Numer. Math. 22, 17--31 (t973) �9 by Springer-Verlag 1973

Iterative Variants of the Nystr6m Method for the Numerical Solution of Integral Equations

Kendall Atkinson

Department of Mathematics, The University of Iowa, Iowa City, Iowa

Received April 4, t973

Abstract. Some iterative variants of the Nystrrm method for the numerical solution of linear and nonlinear integral equations are introduced and discussed. Numerical examples are given; some are for integral equations with singular kernel functions.

I. Introduction

Consider the Fredholm equation

2 x ( s ) - f K(s, t )x ( t )d t=y(s) , seD, (1.1) D

with D a bounded closed domain in IR '~, y cC (D). Assume K (s, t) is such that the associated integral operator 9ff is compact from C (D) into C (D). Also assume 2 is not an eigenvalue of JT" and 2 ~ 0. Eq. (l .t) is written symbolically as (2--o%d) x = y .

The Nys t r rm method for (iA) uses some type of numerical integration to obtain the approximating equation

2 x . (s) - - ~, wi(s)x.(ti)----y(s), seD; (t.2) / = 1

the node points tl, t2 . . . . . t, are in D, and x~(t)~ x(t). The weights wi(s ) are obtained in a variety of ways, depending on the smoothness and form of the kernel function. If K(s, t) and x(t) are reasonably smooth, then usually wj(s) -= ctiK (s, ti), where

f / ( t ) d t ~ ~. ccj/(ti), /eC(D), D ~=1

is a numerical integration formula. If K (s, t) is singular, then special numerical integration formulas must be used, e.g., methods based on product integration [3]. The Eq. (! .2) is written symbolically as (~--9~n ) x~ = y .

Eq. (1.2) is solved by reducing it to the equivalent finite linear system

2~(tr -- ~ wj(ti)~(ti) =y( t3 , i = 1 . . . . . n. i = 1

(t .3) 2 Numer. Math., Bd. 22

18 K. Atkinson

The equivalence is accomplished using the interpolation formula

x~(s) =-~-y(sj +~- ~, wi(s)x~(ti), sOD. (t.4) i=1

This is the " n a t u r a l " interpolation formula for solutions of (1.3); it originated with NystrSm I20].

With many Eq. (t .t) the system (t .3) is too large to be solved directly, mostly because of computer memory size limitations. The purpose of this paper is to consider some iterative variants of (1.2)-(1.4). I t will be assumed that (t.3) can be solved directly for some small number n of nodes; this will be used in defining iterative methods for solving (1.3) with m > n. One of the methods discussed is due originally to Brakhage [8].

The methods are defined and analyzed abstractly in w 2. In w 3 and 4 examples are given based on integral equation reformulations of the Dirichlet problem for Laplace's equation, in three and two dimensions, respectively. In w 5 the theory is extended to nonlinear equations, and w 6 contains an example based on Nekra- sov's equation.

Other iterative techniques can be found in [9, 11, t3, 21].

II. Linear Iterative Methods

The methods will be defined and discussed using the abstract formulation of Anselone [1, 2] for families of collectively compact operators. The hypotheses on { ~ [ n ~ t} and ~ are as follows.

At . ~ and 4 , n ~ 1, are linear operators on the Banach space ~ into ~ .

A2. ~x--->Jg~x as n-->~, for all xEs r.

A3. {~/~} is a collectively compact family, i.e., the set

{ ~ x l n > - - t and Ilxll--<l}

has compact closure in ~ .

For the operators 9K and ~ of (1.1) and (t.2), it can be shown that tlX'-~ll----tl~ll, t. This means the error analysis of (t.1)-(t.4) cannot be based on standard perturbation arguments. An abstract error analysis has been given by Anselone [t, 2]; and it is based on the consequence of A t - A 3 that I ] (3K-~)JCII and H(gK-gt~)~]] converge to zero as n-->o~. If (2--~K) -1 exists, then for all sufficiently large n, say n--> N (2), (2--~K~) -1 exists and is uniformly bounded by c 2 (2). In addition,

]] x -- x, [[ ~ [I (2 - - ~ ) q y - - (2 --~r ]] < c~ (2)Hgffx - - ~ x ]], n > N (2).

This shows x~-+x and gives a rate of convergence.

For later convenience, define

q = sup I~l l , a,, = sup sup ] ] ( x " - ~ ) ~ ] 1 . n>l m~_n 1>1

From A2 and A 3, a . -+0 as n-+oo.

NystrSm Method t 9

Iterative Method (i). Let m > n. F rom (2--~Y(',) x,, = y , write

(k - - ~ ) x~ = y + (oY', --~ff,) x~. (2.t)

Define the i terat ive method by

(k --oU,) x~ +~) = y + ( ~ - - ~ ) x~ ), v > 0. (2.2) For the error,

x. - x~ +'1 = (k _ ~ . ) - 1 ( ~ _ : ~ . ) (x. - x~l), v_~ 0. (2.3)

Generally the error opera tor in (2.3) does not have norm less than one. But by i terat ing (2.3) once, we get a sat isfactory result. We can show

U [(k --.~n)-I (fro __~) ]2 [[ ~ bn (/~)2 ~_ 2 (a, .qL an ) lZl c,(2) [~ +ClC2(2)] ; (2.4)

and is s t ra ightforward tha t b. (2) < t for all sufficiently large n. Using

Ilx,.-xT.+')tl<b,,(k?llx.--x~)li, ~,>=o, (2.5)

we obtain convergence of the i terat ion for all large n: x~ )---> x~.

An equivalent form to (2.2) is

r (~) = y - - (2 - -J( ' . ) x~ I,

x~ +1) = x~, ) + (2 - - ~ ) - l r l ' ) . (2.6)

This makes the method look like a residual correction method, with (2 _ ~ , ) - 1 an approx imate inverse to ( 2 - - ~ ' , ) . But it is t rue generally tha t

II (2 - ~ ) - , - (2 -~,~)-11[_b~ 0 aS n~ m-->-oo .

From (2.5) it is not necessary tha t the errors decrease in a uniform fashion as v increases. This is confirmed with most numerical examples, as will be i l lustrated later. The lack of regular behaviour is a mot iva t ion for the next method.

t Iterative Method (ii). In (2A), replace x , on the right by x , - - - - 2 - Y +

�89 0F.x,~ to give

~ (ar~ - ~ . ) ~ x~. (2.7) ( k - ~ . ) ~ . = y + ~ - ( ~ - ~ . ) y + -~-

Define the i terat ion method as

(2_yd,)xl,+,)_ - - : + tk (.Yd', --~t~.)y + fl (..~,n__~n).~,nX(~), v~O.__ (2.8)

For the error,

II x . - ~+')II =< a. (2)U ~ . - x ~ II, (2--,X~.)-I(.~t",~--~).J~d,~ ~d . (2 )~ C~-[~[-%(2)(a,, +a.). (2.9)

Easily, d, (2) < t for all large n, which proves x~ )-+ x~,.

2*

20 K. Atkinson

An equivalent form to (2.8) is

r (') = y - - (2 - - J('m) x~ ), (2.10)

I ri0+ ~ (2 --~/~.)-l~ffmrl'/.

For computa t ional purposes this form is much superior to (2.8) ; it requires m a n y fewer evaluations of the kernel of the integral equation.

As might be imagined from (2.9), the convergence of x~ ) to x m should be fairly regular; this is borne out with most numerical examples. When viewed using (2.t0), the method consists of first regularizing the error equat ion

(2 --~ffm) (x,~ - - x~ )) = r I'),

in the sense of Kantorov ich [14, p. 5521; then apply me thod (i) to the regularized equat ion with its new r ight-hand side dCm rC'). This was the principal mot iva t ion in Brakhage [8].

I terat ive M e t h o d (iii). Define

_ 1 1 x n = ~ y + ~- (2 --o,',~)-lgff'y ;

it is the result of applying the Nys t r6m method to the regularized form of (2--JT ~) x = y . Then define

Z~ +1) =Xn -~- ~-1 (2 _ _ ~ ) _ 1 (~/.~n)~,.mZ(~), ~>~0.__ (2A1)

This will converge to a point z~ provided

in (2) =- ~' (~) an I~1 < a

because

~- (2--~)-1(~--~n)~m ~/n(21.

The point z~ satisfies

c~lZl IIx-zmll<= l~l-anc~(a) ll(X'--~m)(X'--~n)Xl["

Compare this with

For method (iii) it is not necessary tha t m > n. The me thod is more t ime consuming than methods (i) and (ii). But if certain preliminaries are carried out in advance for a number of values of 2, y, and/or m, the method will be superior.

As an example of the implementa t ion of these methods on a computer , we consider method (i) for (1.1)-(1.4). Given x~ ) at the points {ti,n} and {ti,m}, first compute

r (0 (s) = y (s) - - 2 x~ ) (s) + ~. wi, m (s) x~ ) (ti,,,) 1

Nystr6m Method 21

at the same node points. Define (5 = (4 --J~(',)-lr(~); then x~ +1) = x~ ) + 6. Solve for (5 (ti,~), t <= i _<-- n, using

(5 (t~, ~) - ~ wj , . (t~,~) ~ (tj,~) = r <~1 (t~,~), i = 1, 2 . . . . . n. i = 1

The explicit inverse or an L U decomposition of this system should have been stored earlier. Finally, evaluate

0 (s) = r r (s) + F wj,~ (s) ~ (t;,~) i=z

for s = t 1 . . . . . . . t,~,~.

The number of evaluations of the terms wj (s) will serve as a basis for comparing the efficiency of the methods. For method (i), the count per iteration is m (m + 2n). For method (ii), it is m(2m+n), which is about twice that of method (i) per i teration when m>>n. Since method (ii) will generally converge more rapidly than method (i), the t iming of the two methods for the same order of accuracy is about the same in most cases. For method (iii), these comparisons are not valid. For the case w i (s) =ociK (s, ti), method (iii) requires the evaluation of

b

E.(s, t) ---- f K(s, u)K(u, t ) d u - ~ ~i,~K(s, tj,.)K(t~,., t) a ~=I

at the points {t~,~} and {t~,~} for each iterate. If this can be approximated for use with several values of 4, y, and m, then method (iii) becomes useful, but not otherwise.

In practice we would never t ry to calculate, a priori, whether or not n is sufficiently large to insure convergence. Ins tead we pick an n and t ry it. In practice, if x~ has approximately one digit of accuracy with respect to x, then tha t n is generally sufficient to give an adequate rate of convergence in all of the methods. One of the difficulties of t rying for an a priori estimate on the rate of convergence is tha t the bounds b~ (2,), d~ (;t), and ], (4) are too conservative and too expensive to evaluate. In addition, the rate can va ry great ly with the unknown function, and this has not yet been explained; an example is given in w 4.

III. Laplace's Equation in Three Dkmensions

Consider the problem

A u t O on D, u = / on aD, (3.t)

with D a bounded simply-connected region in IP, 8 with a smooth boundary ~D. We will restrict ourselves to ellipsoidal regions D, al though the methods to be presented will work for virtually any region D with a smooth boundary. We will use the s tandard parametric representation of ~D:

x x = a sin r cos 0, x 2 = b sin r sin 0, x 3 = c cos ~b, (3.2)

for0--<_0=<2~, 0=<_r Let S = [ 0 , 2 ~ X [ 0 , ~ ] in the 0 - - r plane.

The unknown function u can be written as a double layer potential:

' f f u(z) =~-/# sin r 4) (~c--z).y clOdS, z E I n t (O), (3.3) S

22 K. Atkinson

wi th /x the dipole densi ty function; see Garabedian [12, p. 334-34t] . The point x = x ( O , r is given by (3.2), and y(O, r is given b y

yx =bc sin r cos O, y2 =ac sin C sin O, y s = a b cos r

The expression ( x - - z ) . y uses the ordinary dot product , and the norm is the Euclidean norm in IR 3.

Let p = (cr and q = (0, r in S. B y letting z tend to p , we obtain the well- known equat ion for/~ (0, r :

ff #(q) sin ~ [ t - - w ( p ) . w ( q ) ] d q ~t(p) + ~ - .,J i ix(p)_x(q) l l3 - / ( p ) , pES , (3.4)

S

with w(O, r given by (3.2) with a = b = c = l . For notat ional simplicity, we will let

1 t - - w ( p ) . w ( q ) K(p , q ) - - 2zt lix(p)--x(q)l] a "

To define a numerical me thod for (3-4), we will first define a numerical integral opera tor approximat ing the integral opera tor X" of (3.4). Let no > 4 , n , > 3 h o = 2 ~/n o, h, = ~/nr N = (n~ -- 2) n o + 2. The regions

S x ---- [0, 2 ~r] X [0, he], S y = [0, 2 yt] • [~ - - h,, yt],

correspond to the polar regions of 8D. For i <-i <~n o and 2 < i < n , - t, define

l = i + t +( j --2)no, Sl----- [(i --t)ho, iho] • [(j - - t ) h , , ih , ] .

Define 101 = (0, 0), PN = (0, ~) ; these correspond to the poles of 8D. For t < 1 < N, define Pt as the midpoint of S v

Rewri te (3.4) as N

It(P) + X f f sin CK(p , q) t t (q)dq = / ( p ) , pES. (3.5) l = l St

Denote the subintegrals in (3.5) by Il(p). We will approx imate them using a combinat ion of the ordinary midpoint rule and the product integrat ion midpoint rule.

For 1 < l < N and Pr use the ord inary midpoint rule to give

I t (p) -~ sin (Pl)K (p, p~)p (p~) hoh , , (3.6)

where sin p ---- sin r for p = (0, r

For l < / < n and pESt, we mus t take into account the []x(p)--x(q)[[ -1 singulari ty in K. For p = (~,/5) ~ q = (0, r use Taylor ' s theorem to obtain

t - - w ( p ) . w(q) ~ _ [(r _/5)8 + (0 _~r sins/5] = L I ( p ' q), (3.7)

Ilx (v) - x (q)II ~ ~ [ ~ (0 - ~)~ + as (0 - ~) (r - / 5 ) + d~ (r --/5)~]a - - Ls (V, q),

with

d a = [a 2 sin ~ a + b 2 cos ~ ~] sin ~/5, dz = � 8 9 (b ~ - - a s) sin 2 ~ sin 2/5,

d s = [a ~ cos 2 ~ + b ~ sin ~ ~] cos ~/5 + c s sin s/~.

Nystr6m Method 23

Then obtain for t < l < N, p ESI,

iz(p) ~sin(pl)lX(pl) K (p, pl) L~(p, pz) f f Ll(p, q) dq. (3.8) LI (p, Pl) L~ (p, q) Si

T h a t par t of the in tegrand which was smooth was approx imated with the midpoint rule, and the remaining singulari ty is to be in tegra ted exactly. Because of its special form in (3.7), the integral in (3.8) can be evaluated in te rms of simple functions.

For I1 (P) with pr v use

Ii(P)~g(Pl)K(p, Pl) f f sinCdq=2z~[t--cos(h~)]K(p, Pl)/x(Pl). (3.9) S~

This is the midpoint rule for the polar region about (0, O, c) on 8D, corresponding to S r For peS~, use Tay lo r est imates, as in (3.7), and the midpoint rule, as in (3.9), to obtain

r [ ( ~ - fl)2 + 4 ~fl sin2 ( ~ - ) ] dCdO

I~(p)~l~(p~) f f {a2[~cosO_flcoso~]2+b2[$sinO_flsincr �9 (3AO) S

This reduces to a single integral which can be evaluated numerical ly; the evalua- t ion is rapid if the singularities are subt rac ted out. The integral I N (p) can be evalua ted similarly.

Collecting together the preceding results, we obtain a new functional equation approximat ing (3.5),

N

N(P) + =/(V), peS. (3.tt) /=1

This equat ion corresponds to (t.2), and it can be solved as in (t .3)-(t .4). Using the general theory referred to in the first paragraphs of w 2, it can be proven tha t /XN converges to /x uniformly on S. More precisely,

-- NII < B I I ~

with 9ff y the numerical integral opera tor defined implicit ly by (3.5)-(3.1t).

The accuracy of the numerical method was checked indirectly since only one t rue solution # (p) was known. Harmonic functions u were chosen, a solution/~N was produced from (3.t 1), and then a numerical solution u N was genera ted using an approximat ion of (3.3) based on (%6) and (3.9). The solutions u and u N were then compared.

Case (i). a = b = c = t , u ~ t , t* ---- ~-. Denote ae l0 = i (0.2, 0.2, 0.2).

n 0 % N It# -- ~NII ('* - ~N) (~(~ (u -- UN) (X(') (~, -- u N) (x(~))

4 3 6 t . 4 6 - - 2 - - 0 . 0 2 9 9 - - 0 . 0 t 91 0 .154

8 6 34 3.30 - - 3 --0.0142 --0.0129 0.0295 16 t2 t62 1.74 -- 3 --0.00535 --0.00523 --0.00357

24 K. Atkinson

Case (ii). a ---- b = t , c = �89 u (x, y, z) = e * cos (y) + e y sin (z). Le t x (0 = (i/5 If3) (t, t , 0.5).

n 0 n• N (u -- UN) (x (0)) (• - - UN) (x (1)) ('~ - - UN) (x(2))

4 3 6 - - 1 . t l - -1 .19 --0.475 8 6 34 --0.171 --0.144 0.0379

t6 12 162 - -0 .0 t85 --0.0153 --0.00142 u (x, y , z) t .0000 1.1797 t. 3715

The above systems for N = 3 4 and N = t 6 2 were solved i terat ively using method (i) of w 2. For the behavionr of the i terat ive method, we give tables for the above cases with N----6, M = t62.

v I ] / ~ - - , u ~ - l ) II Ra t io U#~-/ ,~ / -1)11 Rat io

I 1 . t 9 - 2 1 .43 7 .6 2 .8

2 t . 5 7 - - 3 5 . t 2 - - t 12 .7 2 .3

3 1 .24 - - 4 6 .5 2 . 2 4 - - t 2 .3

4 1.91 - - 5 6 . 4 9 . 7 2 - - 2 2 .3 5 3 . 0 0 - - 6 4 . 1 8 - - 2

I t should be stressed tha t the numerical me thod of this section was proposed for (3A)-(3.4) to provide an il lustration of an i terat ive var ian t of the Nys t r6m method for a nontr ivial integral equation. Eq. (3.4) is two-dimensional and has a singular kernel, thus showing tha t method (i) can be useful for such equations. In order to obtain a more practical method for (3.4), several changes should be made: (i) Take full advantage of all s y m m e t r y of the region; (ii) Be more careful in the use of the approximat ions (3.6)-(3A0), applying the approximat ion (3.8) to those integrals Ii(p) with "p close to Sl", a l though not in Sz; (iii) Inves t iga te more sophist icated product integrat ion methods for the two-dimensional integrals I,(p).

IV. Laplaee's Equation in Two Dimensions

Consider the integral equat ion

1 2x(s)-- f K~(s+t)x(t)dt=y(s), 0--<s--<t,

0 with

(4.1)

t + y2 _ 2y cos (2~0) ---- 1 + 2 Z 7 i cos (2f7~0), 1

and 0 _ y < 1. This arises from a classical reformulat ion of the Dirichlet problem for Laplace 's equat ion on an ellipse; see [t 5, p. t21].

Define the numerical integral operators ~ , using the ordinary midpoint rule

1

. ,_ ) ,

N y s t r 6 m M e t h o d 25

just as in [8] in which (4.t) was an example for method (ii) of w 2. In the following tables, I = 1 means x(s)--t and I = 2 means x(s)=sin(2~s)/(2+y). For all cases y = 0.8. The index of the final i terate x!~ ) is denoted by N,

For methods (ii) and (iii) of w 2, the tables include

B ~ . ~ _ . . . . . . . -1 )

since such ratios were fairly constant for all i terates x~ ).

M e t h o d 1.

1 ,~ n m N E 1 E 2

l - - 1 . 0 30 60 6 3 . 9 - - 9 1.5 - - 6

t 0 . 9 t 6 64 11 4 . t - - 3 6 .5 - 3

I 0 . 9 9 32 64 11 2 .4 - - 5 1.4 - 4

2 o . 9 9 32 64 5 3 .o - - 7 4 .0 - 7

2 0 . 9 9 32 t 2 8 5 3 .0 - - 7 9 .8 - t 0

M e t h o d 2.

1 ~ n m N E 1 E a R

1 - - 1 . 0 3O 60 4 t . 9 - - 9 1.5 - - 6 800

I 0 . 9 t 6 32 11 1.7 - - 4 1 .6 - - 2 2 .2

1 0 . 9 16 64 11 t . 5 - - 4 1 . 2 - 4 2.3

1 0 . 9 9 32 64 8 8 .7 - - 7 1.3 - - 4 7 .2

2 0 . 9 9 32 64 3 2 .4 - - 7 4 . 0 - - 7 1 5 0 0

2 0 . 9 9 32 t 2 8 4 9 .0 - - 10 5 .2 - - 13 875

M e t h o d 3.

t ~. n m N E 1 E~ R

t 0 . 9 16 32 I t t . 6 - - 4 2 . 0 - - 2 2 .2

t 0 . 9 24 48 8 1 . 4 - - 7 5 . 0 - - 4 9.3

1 0 . 9 9 24 48 t t 4 .4 - - 4 5.0 - - 3 2 .0

2 0 . 9 9 24 24 4 3 . 9 - - 7 2 .4 - - 3 t t 0

2 0 . 9 9 24 48 4 3 .7 - - 8 1.1 - - 5 2 4 0

The ra te of convergence varies with the unknown function x(s); see 2 = 0 . 9 9 with I = t , 2. Thus the bounds b~ (2), d~ (2), and/~ (2) on the rates of convergence of methods (i)-(iii), respectively, are not complete since they are independent of the unknown function.

The ratios of successive differences

D, = I] x~' I - - x~ -1) ]l

26 K. Atkinson

were usual ly not cons tan t wi th me thod (i). As examples of two ex t reme bu t no t un typ ica l cases, we give the following tables.

I = t, 2.=--1, n = 30, m =60 I = 1, 2.=0.99, n =32, m = 6 4

v D Ratio v D v Ratio

t 1.0 6 5.0 - 3 400 2.3 2 2.5 - 3 2.t 7 2.2 - 3 22 3 t .2 -- 3 8 t.0 -- 4 390 0.3 4 3.1 --6 2.1 9 3.3 --4 4.4 5 t .5 - -6 380 to 7.5--5 3.1 6 3.9-- 9 11 2.4-- 5

Addi t iona l examples are given in the technical reports [4] and [5], including results for

x(t)dt 4x(s ) - - l+(s-- t )* = y ( s ) , O<--s<--a.

0

F o r 4 < 0 bu t near zero, me thod (i) often converges when methods (ii) and (iii) fail. F o r o the r values of 4, the examples genera l ly favor me thod (ii) because of i ts regular ra te of convergence. But i ts efficiency is not necessar i ly grea ter t han t ha t of me thod (i).

V. I terat ive Var ian t s of the N o n l i n e a r N y s t r 6 m Method

In order to have a specific form of nonl inear equat ion to discuss, we consider Urysohn ' s equat ion of the second k ind

b

x(s) = y ( s ) + 4 f K ( s , t, x(t))dt, a<--s~b; (5.t) a

symbol ica l ly x = 4 ~ ( ' ( x ) + y . We assume ~ is a comple te ly cont inuous and cont inuous ly dif ferent iable opera to r on some open domain ~ ( s into s a Banach space; usual ly ,~" = C [a, b]. Jus t as in w 1 we have a p p r o x i m a t e schemes

t t

x,(s) = y ( s ) + 4 Y , wi(s, x~(tj)), a<--s<--b; (5.2) i=1

e.g., see Moore [t8, p. 873. These are solved by reducing to the equiva lent non- l inear sys tem

x,(t,) = y ( t , ) + 4 ,~ wi(t,, x~(ti) ), i =- t . . . . . n; (5.3) i=1

the equivalence is a t t a ined using (5.2) as an in te rpola t ion formula. Symbol ica l ly we wri te (5.2) as x , = 4 ~ ( x , ) + y . A comple te theo ry for (5.t)-(5.3) and for more general equat ions is given in [6] ; also see the work in [19] based on Newton ' s

Nystr6m Method 27

method. With appropriate smoothness assumptions, one obtains

II~-x.tl--<~ll~( x ) - ~ ( ~ ) l l , ,,>__N, (5.4)

for some c > 0 and N >-- 1.

We will look at iterative methods for solving (5.2), and these will yield methods for system (5.3). Newton's method (see [t8]) for (5.2) is

x~+al = x~) -- [I --2#g,~ (x~))]q[x~)--2Ot~,~(x~l)--y] , v>=O. (5.5)

We will define two variants of (5.5) by using the methods of w 2. As before, let n be an index for which (5.2) can be solved and for which

[z - ~ . ' (x.)]-i

can be calculated and saved. Let m > n, m the index of the problem (5.2) we want to solve.

Method (i). Use the approximation

- ~yd.(x.)]-

in Newton's iteration (5.5). This results in method (i) of w 2 if ~ is linear.

Method (ii). Use the approximation

[I--a.X",,~ (x!:))]-lm, I + ~ , [ I ' ~ " - a ~ (x.)]- x~ (x,,)

in the iteration (5.5). For ~ linear, this reduces to method (ii) of w 2.

Convergence of methods (i) and (ii) for n sufficiently large can be proven by combining the results of w 2 with the standard contractive mapping theorem for a "modified Newton method", in particular, when (5.5) is first modified by using

-~ , . (x , ) ] - .

The results are not of practical value except in saying that the iterative methods will work for n sufficiently large. The rates of convergence are closely associated with the rates (2.5) and (2.tl) using b~(~) and d~(2).

The cost of method (ii) is likely to be more than twice that of method (i), although it will vary according to the cost of evaluating the derivative relative to evaluating the function. In the linear case, ~ , / (x~)----~; but in general evaluating J~U~ (x~) is not directly connected with evaluating )g',, (x). If we try to avoid evaluating )U~ (x~) more than once, then method (ii) becomes much more competitive; but there may not be sufficient memory space, as illustrated with the example in w 6.

Other iterative variants for (5.2) are possible. The ones given here have proven useful, and they illustrate the general concept of iterative variants of the Nystr6m method for nonlinear equations.

28 K. Atkinson

VI. Nekrasov's Equation

The i terat ive methods of w 5 will be i l lustrated by using them to solve Nekra- sov 's equation,

x(s) = 2 sin L(s,t) x(t)dt , O<_s_<_~, (6.1) ~6 1 + 3 fx(r)dr

0

which we will write symbolical ly as x = 2 ~ (x). This equation arises in the s tudy of wave forms on a moving liquid of infinite depth [17, p. 4t 5]. In the equat ion

I In s i n} ( s+ t ) 2 2 1 L ( s , t ) = ~ sin�89 - - n ~ n sin(ns) sin(nt).

The Eq. (6.1) has x(t)=--0 as a solution for every 2. We are interested in obtaining solutions which bifurcate from the zero solution. Since

Jf'(O)h(s) = f L(s, t)h(t)dt, h.EC[O, ~], 0

the bifurcation occurs at 2 = m for every posit ive integer m. For 2 near m, the solution x(s) acts like a mult iple of s in(ms) ; see Krasnoselskii [16, Chap. 4]. Empirically, for 2 ~ m,

x (s) m l (2 -- m) sin (ms).

To solve (6.1) numerically, we first define a numerical opera tor ~ , with JT', (u) ~ f f (u) for all functions in the domain ~ of (6.1). To aid in dealing with L (s, t), split it as follows :

L(s, t )= I ln{ (s--t) sin�89 } =- (s + t) i2~-- ~ - ~ ) ; ~ ( s - t )

+ ~- (ln (s + t ) + l n (2 :z - - s - - t ) - - l n l s - - t [ )

=Ll(s,t)+L2(s,t), O<--<_s,t<m

L 1 (s, t) is analyt ic for 0 <= s, t <= :~. Let n=>2 be even, h=:~/n, t i=]h for I ' = 0 , t . . . . . n. Let Wo=W~=h/3,

wz i = 2h/3, w2/'-1 = 4h/3, the weights for Simpson's rule. For the integral operator wi th kernel L 1 (s, t), approx imate it by

f Ll(S,t ) x(t)dtt ~ 2 w i x(ti)Ll(s'ti) o t + 3 fx(r) dr i=o l + 3 f f x ( r ) dr

0 0

n ~, wi x (ti)iL~ (s, ti)

i=o 1 + 3 E=o~izx(h)

(6.2)

with/5o, 0 = 0 , fll,o = • 1 , 1 =hi2. For i even, let {flit} be the weights for Simpson's rule on [0, @ ; and for i odd, use Simpson's rule on [0, ti_ 3] and the 3/8's rule on [ti_a, ti].

Nystr6m Method 29

To approx imate the integral with kernel Lz (s, t), we use the product Simpson's rule which was first discussed in [3]- In general,

hi2 t~#

tL,(s,t)i(s,l)dt:~l, f L2(s,t)[(s,t)dt 6 j=l t,l-,

n]2 l~il

~ f L,(s.t){~-I (t_t,s_~)(t_t=s)l(s.t.q_,) j = l ti#-2

t

1 (t - t~ S-l) /(s , r s)} + ~ (t-t~._~) d t

-2 - ~ ( s ) I(s , 6). l=O

( .' ) For ] (s, t) = x (t) / t -t- 3 f x (r) dr , with the denominator approx imated as in (6.2), we obtain "

o 1 + 3 f x(r)dr j=o 1 + 3 Z flflX(tl) 0 I=0

Combining (6.2) and (6.3), define

~ / ' x ( s ) ~ x ( s ) - ~ s i n { i = ~ ~ [wiL~(s, ts)+wi(s)] x(ti) - - - I '

(6.3)

(6.4)

for all xEC EO, ~] for which the denomina tor is str ict ly positive. The weights wi(s ) are generally needed a t just a finite set of points in [0, ,~], and they can then be very efficiently calculated, as was described in [31.

We now wish to solve x~ = , ~ (x~), which can be reduced to an equivalent finite nonlinear system, just as in (5.1)-(5.3) and ( t . ! ) - ( t .4) . For the rate of con- vergence of x, to x, the results of EtO] for product integrat ion can be combined with (5.4) to give

IIx-x.ll <=ch In (6.5)

In [7] it is shown tha t for eigenvalues 40 of mult ipl ic i ty one of i f ' (0), t ha t the bifurcating solutions of x, : A3~,, (x,0 converge uniformly in a relative error sense in the (x, 2) space to the solutions of x =AJd(x ) . More precisely, let x(4) and x~(,~) be the corresponding nonzero solutions of x : ~ ( x ) and x~ =2~d,(x~), respectively. Then for some ~ > O,

Ilx(~)-x'(~)[[-~o as n-~oo. (6.6) sup

This result need not be true for eigenvalues of ~ ' (0) of odd mult ipl ici ty grea ter than one.

3 0 K . A t k i n s o n

Many examples were calculated, including the following of bifurcating solutions for some distance in the (x, 2) plane. But we will give just two simple examples:

(i) 2 = t . 0 5 (ii) ~ = t . 9 5

n I1~,,. - ~ . / , l l R a t i o n II ~,,. - ~/ ,11 R a t i o

4 1.23 - - 2 8 t .65 - - 2 8 1.56 - - 3 7.9 16 2 .49 - - 3 6 .6

16 t . 2 7 - - 4 12.3 32 2 .03 - - 4 t 2 . 3 32 8 . 4 2 - - 6 t5 .1 64 1.38 - - 5 14.7

128 8 .94 - - 7 t 5.4

In all cases the ratios approached 16, which would be implied by (6.5). The result (6.6) was also verified empirically for bifurcation at 2 = 1, 2, and 3.

To illustrate the iterative methods, we will look at the case 2 = 3 . t with n = t 6 and m = 64, with x~ ) = x,,.

M e t h o d (i) M e t h o d (ii)

v I1X(r~ ) - - X(m - 1 ) 11 R a t i o v [[ x(~) _ x ( v - 1)11 R a t i o

t 2 . 5 - - 2 6.3 1 1.1 - - 3 0 .5 2 4 .0 - - 3 0.5 2 2 .0 - - 3 67

3 7.6 - - 3 o .9 3 3.0 - - 5 9 .4 4 8 .8 - - 3 1.6 4 3.2 - - 6 45

5 5.6 - - 3 5.2 5 7.0 - - 8 5.8 6 t . t - - 3 33 6 1 . 2 - - 8 86 7 3 . 2 - - 5 1.9 7 t . 4 - - t 0 2 .8 8 t . 7 - - 5 6 .4 8 5.0 - - 11 9 2 .6 - - 6 4 .2

l o 6.1 - - 7

For a second example of method (i), we look at 2 = t.95 with m ----- t28 and n = 16; method (ii) was not feasible because of memory and time limitations.

[1 g ~ ) - - X~ - 1 ) 11 R a t i o

t 8.1 - - 3 0.7 2 1 . 2 - - 2 4.3

3 2.7 - - 3 3.6 4 7.7 - - 4 t 5 5 5.0 - - 5 1 t 6

6 4 .3 - - 7 1.9 7 2.3 - - 7 7.7 8 3 . 0 - - 8

With both methods (i) and (ii), there is no regular behaviour to the convergence of the iterates, as illustrated above. Thus caution must be exercised with all te rminat ing schemes.

Nystr6m Method 31

Bibliography

1. Anselone, P .M. : Convergence and error bounds for approximate solutions of integral and operator equations. In: Error in digital computation, vol. 2, ed. by L. ]3. Rall, p. 231-252. NewYork: Wiley & Sons 1965

2. Anselone, P. M. : Collectively compact operator approximation theory. Engle- wood Cliffs, New Jersey: Prentice-Hall 1971

3. Atkinson, K. E. : The numerical solution of Fredholm integral equations of the second kind. SIAM J. Numer. Anal. 4, 337-348 (t967)

4. Atkinson, K . E . : Iterative Methods for the numerical solution of Fredholm integral equations of the second kind. Technical Report, Computer Center, Australian Natl. Univ., Canberra

5. Atkinson, K. E.: A survey of numerical methods for the solution of Fredholm integral equations of the second kind, to appear in the proceedings of the sympo- sium "Numerical Solution of the Integral Equations with Physical Applications", 1971 Fall Meeting of SIAM, ed., Ben Noble

6. Atkinson, K. E. : The numerical evaluation of fixed points for completely conti- nuous operators. SIAM J. Numer. Anal., 10, 799-807 (1973)

7. Atkinson, K. E. : Bifurcating solutions for approximate problems. Submitted for publication

8. Brakhage, H. : l~ber die numerische Behandlung yon Integralgleichungen nach der Quadraturformelmethode. Numer. Math. 2, t 83-195 (1960)

9. ]3iickner, H.: A special method of successive approximations for Fredholm integral equations. Duke Math. J. lS, t97-206 (t948)

10. Hoog, F. de, Weiss, R. : Asymptotic expansions for product integration. Math. Comp., 27, 295-306 (1973)

11. Fox, L., Goodwin, E. T. : The numerical solution of non-singular linear integral equations. Philos. Trans. Roy. Soc. London A 24S, 50t-534 (1953)

12. Garabedian, P. R. : Partial differential equations. NewYork: Wiley & Sons t964 13. Hashimoto, M. : A method of solving large matrix equations reduced from Fred-

holm integral equations of the second kind. J. Assoc. Comput. Mach. 17, 629-636 (t970)

t4. Kantorovich, L. V., Akilov, G. P. : Functional analysis in normed spaces. London: Pergamon Press 1964

t 5. Kantorovich, L.V., Krylov, V. I. : Approximate methods of higher analysis. Groningen, Netherlands: P. Noordhoff, Ltd. 1964

16. Krasnoselskii, M. A. : Topological methods in the theory of nonlinear integral equations. London: Pergamon Press 1964

17. Milne-Thomson, L. M. : Theoretical hydrodynamics, 5th ed. MacMillan 1968 t 8. Moore, R. H. : Newton's method and variations. In : Nonlinear integral equations,

ed. by P. M. Anselone. Madison, Wisconsin: Univ. of Wisconsin Press 1964 19. Moore, R. H. : Approximations to nonlinear operator equations and Newton's

method. Numer. Math. 12, 23--34 (1968) 20. Nystr6m, E. J.: l~ber die praktische Aufl6sung yon Integralgleichungen mit An-

wendungen auf Randwertaufgaben. Acta Math. S4, 185-204 (1930) 21. Petryshyn, W. V. : On a general iterative method for the approximate solution of

linear operator equations. Math. Comp. 17, t-10 (1963)

Prof. Kendall Atkinson The University of Iowa Dept. of Mathematics Iowa City, Iowa 52240/U.S.A.