such that 0„ < 4 5 and

o o

0 Q < ^ m i n { c o s 1 ( t T s) | seWnS , t e T o r t e T T ^ f O ) and <J> ( t ) > - j <j)Q } •

the set of directions

Ü = {teS I 0 ( t) <0Q / c}) ( t ) > c f ) o ]■ and i t s c l o s u r e a r e d i s j o i n t f r o m T and n 1 ( 0) n S , so t h a t E j = mi n { I rr 0 ( t ) I t e U ) > 0 , p Q - i n f { r ( t ) j t e( J} > 0 , and similarly e 2 = mi n { || g j ( t ) || | t e d } > 0 .

The positiveness of follows from the fact that, because i = l, the first nontrivial coefficient vector g^t) can only vanish at a regular direction if t e

T

, as shown in the proof of (i).

Whenever t^ e U and < p Q we have by (37) ii , . „ ^ 2 . 2+Am

p . t . -p.g, t. < y p./£

iih3 + i 3+1 i 3 1 ' r 3 i (49)

which implies because g^(tj) e

sin 0(t . ) ^ sin 3 + 1

cos 1 (tT+lg x (tj)/||gi (t^ ||)

min

Xe]R Atj+r 9 i(tj)/llgi(tj

. . . 2+Am.

< Y1Pj/(e2e i ) •

Thus we find that for sufficiently large j

tj e

U

and p^ < p = minjp0 , e j C ^ ^ Y j * sin 0Q

ensures t. e LI . Unless {x.}_ converges quadratically it has by

3+1 3 3-°

(iii) a tangent in

N

which must be regular by assumption. Since

furthermore lim p. = 0 < p all but finitely many iterates must belong to j-K» J

the set

{ x * + p t I t e l l , p < p }

Finally we derive from (49) that

lim inf p . / p . > lim inf -i-x» 3 + 1 3 -i-KO

!|g

1

( t j ) | | - y

1

p

j/£1

2+Am > e.

which completes the proof. ////

Even though it could not be shown conclusively,it seems likely that Newton sequences always have integer order unless f involves

fractional powers. At singularities quadratic convergence from certain initial points is theoretically possible but numerically unstable,as

r o u n d i n g e r r o r s w i l l p r e v e n t the m i n i m a l a n g l e s b e t w e e n the tj and d i r e c t i o n s in

T

from b e c o m i n g a r b i t r a r i l y small. W i t h o u t a n a l y z i n g the na t u r e of such d o m a i n s of q u a d r a t i c c o n v e r g e n c e in any detail, we note that for all Q - s u p e r l i n e a r l y c o n v e r g i n g N e w t o n s e q u e n c e s {x^=x * + P j t )j>0 w i t h lim p /p. + 0 j-xx> 3 J llg(x.) lim sup Tj :---- P • / P • 3 + 1 3 j->co 1 p j + 1/ pj ^ lim sup 0 . (50) l-~°

y

1

E v e n u n d e r the o p t i m i s t i c a s s u m p t i o n that e ach N e w t o n step can be c a l c u l a t e d w i t h a u n i f o r m l y b o u n d e d r e l a t i v e e rror G , the n u m e r i c a l l y e v a l u a t e d new i t e r a t e c ould be any p o i n t in the ball

{xe3Rn I ||x-x*|| < e | | g ( x j -x_.il - ||g(x )-x*||} ,

w h i c h is by (50) n o n e m p t y for s u f f i c i e n t l y large j . We h a v e n o t e d that in g e n e r a l any n e i g h b o u r h o o d of the s i n g u l a r i t y x* c o n t a i n s p o i n t s x = x * + p t w h e r e the N e w t o n step is e i t h e r u n d e f i n e d (6(x)=0) or leads f u r t h e r a way f rom the s o l u t i o n (||g (t) ||> 1) . T h u s s u p e r l i n e a r c o n v e r g e n c e of the u n m o d i f i e d N e w t o n m e t h o d on a s i n g u l a r p r o b l e m seems u n l i k e l y to o c c u r in p r a c t i c e and is n o t e v e n desirable.

CHAPTER 2

STARLIKE DOMAINS OF CONVERGENCE

In document Analysis and modification of Newton's method at singularities (Page 59-62)