Multipoint algorithm applied to nonlinear Chandrasekar integral equatio

Bosarge, W. E., Jr.

Falb, P. L.

English

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A MULTIPOINT METHOD OF THIRD ORDER 
by 
9 ** W. E. Bosarge, Jr. and P. L. Falb 
Center for Dynamical Systems 
Brown University 
Providence, Rhode Island 
r 
c: 
* 
This Research was supported by the National Aeronautics and Space Adminis- 
tration, under Grant No. NGR 40-002-015. 
H 
This Research was supported by the Air Force Office of Scientific Research, 
under Grant No. AF-AFOSR 693-67. 
https://ntrs.nasa.gov/search.jsp?R=19680020726 2020-03-12T10:22:41+00:00Z
A MULTIPOINT METHOD OF THIRD ORDER 
W. E. Bosarge, Jr. and P. L. Falb 
Abstract 
L e t  F be a mapping of the Banach space X in to  itself. A 
convergence theorem f o r  the i t e r a t ive  solution of F(x) = 0 i s  
proved 
$(XI = 
of F. 
on F, 
fo r  the  multipoint algorithm x = x - $(xn) where n+l n 
-'[F(x) + F(x-F'-$(x))] and F; i s  the  Frechet derivative FJ, X 
The theorem guarantees that, under appropriate conditions 
the multipoint sequence Exn] generated by fl converges 
cubically t o  a zero of 
l inear  Chandrasekhar integral. equation 
F. The algorithm i s  applied t o  the  non- 
where a0 > 0. 
discussed and some numerical r e su l t s  are given. 
A discret izat ion of the equations of i t e ra t ion  is 
1. Introduction 
L e t  F be a mapping of the Banach space X in to  i tself .  
Considerable e f fo r t  has been devoted t o  the study of i t e r a t ive  
methods for  the determination of 'zeros  of F i.e. solutions of 
F(x) = 0 
Here, we consider a multipoint method of t h i rd  order for  the  s o h -  
t ion  of F(x) = 0 which i s  based on the  algorithm 
(see, for example, [I], [21, C31, [ G I ,  171 and [81). 
where 
and F; i s  the  Frechet derivative of F. This algorithm requires 
several  evaluations of F and a single inversion of the l inear  
operator Ft a t  each i t e r a t ion  step. We show that, under appro- 
p r i a t e  conditions on F, the algorithm generates a sequence which 
x 
converges cubically t o  a zero of F. Since most higher order methods 
require the computation of correspondingly high order Frechet 
derivatives, we f e e l  t h a t  the algorithm (1) and i t s  generalizations 
([ 71 ) have def in i te  p rac t i ca l  u t i l i t y .  
We apply the algorithm and convergence theorem to the 
Chandrasekhar in tegra l  equation 
where u0 > 0. 
operator equation i n  
The in tegra l  equation ( 3 )  is equivalent t o  the 
.5f( [o, 13) 
(4) F(x) = B(x,x) - Ix + 1 = 0 
P a 
3 
where 
and Oo > 0. 
2. A Convergence Theorem 
We now state and prove a convergence theorem for the algorithm 
(1). 
vergence of Newton's method ( [ 3 ] ) .  
The result is similar to.Kantorovich's theorem on the con- 
TIEOREM. Suppose that F is twice continuously differentiable on 
- 
do the closed sphere S(xo,r) = 5 and that there are constants 
and K such that -
(i) F&-' exists and llF&-lll 5 Bo; 
0 0 
81 
1-7 O 
(v) -7 6 r and B K7 5 5/9  
0 0  -
= do(l + - ). Then the multipoint sequence {xn] - for 
QO 2 where 
F based on x (i.e. the sequence x = x - 9 (xn-,)) converges 
0 n n-1 
4 
cubically t o  a root 
vergence i s  given by 
X* of F(x) = 0 
- 
i n  S - and t h  r a t e  f con- -
where ho = B K T ~  
0 0  
Proof: The proof i s  by induction on n. We s h d l  define sequences 
Bn> dn> vn and hn by se t t i ng  
hn = KB,Tn 
and we s h a l l  show tha t  the basic hypotheses (i) - (v) are sa t i s f i ed  
for  xn re la t ive  t o  the constants Bnydn and K. W e  then estimate 
-x 11 and show tha t  [xn] i s  a Cauchy sequence i n  S. Since ~~xn+m n 
S i s  complete, Ex,] converges t o  a n  element x of which we 
prove i s  a zero of F. 
- 
* - 
We begin with the t rans i t ion  from n = 0 t o  n = 1. Since F 
i s  twice differentiable on zy 
5 
- 
for all x,y i n  S and so, i n  particular,  we have 
[note tha t  [Ixl-xo[l = [l$(x,)[I I qo 6 1-1. It follows tha t  
exis ts .  Moreover, [[F;-’[I - 5 BohA(l-ho) -1 and hence t h a t  F; 
1 1. 0 
so that 
Letting y = x - F I - $ ( ~  
0 0 X 0 
0 
[lF;-$(xo)]l S do 6 qo I r, we 
0 
and noting t h a t  [Iyo-xol[ = 
expand F(xl) i n  a Taylor s e r i e s  
about yo t o  obtain the  inequality 
which, i n  view of (ll), yields the  inequality 
5 * 
6 
and hence t h a t  
(13 ) 
-1 
We now observe t h a t  g1 6 (-)h 701 2 g 6 (-) 5 2 (-)g 701 
5 {(-) (-) + l ]go  S vgo S r. 
6 7, so tha t  512 0 0 g 512 o 
5 2 701 81 We claim t h a t  x2 = x - 71 + 70 9 512 1 
-1 F' [F(xl) + F(x -F1-?I?(xl))] i s  a well-defined element of z and x, 1 x, 
t h a t  [{x -x 11 6 Letting y1 = x1 - F1-?F(x1) 
1 2 1  X 
~ ~ y l - x l ~ ~  = llF;-?I?(xl]l 6 d1 so tha t  y 1 E z, we can 
1 
and noting tha t  
see tha t  x2 i s  
81 5 -7 and since h = KB171 so t h a t  81 17% 17 0 1 Since 
A 
1 
we deduce t h a t  the basic hypotheses (i) - (v) are  sa t i s f i ed  fo r  
re la t ive  t o  the constants B K and dl. 
x 
1’ 
We can show, by exactly the same arguments, t h a t  i f  the basic 
hypotheses (i) - (v) are  sa t i s f i ed  f o r  x r e l a t ive  t o  B K and 
dnl n+ 1 n+l’ 
n n’ 
then the same holds t rue  fo r  x re la t ive  t o  B K and 
. Moreover, we have dn+ 1 
and 
for n = 0,1,2,. . . . It follows tha t  
and 
- 
for  n = 1,2, ... . Since the xn are  in  S and since the ser ies  
tThis follows from (16) and (18) as 
8 
is convergent and since vn 
- 
we deduce that {xn] is a Cauchy sequence in S and, therefore, 
has a limit x* in S with 
- 
for  n = 0, 1,2 ,... . 
It remains to show that x* is a zero of F. Now, the in- 
duction argument yields (compare (13)) the inequality 
for  n = 1,2,. . . , since Bn,l 2 Bo. Since F is continuous, we 
deduce that x* is a zero of F from (18) and (19). Thus, the 
theorem is established. 
We note thzt the numerical estimates in condition (v) and in 
equation (6) are not quite optimum but rather are chosen for con- 
venience of calculation. 
, 
9 
. . .  
3 .  A Nonlinear Integral  Equation 
We consider the nonlinear Chandrasekhar in tegra l  equation ( 3 ) .  
We l e t  X be the Banach space of continuous functions on [O,l], 
g([O, l]), under the uniform (or supremum) norm and we l e t  
be the b i l inear  map of X @ X in to  X given by 
B(* ,  0 )  
wO I t  B ( u , v ) ( ~ )  = p ( t ) J  -v(S)ds s-tt 
0 
fo r  t E [0,1]. Then determining a solution of ( 3 )  i s  equivalent t o  
determining the roots  of the "quadratic" operator equation 
i n  %([0,1]). 
Now we observe t h a t  the operator F i s  twice continuously 
Frechet different iable  and tha t  
F4Z = B(x,z) + B(z,x) - IZ 
F;(W,~) = B(w,z) + B(z,w) 
fo r  any x i n  g ( [ O , l ] ) .  We then have: 
81 
17 COROLLARY. Suppose t h a t  0 < wo 5 .65 and t h a t  r 2 -(l+@oi?n2). 
The the  multipoint sequence {x generated by the algorithm (1) n 
10 
* - . .  . . .  ~ - 
with xo = O(*) converges t o  a solution x of ( 3 )  i n  S(0,r) and 
the rate of convergence i s  given by (6) with 
- - - -
wo an2 
v0 = (1+-) 2 
w- in2 
and -
0 h = U) &n2(1+-). 
0 0 2 
Proof: We simply show t h a t  the hypotheses of the theorem are  
fied. Since x = O(-), we have F' = -I and F(xo) = 1(-) 
0 X 
0 
satis- 
by 
-1 
vi r tue  of (25) and the  def ini t ion of F. Thus, llFr 11 = 1 and 
I[F$-'[l [lF(x,)II = 1 so tha t  we may take Bo = d = 1 and the hy- 
potheses (i) and (ii) w i l l  be sat isf ied.  
t ha t  
X 
0 
0 
0 
A s  fo r  (iii), we note 
which implies t ha t  [IF"[[ 5 K for  a l l  x with K = ao.en2. Now 
X 
$(x0) = -I[l(-) + F(-l (*))]  = -l(*) - B(-L(*) )  + 1(-) i 1(*) = 
1(*) - B ( - l ( * ) ,  -l(*)) and B ( - l ( * ) ,  -l(*))(t) = $1 0 I t  -&is SO 
wokM 0 
tha t  
The hypothesis (v) i s  sa t i s f i ed  by v i r tue  of the  assumptions of t he  
corollary. Thus the corollary follows from the theorem. 
[l$(xo)ll 5 1 + - - and the hypothesis ( iv) i s  sat isf ied.  2 - 7 0  
We now describe the  algorithm (1) for  the  mapping F of (24). 
11 
Setting y = x - F*-!F(x ), we have n n X n n 
and 
for n = 1,2, ... . Thus, the  algorithm (1) is  equivalent t o  the  
sys tern 
which can be writ ten more expl ic i t ly  i n  the form 
OO I t  
2 n  
0 
-X (t)J -X (S)ds - 1 s+t  n 
Q 
P 
The system (30) i s  a pa i r  of "similar" l inea r  Fredholm equations fo r  
and x respectively. We s h a l l  see, i n  the next section, 'n n+lJ 
t h a t  only one of the pa i r  of equations need be solved a t  each s tep 
i n  the "practical" application of the method. 
4. Discretization and Numerical Results 
The corollary quarantees the  convergence of the algorithm (1) 
for  the Chandrasekhar integral  equation w i t h  0 < w S .65. How- 
ever, it i s  clear  from the form of (30) tha t  the pa i r  of l inear  
0 
equations required a t  each i t e r a t ion  step i s  not easi ly  solved. 
Thus, we s h a l l  "discretize" the system i n  an appropriate way and 
then carry out the calculations on a computer. 
of the multipoint method w i l l  then become apparent since only one 
"matrix inversion" w i l l  be required a t  each i te ra t ion  step. 
The chief advantage 
Now l e t  r(t, s) = t / ( s+t )  and l e t  to,. . . , t  be a par t i t ion  m 
(or mesh) on [0,1] with to = 0 and tm = 1. Suppose that 
(rl, ., r, s p  *, Sm> 
the y ls  as weights and the  s as mesh points i n  [O, l ] .  Then, 
we have 
represents a suitable quadrature rule w i t h  
i 
f o r  i = 0,l ,..., m and so, 
1 
provides an m-dimensional representation of 1 r ( t ,s )x(s)ds .  I n  a 
similar way, we can represent (approximate) the b i l inear  operator 
by the m-dimensional b i l inear  operator Bm defined by the "three 
0 
B 
dimensional matrix", Bm w i t h  - 
(6. being the Kronecker del ta) .  In  other words, i f  a and @ are 
m vectors with components 5 ,...,am and B, ,..., B,, respectively, 
then - Bm(sp) 
Thus, if  we s e t  
i n  the discrete  form 
i k  - - 
m m  
i s  the m vector with components c bijkcx$3 j. j=1 k=l  
- x = (x(tl), ..., x(tm))t ,  then we can represent (24) 
where I? is the m x m ident i ty  matrix and lm is  an m vector - - 
with all components equal t o  Applying the  algorithm (1) direc t ly  
t o  (34), we obtain a system of equations ident ica l  t o  the discretized 
version of (30). 
1. 
In  other words, we have 
14 
or, equivalently, 
where M i s  the m x m-matrix given by -n 
(37) Zn = [$n(zn>*) + "(*,zn) - _r"1 
and sn, A are the  m vectors given by -n 
(38) 
m 
-n c = - BYzn,zn) - Am, L& = 2 (EnJn) - En + 1". 
Thus, i f  M i s  invertible,  we have -n 
and so only a s ingle  inversion i s  required a t  each i t e r a t ion  step. 
Although the approximate solution i s  i n  "discrete  form", we can f i t  
, 
cuO 0.1 0.2 0.3 0.41: 0.5 0.6 0.7 0.8 0.9 1.0 
2 2 2 3 3 3 3 4 4 5  
I terat ions 
Required 
i 
an interpolating polynomial t o  the points 
degree of accuracy has been obtained. 
t ha t  the operations of "discretization" and " i terat ion" are  
x (t ) n k  a f t e r  the  desired 
It can a l so  be shown (cf. [4]) 
commutative and an e r ror  analysis, based on t h i s  fact ,  can be 
carr ied out. 
Results of some par t icu lar  computations appear i n  the following 
tables.  The in te rva l  [0,1] was parti t ioned in to  10 subintervals 
of length h = 0.1 
integration. 
quired t o  obtain "convergence"t for various values of cu 
0' 
Table 11, we present the actual  approximate solutions. The r e su l t s  
i n  Table I1 compare quite favorably with those presented by Rall i n  
[?I. We a lso  observe that, although the corollary guaranteed con- 
and Simpson's ru le  was used fo r  the numerical 
I n  Table I, we present the number of i t e ra t ions  re- 
and, i n  
vergence only fo r  0 < w 5 a65, the  actual  computations converged 
f o r  values of coo > .65. 
0 
TABLE I 
'Convergence i s  here construed t o  mean tha t  
l l x  -x 11 s 1.0 10-7 -n+l -n 
11' where II.II i s  the Euclidean norm i n  R 
16 
I 1.5977897 
TABLE I1 
. .  _ _ . .  . 
L t 
0.0 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1.0 
KL 
t 
0.0 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
I. 0 
0.1 
1,0000000 
1.0124406 
1.0230083 
1.0263061 
1.0289211 
1.0186357 
1.0310583 
1.0328445 
1.0343627 
1.0356721 
1.0368137 
0.6 
1.0000000 
1.0916738 
1.1451549 
1.1857929 
1.2285030 
1.2456979 
1.2688045 
1.2887496 
1.3061819 
1.3215723 
1.3352737 
0.2 - 
1.0000000 
1.0257406 
1.0389366 
1.0482969 
1.05.54552 
1.0611705 
1.0658683 
1.0698127 
i.0760908 
1.0786371 
0.7 
1.0731792 
0.3  0.4 
1.0000000 
1.040051G 
1.0611696 
1.0763397 
1.0880966 
1.0975427 
1.1053562 
1.1176071 
1.1225166 
1.1268243 
0.8 
1.11193183 
1.0000000 
1.1133937 
1. a24722 
1.236290 
1.3177642 
1.398840 
1.3779287 
1.4026842 
1.4247322 
1.280475 
1.4445133 
1.0000000 
1.1391964 
1.22853 75 
1.3003721 
1.3608188 
1.4129552 
1.4586525 
~4991760 
1.5354414 
1.5681352 
1.0000000 
1.0555925 
1.0858020 
1.1078805 
1.1251392 
1.1391573 
1.15083 98 
1.1607608 
1.1693124 
1.1767711 
1.1833 400 
0.9 
1.0000000 
1 17253 49 
1.2912197 
1.3909588 
1.47799668 
1.2554475 
1.6252145 
1.7465906 
1.7999086 
1. a86072 
1.8491525 
0-5 
1.0000000 
1.0726499 
1.113 4719 
1.1438493 
1.1679133 
1.1876716 
1.2042799 
1.2184896 
1.2308130 
1.2416200 
1.2411844 
~ ~ 
1.0 
1.0000000 
I. 2441701 
1.4416694 
1.6277685 
1.8075085 
1.9830198 
2.1554050 
2.3253097 
2.4931469 
2.6591930 
2.823638 
References 
1. Altman, M., "An I t e r a t ive  Method f o r  Solving Functional Equations", 
math. astr. e t  phys. vol. 9, pp. 57-67, 1961. 
Serie des sciences 
_ -  
2.. Collatz, L., Funktionalanalysis and Numerische Mathematik, Springer, 
Berlin, 1964. 
3 .  Kantorovich, L. V. and Akilov, G. P., Functional Analysis i n  Normed 
Spaces, Maemillan, New York, 1964. 
4. Ortega, J. M. and Rheinboldt, W. C., "On Discretization and Differ- 
ent ia t ion of Operaters with Application t o  Newton's Method", J. SLAM 
Nmer. Anal., V o l .  3 ,  No. 1, 1966. 
5. Rall, L. B., "An Application of Newton's Method t o  the Solution of 
a Nonlinear Integral  Equation", Technical Report No. 7, Office of 
Ordnance Research, U. S. Amy, Contract No. DA-04-200-177. 
6. Traub, J. I t e r a t ive  Methods f o r  the Solution of Equations, Prentice 
Hall, 1964. 
7. Bosarge, W. E., Jr. and Falb, P. L., "Multipoint Methods and the 
Solution of Boundary Value Problems", t o  appear. 
8. Falb, P. L. and deJong, J. L., Some Successive Approximation 
Methods i n  Control and Oscillation Theory, Academic Press, New York, 
to appear i n  1969. 


A multipoint method of third order

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