CONVERGENCE AND COMPLEXITY OF NEWTON ITERATION FOR OPERATOR EQUATIONS

J. F. Traub Department of Computer Science Carnegie-Mellon University

Pittsburgh, Pennsylvania 15213

H. Wozniakowski Department of Computer Science Carnegie-Melion University

Pittsburgh, Pennsylvania 15213 (Visiting from the University of Warsaw)

March 1977 Revised December 1977

This research was supported in part by the National Science Foundation under Grant MCS75-222-55 and the Office of Naval Research under Contract N0014-76-C-0370, NR 044-422,

ABSTRACT

An optimal convergence condition for Newton iteration in a Banach

space is established. We show there exist problems for which the iteration

converges but the complexity is unbounded. Thus for actual computation

convergence is not enough. We show what stronger condition must be imposed

to also assure "good complexity11 •

1. INTRODUCTION

Numerous papers have analyzed sufficient conditions for the convergence

of algorithms for the solution of non-linear problems. In addition to con­

vergence, we consider another fundamental question. What stronger conditions

must be imposed to assure "good complexity11? This is clearly one of the

crucial issues (in addition to stability) if one is interested in actual com­

putation. We believe it is also a most interesting theoretical question.

We consider Newton iteration for a simple zero of a non-linear operator

in a Banach space of finite or infinite dimension. We establish the optimal

radius of the ball of convergence with respect to a certain functional. There

exist problems where the iteration converges but the complexity increases log­

arithmically to infinity as the initial iterate approaches the boundary of

the ball of convergence. (This phenomenon does not occur in the Kantorovich

theory of operator equations; see Section 3.) We establish the optimal radius

of the ball of good complexity.

In this paper we limit ourselves to the important case of Newton iteration.

In other papers (Traub and Wozniakowski [76b] and [77]) we study optimal con­

vergence and complexity for classes of iterations.

We summarize the results of this paper. Definitions and theorems con­

cerning the optimal ball of convergence are given in Section 2. We conclude

this Section by giving conditions under which the radius of the ball of con­

vergence is a constant fraction of the radius of the ball of analyticity of

the operator.

Complexity of Newton iteration is studied in Section 3. We show that

Newton iteration may converge but have arbitrarily high complexity and conjec­

ture this is a general phenomenon. We establish the radius of the ball of

good complexity as well as a lower bound on the complexity of Newton iteration.

-2-

2. CONVERGENCE OF NEWTON ITERATION

We consider the solution of the non-linear equation.

(2.1) F(x) - 0

where F: D_c -+ B2 and B^, B2 are Banach spaces over the real or complex

fields of dimension N, N = dim(B]L) = dim(B2) , 1 £ N ^ -H». We solve (2.1)

by Newton iteration which constructs the sequence [x 3 as follows :

xi+l = x i " F ,(x i)" 1F(x i)

where x Q is an initial guess and i = 0,1,... . Suppose that F has a simple

zero a, i.e., F(a> - 0 and Ff (or)"1 exists and is bounded. It is well-known

that if F f is a Lipschitz operator in a neighborhood of a and x^ is suffic­

iently close to a then {x. 3 converges to a and the order of convergence is

two, see, e.g., Gragg and Tapia [74], Kantorovich [48], Ortega and Rheinboldt

[70] and Rail [69].

In this paper we are interested in sharp bounds on how close x^ has to be

to a to get quadratic convergence of Newton iteration and how close x^ has to

be to a to guarantee good complexity.

We begin with a theorem which describes the character of convergence of

Newton iteration.

Let F > 0 and J = {x: ^ V). From now on we assume that

a is a simple zero of F,

(2.2) J F1(x) exists for all x € J,

Note that if F n is continuous for x 6 J then

-3-

m d - sup r w 1 ^ ! !

which shows that in this case k^ is a bound on the "normalized11 second derivative.

Let q be any number such that 0 < q < 1

Theorem 2.1

If F satisfies (2.2) and

< 2 - 3 > V * l f e '

(2.4) x Q 6 J,

then Newton iteration is well-defined and

(2.5) lim x i = a9 e 1 + 1 £ qe ±, Vi,

(2.6) e. - £ C. e2. v i+l 1 1

where C± ~ k^ (1 - 2A 2e i) .

If F is continuously twice-differentiable at a then

(2-7) x. + 1 - c-|F ' ( a ) " V(c y)(x i-a) 2+ od^-alf).

Proof

The proof is by induction. Let x^ € J and let

(2.8) R(x;x ) = J^CF'Cx. + t(x-x.)) - F'(x.)}(x-x.)dt.

0 1 . 1 1

Define w » w(x;x^) as the polynomial of first degree which interpolates F(x^)

and F 1 (x^) . Then w(x;x^ * F(xi) 4- Fl(xi)(x-xi) and the next approximation x i + 1 is the zero of w whenever F 1 (x^) is invertible. From Rail [69, p. 124]

we get the error formula

(2.9) F(x) - w(x;x±) - R ^ X j )

for x € J while due to (2.2),

(2.10) |^,(oi)"1R(x;xi)|| * A2|(x-xi|f.

From the definition of we have

| | i-F ' < a ) " V < x ) | | * 2 A 2 | M I * 2A 2 r S - j fg^ < i .

This means that F!(x) is invertible for any x € J and is well-defined.

Furthermore ||F'(xrV(a)|| * 1/(1 - 2A2|(x-a|| ) ,

see Rail [69, p.43]. We can rewrite the polynomial w as w(x;x^) - F 1 (x ) (x-x + )

Set x 23 a in (2.9). Then F 1 (x^ (xi+j"c*) ~ R(»;xi> which yields

- 1 1 A 9 e 5 A 9 r

l h i + H I = Ifr' <xi> F' <«>F' Wx.)II * i ^ a ^ : * i r ^ f e± * q e i

due to (2.3). This proves (2.5) and (2.6).

If Ffl(x) is continuous at ot then

R(of;x±) = \ F»(a)( X i-a) 2 + od^-afp),

F'(x.) = F'(or) + Od^-crll )

which implies (2.7). •

Remark 2.1 Theorem 2.1 states quadratic convergence of Newton iteration since

2 e

i + l ^ Ci ei a n d C i ^ A 2 ^ ~ 2 A 2 e 0 ^ * I(" i s known ( s e e 0 r t e S a a n d Rheinboldt

-5-

[70, p.319]) that if F1 is not a Lipschitz operator then Newton iteration

does not possess, in general, quadratic convergence. For instance, apply­

ing Newton iteration to F(x) - x + x* where a = 5/3 we get

ei+i • ( a - i > e i / < 1 + a e r 1 > # •

Remark 2.2

Theorem 2.1 states how fast Newton iteration converges. The speed of

convergence depends on the value of q. This technique based on q seems to be

general and can be applied for any iteration. See Traub and Wozniakowski [77]

where the class of interpolatory iterations I , n 3, is considered. •

We show that if we want (2.5) to hold then (2.3) is sharp for any value

of q, 0 < q < 1.

Theorem 2.2

There exists a polynomial F satisfying (2.2) for which the following is

true. For any q 6 (0,1) such that A 2 ( O r > q/(l+2q) there exists an x^ so

that for the Newton iterate Xj^ we have e. > qe Q.

Proof

2

Let F(x) « x + x , x 6 K. Then a = 0 and A 2 ( D = 1 . Suppose that

A 2T » r > q/(l .4- 2q) and let x Q - -F. The next approximation x^ constructed

by Newton iteration is equal to x^ - x Q - F f (x^) *F ( *q) ~ ^I(1-2H > qr - -q^Q-

Hence e^ > qe^. •

From theorem 2 . 1 with q -» l" it follows that Newton iteration converges

whenever

-6-

A 2 r < i / 3 .

We show that there exists a problem F such that k^T - l/3 and e Q =* T imply

that Newton iteration does not converge.

Theorem 2.3

There exists a problem F that satisfies (2.2) such that if

A 2(r> r = !

then there exists an x^ 6 J for which Newton iteration is not convergent.

Proof Define

f x + x 2 for x ^ 0 (2.11) F(x) =< 2

I x - x for x ^..0.

Then F(0) = 0, F1 (0) » 1 and F1 is a Lipschitz function with A 2(D a 1. Let 1 1 1 1 r = * and X Q - --j. Then x^ = — and - -g". Hence Newton iteration cycles

and is not convergent. •

Rail [74] showed that assuming the existence of or and using the Kantorovich

theorem it is necessary to assume

A 0F o.l5. 2 4

4"f*2 f2. Thus we have obtained an improvement by a factor of — = 2 . 3 .

We have established above that A^T < l/3 is a necessary and sufficient

condition for the convergence of Newton iteration. Using different tech­

niques Rheinboldt [75] established the sufficiency of < l/3. Indepen­

dently, den Heiger [76] established that it was also necessary.

-7-

We define optimal convergence number. Let X c: 31 be a set of

positive numbers r such that for any F with a simple zero or such that F f is

a Lipschitz operator in J, k^T < r implies that an iteration converges

whenever |jxQ-ar|| f.

Definition 2.1

r c is called the optimal convergence number (with respect to

r - sup X. • c

From theorem 2.1 and theorem 2.3 we have

Corollary 2.1

For Newton iteration

(2.11) * = 1/3. •

Consider the non-linear scalar function.

g ( D = A 2(nr, r * o.

Provided F is not linear, g(D is strictly increasing, g(0) = 0, g(°°) = °°.

Let h denote the inverse function g Then Newton iteration converges

provided ||KQ-<*|| h(r) , r < j.

Definition 2.2

is called the optimal radius of the ball of convergence (with respect

to A 2) if

R c - h ( r c ) . •

Corollary 2.2

For Newton iteration

R = h(l/3). • c

-8-

Remark 2.3

There exists a class of iterative methods whose convergence condition

is of the form A2(Or < r. Examples include the secant method and Newton­

like methods,,both for the scalar and multivariate cases. Let cp and \(r be any

two iterations whose convergence condition is of the form A 2 ( D T < r. Let

r c (cp), rc ( t ) denote the optimal convergence numbers of cp and i|r, and let

R c (cp), Rc ( t ) denote the corresponding optimal radii of the ball of convergence.

Then

Therefore the optimal radii of convergence of two iterations can be ordered if their convergence numbers are ordered.

In general, Newton iteration converges only locally. We give conditions

under which Newtoij iteration enjoys a "type of global convergence".

r c ( c p ) * r c W

implies

analytic in D = {x: |{x:-cy|| < R} e B i=l

where is a Banach space over the complex field.

( 2 . 1 2 ) r ^ r V ^ ^ n ^ i - i

for i = 2,3 , . . • , One way to find K is to use Cauchy's formula

we get W/R & KR (KR) i-1

which yields m/r £ K

-9-

Theorem 2.4 If F satisfies (2.12) then Newton iteration converges in J - {x: ||X-<Y|| T}

where

c i i

(2.13) r - and ^ =s| .

Proof A small calculation shows ||F' (oi)'1?^ || £ f (|(x-or||) where f (s)=s/(1-Ks)

then A 2 ( D T <> K F < \

z (1-KT) J

cl 1 which yields T = ^ p * i t h

c^ — *£• This proves (2.13). •

Note that this result is especially interesting if the domain radius R is

approximately equal to l/K, say R » C 0 / K . Then r = •§ = — R ~-r^- R and Newton z K c^ — oc 2

iteration enjoys a "type of global convergence".

-10-

3. COMPLEXITY OF NEWTON ITERATION

In the previous section we showed that the sequence of errors e^ = ||x£-ctf|j

for Newton iteration satisfies

A 2 2 6i

(3.1) e ± + 1 £ j _ 2 A e whenever A ^ < l/3. 2 i

We want to find x^ such that k is the smallest index for which e^ ^ € ? e o w ^ e r e

ef, 0 < e1 < 1, is a given number. Define c ef so that

(3.2) e k = ee Q.

The complexity of Newton iteration, which is the total cost of computing x^,

k ^ 1, is given by

(3.3) comp == k • c

where c is the cost of one Newton step. (See Traub and Wozniakowski [76a]

for a detailed discussion.) Let c ( F ^ ) denote the cost of one evaluation of

F ^ , j 8 8 0 and 1, and let the combinatory cost d be the cost of solving the

linear system Ff (x^) (x£+i~xi) 5 3 ~ F( X^) • Then the cost per step is given by

(3.4) c » c(F) + c(Ff) + d.

Note that the dimension of the problem N ^+ 0 0. For multivariate problems,

(j) N < +°°, we can assume that each arithmetic operation costs unity and c ( F ^ )

can denote the total number of arithmetic operations needed to evaluate F

and d =* d(N) is the cost of the solution of the linear system

F f (x.,) (x^-x,^ = -F(x ±). Hence d(N) = 0(NP) with P £ 3.

-11-

For the rest of this section N £ +<». Define

A f 2

< 3- 5 ) f i + i = T ^ A 7 T ' f o " V i s s ° ' 1

Clearly, the sequence {f^} is a majorizing sequence of {e^3> i.e., e^ ^ f ,

Vi. Let comp- = k c where k is the smallest index for which f ^ ee n. Of 1 1 1 K. 0

course, comp ^ comp^. Let p » A^EQ - A2 e0* N o t e that comp^ = comp^(e,p).

We derive some properties of comp^ as p approaches l/3 (p < -j. is the necessary

and sufficient condition for guaranteeing convergence)•

Lemma 3.1

Let n be any fixed integer. If p = ~ - 6 with 6 then 4

(3.6) f. = (1 - g i6)f Q where 0 £ g± <z 4 1 + 1 - 4

for i = 0,1,..., n.

Proof

Assume by induction that f i = (1 - S i6)f Q with 0 ^ g ± ^ 4 1 + 1 - 4. Since

fi+i * fo t h e n gi+i * 0 a n d

_ P(l-g16)2f0

Ei+1 = l-2p(l-gi6)

which yields after algebraic manipulations

8i+l * 4 g i + 9 + 3 ( 8 i 6 ) 2 * 4 1 + 2 " 4 + 3<8i 6> 2 " 3 * 4 1 + 2 " 4 '

This proves (3.6).

We are ready to prove

-12-

Lemma 3.2

(3.7) | lg(-|)(l + o(l)) <: comPl(e,p) £ c log(-|)(l + o(l))

where p = ~ - 6 with 6 -* 0 + and lg is the logarithm to base 2,

Proof

Recall that comp^ - k c and we seek bounds on k^. Since

f * A * f 2 y ^ o V " 1

f = / l i l f i V * 1 f

0

r i l-2A 2 f Q

r i - l I l-2A„f J r0 then

k ] _ * i g ^ . i g « / i g ^ i | | ^ = i g i ( i + o ( i ) ) .

This proves the righthand side of (3.7).

Let 6 » l/4 n + 1. From Lemma 3.1 for i = 1,2,..., bi/2J we get

f. * (1 - (4 i + 1-4)/4 n + 1)f Q * (l-2~n)f0 > efQ

for large n. Thus k^ ^ U*/ 2 J - lg("|)(l + o(l)') which proves the lefthand

side of (3.7). •

Lemma 3.2 states the important difference between convergence and complex­

ity of the sequence (f^3. p < assures convergence of {f^} but with only

this condition complexity can be arbitrarily (logarithmically) large.

We believe results similar to Lemma 3.2 also hold for comp (rather than

comp^) of an arbitrary iteration and therefore propose

Conjecture 3.1

For any superlinear one-point iteration $ there exists a function F with

a simple zero a and a number T, 0 < T < +00, such that

-13-

(i) the sequence ^ - $(x^,F) constructed by the iteration $

is convergent to a and ||>ci+^-cy|| < or[|9 Vi, whenever an

initial approximation x^ is any point of J = {x: ||x-cy||< r )

(ii) the complexity comp, which is now a function of x^ and e,

satisfies

lim comp(xA,e) = +» Ve > 0. •

Conjecture 3.1 states that starting from any point of J we get convergence;

however, comp(Xg;e) can be arbitrarily high when x^ approaches the boundary of

J.

We prove Conjecture 3.1 for Newton iteration.

Theorem 3.1

If Newton iteration is applied to

2 CYL '+ x for x ^ 0

(3.8) F(x) =< 2

ix - x for x ^ 0

then for e fixed comp(Xg,e) tends logarithmically to infinity as A2F approaches

l/3 and X Q * T.

Proof

Note that a 8 8 0 and A 2 ( 0 = 1, see (2.11). Applying Newton iteration to

(3.8) we get 2 2 -sign(xj[)xi e±

x i ( 1 s , \ and e i+1 l-2sign(xi)xi

ci+l l-2ei

which is equivalent to (3.5) with A 2 - 1. Let x Q - T =» | - 6. From Lemma (3.2)

we get

-14-

- comp(x.,e) T(l+o(l)) * T — * 1 + o ( 1 ) a s 6 "* °' 4 c l g ( J )

This proves theorem 3.1. "

This phenomenon aoes not occur in the Kantorovich theory. If h * (in

the usual notation; see Rail [74]), the character of the convergence depends

on the multiplicity of or. If a is a simple zero, the convergence is quadratic

and the complexity g^es as lg lg(l/e). If a is a multiple zero, the conver­

gence is linear and the complexity goes as lg(l/e).

Note that F defined in (3.8) is not twice-differentiable at a. "'It

seems to us that comp can tend to infinity as A^T approaches l/3 for twice-

differentiable problems. Therefore we propose

Conjecture 3.2

There exists a twice-differentiable function F with a simple zero a such

that the comp « c o m p ^ D of Newton iteration satisfies

lim comp(A9D = + 0 0 • A 2r-i/3~ 1

Theorem 3.1 states that if k^V is close to l/3 then complexity can be

arbitrarily large. We seek a bound on k^T (stronger than A 2 r < l/3) to be

sure that complexity is not too large. Define

(3.9) t - max(l, lg l/e).

Theorem 3.2

If A 2T £ l/4 then the complexity of Newton iteration satisfies

(3.10) comp £c lg(l+t).

-15-

Proof

Let h i+1 - Kh ± with K = A2/(l - 2A 2e Q) and h 0 8 3 e 0 . From (3.1), e ± £ h ±

._ /-rr _ \ 1 . r\ . i

4 > Vi. In Traub and Wozniakowski [76a] we proved that if w » (Keg)"1 ^ 2 then

the complexity comp(h) of the sequence (hu) is bounded by

comp(h) £ c lg(l+t) •

Remark 3.1

We have chosen the endpoint w = 2 in the inequality w ^ 2 only for con­

venience and definiteness. Any value w bounded away from unity can be used 1/v

to get a good bound on complexity. For instance, w ^ 2 implies comp £ c log(l+vt). See Traub and Wozniakowski [76a]. g

Theorem 3.2 motivates the definition of the optimal complexity number for

a superlinear iteration. Let Y <z SR be a set of positive numbers r such that

for any F with a simple zero a such that F 1 is a Lipschitz operator in J,

A^F £ r implies that an iteration converges and comp ^ c Ig(lH-t), Vt ^ 1,

whenever |JxQ-a|| ^ r.

Definition 3.1

r is called the optimal complexity number [with respect to A 9 and the

complexity criterion comp £ c lg(l-ft)] for a superlinear iteration if

Since comp ^comp(h), (3.10) follows.

r = sup Y.

Compare with the definition of the optimal convergence number r^. Of

course r £ r and from Theorem 3.2 it follows that r ^ l/4 for Newton g c g iteration.

-16-

Theorem 3.3

For Newton iteration

(3.11) r = ~ • g 4

Proof 2

Let F(x) - x + x . Then F(a) - 0, F 1 (a) =» 1 and - F. Newton itera­

tion produces [x. 3 such that

2 X J

(3.12) x . - 2 #

1 + 1 l+2X. L

1 J. Let t - 1. Then comp £ c is equivalent to e

x ^ 2 E 0 # S E T X 0 " F 2 *

From (3.12) we get e ^ - R 2/(l-2D. Thus e 1 £ ~ e Q is equivalent to F ^

Hence A 0 ( T ) R ^ r £ y and from Theorem 3.2 we conclude r = 7. • 2 g 4 g 4

Remark 3.2

It follows from Remark 3.1 that as long as A 2 F is bounded away from l/3

we are assured of good complexity. m

com-We summarize our results on optimal convergence number and optimal plexity number (Corollary 2.1 and Theorem 3 . 3 ) :

W A 2 E 0 < rc = ^ 3 a s s u r e s convergence but complexity can be arbitrarily large

(ii) A 2 e Q £ r g » 1/4 assures convergence and good compl exity.

-17-

Definition 3.2

R is called the optimal radius of the ball of good complexity Twith

respect to k^ and comp ^ c lg(l+t)] for a superlinear iteration if

R - h(r ). g g

Corollary 3.1

For Newton iteration

R g = h(l/4)

Remark 3.3

A remark analogous to Remark 2.3 holds here concerning the ordering of

the optimal radii of good complexity for two iterations. •

The ratio between the optimal radius of the ball of convergence and the

optimal radius of the ball of good complexity,

S a Ml/3)

indicates what stronger condition must be imposed to assure good complexity.

We end this paper by deriving a lower bound on the complexity of Newton

iteration. Let F be any operator for which

(3.13) e 1 + 1 * a 2 e 2

for a positive a^. We show that the class of such operators is not empty. Let x

F(x) • - - a, a > 0, x £ 0. Then

ei+i • a e i -

-18-

Recall that c - c(F) + c(F') + d is the cost of one Newton step where d

is the cost of solving a linear system of size N. Let cT and c denote lower L U

and upper bounds on c. If a2 eg ^ V f c > then Theorem 3.1 in Traub and

Wozniakowski [76a] yields comp ^ c (lgt-lglgt). L

We summarize the complexity analysis for Newton iteration.

Theorem 3.4 if i i

then the complexity of Newton iteration satisfies

cL(lgt-lglgt) £ comp £ C u lg(l+tX

where t = max(l, lg l/e) .

ACKNOWLEDGMENT

We are indebted to H. T. Kung for several important suggestions on this

paper.

-19-

REFERENCES

Kantorovich [48]

Gragg and Tapia [74]

den Heijer [76]

Kantorovich, L. V., "Functional Analysis and Applied Mathematics," Uspehi Mat. Nauk, 3 (1948), 89-185 (Russian). Tr. by C. D. Benster, National Bureau of Standards Report No. 1509, Washington, D.C., 1952. Gragg, W. B. and Tapia, R. A., "Optimal Error Bounds for the Newton-Kantorovich Theorem," SIAM J, Numer. Anal. 11 (1974), 10-13.

den Heijer, C , "Iterative Solution of Nonlinear Equations by Imbedding Methods," Report NW 32/76, Mathematisch Centrum, Amsterdam, August 1976.

Ortega and Rheinboldt [70]

Rail [69]

Ortega, J. M. and Rheinboldt, W. C , Iterative Solution of Nonlinear Equations in Several Vari­ ables, Academic Press, New York, 1970.

Rail, L. B., Computational Solution of Nonlinear Operator Equations, John Wiley and Sons, New York, 1969.

Rail [74]

Rheinboldt [75]

Rail, L. B., "A Note on the Convergence of Newton1s Method," SIAM J. Numer. Anal.. 11 (1974), 34-36.

Rheinboldt, W. C , "An Adaptive Continuation Process for Solving Systems of Nonlinear Equa­tions," Technical Report TR-393, University of Maryland, July 1975.

Traub and Wozniakowski [76a]

Traub and Wozniakowski [76b]

Traub and Wozniakowski [77]

Traub, J. F. and Wozniakowski, H.,"Strict Lower and Upper Bounds on Iterative Computational Com­plexity,11 in Analytic Computational Complexity, edited by J. F. Traub, Academic Press, 1976, 15-34.

Traub, J. F., and Wozniakowski, H., "Optimal Radius of Convergence of Interpolatory Iterations for Operator Equations," Dept. of Computer Science Report, Carnegie-Mellon University, 1976.

Traub, J. F., and Wozniakowski, H., "Convergence and Complexity of Interpolatory-Newton Iteration in a Banach Space," Dept. of Computer Science Report, Carnegie-Mellon University, 1977.

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CONVERGENCE AND COMPLEXITY OF NEWTON ITERATION FOR OPERATOR EQUATIONS

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An optimal convergence condition for Newton iteration in a Banach space is established. There exist problems for which the iteration converges but the complexity is unbounded. We show what stronger condition must be imposed to also assure "good complexity",

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