7)w
get
Xk§ _< [[f'(xk+l)-*f'(xo)[[. <_ s k + 2 -- S k + l <_ t k + 2 -- t k + l .
Thus for every z C U(xk+2, rl - sk+2), we have
That is, Z
E U(xk+l,
rl-
8k+1).
(6.5.a9)
Estimates (6.5.38) and (6.5.39)imply (6.5.33) and (6.5.34) hold for n - k + 1. By induction the proof of (6.5.a3) (6.5.34) is completed. Estimate (6.5.30) follows from (6.5.29) by using standard majorization techniques. It also follows that {s~} (and {tn})is a Cauchy sequence. From (6.5.3a) and (6.5.34) {xn} becomes a Cauchy sequence too, and as such it converges to some x* E U(Xl, rl - / 3 ) (since U(Xl, r, - / 3 ) is a closed set). By (6.5.5), lIF'(xn)ll is uniformly bounded. Hence, by F(xn)
+ F'(Xn)(Xn+l
we get lim X n - X * .
- X n ) - - O,
9
n----+ ( x )
R e m a r k 6.5.2 If equality holds in (6.1.1~) then sn = tn (n > O) and Theorem 6.5.6 reduces to Theorem 3.1 in [3~0]. Otherwise our error bounds on the distances IIXn+l -- Xnll, llXn -- X'It are finer (see (6.5.29)-(6.5.32)) for all n > 1 than the ones in [3~0] using majorizing sequence {tn} instead of the more precise {sn}. We also note that our results are obtained under the same computational cost with the ones in [3~0] since in practice finding L requires finding Lo. The rest of the error bounds in [3~0] can also be improved by simply using Lo in (6.5.7) instead of L see Section 5.1. Instead we wonder if the convergence of scalar sequence {sn} can be established under weaker conditions than the ones given in Theorem 6.5.6.
6. More Results on Newton's Method
224
We showed in Section 5.1 that { 8 n } is monotonically convergent to s* if: (A) If Lo, L are constants then sequence {Sn} given by (6.5.28) becomes
L(sn+l -- 8n) 2 s~+2 - Sn+l + 2(1 - LoSn+l) '
80 - O, 81 -/~.
The advantages in this case have been explained in Section 5.1. (B) There exists a minimal positive number /~1 E [#, R0] satisfying equation -
1- s
L0( )d
r = 0.
(6.5.40)
Moreover R1 satisfies
Z IF~ILo(u)du < 1.
(6.5.41)
Indeed if 8 n + 1 ~ 1~1, then f88n n+l
L(u)du
Sn+2 <_ sn+l +
(Sn+l--Sn)
1-- f ? l Lo(u)du <
asnfSn+lL(u)du utl~n--"
-
1-
fo
# + f:n-~lL(u)du sn+l 1 - fo Lo( )d
foR1 L(u)du
I2~1 __ iI~1 9
Hence, {sn} is monotonically increasing (by (6.5.41)) and bounded above by /~1, and as such it converges to some s* E (/3,-R1]. (C) Other sufficient convergence conditions for sequence {s~} can be found in Section 5.1 but for more general functions L0 and L. Following the proof of Theorem 6.5.6 but using hypotheses (A) or (B) or (C) above we immediately obtain: T h e o r e m 6.5.7 Under hypotheses (A) or (B) or (C) above the conclusions of Theorem 6.5.6 for method (#.1.3) hold in the ball U(Xl, s * - # ) and s* < rl. Note that s* replaces rl in the left-hand side of estimate (6.5.30). Next we examine the uniqueness of solution x* in the ball U(xo, r l ) or U(xo, s*). R e m a r k 6.5.3 Assume: rlLo(rl) < 1,
(6.5.42)
225
6.5. The Inverse Function Theorem
~01Lo[Orl§
( 1 - 0)/~][0rl + ( 1 - 0)/~]d0 < 1.
(6.5.43)
Then under the hypotheses of Theorem 6.5.6 the solution x* is unique in U(xo, rl) if (6.5.~2) holds and in U(xo, R) for R >_ rl if (6.5.~3) holds. Indeed, if y* is a solution of equation F(x) - 0 in ~(xo, R), then by (6.5.5) we get for L - f l F'(y* + -
y*))dO
] I F ' ( x o ) - I ( L - F'(xo))tl <_ <_
Lo(u)dudO
~001Lo[O]lx*-
Xo]] + (1 - O)lly* - Xo]]]
9 [OIIx* - Xol] + (1 - O)lly* - Xo]]]dO < 1 by (6.5.~2),
(6.5.44)
if y* c U(xo,rl) or (6.5.~3) if y* E U(xo, R). It follows by the Banach Lemma on invertible operators and (6.5.~) that linear operator L is invertible. Using the identity 0 - F(x*) - F(y*) - L(x* - y*) we obtain in both cases x* - y * , which establishes the uniqueness of the solution x* in the corresponding balls U(xo, rl) and U(xo, R). Replace r~ by s* in (6.5.~2) and (6.5.~3) to obtain uniqueness of x* in U(xo, s*) or U(xo, R) under the hypotheses of Theorem 6.5.7. The corresponding problem of uniqueness of the solution x* was not examined in Theorem 1.5 in [3~0].
R e m a r k 6.5.4 Assume operator F is analytic and such that IJF'(xo)-lF(~)(xo)ll _< n!"fn-1
(n ~ 2).
(6.5.45)
Smale in [312] provided a semilocal convergence analysis for method (~.1.3) using (6.5.~5). Later in [177] Smale's results were improved by introducing a majorizing function of the form f (t) - / J -
t+ ~
7t 2
1 -
7t
t e [0, R],
(6.5.46)
(see also Section 6.~). Define function L by
27
L(u) - (1 - 7 u ) 3"
(6.5.47)
Then conditions (6.5.3) and (6.5.~) are satisfied and function f defined in Remark 6.5.1 coincides with (6.5.~6).
226
6. More Results on Newton's Method
Concrete forms of Theorems 6.5.1 and 6.5.5 were given by Theorems 5.1 and 5.2 respectively in [340]. Here we show how to improve on Theorems 5.1 and 5.2 along the lines of our Theorems 6.5.1 and 6.5.5 above. Indeed simply define functions f0, L0 corresponding to f, L by fo(t) - / 3 -
~/ot2
t+
1 -
t E [0, R]
70t
(6.5.48)
and Lo(u) (1 -
270 7oU) 3"
(6.5.49)
Again for this choice of Lo, function fo given by (6.5.10) coincides with (6.5.48). Moreover condition (6.5.5) is satisfied. Clearly ~o ___~
(6.5.50)
holds in general. If strict inequality holds in (6.5.50) then our results enjoy the ones corresponding to (6.1.14) benefits (see Remark 6.5.1). In particular as (6.5.51) the zeros of fo are 1 + ct -
r l~
470
rO= l + a + v / ( l + a ) 470
,o _
8a
V/(1 + a) 2 -
(
2-8a
1 ) 1 1- G ?-/o' R o -
1 2~o
and b o - ( 3 - 2 v / 2 )
1
~o
Then the concretizations of e.g., Theorems 6.5.1 and 6.5.5 are given respectively by: T h e o r e m 6.5.8 Let 7o > 0 be a given constant. Assume operator F satisfies 1 ___ (1 -~ollx - xoll) ~ - 1
[[F'(xo)-l[F'(x) - F ' ( x o ) ] ] l for all xcU
[ ( 1)1] xo,
1--~
-U.
Then F t ( x o ) -1 E L ( Y , X ) , and is diyerentiable in the ball
1)
7ollF,(xo)-i 1[ 9
(6.5.52)
227
6. 6. Exercises Moreover
Uo C U
(6.5.54)
and the radius in Uo is the best possible.
Theorem
6 . 5 . 9 Let 7o > 0 be a given constant, and
a - ~'~o <_ 3 -
2v/-2.
A s s u m e operator F satisfies 1
I~t(Xo)-l[Ft(x) - zb--'t(X0)]ll ~ (1 - ~ollx - x01 )2 - 1
(6.5.55)
f o r all x E -U(xo5 r), where r ~ < r < r~ if a < 3 - 2v/2, or r - r ~ if a - 3 - 2v/-2. T h e n equation F ( x ) - 0 has a unique solution x* satisfying (6.5.19) in the ball
U(xo,r) The rest of the results in [340] can be improved along the same lines. However we leave the details as exercises to the m o t i v a t e d reader. Iterative m e t h o d s of higher order using hypotheses on the second (or higher) Fr~chet-derivative as well as results involving outer or generalized inverses can be found in the Exercises t h a t follow.
6.6
Exercises
6.1 Let F 9 S c_ X --. Y be a three times FrSchet-differentiable operator defined on an open convex domain S of B a n a c h space X with values in a Banach space Y. Assume F ' ( x 0 ) -1 exists for some x0 C S, IIF'(x0)-lll <__ /35 IlF'(xo)-lF(xo)l[ _< w, IIF"(x)ll ___ M, LIF'"(x)II _< N, IIr'"(x)- F'"(Y)II _< LI x - Y L for ~1] ~, y e S, ~ d U (Xo, r~) c_ S, wh~r~ A-
M~],
B-
N/3~] 2,
ao - l - co 5 b o - - - 52A -5
Cdo-
an an+ l a n + l -- 1 - A a n ( c n + d n ) 5 _
_
-
32 Cn+ l = 2187
L/~r]35 7A( 1 +
A) ,
2A "-5-an+IOn+l,
-
a~+18ABan4-17c
~4(1§ 3
d n + l - 3bn+l (1 + :3b n + l ) c n + l
an+ld45
a
(u >_ o),
1 B - [0, 1 ( P (A) - 1 7C)] &rid and r - limn E i =n o (ci + di) If A E [0, ~] P(A) 1 C E 0 5 17 j5 where P (A) = 27 (A - 1)(2A - 1) (A 2 + A + 2) (A 2 + 2A + 4). ---+ (:X)
"
5
6. More Results on Newton's Method
228
Then show [186]" Chebyshev-Halley method given by yn-xn-F'(xn)-lF(xn) 2(yn-Xn))-Ft H n -- r ' (Xn) -1 I f ' (Xn -5 "5
(Xn)]
3Hr~ [ I - -~ 3Hn] ( Y n - Xn) Xn+ 1 -- Yn - "~ is well defined, remains in U (x0, rr/) and converges to a solution x* E 5 (x0, rr/) of equation F (x) - 0. Moreover, the solution x* is unique in U x0, ~
- r~ .
Furthermore, the following estimates hold for all n _> 0 (X3
[[xn - x* [I -< ~
OO
A (1 -5 A)] ~bl E ~'4i-1 /3 (ci -5 di)r/ _< ~ [1 + -ff
i=n
i=1
b_x
where 7 - bl"
6.2. Consider the scalar equation [185] f (x) -
o.
Using the degree of logarithmic convexity of f
L f (x) -
f (x) f " (x) f' (x) 2
the convex acceleration of Newton's method is given by
f (Xn) f'(Xn)
Xn+l = F ( X n ) - - X n -
1-5
Lf(xn) 2(1-Lf(xn))
1 (n > O) -
2k--1 f(b) < for some xo C R. Let k > 1754877, the interval [a, b] satisfying a + 2(k-Y) f'(b i --- ff((bb))9 If ILf (x)[ _< 1 and b and x0 E [a, b] with f (x0) > 0, and x0 > a + 2~-})
,
- ~
in [a, b], then show" Newton's method converges
to a solution x* of equation f (x) - 0 and x2~ >_ x*, X2n+l <_ x* for all n >_ 0. 6.3 Consider the midpoint method [681, [158]" -
-
(.n),
rn -
F' (*n)-',
Zn -- Xn -5 "~ Xn+l -- Xn -- Pn F ( X n ) ,
Pn -- F ! (Zn) -1
(Tt > 0 ) ,
for approximating a solution x* of equation F (x) - 0. Let F 9 f~ C_ X + Y be a twice Pr6chet-differentiable operator defined on an open convex subset of a Banach space X with values in a Banach space Y. Assume"
229
6.6. Exercises
(1) Fo E L (Y, X) for some Xo E f~ and IlColl < 9;
(2) Ilrof (~o)11 ~ ,; (3) IIF" (x)ll < M (x E fl); (4) IIF" ( x ) - F" (Y)It < K IIx- Y[I (x, y e a). D~not~ ao - M g ~ , bo - K g ~ ~. D ~ f i ~ s ~ q ~ c ~ an+~ - a n f (~n)~g (~n, bn),
bn+l
-
b n f (an)3g (an, bn) 2, f (x) -
96(1-x)(1--2x)
7(2-x)~
R-
2
x ~-
i2-x)~ + ~ "
If
where
0
h (~) =
2-3x2-X, and g (x, y) -
1
2 - a o 1--A ~
, u (x0, R~) c ~, -
A - f (a0) -1
then show: midpoint method {xn} (n _> 0) is well defined, remains in U(xo, Rrl) and converges at least R-cubically to a solution x* of equation F (x) = 0. The solution x* is unique in U(xo, 2 _ Rr/) N f~ and IlX~+l
X* -
3~-1
2
II <- ~---~o7
~
An
l--Z~.
6.4 Consider the multipoint method in the form [68], [158]: yn - zn - r . F
(zn),
rn - F'
(Xn) -1 ,
z~=x~+O(y~-x~), Hn-
1 -grn [F' ( X n ) - F I (Zn)] , 0 C (0 , 1] , 1
Xn+l -- Yn 2v -~Hn (Yn - Xn)
(It > 0),
for approximating a solution x* of equation F (x) = 0. Let F be a twiceFr6chet-differentiable operator defined on some open convex subset f~ of a Banach space X with values in a Banach space Y. Assume: (1) Co e L (Y, X),
for some xo E X and Ilroll ___/~;
(2) IlVoF (~o)11 ~ ~; (3) lIF" (x)ll _< M, (x e a); (4) IIF" ( x ) - F" (Y)II - K lix- yll ~,
(x, y) e ~, K > o, p e [o, 11.
Denote ao - M/3rl, bo - K/3rl I+p and define sequence a~+x -- a,~f(a~) 2 90 ( a n , bn) , b~+l - b n f (a~) 2+p go (an,
bn) I+p ,
f (x) - 2--2x-x 2 2 and 90 (x y) = x~+4x~ [2+(p+2)O~ly , 8 + 2(p+l)(p+2) 9
230
6. M o r e R e s u l t s on N e w t o n ' s M e t h o d
Suppose ao E (0, 89 and bo < hp (ao, 0 ) , where (p+l)(p+2)
hp (x, 0) - 412+(p+2)op] (1 - 2x)(8 - 4x 2 - x3). Then ' if U-(Xo' R~/) c-- ~, where R - ( 1 + ~ ) 1 ' A -- f ( a o ) - i show" 1-'),A _ iteration {x~ } (n > 0) is well defined, remains in U (x0, R~) for all n _> 0 and converges with R-order at least 2 + p to a solution x* of equation F (x) - 0. The solution x* is unique in U(x0, 2 - Rr/) N ft. Moreover, the following estimates hold for all n > 0
[Xn __ x*ll
l+p ) ~/ ((2+p)n--1 l+p ) < [1 + ~'~")/( (2+P)n--1 --
An 2 1 - 7(+P)
n
A
r]
'
where ~ / - ~al. 6.5. Consider the multipoint iteration [191]" Y n - - X n -- F n y
Fn - F ' (Xn) - 1 ,
(Xn),
2
Zn--Xn-sFnF(xn
),
H~ - r n [F' (z~) - F ' (x~)], 3 -1 3 [ I + ~Hn] Hn(yn-x~) Xn+ 1 -- Yn -- -~
( n >_0 ) ,
for approximation equation F ( x ) - 0. Let F 9 ft c X ~ Y be a three times F%chet-differentiable operator defined on some convex subset f~ of a Banach space X with values in a Banach space Y. Assume F ~ (xo) -1 C L (Y, X) (xo E f~), IFoll _< c~, IlVoF (xo)ll _
ao-co-1, an+l
__
- - 1 - Oaan n d n ~
Cn+l =
do-
bn+l - -
8(2-3bn)4 [
2-o 2(~-o),
20an+len+l -3
a~
03
(4_3b~)4 (2_3bn)2
wan
4-3b~+1 4-6b~+1 C n + l
( n > 0) _
9
Moreover, assume: U (xo, R/3) c_ ~, where
n R-lim
n---+OQ
d~,
0E
0,7
]
4
+ I-~S5 + 3(2-3bn)0 a n + l d n
and
dn+l --
'
,
0 <--5 <
i--0 27(20-1)(03-809. + 160-8) 17(1_0)2
0 __~ ~ <
3(20-1)(03 -802 + 160-8) 40(1_0)2
175 360 "
6. 6. Exercises
231
Then show: iteration {xn} (n _> 0) is well defined, remains in U (xo, R/3) for all n _> 0 and converges to a solution x* of equation F (x) - 0. Furthermore, the solution x* is unique in U (xo, 5-~ 2 - R/3) and for all n _> 0 (2-0)
1
/3
Ilxn - x* II < ~ gift <_ ~ 2(1-0)3,1/3 Z ")/4i+1 ikn
j>_n
bl where ~ / - 7o"
6.6. Consider the multipoint iteration [157]" y ~ -- X n -- r '
(x.)
~ r (xn)
a . --IF' (Xn + p (Yn -- Xn)) - r ' ( ~ . ) ] (Yn -- ~ ) . x~+i-y~-~
1F' (Yn) - 1 a ~
p E (0, 1],
(n>O)
for approximating a solution x* of equation F (x) - 0. Let F 9 t2 c_ X ~ Y be a continuously Fr6chet-differentiable operator in an open convex domain gt which is a subset of a Banach space X with values in a Banach space Y. Let Xo c f t such that Fo - F ' (xo) -1 C L (Y,X); Ilroll < ~, Ilyo- xolt _< ,, 2 P - 5, and IIF' ( x ) - f t (v)ll <_ K II~- vii for all x , y C f~. For bo - K/3r], define bn - bn-~f (bn _ ~)2 g (b~-l), where 2(l--x) f (x) - x2_nx+ 2
and
g(x)
-
x(x~-~+~) 8(1--x) 2
If b0 < r - .2922..., where r is the smallest positive root of the polynomial q ( x ) - 2x 4 - 1 7 x a + 4 8 x 2 - 4 0 x + 8 , and U(xo, KA---3)C_ f~, then show: iteration 1 and converges to a solution {x~} (n >_ 0) is well defined, remains in/F(xo, ~--~) x* of equation F(x) - O, which is unique in U ( x o , - ~ ) . 6.7. Consider the biparametric family of multipoint iterations [159]"
w-x~-r~F(~).
~-~
+p(y~-x~).
p e [0.11.
1 n [[gt (Zn) -- F I (Xn) ] , H n -- -~r 1 X n + l -- Yn -- -~Hn ( I - ~ - o z H n ) (Yn -- X n ) ,
(?~ ~ O)
where Fn - F ' (Xn) -1 (7t ~ 0) and a - -2/3 c IR. Assume Fo - F ' (X0) -1 C L ( K X ) exists at some xo E fro _c X, F : fro c_ X --+ Y twice Fr6chetdifferentiable, X, Y Banach spaces, IIFoll <_/3, IIFoF (zo)[[ _< r/, liE" (z)tl _< M, z E fro and I I F " ( x ) - F " ( y ) l l -< K l l x - y l l for all x , y C t2o. Denote ao = M/3~, bo = k/3~ 2. Define sequences
a n + 1 - a n f (an)29(an, bn),
bn+l - bnf (an)ag(an, bn) 2,
6. More Results on Newton's Method
232 where f (x) - 2 [2 - 2x - x 2 - I ~ 1 x ~]
-1
and
g (X, y) --
X5 -t- L~--x4 +
1+4[a[ 8 X 3 _t_ II+c~[ 2 X2
-'t- ~ -P - x y -~ ~2 +131p ~
for some real parameters c~ and p. Assume:
1 a0C (0, g),
3(8--16ao-4a~ +Taa+2a4) 3ao+2 '
bo
[c~] < min {6, r}, p E (0, 1] and p < h (]ct[), where r is a positive root of h ( x ) - - 6 b o ( 11- a o )
[ (24 - 48ao - 12%2 + 21a 3 + 6a 4 - 2bo (3ao + 2))
+ 6ag (2ao~ + 3 4 - 6ao - 2) 9 + 3ao~ (2ao - 1)x~], C (xo, Rr/) c_ fro, R - [1 + ~ (1 + [c~[ao)] 1-vA,1
~/ __ ~oo'al A - -
f (ao) -1 .
Then show: iteration {xn} (n _> 0) is well defined, remains in U (Xo, Rr/) for all n _> 0 and converges to a unique solution x* of equation F (x) - 0 in g(xo, 2 - Rr/) N fro. The following estimates hold for all n >_ 0: A n
1_~,3n A 1]"
6.8. Let f be a real function, x* a simple root of f and G a function satisfying G ( 0 ) - 1, G' ( 0 ) - 71 and IG" (0)l < +oc Then show: iteration f (xn) f'(Xn)'
Xn+1 - x n - G ( L f ( x n ) )
(n > 0) -
where
L f (x) -
f ( x ) f " (x) f'(x) 2
is of third order for an appropriate choice of Xo. This result is due to Gander [167]. Note that function G can be chosen
a (x)-
1+
a(z)-
1~ 2-x
(Chebyshev method); x
x
G (x) - 1 + 2(1---x)
(); (Super-Halley method).
6.6. Exercises
233
6.9. Consider the super-Halley m e t h o d [68] for all n > 0 in the form:
r(Xn)+F'(~)(yn-~n)-O,
3r (x~) + 3F' (~) [x~ +
2 (Yn -- Xn)] (Yn -- X n ) -~- 4 F ! ( Y n ) ( X n + l
- Yn) -- O,
for a p p r o x i m a t i n g a solution x* of equation F (x) - 0. Let F " ~ c_ X ~ Y be a three-times Fr~chet-differentiable operator defined on an open convex subset ~ of a B a n a c h space X with values in a B a n a c h space Y. Assume" (1) Fo - F ' (xo) -1 e L (Y, X ) for some x0 e ~ with [tF0J} < fl;
(2) 11roF (xo)Jl _< ,; (3) lit" (x)II <_ M (~ e ~z); (4) [[F'" (x) - F ' " (Y)t] -< n [Ix - y[[ (x, y e ~ ) ,
(L > 0).
Denote by a0 - M/~r/, co = L/3~/3, and define sequences
an+l -- a n f ( a n ) 2 g(an, Cn) , Cn+l - - C n f ( a n ) 4 g ( a n ,
Cn) 3 ,
where f (x) = x2_4x+~
g
(i-x) ~ + ~TY 9
Suppose: ao E (o, 89 co < h (ao), where
h (x) = ~(~x-~)(~-~)(x-~+e~)(x-~-e~) ...........
_
(x0,R~/) c ~,
17{i,/:)~ . . . . . . . . . .
[1 +
R-
,
a o ] l t--A'
2('i-a0)
and A - f (ao) -1. then show: iteration {Xn} (n > 0) is well defined, remains i n / ] (Xo, R~/) for all n _> 0 and converges to a solution x* of equation F (x) = 0. the solution x* is unique in U(xo, ~ - R~/) A ~ and
[ IfXn_X,t[<_
a0~/
1+2(
4n--I a~) ] a
1-
4~ - 1
An
.y---r-l_.y4nAT/
(n>0),_
ao
where 7 - ~ -
6.10. Consider the multipoint iteration m e t h o d [161]:
yn --- Xn --12' (Xn)--t12 (Xn) Gn --- F ' (Xn) -1 [F t (Xn Jr- -~ 2 (Yn -- Xn)) -- 12' (Xn)] , Xn+ 1 -- Yn
--
~3 G n [ I -
3 ~Gn] (Yn- Xn) (Tt _> O)
for a p p r o x i m a t i n g a solution x* of equation 12 (x) - 0. Let 12 " ~ c_ X ~ Y be a three times Fr~chet-differentiable operator defined on an open convex subset ~ of a B a n a c h space X with values in a B a n a c h space Y. Assume"
6. More Results on Newton's Method
234
(1) F0 - F' (Xo) -1 e L (Y, X) exists for some Xo Eft and IIr011 _ 0). Denote by ao - M/3~/, bo - N3~/2 and co - L3~/3. Define the sequence
an+l =an/(an) 29 (an, bn, Cn) , bn+l =bnf (an)39(an, bn,cn) 2 Cn+l =Cnf (an) 4 g(an, b~,cn) 3
where 2
f (X) -- 2_2x_x2_x3 ~ and .q (X, y Z) --
1 [27X 3 (X 2 + 2X -4- 5) -4- 18xy -4- 17z]
1 If ao E (0, 7), 17Co + 18a0bo < p (ao), where
p (x) -- 27 ( 1 - x) ( 1 - 2x) (x 2 + x + 2 )
U(xo, RII) C_a, -
R-[1-4--~-(1-t-ao)]
(x 2 + 2 x + 4 ) ,
I_~A, 1
~
_ _
al ~oo'
/~ __ f ( a 0 ) - I
then show: iteration {zn} (n _> 0) is well defined, remains in U (zo, Rr/) for all n >_ 0 and converges to a solution x* of equation F (z) - 0, which is unique in U(zo, 2 - Rr/) Cqft. Moreover the following estimates hold for all n>0:
]lXn -- X* [I ~
1 + 5ao7 ~x=-! (1_ +_ao7
~_1)] -3-
4~_1
,-)/ 3
~
1_,,/4n A T]"
6.11. Consider the Halley method [68], [131]
Xn+l -- Xn - - [ I -
LF (xn)] -1
F ' (Xn) -1 f (Xn)
(Tt ~_ O)
where
Ls (x) -- F ' (x) -1 F " ( x ) F ' ( z ) - l f
(X),
for approximating a solution x* of equation F (x) - 0. Let F 9 ft _C X --~ be a twice Fr6chet-differentiable operator defined on an open convex subset of a Banach space X with values in a Banach space Y. Assume:
6. 6. Exercises
235
(1) F ' (x0) - 1 E L (Y, X ) exists for some Xo C ft;
(2) IIF' (X0) -1
/[7' (X0)II
~ 9;
(3) I[F' (X0) - 1 Ftt (xo) II ~ "1';
(4) IF' (X0) -1 [F tt ( x ) -
F tt
(y)] II <-- M IIx- Yll (x, y ~ ~ )
If
212V@2+2M+'Y] [_7/(Xo,rl) C_ f~, 3 ~ 3[V/3,2+2M+712, M 3, (rl ~ r2) where rl, r2 are the positive roots of h (t) = / 3 - t +_~t 2 + -g-t then show: iteration {Xn} (n _> 0 ) i s well defined, remains in U (x0, rl) for all n >_ 0 and converges to a unique solution x* of equation F (x) - 0 in U (x0, rl). Moreover, the following estimates hold for all n _> 0:
IIX* -- Xn+l II < ( r l
l~n+l) /~ IIx*-x'~]])\ 3
(~20)3n A10 ~2_(~20)an (r2 -- r l ) < r l -- tn ~ A1_()~I0)an (r2 -- r l ) , /
0-
~1
A1 - ~/(~~176
r2~
< 1
V (ro--rl)2+rorl --
A2-
~2
- t o is the negative root of h, and tn+l -- H (tn), where H (t) - t -
h (t)/h' (t)
Lh (t) --
1 - ~ 1Lh (t)'
h (t)/h" (t) h' (t) 2
6.12. Consider the iteration [68], [158]
Xn+l -- Xn -- [I d- T (xn)] F n F (Xn)
(n ~> 0),
w h e r e r n -- F ' (Xn)-X a n d T (x n) - ~Xr n A r n f
(Zn) (n _> 0) , for approximat-
ing a solution x* of equation F (x) - 0. Here A 9 X x X --, Y is a bilinear operator with IIAI - a, and F " f~ c_ X --, Y is a Fr~chet-differentiable operator defined on an open convex subset ft of a Banach space X with values in a Banach space Y. Assume: (1) F ' (xo) -1 - Fo C L (Y, X ) exists for some xo E ~t with llFol[ _< ~;
(2) IIr0r (~0)ll _< ~; (3) lit' (x) - r ' (y)ll <_ k IIx - yll (x, y e
~) .
Let a, b be real numbers satisfying a E [0, 31) , b C (0, or), where
~[2a~--aa--X+,/l+Sa--4a~] (7 --
a(1-2a)
"
6. More Results on Newton's Method
236
b Define sequence 1, c o - 1, b o - 7b and do - 1 + 3"
Set a o -
an
a n + l = 1-aandn bn+ l __
Cn+l -- - ~
b "~an+ l Cn+ l
dn+l -
[
hi2
a + (l+bn) 2 dn,
(1 + b n + l ) C n + l
N-~n+l
and rn+l z_.,k=0dk (n_>0). I f a -- ~ r / E [0,89 5 ( x o , rr/) C_ ft, r = limn__~ rn, ct E [0, ~ ) , then show: iteration {xn} (n >_ 0 ) i s well defined, remains in V(xo, rrl) and converges to a solution x* of equation F (x) = 0, which is unique in U(xo, ~-~ - rri). Moreover the following estimates hold for all n > 0
IlXn+l -- Xnll ~ d ~ and 0(3
IIX*
-- Xn+ll]
~ (r -- rn)7]
E dkr/. k=n+l
6.13. Consider the Halley-method [68], [175] in the form: y~ - x~ - F ~ F (x~), Fn -- / ; ' (Xn) -1 , 1 X~+l - Yn + 5CF (Xn) H~ (y~ - Xn) (~ >_ 0),
LF (x~) = F n F " (Xn) F n F (Xn) ,
H n = [I - L r (Xn)] -1
(n_>0),
for approximating a solution x* of equation F (x) - 0. Let F " ft C_ X ~ Y be a twice F%chet-differentiable operator defined on an open convex subset ft of a Banach space X with values in a Banach space Y. Assume: (1) F0 E L (Y, X) exists for some xo E f t with ]tF011 <_ ~;
(2) IIF" (x)lt <_ M (x Eft); (3) IIr" (~) - F" (Y)II -< N Itx - Yll (x, y e a); (4) IlroF (xo)ll _< ,; (5) the p o l y n o m i a l p ( t ) - 7k t 2 - ~ 1t + ~ , roots rl and r2 with (rl < r2).
where M 2 + ~N _< k2 has two positive
Let sequences {sn}, {t~), (n _> 0) be defined by
8n--tn
p (tn) p'(tn)'
tn+l
--
Sn
1 Lp (tn) § 21-Lp(tn)
(Sn-
tin)
(It > 0 ) . -
If, U (x0, rl) _cft, then show: iteration {Xn} (ft ~ 0) is well defined, remains in U (x0, r]) for all n _> 0 and converges to a solution x* of equation F (x) - 0.
6. 6. Exercises
237
Moreover if rl < r2 the solution x* is unique in U (x0, r2). Furthermore, the following estimates hold for all (n > 0) (r2 -- rl) 0 4~ 1_04 n ,
0 - - ~r2
6.14. Consider the two-point method [68], [153]" yn=Xn-ft(Xn)-lF(xn), Hn -- p1 Ft (Xn)-i [F t (Xn -]- p (Yn -- Xn)) -- F' (Xn)] , X n + l -- Yn -- ~1H n F -~- H~] -1 (Yn - Xn)
P C (0, 1],
(n > 0 ) ,
for approximating a solution x* of equation F (x) - 0. Let F " f~ C_ X --+ Y be a twice-F%chet-differentiable operator defined on an open convex subset f~ of a Banach space X with values in a Banach space Y. Assume:
(1) F0 - F' (x0) -1 E L (Y, X) exists for some x0 E f~ with liP011 _< #; (2) llroF (~o)ll _< n;
(3) [IF"(*)II< V (. e ~); (4)
IIF"(x)
-
F"
(v)ll _ K 11~- vlt,
x, y ~ ~.
Denote by ao - M/3r/, bo - K/3r] 2. Define sequences
a~+l =a~f (an) 2 gp (a~, b~) , bn+l =b~f (a~)3gp (a~, b~) 2 , where f (x)
-
2(l-x) x~--~-xg2 and
gp (X, y)
-
-
3xa+2y(1-x)[(1-6p)x+(2+3p)] 24(1_x)2
If a0 C .
(0, 2)' bo < hp (ho), where hp (X) -- 3(2x--1)(x--2)(x--3+v/'5)(x--3--V/-5) 2(1-x)[(1-6p)x+2+3p] then show: iteration {Xn} (n > 0)is well defined, remains in U(xo __v_.)for all n > 0 and converges to a solution x* of equation F (x) - 0, which is unique in U(xo, ~). Moreover, the following estimates hold for all n > 0: --
'
3 n --1
llx*-Xn]}_< 1 + 2 ( 1 _ ; o ) where 7 = ~a o and kx = f (ao)- 1
7~~r/'l-A
ao
238
6.15.
6. More Results on Newton's Method
Consider the two-step method: yn-xn-Ft(xn)-lF(xn) Xn+l -- Yn -- Ft (Xn) - 1 F ( y n )
(?% ~ 0);
for approximating a solution x* of equation F (x) - 0. Let F 9 f~ C_ X C_ Y be a F%chet-differentiable operator defined on an open convex subset f~ of a Banach space X with values in a Banach space Y. Assume: (1) F0 - f ' (Xo) -1 C L (Y, X) for some x0 E f~,
Ilrol ~ ~;
(2) IIC0F(x0)ll (3) IIF' (x) - F' (Y)I[ <- K
IIx -
yll (x, y
C a).
Denote a0 - k~r / and define the sequence an+l - f (an) 2 g ( a n ) a n (n >_ 0), where f ( x ) - 2-2x-x2 ~ and g (x) - x 2 (x + 4 ) / 8 . If ao E (0, 89 , O (xo, R~]) c_ ao ~'~' /~ -- ll-+--ff---~,A,")/ -- aoa'! and A - f (a0) -1, then show: i t e r a t i o n
{Xn} (Tt _>O)
is well defined, remains in ~r (x0, Rr/) for all n > 0 and converges to a solution x* of equation F (x) - 0 , which is unique in U(xo, ~2 - Rr/) A ft. Moreover, the following estimates hold for all n > 0
IIXn- X* tl ~ [1 § 9~/
3 n.2:--1 ]
3n --I
An 1 - 73~A rl (n > 0).
6.16 (a) Assume there exist non-negative parameters K, M, L, g, p, ~/, 6 E [0, 1] such that" L<_K,
(1)
g + 2# < 1,
(2)
and hs-
( K + L S + T _ 4M) ~ ~]+M+2#<_5.
(3)
Show" iteration {tn} (n _> 0) given by to -- O, tl -- ?7, tn+2 -- tn+l +
K(tn+l-tn)+2(Mtn-t-tt) (tn+l -- tn) (n > O) 2(1-g-Ltn+m)
is nondecreasing, bounded above by t**, and converges to some t* such that O <_ t* <_ 22~_5- t**.
Moreover, the following estimates hold for all n _> 0
0 < tn+2
tn+l < ~(tn+l
tn) < (25-) n+l
6. 6. Exercises
239
(b) Let F" D C_ X + Y be a F%chet-differentiable operator. Assume: there exist an approximation A(x) E L(X, Y) of F'(x), an open convex subset Do of D, x0 E Do, a bounded outer inverse A # of A(xo), and parameters > 0 , K > 0 , M>_0, L > 0 , p_>0, C _> 0 such that (1)-(3) hold
IIA#F(x0)ll ~ W, IIA#[F'(x)- F'(y)]II ~ KFIm- YlI, I[A#[F'(x) - A(x)]ll _< M l l x -
x011 + ~,
and [[A#[A(x) - A(xo)]ll _ L l l x - xoll + e for all x, y E Do, and U(xo, t* ) c_ Do
Show: sequence { X n } (n ~ 0) generated by Newton-like method with A (Xn) # - [ I + A # ( A ( x n ) - A(xo))]-lA # is well defined, remains in U(xo, s*) for all n _> 0 and converges to a unique solution x* of equation A#F(x) - O, (7 (xo, t*) N Do Moreover, the following estimates hold for all n > 0
II~+~ -
xnll ~ t n + l -- t n ,
and IlXn -- X* I] ~ t* -- tn.
(c) Assume: there exist an approximation A(x) E L(X, Y) of F'(x), a simple solution x* E D of equation (1), a bounded outer inverse A1# of A(x*) and non-negative parameters K, L, M, #, f, such that"
-
IIA~[F'(x) - F'(y)]11 ~ K l l x - Yll, [IA~[F'(x) - A(x)] I! ~ ~r[Ix - x* II + ~, and IIA#~[A(x)
for all x, y E D;
-
A(x*)]ll _< LIIx- x*ll +
6. More Results on Newton's Method
240 - equation
(K-7- + A I + L ) r + / 2 + t ~ -
1=0
has a minimal non-negative zero r* satisfying
L r + t ~ < 1, and
U(x*,r*) c_ D. Show: sequence {x~} (n > 0) generated by Newton-like method is well defined, remains in U(x*, r*) for all n _> 0 and converges to x* provided that Xo E U(x*, r*). Moreover, the following estimates hold for all n _> 0:
tlX* -- Xn+lt[ _< 1
[~X*
]
*
- Xnfl
< (++/~)~*+~ t-Lr* - [ IIx* -- X~ It 6.17 (a) Let F " D C_ X ~ (m _> 2 integer). Assume:
Y be an m-times Fr6chet-differentiable operator
(al) there exist an open convex subset Do of D, x0 c Do, a bounded outer inverse F' (x0) # of F ' (x0), and constants ~ > 0, ai > 0, i - 2, ..., m + t such that for all x, y C Do the following conditions hold:
[tF'(xo)#(F ('~) ( x ) - F (m) (xo))II _< e,
c > o,
(6.6.1)
vx E u (xo, ~0)and some ~0 > O.
t[F' (xo) # r (xo)t[ _< ,~, [fF' (xo) # F + (~o)II - ~ the positive zeros s of p' is such that
p(~) ___ o, where, a2t 2
p(t) -- 71-- t + --~. + ' ' ' +
am+~
m!
tm.
Show: polynomial p has only two positive zeros denoted by t*, t** (t* _< t**).
6.6. Exercises
241
(a2) (;(xo, (~) q Do,
(~ -- max{ (~o,t*, t** }.
(a3) (~o E [t*, t**] or (~o > t**. Moreover show" sequence {an} (n >_ 0) generated by Newton's method with F'(xn) # - [I + F ' ( x o ) # ( F ' ( x n ) - F ' ( x o ) ) ] - l F ' ( x o ) # (n _> 0) is well defined, remains in g(xo, t*) and converges to a solution x* E U(xo, t*) of equation F ' ( x o ) # F ( z ) - 0; - the following estimates hold for all n > 0 IlXn+l -- Xn]] ~ tn+l -- tn
and
where {tn} (n _> 0) is a monotonically increasing sequence converging to t* and generated by t o -- O,
t n + l -- tn -- pP~(tn) (t~) 9
(b) Let F" D C_ X ~ Y be an m-times Pr~chet-differentiable operator (m >_ 2 an integer). Assume: (bl) condition (1) holds; (b2) there exists an open convex subset Do of D, Xo E Do, and constants c~, ~, r/_> 0 such that for any z E Do there exists an outer inverse F' (z) # of F'(z) satisfying N ( F ' ( x ) #) = N ( F ' ( z o ) #) and
IIF'(zo)#f(zo)ll_ W, IlF'(y)#~lF"[z+t(y-x)](1-t)dt(y-z)21] < [ am,+e
-L
m, IIv
_
_
zll
m-2
<
I
for all x, y E Do, am+a?]m-2 m' +...+~]r/<
1,
and D
U(Xo, r) c_ Do
with r - m i n { - - ~l - t o ~5o}
where, ro __ [am-t-e m! Tim-2 + ' " + 7 ] ~c~2 /9
-zll 2,
6. More Results on Newton's Method
242
Show: sequence {x~} (n _> 0) generated by Newton's method is well defined, remains in U(xo, r) for all n > 0 and converges to a solution x* of F ' ( x o ) # F ( x ) - 0 with the iterates satisfying N ( F ' ( z ~ ) #) - N ( F ' ( z o ) #) (n _> 0). Moreover, the following estimates hold for all n _> 0 I1-~+, - *~ii ___ ~g ri*, - .011, -
1_-~o I1~1 - *0fl,
and 1--r~z 1--r~ IIz~ - zolf _< ~--:~-o ilz, - zoll _< 1--=~o ~ - r.
6.18 Let X and Y be Banach spaces, and let L be a bounded linear operator on X into Y. A linear operator M : Y --+ X is said to be an inner inverse of L if L M A = L. A linear operator M : Y --+ X is an outer inverse of L if M L M = M. Let L be an m • n matrix, with rn > n. Any outer inverse M of L will be an n • rn matrix. Show: (a) If r a n k ( L ) = n, then L can be written as
0
'
where I is the n x n identity matrix, and A is an rn x rn invertible matrix.The n x m matrix
M--[I
B]A
-1
is an outer inverse of L for any n • ( m - n) matrix B. (b) If rank (L) = r < n, then L can be written as
L_A[I
0
0 0
C,
where A is an rn x rn invertible matrix, I is the r x r identity matrix, and C is an n x n invertible matrix. If E is an outer (inner) inverse of the matrix [ I L 0
0 ] then t h e n x m m a t r i x 0 ]'
M = C - l E A -1 is an outer (inner) inverse of g. (c) E is both an inner and an outer inverse of [ I0 be written in the form
E-
C
CM
-
C
[I
M].
01ifandonlyifEcan
6. 6. Exercises
243
(d) For any ( n - r) x r matrix T, the matrix E inverse of [ I [ 0
0 0
I T
0 ] is an outer 0 J
"
6.19 Let F 9D C X -+ Y be a F%chet-differentiable operator between two Banach space X and Y, A (z) E L (X, Y) (x E D) be an approximation to F ' (z). Assume that there exist an open convex subset Do of D, xo E Do, a bounded outer inverse A # of A ( = A (zo)) and constants r/,k > O , M , L , # , I > 0 such that for all x, y E Do the following conditions hold:
IIA#F(xo)II
~ ~,
IIA# (F' (x) - f ' (y))II ~ k IIx - YlI,
IIA # ( f ' (x) - A (x))[[ _< M I I x - xo[[ + #, IIA# (A (x) - A)II - L I]x - xoll + z,
b . - ~ + 1 < 1.
Assume h - ~r] _< ~1 ( l - b ) 2 ~ a "- max ( k , M + L), and U -
U(xo t*) _c
Do t* -- l-b-v/(l-b)~-2h. Then show (7 (i) sequence {Xn} (n _> 0) generated by xn+l - x n - A (Xn) # F (xn) (n _> 0) with A ( x n ) # - [ I + A # ( A ( x n ) - A ) ] - 1 A # remains in U and converges to a solution z* E U of equation' A # F (x) - 0.
(ii) equation
A#F(x)
-
0
has a
unique solution in /)Cq
{R(A #) + xo},
where
(xo, t*) ClDo if h - 7 1 (1 _ b)2 U(xo,t* 9 )VIDo if h < 17 ( l - b )
-~_ U
2,
R ( A # ) + xo "- {x + Xo " X E R ( A # ) } , and t* =
(iii) [Izn+l
1-b-+-4(1-b)2-2h
- xn[I I(t~)
_< t n + l
-- t n , ~ 2
tn + g-UTi-Y~)' f (t) - gt
Ilx* - a n I I
<-- t* -
tn,
where
to -
O, t n + l
--
- (1 - b) t + rl, and g (t) - 1 - Lt - g.
6.20 Let F 9D c_ X ~ Y be a F%chet-differentiable operator between two Banach spaces X and Y and let A (z) E L (X, Y) be an approximation of F ' (z). Assume that there exist an open convex subset Do of D, a point z0 E Do and constants r/, k > 0 such that for any x E Do there exists an outer inverse A (x) # of A (x) satisfying N ( A (x) #) - N ( A # ) , where A = A (xo) and A # is a
244
6. More Results on Newton's Method bounded outer inverse of A, and for this outer inverse the following conditions hold:
IIA#F (Xo)11 ~ r], IIA
(y)~ (F' (x + t(y - x)) - F' (x)) II ~ kt IIx - yll
1 for all x , y c Do and t E [0,1], h - 5kr/ < 1 and U(x0, r) c_ Do with r = l-h" ~ Then show sequence {xn} (n _> 0) generated by xn+l - x n -
A (x~) # F (x~) (n > 0) with A (x~) # satisfying N ( A (x~) #) - N (A #) remains in C (x0, r) and converges to a solution x* of equation A # F (x) - 0. 6.21 Show that Newton's method with outer inverses Xn+l - x n - F ~(x~) # (n _> 0) converges quadratically to a solution x* E U Cq { R ( F ' ( x o ) #) + x0} of equation f ' ( x o ) # f (x) A (x) - F' (x) (x E Do).
0 under the conditions of Exercise 6.4 with
6.22 Let F " D C_ X ~ Y be Frhchet-differentiable and assume that F ' (x) satisfies a Lipschitz condition
IIF' (x) - r ' (y)II ~ L IIx - YlI, x, y c D. 1 Assume x* c D exists with F (x*) = 0. Let a > 0 such that U (x*, a) C_ D. Suppose there is an
F ' (x*) # C s (F' (x*)) - { B E L (Y, X ) " B F ' (x*) B - B, B 7/= 0} such that ]lF ' (x*)# ]i ~ a and for any x E U (x*, 3-~a)' the set ft (F' (x)) contains an element of minimum norm. Then show there exists a ball U (x*, r) G D with cr < ~ 1 such that for any Xo e V (x*, r) the sequence {Xn} (n _> O) Xn+l -- X~ -- F' (Xn) # F (X~) (n >_ 0) with F ' (xo) # E argmin {tIBIt IB EFt (F' (xo)) } and with F' (x~) # - (I + F' (xo) # (F' (Xn) - F t ( x o ) ) ) - l F t (Xo) ~ converges 1 quadratically to 2* E U (xo, Z-~) V~{ R ( F ' (xo) #) + xo}, which is a solution of equation F ' ( x o ) # F ( x ) R ( F ' (xo)e)}
- O. Here, R ( F ' ( x o ) # ) + x o
- {x+xo
"x C