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AN ITERATIVE METHOD FOR SOLVINGNONLINEAR SYSTEMS OF EQUATIONS
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Authors Bryan, Charles Allen, 1936-
Publisher The University of Arizona.
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BRYAN, Charles Allen, 1936-AN ITERATIVE METHOD FOR SOLVING NON-LINEAR SYSTEMS OF EQUATIONS.
University of Arizona, Ph.D., 1963 Mathematics
University Microfilms, Inc., Ann Arbor, Michigan
AN ITERATIVE METHOD FOR SOLVING
NON-LINEAR SYSTEMS OF EQUATIONS
by £
Charles,Bryan
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF MATHEMATICS
In Partial Fulfillment of the Requirements For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
19 6 3
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
I hereby recommend that this dissertation prepared under my
direction by Charles Bryan
entitled An iterative method for solving non-linear systems
of equations
be accepted as fulfilling the dissertation requirement of the
degree of TTnnt.ny of Philnanphy
/ ) / A { e , / 3 tion D/rfrctor 7Date/ **7 / DissertationDfrector
/ /
After inspection of the dissertation, the following members
of the Final Examination Committee concur in its approval and
recommend its acceptance:*
^ Q. .SgAui.u x/—- 3 ) S Q / 6 3
a/j7y j ^ OR^F)
6.2
•This approval and acceptance is contingent on the candidate's adequate performance and defense of this dissertation at the final oral examination. The inclusion of this sheet bound Into the library copy of the dissertation is evidence of satisfactory performance at the final examination.
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in The University Library to be made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in their judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGNED:.
ACKNOWLEDGMENTS
The author would like to express his appreciation
to Dr. H. Melvin Lieberstein, whose kindly guidance has
helped to sharpen many of the results of this paper. This
paper was prepared under the sponsorship of the National
Science Foundation Grant NSF-G16249.
ii
TABLE OF CONTENTS
Title: An Iterative Method for Solving Non-Linear
Systems of Equations.
Page
1. Introduction .... ..... 1
2. Some consequences for finite dimensional Banach spaces 4
3. Conditions for convergence in Tchebycheff norm . . 13
4. Choice of to 24
5. Conditions in a deletion space 29
6. Applications 43
Bibliography 50
iii
AN ITERATIVE METHOD FOR SOLVING
NON-LINEAR SYSTEMS OP EQUATIONS
1. Introduction
The purpose of this paper is to establish suffi
cient conditions to guarantee that the iteration
(11) (k+1) (k) i* 1 '""Xn > ji = 1,2,...,n7 u - } x i - x i w : r u n r n ^ u = 0 , 1 , 2 , . . . ] »
i, i 1 ' * * *' n '
•\ f u> a real number, and f_. ° i , will converge to a solu-1,3 TT7
J tion (unique in some neighborhood) of the system
(l»2) f (Xj ,Xg t • • • t) = 0 , i = 1,2,... ,n
where the functions f^ are real-valued functions of real
variables x., j = 1,2,...,n, and to establish computable 3
error bounds for the iterates. This iteration method was
first proposed by H. M. Lieberstein 193. The method is a
generalization of the Jacobi methodfl23 for solving a
system of linear equations. Along with this method Lieber
stein also proposed the method
f ^(k+l) Jk+D Jk) (kK ,, (k+1)_ (k) *^*1 >xj »* * * »xn > (1.3) x. - x. -u>- ( k + 1 ) (k+1) (k)TkT7
i ,i 1 '•••'xi_l »xi »•••»xn )
which is a generalization of D. Young's overrelaxation for
linear systems [l3j.
1
2
The advantage of methods (l.l) and (1.3) over
Newton's method is that the correction formulas are expli
cit, whereas in Newton's method a linear system of equations
must be solved at each step of the iteration; this often
leads to approximating the solutions of the linear systems
by iterative techniques and hence applying iterations within
an iteration and the calculations necessarily become lengthy
and time consuming even when carried out on high speed com
puters. Moreover in many cases, such as finite difference
analogs for differential equations, all calculations are
based on a single correction formula. Because of these
features, the method proves to be very efficient for use on
computers.
In section 2 the problem is reformulated in terms
of finite dimensional Banach spaces whereby the problem be
comes that of finding conditions under which a certain
mapping has a fixed point obtainable by Picard iteration.
In section 3, using the comparison method developed by L. V.
Kantorovic [63 and analysis techniques similar to those • •
used by J. Schroder 111} , sufficient conditions are ob
tained for the case when the space has a Tchebycheff norm.
In section 4 it is shown that under an appropriate defini
tion of "best" the optimum value of co in (l.l) is one,
whereas it is well-known that this optimum is not neces
sarily one in (1.3) when the f^ are linear. In section 5
3
the conditions are generalized to apply to so-called dele
tion spaces which include finite dimensional spaces with
Tchebycheff and ^ norms, thus yielding conditions other
than those of section 3. Uniqueness of the solution is
also shown here. Finally, in section 6 the application of
method (l.l) to finite difference equations is considered.
By using (1.3), which experimentally seems to converge
quite rapidly, to obtain a point at which the theory be
hind (l.l) is applicable, we obtain by the theory of (l.l)
the existence of a solution of a system of finite difference
equations for a non-linear two-point boundary value problem.
A maximum error bound for the approximation is obtained of
the order of magnitude of 10~^ where the solution itself is
of the order of magnitude of unity.
2. Some consequences for finite dimensional Banach spaces.
Let X be a real n-dimensional Banach space with a
basis e.^ , j = l,2,...,n. Corresponding to each element
e. of the basis consider the linear projection operator P. J J mapping X into X such that
(2.1) P4U) -Pj( - orj.j
and to each linear operator A mapping X into X there
corresponds the operator DA such that
(2.2) DA(x) = ZJ=1 [A(P.(X))J.
Since every linear operator A mapping X into X can be
represented as a matrix (a^.) it follows that
PiUWl - a^U)
and
C DA(X) 3 = tA(P.(x))] = Xi=l aiiPi(Pi(x))
= a..P.(x) IX 1 '
or equivalently, DA can be represented as a diagonal matrix
with the same diagonal as A. Finally if we consider any map
ping F(x) mapping X into X we may examine the mapping for
its differential properties. If F has a Frechet differen
tial at x it is represented by
F'(x)fh] = lim ILx+fh)-F(x) PtO A
where x and h are elements of X and (5 is from the scalar
field C2» Definit ten, p. 490; 10, Thm. 1 , p. 185 J •
With "these definitions and remarks in mind we re
formulate the problem of convergence of (l.l). Let X be an
n-dimensional Banach space and F a Frechet differentiable
operator on X into X. Consider the mapping
(2.3) T(x) = x -WD^F'U^FU)]
where F'(x) is the Frechet derivative of F(x) aiid we write
for short
D-1F1 (x) = (DF'U))"1.
If elements of X are represented by column vectors and F(x)
by the column vector
^f (x11 t • /*!<*> \
f 2(x)
F(x) =
;n(x)/
'V\
f2 xl*x2'* n )
\ fn(xi»x2»•••»xn)/ then the Frechet derivative F'(x) is represented by the
Jaeobian matrix
F*(x) =
(fl,l(x)
f2,l*X
fl,2^
f2,2^
\fn,l*X* fn,2 X*
f2,nW
r UV n,nx 1
and hence
DF'(x)=
f2,2^
If D F*(x) exists - that is, if f . . (x) =j= 0 for i = 1,2,..., 1»1
then.
V 0 • (x)/ n,nx '!
D_1F'(x)=
fl,lU* 0
1
2,2(x)
0
0
\
f—UV n,nx 'I
It is then seen that the iteration (l.l) is equivalent to
Picard iteration on T, where T is defined as in (2.3). We
shall use the following lemmas:
Lemma 1. Let A and B be any two linear operators on an
n—dimensional Banach space X into X. Then
DA -DB = D(A - B).
Proof• D(A-B)(x) » -J2=l I' i{(A-B)(P1(x))} =
- 1 U p i f A ( p i 2 i - i
= DA(x) - DB(x).
Lemma 2. Let X be an n-dimensional Banach space and F an
operator on X into X which is twice Frechet differentiable.
Then
(2.4) D[F»(x)(h1,h2)] = (DP*(x)Lh2l)'[h^I ,
where (DF1 (X) [h2L ) 1 .is the Frechet derivative of DF' (x) [h^L
with respect to x, F"(x)(h^,h2) .is the Frechet differential
of F'(x )[hj] with respect to x, and
D[F» (x) (hx ,h2)] = ^ 1Pi F"(x)(h1,Pi(h2)].
Proof:
(DP'(x) h )' [h ] = ai™ -±- JU"(x+(>h )[h 1- DF'(x)[hJ
—v H^J
= Zi=lPi{F"(x)(ri(h2>'hl)]
and since F"(x) (P±(h2) ,1^) = F" (x) (h^ ,P^(h2))}
(DF1 (x) £h^J)'[hjl = 5 i=lPilF"(x) (hl'Pi(h2) ]l
= Dp-'UMh^)] .
Two properties of the spaces under consideration
will be assumed and spaces with these properties called
deletion spaces.
Definition: An n-dimensional Banach space X will be called
a deletion space if lor all linear operators A on X into X
the following inequalities are satisfied:
(2.5) IIDAII 5^ ||AII
(2.6) IjA-DAll $ IJAll.
Lemma 3. Finite dimensional Banach spaces with the Tcheby-
cheff and norms are deletion spaces.
Proof:
Case 1. Tchebycheff norm.
8
When A is represented by the matrix (a j)
Thus
and
»•411 = 14iin 2j=l
= 4n - "All
lU-DAll = ?" ->n , .la. I £ IIAlt* liiin <£-j=1,j*i' ij '
Case 2. ^ norm.
Under the matrix representation for A
and in a manner similar to that of Case t,l conditions (2.5)
and (2.6) are satisfied.
It should be noted that Si spaces, l<p<-*°, are not
in general deletion spaces as can be seen by considering
the following example.
Let X be E with the 5^ norm. If A is a linear
operator on X then
II All = (p|ATA|)*
where piAt is the modulus of the eigenvalue of A with
largest modulus Cl2l. Let
1 -2 2 \
A = -2 1 2
\ 2 2 1
then
>T f> |ATA I = 9
/
and
p| (A—DA)T(A-DA)| = 16.
Therefore IIA-DAll > IIAll and hence X is not a deletion space.
Ve will rely heavily on two well-known theorems of » •»
functional analysis which are stated here without proof,
i) Mean value Theorem L4, p.155^. Let X and Y be real
norraed linear spaces and y = T(x) be an operator with domain
DCX and range ECY where D is a convex open set. Let T be
Fxechet dif'f erentiable on D. Then
(2.8) l\ T(x+h)-T(x)II ± l/h»sup[llT'(x+th)ll ,Ottil].
(ii) Let X be a Banach space and A a linear operator
from X into X. If llAll^-1, then (I—A) exists and has domain
X where I is the identity mapping. Moreover C 8, p.1023
(2.9) \\ (I-A)_1ll i I-iXir .
The statement and proof of the following were dis
covered by the author after examining a similar theorem of • •
Schroder Lll, p.l90l. However, the author feels that it is
quite probable that Kantorovic has established the same
result, but no reference has been found in the available
Russian literature.
Lemma 4. Let X be a Banach space, xQ an element of X, k a.
positive real number, and T a continuous operator defined on
S(xQ>k) = x|xeX, llx-xoll * k^
into X. The sequence {xn$ defined by
xn+l = T(*n>
10 £
is well defined and converges in norm to an element x be
longing to S(xQ,k) i£ there exist a positive real number
M, a set B, a^ mapping G from 0 into the reals, a number pQ
belonging to [0,M), a mapping 7 from Co,M) into the reals,
and a sequence of real numbers such that
^ B = i (p1, p2, p3, jo4, jo5)| p* real, Of i£5, Otfc M,
fL+ A M],
(2.10) ||T(x)-T(y)||* G^Jlx-y/l, llx-xjl ,lly-xQH ,UT(y)-xf/f/lT(y)-yli]
whenever both sides are defined,
(2.11) 0£G((o1,p2,p3,p4,p?) ± GU1,*2, 3, 4,^) for O.y 1
and (j^+ M '
(2.12) ||xo-T(xo)|U T(po)- po ,
(2.13) G(p-tr,p-£,tf-po,T(\r)-f,T(<r)j*) ± T(f»)-T(<r)
for whenever both sides are defined.
<2-14> Pn+l = T<Pn> '
(2-15) Pn+l '' fn '
<2-16> ill • r* A M
* and p - pQ k. Moreover
ii * ii / * Ux -xn|/ 4 f - fn.
Proof: By induction we show that
(2.20) (a) xn belongs to S(xQ,k)
(b) llxa-H-xn" 4 Pn+l-Pu
(c) »xn+l-xnll +K-xol14 fn+1" Po
where p^- pQ < M for all myO,
XI
By hypothesis (2.20) holds for n = 0. Assume that
it holds for n* p-1. Then for n £ p-1
" xrn-l-xn" ~ Pn+l~fn
and hence
(2.21) l|xp-xol|f "xk+l-xk" - STiLl'Vl"^ ~
and therefore x^ belongs to S(xQ,k). Furthermore
II x -x ,1/ + II x ,-x IK M p p—1 p—i o
so that
llx .-.-x. Uil|T(x )-T(x ,)lli p+1 pM p' p-1
± G^llx-x .11 ,llx -x II , llx ,-x || ,|lx -x ll ,Mx -x ,|l} p p-l p o p-1 o ' p p p p-1 "*
or by applying (2.11), (2.13), (2.20b) and (2.21)
^ xp+l"~xp" " Pp"" fp-1' fp" Po' fp-l~ fio' fp' Pp~ /p-1^ 4
4 T(fp)-T(Pp_i) = pP+i-pP
and hence (2.20b) holds for n = p. Therefore
11 Vl-xpll+llVxo" -< fp+l-PpVp-Po = Pp+l"ft> and the induction is complete which proves that the sequence
{x^} is well defined. By the triangle inequality
(2"22> Wn" Since $ pnl is a convergent sequence, it is a Cauchy se
quence; hence {x^ is a Cauchy sequence in X and therefore *
has a limit x in X. Since
IIX -X || £ P -Pi P - P n o ~ • n • o ' • o
for all n * 0 it follows that
II x*-xQll & p*- fQ <• k
* / \ and hence x belongs to S(xo,k). Since T is continuous
x* = iiixn+l = ***» T(xn* = *u) = T<x*)»
Finally since
^ xn+p~xn'^ ~ P n+p~Pn ~ ^ ~ n
for all n and p greater than or equal to zero it follows
that
ll*#-xnIU p*-pn
for all n^O and the proof is complete.
3. Conditions for convergence in Tchebycheff norm.
We examine first the case in which the norm is
given by
.. max | i " x , , = U U n U ± {
using the comparison method. This yields results which
allow us later (section 5) to obtain a somewhat more
general theorem using simple induction.
Since D^F^x) can be represented as a diagonal
matrix, the mapping T of (2.3) satisfies
P±(T(x)) = P.(x)-P. iD~1F'(x)CF(x)]} =
= P.(x)-D-1F'(x)^Pi(F(x))j .
Theorem 1. Let X be a finite dimensional Banach space with
Tchebycheff norm. Let F be a map ping of a set AtX into X.
The iteration
6.1) *n+l = V C F(xn>J
is well defined and the sequence {*x "j; will converge to x , *.
such that F(x ) = 0 if the following conditions are satis
fied. There exist Xq belonging to A and positive real num
bers r ,N,(3 .0 and H such that S(x .r )cA, F is twice —— o O O —— O ' 1 o o —
Frechet differentiable on S(xQ,ro), D"*^F,(xo) exists, and
(3.2) llF"(x)ll 4 N for x in S(x0,rQ),
(3.3) IID~1F' (Xq)|| 4 0o ,
(3.4) llD"1F«(xo) F(xo)ll 4 QQ ,
(3.5) l)D"1F,(xo)(F'(xO)-DF'(xo))l| i HQ,
(3.6) h(«) = H + )l- -i. ,
13
14
( 3 . 7 ) ( 1 - Mh ( io ) ) 2 > 2 ( l+ t*r i f e iwO« u» l Q 0 N|3 ,
* (to) = max(l, 11- j-j)),
and
(3 8) 1—|M»|hi(to)—4(1—t«i»ili(u>) ) —2(1+ '«*»'fc(w))i«/iQoN(3q
(1+ iwilUoJJNp < ro *
Moreover
(3.9) llxn-x*IU I^LPWQ0
where
h(u/) + ll(u))( 1-^(1—Hdh(to))2—2(l+l?iu>)M )i«; Q N0 d(w)= 1 0 ° < l.
h(u>) + fc«*»)+ ,~jy( 1—Mht««>))2(1+ IftwWmO wQ N0
Proof! j
Let
(3.10) H(x) = D~1F'(XO)LP(X)] ;
then if x and y are elements of S(xo,rQ)
H'(x)-H'(y ) = D" 1 F' (x o ) tF ' (x ) -P ' (y ) ]
and by statement i) section 2, and (3.3)
(3.11) 11H ® (x) —H • (y) i I sup^llF" (tx+(l-t)y)ll ,0±t*.l^/|x-y lit
i0QNHx-y//.
Let fl N
5(f) ~\?2»
then E (p ) is a monotone increasing, concave (H"(p ) > 0 )
function on t0,«*>) into E0»°°)» and
(3 .12) l lH«(x) -H' (y ) l l i H' (nx-yn) .
The function
= l-Nft f T
15
exists and is a bounded linear function on £0,00) for each
p such that 0 < jyjj- .
Since X is a deletion space, we obtain by Lemma 2
D-1F' (Xq)(DF1 (Xq)-DF' (x)) II £ 0ollD(F' (xo)-F« (x))||i
f ^IIF* (XQ)-F'(x)ll 4; @oN)|x-xoll
which is less than ofie if l|x-xoll <• k where k = min(rQ, jj j-) •
Hence by statement ii), section 2
(3.13) Rx = I-D'1F'(xo)tDF'(xo)-DF«(x)]
has a bounded inverse for each x which is an element of
S(xQ,k) such that
<3'14> K1"* l-MPJx-x0„ •
Now
DF'(x) = DF«(x )(l-D_1F«(x )D>F'(x )-DF'(X)}) =
- DF' (x )B o' x
so that
(3.15) D^F'U) = R~1D"1F,(xo)
for all x in S(xQ,k), or equivalently, T(x) is defined for
all x in S(xQ,k). Now by (2.3) and (3.15)
T(x)-T(y) = wD"1F'(x)^-F(x)- £ DF»(x)lT(y)-x]+
+ h DF'(y)[T(y)-y]+F(y)} =
= uR~1|-H(x)- i DH'(x)lT(y)-x] +
+H(y)+ i DH'(y)lT(y)-y]} = R fui
where
(3.17) f = -H(x)- jjjDH'(x)[T(y)-x]+H(y)+ £dH« (y)I T(y)-y] .
16
We have
(3.18) P.(f) = -P.(H(x))+P.(H(y))-
- i P.^H'(x)[Pi(T(y)-x)J-H'(y)tPi(T(y)-y)]}.
Putting
(3.19) Pi(f(s)) = -P.(H(x))+P.(H(y))-
x ^ r H(x+s(Pi(T(y)-x))-H(x)
" w (, s
H(y+s(P.(T(y)-y))-H(y)
we have
(3.20) P.(f) = Pi(f(s)).
Denoting P^(x) by x^ we obtain
(3.21) fi(s) = -H.(x)+H.(y)- H.U+sU.-x.M.U)-
Hi(y+s(zi-yi))-Hi(y)J
where z = T(y). It now follows (fundamental theorem of
integral calculus) that rl
(3.22) f.(s) =- • (x) (x.-y. )dt-
- RTR ^j=iHi,j®(xryo)dt
where
(3.23) x = tx +*tl-t)y and i = j *
Further,
17
(3.24)
- (1-A_j)uryj)dt" UJ
- 2:"=iHi,ju<.)(1-ii)(xryj) U9
so that
(3.25)
j#=i
since s can be restricted to be less than one.
Now since the space has a Tchebycheff norm, and remembering
(3.5), (3.6) and (3.9)
lf-(s)[^ 1 |h .(x+s(z.-x.) )-H . (x)| ||x-ylidt + 1 siu>i->0 J ,J '3
+ J X j=i Hi,j (*)~Hi^ j (x0)l fc<«»llx-yUdt+h(uj)|f x-yl/ o
and
* lilUUa [IW Hi,3("s(zi-"i) '""i.j®' Hx-y,ldt +
+l*i* n fej.ll Hij <=>-*!,, 'x0'lkl"lllx-yll<li+h(*i)llx-yll.
Since the sums are finite there exist N and N1 such that
rtejHN, j < <2n-*N>>"hn,j(x) 1Hx-yldt + Jo
(3.26) £ 1 (x+s ( Zjj-Xjj) ) -H • (x)ll nx-ylldt + O
+ (llH' (x)-H' (x )llta»l|)c-yMdt+hMlx-yl| 0 °
and applying (3*12) to the integrands, we obtain ^
(3.27) (slIZjj-XjjlDllx-ylldt + £
1 +(H'(\IX-X II )lt(*>llx-ylldt+ h(U>) llx-yll. Jo 0
Now let p1 aiix-yll, p^=|lx-xol|, p^=»y-xQl|, p4=l/z-x/l,
and ^j^=llz-yH; then noting (3.23)
l|x-xollt tf1 +f?
U zN-xN«=|(zN-xN)t+(l-t)(zN-yN)|4 p4t+(l-t)p5,
and, since H'(p) is^increasing,
"f (s)IU^fl' (s^t^+d-t)^ )pXdt + i «
+ \ 1' (tp1+fJ)lfM^dt + hO^f1, Jo
or applying (3.10) t* 1 12
(3.28) ||f(3)||4Si,j(f4+f?) t- + mV.KPV+ •SV_)J +h >f1-
Therefore since the right hand member is independent of a
and the norm is continuous it follows from (3.20) that
(3.29) ,tfil^o[(PV) £_ +lw|lKv»)(f3p1+ (p*)2 )] +h(c*>)p1,
or applying (3.14), (3.16), and (3.29)jtherefore,
+
,1 \ 2
19
(3.30) l|T(x)-T(y)|l 4 HuiHe 1# . |Jf||*
6 G li x-yll, l|x-xQ|/ ,l|y-xo|| ,l\T(y)-xll, llT(y)—ylfl
where
(3.31) olf1, P p5] = —(pW 1-NB p C
^ 2 + Rol kh») N(io(p3p1+ )+iMh(u»)p1 j.
Since M , N(3q, 1? (•«), and h(ui) are all positive it follows
that G satisfies (2.10) and (2.11) in Lemma 4 for 0* p*"/ jj75~« o
Now let p1 =p -0% p2 = f , p3 =<T , p =f-p, f5 =t-<rj
then noting (3.30), (3.31) becomes
(3.32) Gfcp-<r,p,0-j-p,f-<r] = I_Npi "IT*sT-p-q-)(f -<r) +
+ MV<hiNQo(T(p-a-)+ (p-cr)2)+Mh(w) (p-tf)J.#
Let
(3.33) T(p) = p + x.N^ jd + (l+M*»«>)-^ p2_ p -utol P hcu>)}
where C is any constant. Identifying "J as T(^) we obtain
in a manner similar to that used for (3.16)
i r N0 (3.34) f(f)-T(«r) = !_N p\-T- (2f-Mr)(f-<0 +
+ M l?l«o)N0 (^(p-t)+ (P-<*)2) •+• Mh(w)(p-ff )1 = 0 2
= cfa-'V , ,f(o-)-p,T(Tr)-a].
Since T(0) = C , if we choose C = l*»l Qq we obtain
(3.35) J|T(xo)-xo!| £ M Qq = T(0)-0.
Hence by applying (3.34) and (3.35) we see that (2.12) and
tr
20
(2 .13 ) are satisfied for 0 = foST'f 4 H£" It only
remains to be shown that
<3-36> Pn * Pn+1 T T<Pn>
and
(3.37) P = P C min(r , 172--) n*«© «n 1 x o*
in order to satisfy the hypotheses of Lemma 4. .
For this purpose let us examine T(p). The fixed
points of T(p) are
1 — |u»\h(<«0~|( 1—i«w(hC««))2QqIu*\ (l+lu>l l?tw})N0o
1,2= (1+itoi I?(M)N0O "
by (3.5) and (3.6) these two roots are real und positive.
Let ^ 1— iwhM—A (1—Mh(W) )^—2Q lu>l (1+Ik«m»1|u»I )N{3 1
(3-38) P " II+»i»..))nao
and consider the image of the closed interval CO, p ] under
the mapping T(p). *
Since pi, p, the minimum of the two roots of the quadratric,
it follows that
(3 .39 ) q(p) = Q Q M + (l+I"l Uf*>) f — (1—Mh(<«) )p > 0.
Therefore r- _ (1-NB p)|M ll(w>N0 P+Mhtu>)J +Nfl q(p)
(3.40) T»(P) = 2 E J 2 1 ^ 2—L >0 (l-Nfl^r
* 1 for p^p l. and
T(p) J* °T(p*) = p*
21
so that T maps \[.0,f* 3 into L P > P - l « , Furthermore with
Pn+1 = (Pn)' since 0 forp*P*, it follows that
T(f)7f or that (3.35) is satisfied. Now
„„ r u . ,lt2"V, x-21-hcil Nfi0?'D _
i gr" j+ u.n|JoF)3 >
Therefore, #
(3.41) Qtftp* T'(P) = T'(p*) = * lto>hCUj) = 01 (UJ) =
Ch (w) +h (<" )3 -k (uj ) 3
DI(U;) + *(AU)L+
where
(1- i<4h(to) )2-2(l+ MV (I« ))MQ0N60 .
Therefore
(3.42) 0-t d (UJ) < l .
Now by the mean value theorem and (3.41)
(3.43) ^T(p)-¥(^-)| £ ck («o) Ip-^l
and hence T(p) is a contraction map on Lo,p*3and the itera
tion (3.35) will converge to the unique (by Banach's fixed
point theorem C10, p. 27l) fixed pointp . We have already
(3.38) shown that
* P * — r N o
/ \ * and by (3.7) f> < rQ ,
(3.44) p £ min( ,rQ)
r
22
and the hypotheses of Lemma 4 are satisfied for T(x).
Thus there exists x such that T(x ) = x , and by (2.19)
(3.45) ||xn-x*|| < f* -fni •
£ It remains to be shown that F(x ) = 0. Solving (3.1) for
F(Xr) we obtain
(3.46) -F(xn) =>P'(xn)(xn+1-xn).
Now by applying statement (i), section 2 and the triangle
inequality
) I F ' ( X q)II - t l F ' ( x O )U £l]F'Un)-F'(xo)//£
i II xn-xoll sup [llF" (xQ+t (xn-xQ) )U, 0 it£ lji rQN
so '»
(3.47) J| F' (xn)jl i ||F• (xQ)|| +rqN.
W"e then apply Lemma 2 and obtain
(3.48) !|j)F'(xn)l! * llF»(xn))) i J/F'(xo)//+roN.
Using this, from (3.46) it follows that
II F (xn )il iilI xn+1-xnl| (1/ F1 (xQ )ll +roN).
Finally, since the norm is continuous
II F(x*)|| = 0
or F(x*) = 0
(which is immediate from (3.46) if one has first shown that
•DF'(x) is continuous).
Uniqueness is covered by the proof of the more
general Theorem 3, section 5.
23
It is noted that the convergence of the sequence^pn}
to p* follows directly from (3.36) and (3.41) since p n
will converge to a fixed point of T(p) less than or equal * *
top . Since p is the only such fixed point the sequence *
must converge to p • However, no error bounds are obtained
by this argument.
4. Choice of to .
In this section we will show that the optimum
to the convention that <o, is a "better" value of Ui than
U) 2 if o((^) ' (v) 30 "^at error bounds are cor
respondingly related. It then follows that the best choice
of (O is that value which will minimize o( (w). To determine
this value we note the following
(I) Condition (3.5) can not be satisfied if
u) <, 0 and can be satisfied for 0 only as follows
value of <o is one. In order to do this we will agree
i f H 4. 1, o '
l'-"'2 lf H, < ~ •
(II) We have from (3.6) that
- 1 f°r 06UJ £ i
* ( « > =
1 for 1 i W ,
(III) By consideration of (I) and (II) we see
that (3.6) is satisfied if
(4 .1) 0 NO for 0 * t 2
o
24
25
(4.2) g°N(3° L —f±£—* 1 (l-H )2 for £<*><1, (l-H,)2 2(1+I4
(2-«»(l+H ))2 , (4.3) Q NA < 2 < i4l-H )z for u> * l.
0 0 2(l+w) " 0
Furthermore for Wi { and (3.5) and (3.6) satisfied it
follows that
^ 4 , 4 ) 2N(3 Q * N0 O U-C«J
2N0 ^ since 0 4. rrpz—rr \ by (4.1). Therefore if (3.7) is
(l-Ho)2 2{2~uf)
satisfied for u) £ \ it is also satisfied for V = 1. Simi
larly, if ——2(l+ui) ~ * *OT ^ "t0 ~ fol -ows
that
(4.5) l-H -o (1-H„)2—4B0Np0 ^
2N o ^ ( 1-HQ ) -/a? ( 1-HQ ) 2-2QON£OFC»( 1+TF)
^ ,
and hence if (3.5), (3.6), and (3.7) are satisfied for
jiw* 1, then they are also satisfied for 10= 1. Finally,
since
2- t o ( 1+HN) - J [ 2 -U>( 1+HQ )] 2-2QON0N6,( 1+ U+«)NB (4'6> rM = —^
is an increasing function for W?,l, if (3.5) and (3.6) are
26
satisfied for some value of <o?l then
(4.7) (1-H ) - V ( l -H )2-4Q N0 ^ = r ( l ) < r ( » ) .
We have shown
Lemma 5 • If. co nditions (3.5), (3.6), and (3.7) are satis
fied for any value of OJ between 0 and 2, they are satisfied
also for (o = 1.
With this lemma in mind we arrive at
Theorem 2. .If the conditions of Theorem 1 are satisfied
for any value of w between 0 and 2, then the best value of
(JJ jls one.
Proof:
We need only show that
<«•«> —a).
To do this we will consider three cases:
Case I. If (Hu>i£ , then
l-a»(l-H ) + (l-«/)-(1 (1-H ) 2-2kw(2-w) (4.9) clM = 2 °
(1-^) +l-uH l-Ho) + Vu? (l-Ho) 2-2ka/( 2-u>)
where k = and
(4.10) u(u+2(H -i-|)J^(l-HQ)g2k(l-..)](2^Ho-2)>2—«) CX(U>) = u
(2+ (Ho-2)+u)'
where
u = Ju»2 ( 1-HQ ) 2-2ku) (2-w) .
Now
v/o>2(l-H )2-2kw(2-*) * w(l-H ) 2(l—H ) ; O v <P
so
2 (Hq-1 ) + Jw2( 1-Hq ) 2-2kui( 2- ») 0.
Since by (4.1)
k * v i &
(1-Hq)2 2(2-<*) 2 (l-«u)
it follows that
(4.11) u/(l-Ho)2-2k(l-u>) > 0.
Also 2-" 7 0
and since H ? 0, \ 0. o
(4.12) 2+ ui(Hq-2) » 0 ;
therefore,
(4.13) oM 4 0
and oC (u>) ^o((j) for Q*uj< £ .
Case II. If i, then
ojH +1- . o» 2 (L-H )2-2kw(l+ai) *(•») = —2 " ° .
H +1+Jw2 (1-H ) 2-2kw( l+«u)
and
(4.14) ot'(w) = -pa-ii0)2 -2ka»(l+fci) (/w2(l-Ho)22kw(l+«)+l
(u>2(l-H )2-(l+2u»)k) M +1) (w+1) ~ o o •
\ju>2( l-Ho)2-21uu(l+M)
Now by (4.2) and (4.9)
28
4. ——— / ——— j therefore, (1-HQ) 2 2 ( l + t o ) l+2w
to(l-Ho)2 - k(l+2u/) 7 0,
and
(4.15) ot' (">) < 0.
Now c((S)><rf (£)? °< (l) for
Thus if (3.5), (3.6), and (3.7) are satisfied for some
(o^.l, then u> = 1 will minimize o( (<*).
Case III. If u> Tf. 1, then
UH +2)-l-wj(2-w(l+H ))2-2kw (l+o/) (4.16) =
(H +2)-l+ i(2-w( 1+H ))2-2kw(M (*} ' o
and [W(HQ+2 )-l](«+l fe—(1+Hq))(l+H>-k( 1+2^)3
/ x • u(H +3-u)+ ~ (4 .17) O((U*)= 2
(w(HQ+2)-1 + u)2
where
u = \[(2-m(l+Ho) )2-2kw(l+to)«»
Now,
H o +3_j(2-iu(l+H ))2-skw(l+w) 7 2-w(H +1) -
->K2-w(1+Ho)) 2-21uj(1+*J) >0,
and 2-w(1+Hq) > 0. By (3.5) it then follows that ©<• (<o) > 0,
and we have for 1 < 2
(4.18) o(( 1) 6 «< (cu).
5. Conditions in a deletion space.
In section 3 conditions were obtained that would
guarantee the convergence of (l.l) when the norm associated
with the space was the maximum or Tchebycheff norm. In this
section it will be shown that the conditions are also suf
ficient in any deletion space, in particular in a space,
and uniqueness will be obtained.
Theorem 3. Let X be a deletion space. Let F b£ a mapping
of set ACX into X. The iteration
x i = x -D-1F•(x )[F(x )] j n+1 n v n' v n'J _ , *
is well defined and the sequence will converge to x . #
such that F(x ) = Q if the following conditions are satis
fied. There exist xQ belonging to A and positive real num
bers r,N,(3 ,Q ,H such that S(x_,r)cA, F is twice Frechet —• O O O — — o 1 --im • • ——•
differentiable on S(x ,r), D~"'"F,(xo) exists, and
(5.1) IIF" (x)H 6 N for x in S(x ,r)t
(5.2) llD"1F'(xo)l|4 0Q,
(5.3) llD-1F'(xo)[F(xo)]|U Qq,
(5.4) (|D"1F»(xo)[F'(xo)-DF'(xo)]I| £ En < 1,
g Q N (5.5) b =
0 (1-H )2
and Qn ,
(5.6) r = ,° • (l-Jl^fb.) < r. 2b.(1-H.) u-"-,0o'
29
"\
30
* / \ Moreover x belongs to S(xQ,ro) and
(5.7) )lx-x*\U _e£_ Q 1- d °
where
* « = Ho+1-(1-Ho) 4E7
H +l+(1-H )Jl-4b A * A ' * / * .
and x is unique in every closed sphere S(x^,k)C S(x^.r)
where
S(xQ,k) x|x« X,l|x-xQ\l £ kj
and k is. such that
k * 2b (I-H ) u+n^r0). o o
We will first show that the sequence is well
defined. To do this we show that there exist positive real
numbers r ,0 -Q and H such that n' n7*n n
(5.8) where f^ = DF'(xn),
(5.9) |lf-1[F(xn)]|j £ Qn,
(5.10) llf 1(F'(xn)-fn)|l i Hn < 1,
N0 Q (5.11) b = a a < i
n (1-H )2
and
(5.12) = rn, SUa,rn) C SU„,r0) . n n
These statements are true for n = 0 by hypothesis.
Let us assume that they are true for n = p; then
is defined, and by (5.10) and (5.11).
(5 .X3) f i p = r p < r p 2 "p < r p .
1- l-4b 1- l-4b P P
Thus, there exists a positive number rp+ such that
s(Vi'Vi)csUP,rp)CA •
By (5.8)
|,fp1(fp+i~V" - WW
and hence by statement i), section 2
(5.14) ||fp1(fp+1-fp)jk(Jp||xp+1-xp||sup^®F' (xp+t(xp+1-xp) )J 'I), ,0*t*lj .
Applying Lemma 2 we obtain
(5.15) Hf^Up+i-y14
< &p#*p+l-*p# sup^((DP"(xp+txp+1)ll, 0 i ti 1 j ,
and, since X is a deletion space,
(5.16) sup |||P"(xp+txp+1)H,yo it! lj*
9p2pN = bp'1-Hp»2 < i •
Therefore the bounded linear operator
(5.17) hp =
has an inverse h~* such that P
(5.18) Hh^llS - "J • P 1-\(1-HP)
Moreover
Wx# - fP+i(x)'
and hence
(5.i9) #! = h;1*;1
\32
exists and
(5.20) llfriJU — . P+1 l-b (l-H )2 VP+1
p p
Let
(5.21) Gp(x) = x-f'^FU)];
then
G (x ) = x ,, P P P+1
(5.22) G^(x) = fpX(fp- P'(x)),
and
(5.23) G"(x) = -f"V(x) . f Jr
Thus by statement i), section 2,
(5.24) H'p1(F(Xp+]^) ))l= Il0p(xp)"0p(xp+1)|l 1
i"VVi'sup{"°i(Vt(Vl"3Cp)"' ° 4 **
and by applying it again
(5.25) >l^(xp+1+t(xp-xp+1)l|4||a^Up)// +
+ |l-t|llxp-xp+1ll3up^||G»(xp+s(l-t)(xp+:L-xp)j(/,0ssal5
or for OStSl and by (5.25), (5.10), (5.9), (5.8), and (5.1)
(5.26) U^(xp+1+t(xp-xp+1))|li Hp+(l-Uep0pN£Hp+bp(l-Hp)2.
Therefore, by (5.24) and (5.26),
(5.27) llf;1 P(xp+1))l 4 Qp(Hp+bp(l-Hp)2)
and by (5.18), (5.19), and (5.27)
(5.28) llf;^ p(xp+1)|l =iih;1f;1(p(xp+1))i|t
n H +b (l-H )2 „ £ St, P P El =. i •
l-b_(l-H )
33
Now by (5.19)
(5.29) =llhp1(fpl(P'(Vl)"Vl))"4
Let
(5.30) gp(x) = f"X(F'(x)-DF'U)) ;
then by the triangle inequality, (5.8), and Lemma 2
(5.31) llgp(xp+1)|U||gp(xp)|/ +//fp1{f(*p+1)-F'(xp)+DF'(xp)-
-M?'(xp+1)fl|i
i llgp(xp)U +Pp||F'(xp+1)-F'Up)-D(F'Up+1)-F'(xp))|l
and. since X is a deletion space by statement (i), section 2
we have
(5.32) Hgp(xp+1)IU llgp(xp)||+0pHF'(xp+1)-F'(xp)lli
il|gp(xp)ll+?p||xp+1-xp\| 3up^llF"(xp))l, 04Ul]
where -x = x +t(x ^,-x ). Therefore by (5.1), (5.9), and p p p+1 p
(5.12) we have
(5.33) llgp (xp<) |1 i|| gp (xp )|| + 9pQpN.
Therefore,
(5.34) <xp+1)-fp+1)ll i Hp+bp(l-Hp)2
and by applying (5.18) and (5.19)
(5.35) (vi'V111'- yv1-"/ • Hp+i • l-bp(l-Hp)2
Now
(5.36) 1-H j = (1-H ) (l-2hp(l-Hp)) > 0
l-bp(l-Hp)2
34
• ince b * i and H * 1 or P P
(5-37) Vi41-
Also by (5.11), (5.20), (5.28), and (5.37)
(5.38) t V^U-"D)2
(i-Vi)2 ri-2bpd-Hp)]2
and
(5i39) l-4b , = 1~4bp >0
[l-2bp(l-Hp)]2
so that
(5.40) bp+1< i .
By (5.12) and the induction hypotheses
(5.41) SUp,rp) C SU0,r0) .
Applying (5.28), (5.36), and (5.39) to (5.12) we obtain
(5-42) VI • H*- M- -P+I p+I
= 2b (l-H ; P P *
then if *x-xp+1|| 4 rp+1)
(5.43) ||x-xpl|il|xp-xp+1|| +||xp+1-x|/±
£ Q +r -Q = r P P P P
or
(5.44) S(xp+1,rp+1) c S(xp,rp) c SU0,r0) .
Therefore the iterates x are well defined and belong to n
S(xo,rQ) . Since S(xQ,rQ) is closed we need only show
that (xn] is a Cauchy sequence to show that it has a limit
35
in s^o'ro>-
For this purpose let
H +l-(l-H )Jl-4b (5.46) of = n
H +l+(l—H )J l-4b n v n'^ n
then applying the inductive definitions of H and b we n n
see that (5.47), o(n
for all n, or
(5.48) o(n =o<0 = «X
for all n.
Now by (5.28)
2n+l = 2nHn
and by a set of calculations
0 £ 2b (l-H )+Jl-4b < 2b +4l-4b < 1 " n* n' n n n
0 4 l-2bn(l-Hn)-n^bn " .
0 < (l+IIn)(l-Hn)[l-2bii(l-Hn)-h-4bn] =
= (l-Hn)[l-Hn-bn(l-Hn)2J -
" (1+Hn )t (1-Hn) fTUbn-bn (1-Hn) 2J
A (1+Hn) (l-bn( 1-Hn):2) - (1-Hn) n^4bn+bna-Hn)jF 1
Hn( l+Hn+( l_Hn) 4T„) fbn (1-Hn)2 (1+Hn) +bn( 1-Hj 3JI^b; <
< (1+Hn) (l-bn(1-Hn)2)-(1-Hn) JT5bn+bn( 1-Hn) 3 T4b^
(l+Hn+(l-Hn)n=4b;)(Hn+b0(l-Hn)2)
<• U-bn(l-Hn)2)(l+Hn-<l-Hn) g l-4ba)
(5.49) Hn< «„ = .<„ = <*.
Therefore,
(5.50) Qn 6 ocnQ0»
and since «(< 1,
(5.51) tlxn+ -*nl| i Hk=l2n+k-l-^k=l °<n ±Qo = 2o A~C\
so that {x^ is a Cauchy sequence which converges to x
in SU0,rQ) .
Since by statement (i), section 2,
(5 .52) Hfn-f 0!l*«*n-xo\l sup{|lDF» (xo+t(xn-.xQ) )|| ,0iUl] i rQN ,
it follows that
(5.53) l|fn)|i llf0l|+r0N .
Now
(5.54) F(xn) = fn(xn+1-xn)
which implies that
(5.5$) UF(xn)J| t (tfl)+Nr0)Qn
or
(5.56) l|F(x*)l| = 0,
which is equivalent to
(5.58) F(x*) = 0.
The error bounds are obtained by letting p tend to
infinity in (3.51) yielding
(5.59) Ux*-x ft l erf11 Q . n ~ l-<* °
The uniqueness of x in every closed sphere S(xQ,k)
S(x.,r.) with iu (l+>Jl-4b~) is proved by contra-2b0(l-H0)
37 *
diction. If u different from x were a solution in
S(x .k), then o '
(5.60) llu-x.li = ® n—— 0 2b (1-H )U^l-4b ), 0<6<1, 0 0
By induction it follows that
(5.61) ||u-xnl| £ m ^n (1+ 4l—4b^)» m +(l-rf)©< 1.
Obviously this is true for n = 0. Suppose it is true for
some n = 0, then Gp(u) = u where Gp is defined by (5.21).
Then by an argument similar to that used to obtain (5.28)
(5.62) llxp+1-u|| = ||Gp(xp)-Gp(u)|/i||xp-ul/(Hp+3pN//xp-u//).
Since by (5.28), (5.36), (5.38), and (5.39)
-P*1 i tt (l+Jl-4b ,)= 2VI(1"VD P+1
Qv (l+Jl-4b -2b (1-H )). 2b (1-ri ) P P P P P
2bpU-Hp)
"D 2 it follows that by adding and subtracting (m ^p(l+<ll-4b.u)
to (5.62) that
<*•*»> "vi"u//< r s - + p+i p+i
+ 2pHp C l-<ll-4b 1Tn>P^-(mp®)21 . U-H ) PJL 2bp
From the definitions and properties of Hp+ , Qp+ » and
b it follows that p+1
(5.64) 0 i(1-H )(1+H )(l-2b (l-4b )) = Jtr ir Jt Jt
=(1+Hp)(1-Hp)(l-2bp)-(l+Hp)(1-Hp)(l-4bp)
38
or by a set of manipulations
Hp(l+H
pMl-Hp)Hp(l-4bp)*
i (1+H )(l-2b (1-H ))—(1—H )(l-4b ) if jf }r r
HpC1+Hp+(1—Hp)(l-4bp)+2jl-4bp]< (1+Hp)(l-2bp(1-Hp)-
-(1-H (l-4b )+2>ll-4b (H +b (l-H )2) P P P P P V
Hp(l+Jl-4bp) (l+Hp+(l-Hp)\Jl-4bp) *
£ (1+H -(1-H )i l-4b )(l+U-4b -2b (l-H )) P P P P P P
H (1+ l-4b )<, cx (l+Al-4b -2b (l-H )) = P P P P P P
= cx(l+>Jl-4b -2b (1-H )) P P P
(5 .65) , QA(1^) „ g |( 1 , ^ l r - )
2£pu-Hp) - 2b—rnr^T u+u p+i •
Thus since m% < 1 it follows that
mp© -(mp©)2 0,
and from (5.63) and (5.65)
IIu-x nlu p+1 ^ ((m p9 ) 2(l-o()+ o c m p G ) p "2Vi^-Vi'
or
l|u-x ju mPe2^i'1+J1-4Vi) («•+(!-.<)© ) =
P 2ViU"Vl'
= mP+le . 2VIU"VI'
Therefore mn0 Qn(l+fl-4bn) mnS (l-Hj mn© (1-H0)
Uu-V4 2bnH-5n) £ 5T) 4 fTI
39
or
(5 . 6 6) i i lnx n -u | l= 0 .
Since f*nl converges to x this last statement implies *
that u = x which is a contradiction.
It must be noted that another error bound can be
obtained from (5.42), (5.44), and the remarks following
(5.44). Since n is arbitrary,
(5.67) llx -x II i ^n (l-^l-4b ) , for n ^ 0, n ~ 2b (1-H ) n
n n'
and this error bound is the one used for calculation in
Section 6.
In order to aid in possible computational applica
tions we restate the above theorem in terms of the Tcheby-
cheff and norms, respectively.
Corollary 1. Let f (x^ ,... ,xq) be^ a real-valued function
H \ defined on ACE for i = 1,2,...,n, the sequence of Vectors
x(k) = (xj^ ,... >x^^) defined by (l.l) converges to a * * *
solution x = (x^,...,xn) of * *
f (Xx ,. • *,xn) = 0 3 i = 1 »2 ,... ,n ,
if there exist Xq = (x°, x^,... »x ) belonging to A and positive
real numbers r,N,0 ,Q , and H such that • ———— ' 0 0 o —— ———
S(xQ,r ) = £x|x«En, A,
f^ has continuous second order partial derivatives on A, and
(5.68) J",n jkW|« » for x in S(x0,r),
40
<»•«> EM f-~ury| P„ .
(5-70) 11 ii n ] I~r^T I 4 So .
& Q N (5.72) b = -2-2— < i ,
(1-Hq)
and
(5-73) 2b (i-B ) = ro< r-o o
Moreover, x is in S(x ,r ) C. A and
max | (k) *W of Q
1* itn I l i i 1— oC o
where
1+H -(1-H )4l-4b o o o
1+H +(1-H )\|l-4b o o' * o * .
and x ijs unique in every closed sphere S(xQ,k) C S(xQ,r)
where
SUo'k) = \ x|x€En» if iin ixi-xil"kl
and
k " 2b0(l-H0) <l+^b0) •
Corollary 2. Let f^(x^,...,xq) Jbe a real-valued function
defined on A<- En for i = 1,2,..., n, the sequence of vectors
x(k) _ (x(k)ft#,x k ) defined by (l.l) converges to a solu-
"H* "M" tion x = (x^,..•,x^) of f^(x^,...,) = 0, i = l,2,»««,n,
if there exist xQ = (x°,...,x° ) belonging to A and posi
tive real numbers r, N, , Q , and H such that — • ' o o ~ o • ———
41
S(x o , r ) = ^x lx« f i n , 2 * r J c A »
has continuous second order partial derivatives on A, and
(5.74) *** 2 j=llfi,jk(x)M N f££ x in S(xo,r),
< 5 - 7 5 > To«» I r r fe l * s .
(5.76) z»
J . J O ' ' O
. 1 ° f. •(x ) JFJ O'
* Q » s *0
(5.77) ( 3Ej=i |f j — i , i xo )\- H0 x0)|» i-l»2,...,n,
M N (5.78) b = 0 0
° * i »
and
('•79> ro = 2b (LH ) U-fl=4b-0) < r. o o'
Moreover, x* is in S(xo,rQ) and n |x^-x*|i Q(
where
©< =
H +l-(l-H )U-4F o o 1 o
H +1+(1-H M l-4b_
and x i£ unique in every closed sphere S^xQ,k) C £>(xQ,r)
where
and
Stx0,kJ = £x|x«JSn, 2:°=1|xrx°| 4 k]
k 4 2b U-H ) •
42
Thus through the corollaries we obtain two different
sets of conditions which will guarantee that the iteration
will converge (all norm topologies in finite dimensional
B-spaces being equivalent). The condition pertaining to
Hq in both cases is analogous to a condition of diagonal
domination which, as is well-known, is sufficient to guaran
tee convergence when the f^ are all linear. It is this
condition which may somewhat limit the applications of the
method, even if a solution is known to exist.
6. Applications*
The applications considered in this paper will be
to the finite difference analog to the two point boundary
value problem for a second order non-linear ordinary dif
ferential equation. A few remarks will be made concerning
the finite difference analog to the Dirichlet problem for
a non-linear second order partial differential equation
with two independent variables.
Consider now the problem
(6.1) G(y"(x),y'(x),y(x),k) = 0 for a f x i b
(6.2) y(a) = yQ , y(b) = I
and along with it the finite difference analog with central
differences (y(x.) is replaced by y., y*(x.) is replaced by J J J
- 1 +l~ 1 -1 t and y"(x.) is replaced by +1~ .1 con-2h b2
sisting of the following set of equations
(6.3, t. (y Xb, . . ilSaZlzL,, . , J n 2h J J
for all j = l,2,...,n, where
" = fef • xo = a' x3 - yo = and yn+l = J-
We will now see what conditions on G will insure that con
dition (5.71) (the "difficult" condition to satisfy) will be
satisfied. The following notation is adopted
43
44
(j) a(y l'2y yj-l , yj*l'yj>-i,y ,x ) W1 " d IT 2h '
) y«
.0 . .0 >U)=
V h
(<i>. 3 Jy h
2h
.0 o
J J
2h J J
Then
k=J+1» j*n
L ( j )
f j 'k^l»* * * >yn^
G*J/(- h2 +G3 ( j ) ; k=j
0ii)(-T5)-82J)(-k)» j"1' j*1 n
^0 for all other k.
Now we see that in order for (5.71) to be satisfied in the
space with Tchebycheff norm
^j=ilfi,j(yi'*,,,yn)l 4 lfi,i<yJ»---»yJl>l
FOR X — X|2Y«**YU
or
(6.3) o(d)(_|)+GU)a))+jG(d)(^)+GU)(. i_)j4
jGp)(—§)+0 )) j=l,j$n
:MX,J n 1 n
(6.5) |e<n)(- )-G<n) |'|6in)(- •
Now if G is such that j=l,2,...,n, that is
(6.4)
45
G^ and G^ have different signs and if h is chosen
such that
0 < h< 2 3 ,min
then
|G(J>+ | CF(J)|+ |G<J )_ | 0<J>|, JAO'J'-H^'}
for j = 1,2,...,n and hence (6.3), (6.4), and (6.5) are all
satisfied. If we restrict ourselves to i = l,2,...,n-l, we
see that these conditions will be satisfied if and only if
fied. Thus it is possible to examine the differential equa
tion and predict whether or not (5.71) will be satisfied
when the space has a Tchebycheff norm. These conditions are
compatable with the conditions that Henrici uses to obtain
the existence arid uniqueness to the two point boundary
value problem of the class M. £5, p. 3473
Now in a similar manner we consider the Dirichlet
problem on the rectangle A:a x*b,c*y<d for the operator
of the form
where "&A is the boundary of A. We will use a uniform mesh
size on the rectangle such that nh = b-a and mh = d-c. This
will yield mn points where u must be determined by the equa
tions^
G^^G^^* 0 when the mesh size is such that (6.6) is satis-
G(uxx>uyy>%uy,u,x,y) =0 in A
u = g(x,y) on "iA
46
f j (U]_ »• • • »umn) =
u. , —2u.+u. , u. —2u.+u. u. , —u. , _ i _JL±S i IzS .1+1 izi,
,2 ,2 2h
u., -u. •l+m -l-m
» 2h .<yVy.i) = o,
j 'n, j + kPJ+1, jfcton, j* (m-l)n,
with appropriate changes to take care of the boundary con
ditions. If we again consider the maximum norm we see that
the conditions G^^ 2^^ G^*^* 0, j = l,2,...,mn,
and
2 * mm (
Gu> 4 |G33>
opT 2
• |a(j) )
will imply that (5.71) is satisfied where the notation GM
is analogous to that used above.
We will now consider a specific problem, the two
point boundary value problem
y" + (yl)2__JL- + ^ =0, y 2"* ly ' 2"* <y+l)2 '
y(o) = 0, y(1) = 1. (j)
Ve first note that in the above notation G^ =1 for all j
0(S> 2. y<
, 3 " s i r " ( ^^J"1)2 (yj+i)J v 2h
GU> . (iZi, 2 2-x. h
J
Then if y?7-1 Gj G^ < 0 for all j,
47
2 —T- :(1-y.i)(
y.i^-yi-l). ST3T h '•
2
Hence as a first approximation let y. = x. ; then
8 ( J ) J J
2 "TJT > 2 I = 2h > h for all j. G1
Therefore we will let h = .05, n = 19 and start with an
approximation y = x for the finite difference analog.
After calculations for determining the other constants we
see that the other conditions are not satisfied for any
r>0. Nevertheless by performing the iteration designated
by (1.3) with o» = 1.6 it was found on the IBM 7090 at the
U. S. Army Electronic Proving Ground, Ft. Huachuca, th tL± ict
y° = (y°,y°,...,y°) was obtained such that the requisite 1 2 n
conditions were satisfied and error bounds for y were
obtained from (5.67) by letting n = 0.
The point y° and associated Tchebycheff error
bound are here listed:
yi = 0.05833235 r = .001
y°2 = 0.11414186 20 = .00000004
0 y3
= 0.16785497 0o = 0.49931097
o *4
= 0.21983032 H 0 = 0.99862184
o y5
= 0.27037553 N = 1.1
*6 = 0.31975935 b 0 = .001156299
y7 = 0.36822072 and
y8 = 0.41597560 lly*--y0ll
y9 = 0.46322248
y10 = 0.51014680
yll = 0.55692472
y12 = 0.60332645
y13 = 0.6507193
y14 = 0.69807087
y15 = 0.74595187
y16 = 0.79453991
y17 = 0.84402329
0 yl8
= 0.89460586
0 y19
= 0.94651308
This same problem was used by Lieberstein ["97 as
an example but he had no rigorous convergence analysis
and hence he did not exhibit either the existence of a
solution or error bounds.
49
It is to be noted that (5*67) gives a better error
bound than (5.7), and that in either case difficulties with
accumulation of round-off error can be eliminated. Calcu
lations for H , (3 , Q , N and b can be made by hand by n * n n n
assuming that xn is the initial point. This error bound
is actually obtained by the statement of the existence
theorem; that is, x must belong to S(xo,rQ) where
ro • 2bo(l-Ho)<1
and hence the error bound which we have quoted (5.67) arises.
Had we carried out another iteration on the trial problem
to obtain a point yW ve could use the error bound (5.7)
for the Tchebycheff norm but the bound is not as good as
that which we obtain in less time using (5.67). Also,
one should be warned about use of (5.7) unless the next to
the last iterate is taken to be y^°^ since otherwise a
serious unaccounted for error due to round-off may be
involved. In either case the computation for the error
can be monitored and, if it is needed, multiple precision
arithmetic can be used.
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3. Bartle, R. G., Newton's method in Banach spaces, Proc. Amer. Math. Soc., 6 (827-831) (1955).
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50
