CENTRAL INSTITUTE OF PHYSICS

INSTITUTE FOR PHYSICS AND NUCLEAR ENGINEERING

Bucharest, FOB MG-6, ROMANIA

triNi - FT-207- J98 J Pecembe* «

DISCRETIZATION ERRORS OF PERTURBATION NUMERICAL METHODS FOR INITIAL VALUE

PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS

6H. ADAM*1

e Laboratory of Theoretical Physics, JINR, Dubna, Head Post Office 79,

10100 Moscow, USSR

Abstract. Algorithms are discussed, which are obtained within the perturbation numerical (PN) approach to the solution of initial value problems for linear ordinary differential equations arid max-norm bounds are derived to their accumulated discretization errors. The importance is emphasized of the realization of piece-wise polynomial approximations for the coefficients of thj differential equation by means of truncated local Legendre series. On one hand, this is proved to ensure superconvergence of the zeroth order approximation* of perturbation theory with minimum smoothness requirements on the original coefficients. On the other hand, convenient cut of ft of the Legendre series can be defined which ensure the implementation, in the p-th order approximation of perturbation theory (p - 1, 2, ...) of finite PN algorithms that preserve the highest theoretically attainable order of accuracy at given p. The error analysis also shews that, beyond the zeroth order approximation of perturbation theory, an increased efficiency of the PN methods is got provided the diffe* rential equation to be solved is first br< ught to the normal form.

*J On leave of absence from Section of Fundamental Physics,

Institute of Physics and Nuclear Engineering, P.O. Box S206, Bucharest, Romania

1. Introduction *

We want to consider tLc Cauchy problem

k-k

(1.1a) y(k)(x) - I ^ . W j l i , ( i ) • b(x), ko - 1, 2 k,

*€(*0.x,,J. - - < x0 < *H < • -, *i» »£c[xo,Jtg],

(1.1b) y(j)(x0) » nj, j - 0, 1. .... k-K

and to derive max-norm bound* to tbe discretisation errors

associated with its numerical solution vitbin the perturbation

numerical (Pg) approach.

If.a partition of the domain \x ,x_] is introduced,

(1.2) t • \x'Q < x «« ... < xQ < ... < x M L

h„ • x - x , , h • aax hy , 1 i v i n , n • 1, 2, .... •

v

a PI method for the solution of (1.1) is implemented in tvo

stagss as follows, first, » seroth order approximation of

perturbation theory is defined* vbicb admit» a piesevime analytic

solution,

0.3») t o U )(x) • îiîl(x)*°(i)(x) • b(x) , *£(*„.»,].

(1.3b) t°(i)(x0) » ţj , i • 0, 1 k-1,

«bare *.(x), b(x) are pieeavise continuous approximations to tbe coefficients of (1.1) over tbe partition (1.2), while tbe quantities (? approximate (or possibly ooincide with) their corresponding- quantities nţ of (Lib). In (1.3) dad sjl sub-sequent equations, J, mean» summation orer tbe specified variable i vstB tbo suaaation limit» of equation (1.1a). At the second •tags, tbe algorithm of tbe serotb order spprsstiastioc is improved

by adding corrections vitb respect to the perturbations

(l.a) Aaj n(x) - ajtx^ - Z±(x) , ao^x) - b(*) - b(x\.

*.e(?n-1,xn* » n » i , 2 , . . . , I . From the computational point of view, the most idrpcrtant

nev results established by the present error analysis can be formulated as follows.

(i) Oiven the partition (1.2), if on (xn_i»zn)> n " '• ..., II, the coefficient» of the differential equation are appro­ximated by truncated Legendre series transformed to ^..jii.)» tbia yields the highest possible bounds to the discretisation errors is the seroth order approximation of perturbation theory vitb uivţtfam smoothness requirements on the coefficients of tbe given differential equation.

(ii) Beyond tbe zeroth order approximation of perturba­tion theory, the accuracy of the FN algorithms increases propor-

» tionally vitb tbe parameter k^ which enters equation (1.1a). As (k-"l fk-k -ll k is related to the number of derivatives yv* ', ..., yx* *o " O 0

vhicb are aissiag in (1.1a), tbis shows that, if kQ • 1, it is t

advisabla to first bring the given differential equation to wbe normal form and only afterwards to solve it by a PR method.

(iii) Vfae local approximation of tbe coefficients cf tbe differential equation by truncated Lege idre series makes possible an easy and flexible implementation of finit* Pw algorithms in any order of perturbation theory with firm control of the information an tbe coefficients. If tbe p-tb order approximation of perturARtien theory la implemented, then tbe highest attainable-order o/ accuracy of tbis approximation *> reached provided tbe cut off M of the Legendre aeries satisfies Mmia (^ * H * J W > ) ' M»in(*H)» v i t h Mmin(1) " * * IU./***» .

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[[ko/2j] • the least integer vhieb is greater or equal to k /2, Main(p) • m p *• (p-l)(k0*1> for p i 2, where • is the cut off of the Legendre series in the zeroth order approxiaatioc ot perturbation theory.

Rigorous derivations of max-norm bounds to th* discretisation errors associated with PR methods have been previously published for two-point boundary vaJ u<s problems for ordinary linear differential equations. Pruess has derived Tesults for the convergence of the zeroth order approxiaation of perturbation theory for the regular Sturr-Liouville problea ţi] and for equation (1.1a) with k • 1 [?.] . Smooke has derived error bounds for PR algorithas obtained beyond the zeroth order approxiaation of perturbation theory considering equation (l.la) in the special case k • k • 2 (the radial Schroedinger equation) [3] and then extended the analysis tc the systems of coupled radial ftchroedinger equations [»J .

The studies [1-1»] have aainly been stimulated by the * development of several PR algorithms for the olution of the radial Scuroedinger equation, which proved to be highly efficient for the numerical solution of quantum chemistry and quantum physics problems (Oordon et al [5-8], Ixaru, Adam, et al [9-13], Andresen [iW], Saookc [15] etc.). All these algorithms solve ,in essence Cauchy problems of the type (1.1) and then get the solution of the involved eigenvalue problem* by shooting [16]. A max-nora error.analysis which is specifically designed for the Cauchy problem (1.1) is thus relevant to a large extent for the eigenvalue problems as well.

Tn# approxiamtion of a function by means of truncated Legendre serie* is known for a -long time (Jackson [17]) to represent the solution of the best least squares polynomial

9

- a -

approximation, but this was apparantly ignored in the previous

studies of the FN methodo. The use of truncated Legendre series

for the piecevise polynomial approximation of the coefficients of

the differential equation is discussed in section 2.

The technique used to derive nx-nori bounds to the

discretization errors is standard (Henrici [l8] , Gear [19]):

bound? to the local discretisation errors are first derived and

then used to solve for the accumulated discretization errors. The

principal results of the paper. Theorems 3.1 to 3.1», are

collected in section 3. The proofs of these theorems, given in

sections 5, 6, and 7, are heavily based on the integral

representation of the solutions' of the linear ordinary

differential equations (Coddington and Levineon [20, chap. 3])

and on the exploitation of the algebraic structure, established

in section U, of the solutions, called local propagators, of a

set of homogeneous equations associated with (1.3) for 6-

Kroenecker iftitial conditions.

2. PLecewise polynomial approximations to coefficients

The <*)fi»tenee and uniqueness of the solutions of the

Cauchy problems (i.1) *nd (1.3) is throughout assumed under the

hypotheses specified below.

In (1.1), suppose In.I < • and a,, b €. Cr[xo,xH], r i 1.

Then there existe a uniqt» solution of this problem, ytx)fcC

[*0.x«].

Let D [x ,x.] denote the space of piecevise continuously

difftrentiabl* functions with the understanding that for any

f€X>[xofxM], f(x) » j(f(x") + f(xf)). In (1.3), assume |;°| < •

and a., b€ 0 (x0tX«]> *" £

l1. Then, thi» Caucby problem admits a

unique solution, z(x)€ C*"1 [xo,»„]fl Dk*r"1 [x^x,,] .

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Fraa the very definition of (1.3), the initial data and

the coefficients of this problea are perturbations, within some

giren t, of the corresponding quantities of (1.1). Then it is

known (Yoshids [zij) that Ijy'^'-s'^ll, j • 0, 1, .... k-1, are

0(c) at most. Hare and in the following, ||*|| denotes the sup

nora.

for computational purposes, the coefficients (1.3) are

usually restricted to the space P of the piecevise pclynoaial

functions of degree at most •[l-13,)5], a hypothesis which vill

he'also adopted in the present paper.

Oiren a function g(x)£C [*,*„], r 2 1, a piecevise o • .

polynomial approxiaat ion of degree a , g ( x ) , t o g (x ) on the

p a r t i t i o n ( 1 . 2 ) im generated aa f o l l o w s :

(2.1) ?(x) •«*(*) , *£(*„_!»*„) , n » 1, 2, .... M,

r(i> <*0) - «J,,(»0>» i

(i><«g) - ^ ^ ( « i ) . J " 0,1.....k-i. 4

• i « ( j ) o ^ - l [ i ( i ) ( * ; ) • • ( j ) u B ) ] . o - 1 . 2 , . . . . N-i,

where g*(x) is an approxiaating polynoaial of specified degree a.

* Prutis [l,2] baa discussed several realizations of the

approxiaating polynomials g*(x). Among then, particularly

important is the a-th degree polynoaial best approxiaatioo in

the least squares tense. First, it ensures [1,2] superconvergence

of .the ztrotb order approxiaation (t.3) at the knots (1.2).-

perturbations ot 0 (O, 1 • b* * in the coefficients induce

perturbation! of 6nly 0 (« ) in y*J (xa), j • C, 1, ...; k-1,

""• —ml

n • 1, 2, ..., I. Second, a cnange froa gn(x) to »„(*)« M > m,

does not aodify, in the expression of C_(x), the already calcu-• a

lated coefficients for g*(x). This has ths consequence that a n

remarkable manageability is reached vBen one is dealing vith the implementation of perturbation corrections to the xeroth order approximation (1.3). The latter point becomes obvious if g is carried out according to the following fundamenta? lemma (Jackson [iT, pp. TT ff. and pp. 9£~9,6), Ljashko eS ai~.[a2,-p».5^8]):

LEMMA 2.1. Given the function g(x)€C[x. ,x ] , its best m-th degree polynomial r.pproximation in the least squares sense, g*(x). x€(x ,,x ) , consists of the sum of the first m + 1 terms of the Legcndre series expansion of g(x) on (x _,»* ).

(2-2> H<*> --îî-oVl<t(«»-where

(2.3) t(x) - 2(x-cn)/hn, cn - ?(«B_,•«£)•! t€(-l,l).

Here, P^ is the usual Legendre polynomial normed to 2/(21+1) and

(2.») «i • (1 • £)J c(xt*m)>x(t)dt, x(t) - cB *~ş\t.

For the subsequent error analysis, ve have.to consider the reminder

(2.5) Ag»(x) - g(x) - g""(x) , I € ( V , ( I D ) ,

for which we need orthogonality relations, Legendre polynomial expansion, and upper bounds to the sup-norm ||&g II.

LEMMA 2.2. (Ljashko et al. [22, p. 51»]). If the m-th degree polynomial g.(x) is constructed aexforling to Lemma 2.1, then the reminder Ag"(x) defined by (2.5) satisfies the orthogo­nality relations

(2.6) fxn (x-c^'ogjtxjdx - 0 , p - 0, 1 a, n— 1

where c„ is defined in (2.3). n

- *yi

LEMMA 2 . 3 . (Ljashko e t a l . [22, pp. 56-5T] ) . If the m-xth

degree polynomial s*_(x) ia constructed according to Leaaa 2 . 1 ,

then the reminder Ag (x) defined by (2.5) can be expressed as a

s e r i e s ,

(2.7) Ag^x) - G i * 1 - n P l t t ( « ) ) . *€<*._ , .*„) ,

where the r.h. side quantities have the same meaning as in Lemma

2.1.

Upper hounds to the sup-norm ||Ag )| can be given, vhich

depend on the differentiability properties of g(x) . For g(x)

£C [x .,x J, 1 a r $ m, detailed results are derived in the

book by Jackson [l7, pp. 13-18 and 25-32], and they will be

henceforth understood without further specification. If, however,

g(x)€Cr[x .,x ], r i m+1, then the- following lemma will be used

(Pruess [1,2], Smooke [3]).

LEMMA ?.k. It g€.CB*1[x . ,x ] and g* is the approximat­

ing polynomial of (2.1), then there is a constant K » K(n) > 0

such that

(2.8) V HAg;||<Kh-+1.

Remark 2.1. For the Cauchy problem (1.3), the question

arises whether to use the same degree m for all the piecewise

polynomial approximations a., b (ss Pruess does [2]), or not. We

shall require that all the polynomials a" n, i - 0, 1, ..., k-kQ,

be of the same degree a. According to the philosophy of the PI

approach, the only substantial restriction to a comes from the

requirement that equation» (U.I) of the local propagators

associated with (1.3) admit analytic solutions. As for the free

tern b , it enters the solution of (1.3) only through the

integral giving the particular solution, and the feasibility of

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analytic expressions for this integral is in no vay subject to

the restriction m* • m. Intsectians 5, 6, and T, we shall Keep

this flexibility of the PN approach. However, for th*e sake of

simplicity, the Theorems 3.1 and 3.2 of section ,3 are formulated

requiring a common cut off •' » m. The same remark applies to the

thresholds M and M' defined in section 7 for the perturbations

(l.*) and to the Theorem 3.1».

3. Max-norm bounds to the accumulated discretization errors

Within the PH approach to the numerical solution of the

Cauchy problem (l.l), global one-step methods are obtained, tbat

is, piecewise analytic approximations z ** (x), n * 1, 2, ..., E,

are constructed for y J (x), j * 0, 1, .... k-1, over the

partition (1.2), making use, at every n, only of the previously

calculated quantities zn«j(*n_i)•

T n e accumulated discretization

error associated with z J (x) is, by definition,

(3.1) Tn/

J<*> " yU)<*> - 2nj)(x),

*£[*„_!#*„] t n • 1, 2 H , i - 0, 1, .... k-1.

For all the algorithms which will be subsequently given

for z J (x) the accumulated discretization errors (3.1) are found n

to satisfy propagation laws of the type

(3.2.) T^><X) - i?;WJ!,n<vi^)Ti-lK-i> • 'iJ,<*>.

"S^n-I»*,,! . n • 1, 2, .... 8,

(3.2b) ToJ)(x0) • to

4) • nj - «J , 'j-- 0, 1 k-1.

Ai)'

where the quantities t*" (x) are the local discretization errors

of the method and D Î ^ n^xn-1'*)» t h e ProP*S*tors of the accumu­

lated errors.

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I f in ( 3 . 2 ) x € « , then bounda t o T J ^ f x ) are y i e l d e d by n * n

the following leaaa which is a routine generalisation of some

well-known result* for recurrence relations (Henrici, [l8, pp. 18

-19] and [23, chap. 16, aection U]).

I.EMMÂ 3.1. If the quantities T " ' satisfy, over the aesh

(1.2), the difference system

13.3a) Tn - li.0»i+1fBTn_1 • *B » n » 1, 2, ..., H,

(3.3b) ,TQi) - t£j) . j • 0, 1, .... k-t,

such that the absolute values of the coefficients D H ; _ are

upper bounded by

(3.*a) |Diiî,J sî I • hf , f • f(n) > 0,

(3.*b) K H . n l ^ « . • • •<«>> > 0 , i 4 i,

then the aoduli J^_J I »x*« upper bounded by

(3.5) |T£ J )| a k- 1X^[(8 n-l)/. * (k#jl-l)(rf,-1)/«]|t(1,|/h

• *"liîi[»"Mi*J1.i)«B]|tii,|.

n • 1, 2, .... I , j • 0, 1, .... k-1,

where

8 • t • h[f + (k-l)e] * 1 • hs,

Q • 1 • h(f-e) I 1 • hq.

Ree*rk 3.1. For the special case k • «o • 2, the aystea

•(3.3) «an be solved for better estiaatss than (3.*), thus

yielding slightly aore realistic bounds to j t ^ H (Adaa, Adaa,

and Core iovei [2w]).

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Reaark 3.2. equations (3.2) and Leaaa 3.1 are the basic

relations froa which aax-nora bounds are found for the accumulated

Aiacretisation errors everywhere on"[x ,x_J once aax-nora bounds

are known for the propagators D H : -n*^xn-1'z^ *nd for t h e l o c* 1

discretisation errors t'J'(x) at I £ ( I , ,x J , n • 1, 2, ..., N. n n— I n*

The asin results are collected below and the proofs are giren in

subsequent sections along the lines sketched in the present

reaark.

In section $, the seroth order approximation (1.3) of

perturbation theory is considered for which two thepreas are / • \ •

established. In this case* .in (3.1)* x (x) is to be identified n

with the solution t°^'(x) of (5.2) and T^'(x) is denoted by n n

THEOREM 3.1. Let the following conditions hold:

(i) in (1.1), a.£ c"*1 [x -xJ , i • 0, 1 k-k ,

bCc"^,,,]

(ii) in (1.3), Cj • »>0, Vj, and ăţ, b are a-th degree piecewise polynoaial approxiaations to a., b* generated as in (2.1).

Then, for any partition (1.2)* as h * 0,

(3.6) ||T°(J,|| • 0 (h"+1) , j • 0, 1 k-1, n • 1, 2,.,.,I.

Remark 3.3. This result obtained for the Caueby problea (1.1) is the equivalent of Theorea 2 by Pruess [2] for two-point boundary value probleas.

THEOtEH 3.2. If the following conditions are satisfied:

(i) in equation (1.3), %• • H-, Vi» *<» » *r« »-th degree piecewise polynoaial approximation to,a., b generated as ia (2.1) and satisfying Leaaa 2.1,

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( i i ) in equat ion ( 1 . 1 ) , a . ^ C r [ x o , x H ] , i • 0 , : , . . . .

k - k 0 , b £ C r [ x o , x s ] . r • « a t ( T , '•• - ko • 2 ) ,

t h e n , for any p a r t i t i o n ( 1 . 2 ) c h a r a c t e r i z e d by a s u f f i c i e n t l y

small h , there e x i s t the f i n i t e p o s i t i v e constants A. « A . ( n ) ,

Ai » A!(n) , i « 0 , 1 , . . . , k-k , B * B(n) , B* * B'(n) such i i o

that

( 3 . 7 a ) | T ° ( J ) ( X ) | $ hm+\liAi\\&*mi(zi)\\ * BHAb"(n>||) , x t «

(3 .7b) J T ° ( j ) ( x ) | S h U ( L A ! | | A « J ( n ) | | • B' |»Abm(n) ||) , x £ w ,

where u • min(m+1, k - j ) , and

( 3 . o a ) llAa"(n)ll « max suplm.(x) - aT (x) I , 1 v x ' * x* '

(3 .9b) llAbB(n)| | - max s u p | b ( x ) - b ^ ( x ) | , x e ( x v _ 1 , x v ) , 1 SvSn. m

Remark 3.4 The convergence of the least squares

approximation of Lemma 2.1 under very general conditions on the

smoothness of the function g(x) ensures the validity of the result

(3.7) under the weakest possible differentiability conditions on

the coefficients a., b in the Caucby problem (1.1). If the more

stringent differentiability conditions of Lemma 2,k are imposed,

then the following corollary holds,

COROLLARY S.J. if the hypotheses of Theorem 3.2 hold with,

however, the smoothness conditions on the coefficients a., b

replaced by a ^ C * lxo,xs], Vj. * £ Cr [xo,x„] . r' • itax(in-t-l ,

m-k +2), then the finite positive constants C • C(n), C «-C'(n) o

exist such that

(3.9a) |*;(j)<«>| . Ch2"*2 . ,£w,

(3.9b) | T ° ( J ) ( X ) | a Cu"*1*", '**» , u * min(»*1,k-j) ,

n • 1, 2, ...', S , j • 0, 1, ..., k-1.

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Remark 3.5. The accuracy shovn by equations (3.9) for the PN solution of the Cauchy problem (1.1) coincides vith that obtained by Pruess [2, Theorem 3] for the two-point boundary value problems for equation (l.la) in the zeroth order approxima­tion of perturbation theory and under similar assumptions on the coefficients a., b.

In section 6, the p-th order approximation (p * 1, 2, ...) of perturbation theory is investigated. Perturbation corrections

are added to the "dressed" zeroth order approximation (6.2) as Neumann series vith respect to the perturbations (1 .1») in the

coefficients. This technique leads to the algorithm (6.6) whose accumulated discretization errors are denoted by Tp J'(x).

n THEOREM 3.3. Let the following hypotheser hold:

(i) in the dressed zeroth order approximation (6.2), ţ. •

• n-, V-; a. , i • 0, 1, ..., k-k , and b_ are polynomial

approximations, generated as in Lemma 2.1, to the coefficients

a., b of (1.1), of the m-th degree and of the a'-th degree

respectively;

(ii) in the original problem (1.1), a., be c[x .xJ.

Then, for any partition (1.2) characterized by a sufficiently «mail h, tuere exist the finite positive constants A. • A.(n), i • 0, 1, ..., k-k , and B • B(n) such that for every

(3.10) |Tj(j)(x)| * h °||AaB(n)||P(JiAi||Aa;(n)!; • Bf|Ab*'(n)f| )

n * 1, 2 , • • « , M , i • 0 , 1, , . . , ! . - 1 ,

v i t h | | o a * ( n ) j | , | |nb" ( n ) | | d e f i n e d by e q u a t i o n s ( 3 . 8 ) and

(3 .11) !|A».ffl<n)j - Ms|Ha«£(a)N | i • 0 , 1 k-k 1 .

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COROLLARY 3.t. If the conditions of Theorem 3.3 are modi­

fied such that m' « m and a., b€ C [x ,x_], then a finite posi­

tive constant C • C(n) exists for vhich

(3.12) | T ; ( J ) ( X ) | . Ch* 1^ . u(p) - <m+l)(p«-l) + pko,

x ^ x n - 1 X J » n - 1, 2, ..., • , j » 0, 1, .... k-1.

Remark 3.6. por the special case k • k " 2 ana b(x) • 0,

error bounds to the PH solution of the boundary value problem for

equation (1.1a) have been reported by Smooke [3]. His sup-norm

result associates vith |T p* JMx)| in (3.12) an exponent

u'îp) • U * 2)(p • 1).

whereas the present exponent u(p) for k • 2 reads,

u(p) » (m+3)(p*l) - 2.

We have u(p) - u'(p) • p - 1, therefore, for p > 1, the technique

.of reference [3] should still be improved to reach the uniform

bound (3.12).

Remark 3.7. If the conditions of Theorem 3.3 apply, then,

at x • x the error bounds (3.10) can still be improved by unity

at special p values vhicb dep'end on the parity of m. Thus, beyond

the teroth order approximation of perturbation theory, the, u*e of

truncated Legendre series no more leads to a spectacular increase

in the order of accuracy of the PR algorithms. Hovever, there are

whole classes of higher order perturbation corrections vhich

vanish it 1 « 1 , Ve have not succeeded in formaliiing this n

empirical observation in a theorem, .

In section 7. the problem is solved of deriving a finite

algorithm in «be p -tb order approximation (p • 1, 2, ...) of

perturbation theory. The algorithm (7.9) i» obtained which starts

vith the dressed seroth order approximation (7.1) and gets the

perturbatlofl corrections as Reumaon series vith respect to

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perturbation corrections as Neumann series vith respect to

modified (finite) perturbations to the coefficients, of the type

Ag™'M(x) of equation (l.U). n

THEOREM 3.4. Let the following conditions be satisfied:

(i) in the dressed zeroth order approximation (7.1), C- • w

* nî» V;i •' * •» a « a n d D a r e n~th degree polynomial

approximations to a., b which are generated on (zn-i*xn^ **r Lelu>a

.2.1, (ii) the coefficients of (1.1a) satisfy a.£ C*[x -x„],

x o « i - C, 1 k - k Q , b € C H [ x o , x H ] , Q - max (M+1, M-kQ+2) ,

( i i i ) i n the a l g o r i t h m ( 7 . 9 ) , N' » M, wi th

( 3 - 1 3 ) M B i n ( p ) a M a « B a x ( p ) - « ^ ( p + l ) ,

where

(3.1»»a) M„in(0 » a «• [[*0/2]], [[kQ/2l] • the least integer

which is greater or equal -to k /2,

(3.1»»b) HBin(p) - ap • (p-l)(k0+l) , p l 2 .

Then the order* of accuracy v(M,p) of the finite algorithm (7.9)

is related to the value u(p) • v(»,p) predicted by Corollary 3.2

for the infinite algorithm (6.6) as follows:

(3.15) v(M,p) - u(p) - (B+1)(P+1) • pke , i d ,

(3.15b) v(M,p) • »in(u(p), k-j+M) , x£» , j - 0, 1, .... k-1.

In (3.15b), v(M,p) • u(p) for N . M M X(p).

Reesrk 3.8. An empiric realisation of finite Pa> algorithm

in first order perturbation theory is given in reference [9]. Its

rate of convergence is that predicted by Theorem 3.*» however, 1 '

the coefficient •»_ of the Legendre series (2.7) it expressed by n

first irder finite differences, which represent the lowest order

- 15 -

approximation of the exact toraula (2.a). J» Other empiric xaalisations of finite PI algorithm», in

second order'perturbation theory (p • 2), are those of references [10] and [l3]• There, the Schroedinger equation is solved (k"k »2) «tartine vith a atep function approximation oft the potential (m • 0 in (1.3). These algorithms use a cut off M • 2, vhile Theorem 3.* requires M i 3. therefore the algorithms'are inconsistent in that higher order contributiona are included which are of the order of magnitude of deleted terms. On the other band, Lemma 2.1 is not satisfied by the ţalynoaial approximations used in [l0,13] and thus part of the potential capabilities of the PI approach is lost.

A modification proposed in [12] for the algorithm of [10] fits Theorem 3-** with, however, the drawback that Lemma 2.1 is atill not satisfied.

Remark 3.9. The finite algorithm (7.9) still contains perturbation corrections the order of magnitude of which is greater than the order of accuracy v(Hvp) of Theorem 3.U. As the number of the perturbation terms increases-in geometric progression with p, it is very important to rule out from (7.9) all the terms of*order h , a > v(lf ,p) to ensure an economic- and efficient algorithm. A formal derivation of the most economic algorithm in the p-tb order approximation of perturbation theory, p • 2, 3, ..., together with technical details, concerning its implementation in a computer program are planned to be reported elsewhere.

4. Local propagators of the differe tial equation

In the proofs of the theorems given in section 3 ' central role is played by the local propagators dţ ,J'*X>*

- 16 -defined ti soluţiona of the Cauchy problems

U.la) (.k/»xk)diB(a,x) - \ . ^ t W ( > J / > i j ) d i B ( i , i ) .

Xg.ţ i s * x 4 x , i • 1, 2, ...» k , n • 1, 2,, .... I.

(4.1b) (»1/»*1)*itn(».x)|x«8 - V l . i . 1 - 0. 1 k-1,

where tbe coefficients ă? _(x) are those of equation (1.3). Tbe propagators d. (i,z) are functions of class C ~ [i ,,x ] | | c (*. «»* ) vith rotpoct to tooth tfeo variables a and x. n—i n

LEMMA 4.1. If d, _(»,x) are solutions of the Cauchy i,n t->rofclea.s (k. 1) , then tbeir partial derivatives of order j, j • 0, 1, ..,, k-*, vith respect to x have the following algebraic structures:

U.2a) (9j/axj)d. n(s.x) - [(x-i)i'J"V(i-j-l)î]ei>n(j;s,x),

for i • 1, 2, ..., k , and j » 0, 1, ..., i-1, with

(»».2b) eiB(jis,x) - 1 • [(x-s)ll-i*,/(k-i*l)tjfifn(j;a,x)

where f^ n(j;s,x) are finite arbitrary functions satisfying

(A.2c) li» f. „(jjs.x) • 5, (s), x«»s 1 , n ii.»

and respectively

(k.3a) (»J/»*J)«1(B(«,s) - [(x-s)k"J/(k-j)!r«ifII(j;«,x) ,

for i • 1, 2, .... k-1 and j • i, i+1, ..., k-1 where e, „Ui»,x) ara finit* arbitrary function» as*' •'*•*

I »n

U.3») li« e. (J;s,x) - i\ „(' ' x-s l , n i '

Proof. We can **"*<.* from .'*•*••)#

- 17 -

U.U) (ik/a»V.jn(e,x) - •itn<k*,»x> • V - '• 2» •••• k •

where a. (k;s,x) ara finita arbitrary funcţiona satisfying to * «n

the restriction lis a. (k;s,x) • a. (6)." x+s 1 , D 1-1,a

An argument which us\es integration of botb aides of (*.•») with reapect to x oa (s,x), the uitial conditions (a.1b), the ••an Talua theorem, and adequate notation for the existing arbitrary functions laada to equations (a.3)l for i » 1, 2, .... k-1 and j • k-1. The other results (4.3) are then obtained by oaplete induction.

To get equatioaa (4.2), w* start with tba relations d.3) for i • 1, 2, ..., k-1, j • i, and (a.a) for i • j » k and repeat point by point the procedura which has lead to the equations (*.3), Q.I-.D.

S. Discretiiation errors in zeroth order perturbation theory

5.1. algorithms, propagators of the accumulated discretisation errors, and local discretisation errors

The sequence of solutions s°*" (x), x£.(s_ ,,i.), n • i, n n— I n

..., I, of the terotn order approximation of perturbation theory

is obtained fro* the loeal Caueby probleme

(5.1a) sJU)<«) • Ii*îtB<*)*J(i)^> ••5J,(«). o • %• 2 ».

(5.ib) «;u)(x;.,) - «;iiî(vi> . ' • • « . 3 » .

viier* the coe f f i c i ent s a? . , b* are generated a» in ( 2 . 1 ) . * f B D

«•king use of «be loeal propagators dţ „(••*) «e'ined in section *, the solution of tbs Ceuchy problea ($.1) can be

- 18 -

written (Coddington and Lovinson [20, eâap. 3]),

(5.2) ° ( j )(x) - z^ihx) * !* d. b"'(s)(»J/9xj)dk (s.x) Q n,n n-1 *» n

x€[xn_1',xn], n - 1, 2, .... N , j • 0, 1, ..., k-1 ,

where a £ (x) is th^ solution of the homogeneous equation n,n

associated with (5.1a) and the initial conditions (5.1b);

Local discretisation trrora associated with (5.1) are

defined by

(5.M t°(j)(x) - y(j,(x) - rJhx) , »€[«„.,.«„] ,

n • 1, 2, .... M , j • 0, 1, ..., k-1 ,

where y 3 (x) is found from the original problem (1,1), while the

quantity vlJ (x) is obtained from the local Cauchy problem

(5.5a) vnk)(x) • Ii«J B(*>»i

i<*) • ^'<*>.

(5.5b) »iJ,U*_,> - y(j,(x^,) , J - 0, 1, ..., k-1 ,

which admits a unique solution r (i)£C ~ fx ,,1.1(10 (x„ «,x„).

n n—i n n—i n

The solution ot the Caucby problea (5.5) can be expressed

similar.to (5.2), as

(5.6) »ij,(x) - vl*hx) • f" d. £*'(•)(»*/»«')*.. (•,,),

where

(5.7) T » > ( « ) - I ^ U ) ( . ; . i " , J ^ « i ' * i » i . . < S : i ^ -

Using (5.6), the solution of the Cauehy problem (1.1)' oa

l*n-1»*J U f0UBd *•

- 19 -

(5.8) rU)U) -v<J>(*> • £ *-!«! TjCi^.^x)^1^.») .

+ OV8^*^ * j " °» '•• •*•• k_1 • vhere the following notation ţas been used,

(5.9a) ^(i^is.t) - Aa= ^ U K . * * " 1 / » ! 1 * ' 1 ^ (.,*), q

<f.!>b) «!'(i ;».t) .tt;,(B](9l*/»tV;(i,t), q - 1 , 2 , ... , i o - j .

For the solution z°*J'(x), equation (5.2), let the corresponding accumulated discretization error (3.1) be denoted by To("'(*x). Addition and subtraction of T ^ U ) to the n n definition (3.D of T°^'(x) and use of (5 .2-(5.»») and (5.6)-

• n (5.8)*lead to propagation equations of the type (3.2) for T°(j)(x), vita the propagators

and the local discretization errors

(5.1D A i - u ; d . / ^ i , » . . x ) / x , ) ^ , ) * *n-1-

a

5.2. Proof of Theoreo 3.1

Bounds to the propagators 0°\y (x .,x) are immediate 1*1(11 îl™'

froa Learn» U.I. Separate resu l t s for x • x and x £ ( x _ , ,x_) are n n— > n

given as corollaries of Lean a I». 1,

C080LIARV 5.1. oi tren the local propagators d, „(s,x),

«olution» of tJ>» Caueftjr problems (».1)/ the following upper

bounds exist and are finite for n • 1, 2, ..., N,

» 'V

- 20 -

(5.12a) hf° - «•*|(»j/»*j)*j+1fVUv>,.*)!,«» *l|. v«j v

1 S v a" n , 0 a j < k-1 ,

(5.12b) he0 - u i J(' j/^ j)d i + 1 v^v-1'x)L-x v,i,jVi) * '

1 i v $ n , 0 £ i, j(#i) £ k-l.

CORtfUARy 5.2. Given the local propagators d. (s,x), l, n

solutions of the Caucby problems (4.1), the folloving upper bounds 0

hold for the propagators (5.10) at xC*(x ,»*_)»

<5..3a) l ^ ^ ^ ^ . x ) ! $ [ (« -« B . 1 ) l " i / ( i - j ) * ] * l + 1 , I 1 ( j ;n

for i • 0, '<, ..., k-1 and j " 0 , 1, . . . , i ,

and respectively

<>••*> I D W ! . ^ . - I - » ) I * [<»-»»-,>k'J/(^)oii.l+1ill<i)ii for i • 0, 1, ..., k-2 and j • i+1, i+2, ..., k-l,

vhere the sup-noras are taken over the open interval (s ,,,x).

The n-xt step of the proof requires the derivation of

bounds to the local discretisation errors (5.11).

LEMMA 5.1. If the coefficients ă? (x), i » 0 , 1, .... * ***

k-k . and b (x) of equation (5-la) are approximating polynomials for «be coefficients a., b of equation (l.1a() in the sense of equation (2,1),, then the following upper hounds to the local discretisation errors (5.11) bold uniformly on (x_ ,,x_] ,

n~ i n" (5.15) |tj;(i)(x>| * [(*-«n.1>k''3/(k-j)i](Ii«Aatfnimy(,)ll

• llab 'lllle,, „(j)ll) t n - 1, 2, ...,», j - 0, 1, ...,k-1

vn«r« theieup-norms in the r.h. side are tak«n over (*_ .,x).

Proof . in the r.h. «id* ef equation (5.M), Leaaa U.l is used to express the quantities IT* and •* and the integrals are

- 21 -

evaluated by the aean value theorea. Then the modulus is taken on

tooth sides of the resulting equation and the r.h. side quantities

are upper bounded by their sup-noras on (x ,xi, Q.E.D.

The proof of Theorem 3.1 is completed if in (5.15) the

restriction a' * a is iaposed and the conditions of Lemma 2.4 are

satisfied by the coefficients a., b of (l.ia).

5.3. Proof of Theorem 3.2

Bounds to the propagators (5.10) are those given by the

Corollaries 5.' and 5.2 and bounds to the local discretization

errors (5.11) at xe(x„ , ,x ) are given by Leaaa 5.1. At x • x_,

n-i n n

bovever, the bounds to t° J (x) are improved using tha folloving

lemma.

UEMMA 5.2. Let

( i ) the parameters k and k have the meaning given in

equation ( 1 . 1 a ) , a, tbe meaning given in equation ( 2 . 1 ) , «a l j •

• 0, 1, . . . » Jt-1

( i i ) > 2 a . where * o /

(5.16) a 0 - k - 2 ,

( i i i ) I n ( j ) - / * n d . [ ( x B - . ) k - J _ 1 / ( k - j - 1 ) i ] A « ; ( . ) w ( s ) , B~1 '

f\ ka*'"

)

where vi*)C c[xu^,xji\ C ° (*n-l'V' r * »»*( ' t " ' V 2 ) '

and lemma 2.2 applies to og*(a) .

Than the fo l loving es t iaate t are va l id ,

If the «oadition ( i i i i s replaced by

- 22 -

(ii*) 0 S i < mQ, with aQ given by (5.l6),

then the estiaatea (5.17e) are still valid at J • a -»+',

ao~a+2, ..., It—l, vhile at j » 0, i, ..., m

0~**

(5 .i7h) i a ( i j - [h^/(k-j) ! ]Ag;(C ; ) w («n ) . « ; « ( s - . . * « > -

. Proof. The function wis) is expanded in a Taylor series

around cn - J(*n_ • » n),

and this expression ii replaced in the integral for I (j) giring,

V » " V&^l'jf* 4.[(V0H"/N-')!l((.-c//t:]M;(.)

n~ 1

• 4 n * » [ ( v ) k " j ' 1 / ( k - j - i ) t ] [ ( - c n ) M / M t ] f l g ; ( . ) W( M > ( o l l ) .

n —1

In view of Leaaa 2.2, the firat integral vaniahea. To

evaluate the latter integral, we write a-c • (e-x_ ,) - •= h .

use tbe binoaial rule for the expansion'of (a-c ) , and then the

aean value taeorea and get the eatiaate (5.17a).

If it ia the condition (ii') which la fulfilled instead

of (ii), then a splitting of the values of j ia introduced by

Leaaa 2.2 at k-j-1 » a. for x-j-1 i a, i.e.» for J • ao-a*1,

mo-a>2, .... k-1, Leaaa 2.2 holds and the eatiaate (5.1Ta) ia

obtained. For k-j-1 » a*1, i.e., j • 0, 1, ..., i i , the O

eatiaate (5.1Tb) eacrges for In(j) using the a«an value -taaorea.

CQHOLUM 5.9. Let the following hypotheaea Be fulfilled

by the local Ceuctay prpbleaa (5.1): (i) a' • a l • with a„ defined by (5.16), o • o

(ii) a" (*), i m o, 1, ..., *-*„, and b*(») ar* a-tb i at* v n

- 23 -

degree polynomial approximations to the eoeffieianta a-, b of

•«.nation (1.1a), vhich ara obtained according: to Lemma 2.1.

Than the sharpened bounds to the loeal discretisation

errors (5.11) are Talid it x • x , D

(5.i8) | t ; « > ( S ) | . [hr2/(-2)..]i;.02-l(-;2)

" (U,|Aa",aH,,,,i"n(j»i)H * «•*;ilM»^B(j)ll.

* • "-»0*j . j • 0, 1, .... k-1,

vhere the sup-norms are taken orer (*_ «» *„). and

v£B(j.i;..x) - (•"/••">», XJ.il».»>!.., , •„*(*„-,.*„) ' n

•£*n(j;-.*) - (»"/»»")«k,B0*».«)|..,i . •;€(«.-,.«.).

ykn(j,ija,x) • ek>n(j;s,x)yU'(x) ,

vith the quantities e. (j;s,x) defined in Lemma ».1. . x,n

If instead of (i) it la the hypothesis

(i1) 0 • • < mo ,

vbich is fulfilled, then the sharper bounds ($.18) hold for j •

• m -m+1, m -m+2, ..., k-1, while the uniform bounds of Lemaa 5.1

are rslid it i » x. for j • 0, 1, ..., m -a. n o

The proof of Theorem 3.2 i« nov immediate, Q.E.O.

6. Discretization errors of the standard algorithms in p-th order

perturbation theory

6.1. Algorithms/, propagators of the accumulated discretization

errors, and loeal discrstization errors

Several equiralent procedures have been proposed [3-8,10,

13,15] for the derivation of the p-th order approximation

- 2» -

s_ ( x ) . p • 1, 2 , . . . » of perturbation theory on (z , ,x ),' n • B B*"l B

• 1, .... I. In essence, they consist ia the supplementation of

the xeroth order solution of perturbation theory vith the first p

terms of the leuaann series development of the solution <ft the

original equation vith respsct to the perturbations (l.s). The

present derivation, which formalises the approach proposed by

Gordon [$-8]• yields simultaneously algorithms for the calcula­

tion of the auaatities tp J (x), j • 0, 1, .... k-1, and propege-n

tion equations for their associated discretisation errors, T*(j)(x>. n

We begin by definning a "dressed*' xeroth order approxima­tion on (x„.,iX«).

n— • n

(6.1a) i|(k)(x> - [^^(xJi^'fx) • *"/(,) ,

(6.ib) « ; U ) (x ; . , ) - •;i )<*„.,). » - 2 . 3 ».

„herefrom l°a(x)£ ^'[«„„.ijflc -^.,,»,).

Similar to (5.2). we get

(6.2) .J«><«) - ll\lUm) • Si *• *n,<»><»J/a*d)Vn(''*>'

vith

From the equation» ($.6), (5.7), (6.2), (6.3), and the

definition (3.1) of T p ( j )(x), ve get n

(6.») *«>(,) - .;">(«> * lK«Biî)(«..|)<|J/»»J)«fi.»(».-l-«)-

Then, p tines iteration of (5.8) and use of (6.1») yields

the algorithm

- 25 -

(6.5) * n j )U) - i;.0Fqin(-i;io;.o) • SJ., V V . « ° ' o ^

«CCx^ţiX,,]» n » 1 a, j - 0, 1, .... k-1, p » 1, 2, ...,

and an equation of propagation of the type (3.2) for the accumulated discretisation errors T* J (x). with the propagators

<6'6> D?li!n<*n-1.*> * » & ! . < W " > * Ij-I Vn<°> V o > -and the local discretisation errors

(6.7) tj^>(x) - Vlf.<B.X0..0> • F...CUX..-.)'

«here ufi^.C»,, .,«) i» ai*" *y (5.10), whiie

(6.8) ^tn(«il0;.0) • h r;§||(Iq.Bq)0(.;iai.q). c - -1. 0, 1, 2,

vith

o(V (6.9a) G(-l;iqi.q) - .^ * < a q ) ,

(6.9D) 0(0iiqi.q) - (> V M Vi^.pfr.-l",)'

(6.9.) 0(l»iq».q) - X,* .Vl'tVVl'V' / • \

(6.9d) 0(2iiqi«q) - y ** (»q).

In equations (6.5)-(6.8), the quantity I stands for the collection of indices

I_ • (i. • 0. 1 k-k.ll • 1. 2. .... (6.10) I0 - ie « jj Iq • ij - 0, 1 k-k0|l - 1 , 2 , ...J. q>,

o. • 1, 2, .... the quantity 8 stands for the collection of variables

(6.11) 8p • %0 • xj 8q • (»0, Sj, .... s q), q • 1, 2, ....

vhLie the quantities r" „(I„tS,.) denote the operators 4th 9 q'

- 26 -

o. - 1, 2, ...

6.2. Proof of fheorea 3.3

To derive bounds oc (x_..,x ] to the propagators (6.6)

and to the local diacretisation errora (6.7), ve need some

preliminary results.

He define the auxiliary functions

• q£ [«„_,.•„]. iq » 0, 1, .... k-k0, «„(i^i-^GCCx^^s^, V i q .

Here, X - 2 if e(i ) » 0, Vi ;

X • 1 if a(i ) - k -

X • 0 if e(i ) • i -• » q'

and « ( O • k -4

*• Viq;

for i • 0, 1, .... i,

for i^ • V 1 » •••» «"«o» *

where "0 • i < k-k . o

He also introduce the auxiliary function

(6-1k) ' q.n<Vo> * *Iqro!,n<VV * n ^ \ >' V>€ K-1.*„] ,

with I , S . r" defined in (6.10)-(6.12) respectively. 9. 9. , 9,.n

LEMMA 6.1. If a^€ C[x0,x„] and if e\ are a-th degree

piecevise polynoaial approximations to a^( generated as in (2.1),

then upper bounds to the functions F defined in (6.11») are 4,n

given respectively by

(6-i5) l f i ,n(Vo>l * •¥*i-*"1 /<v1 * *k0)ij„.kfB(io)„

»||A»"(n)||||g*|UA(l> for X - 0, 1,

- 27

and by

( 6- l 6 ) l> i . . < v . > l * • °(**"1/^.%i)H.k;ll(i0)ii

* î i» â »î n»M«nU)ll for A,- 2,

where

( 6 . 1 7 a ) 6 . s - x , , o n—i

(6.17b) vx " k ~ i o * Xko •• A " °» 1 f *•••

( 6 . 1 8 ) A - 6 ° e6 | | A a m ( n ) | | lie. J | ,

( 6 . 1 9 ) X x (1) • e x p ( ( 2 - *")«],

( 6 . 2 0 a ) | |e J i - aax | |e . ( i ) | | | i - 0 , 1 , . . . . k - k ) ,

(6 .20b ) Its*» - »axCHg*(i)|» i - 0 , 1, . . . . k - k 0 ) ,

the quantities ||Aa*(n)|| are defined in (3.11) and the occuring sup-noras l|Aa? ||, He (i )||, and |(g (i)ll are taken over the i. tn x ,n i n open interval (x„ , ,s^). n~i o

Proof. In the r.h. side of (6.14), use is made of the

definition (9.9a) suppleaented with Lemma 1». 1 for the quantities »«

T , of the definition (6.13) for G , and then, of the mean n n

value theorem for each of the existing integrals. It results,

F î . » < V o > " Il4<*',*/''q

!)«î.l[-»î1.»(«l)«k.n<il-1»«l'«1.l)]

* «n<W' 'l^Vl-'p) » 1 " 4 • *o " 8<

where Mq - Ij;J(k - i^ • a(iq).

Taking the moduli on both sides of this tquation, upper

bounding the r.h. side quantities 4», . eb _, g„, by their sup-i^ |D A fii n

norms 6n (xn_1,s.), and using the bounds (?.1t) and (6.20) for

- 28 -

i i e V i , w g e t*

I * q\n<Vo>l * H«k,n(i0)H(«^"(-)» IK,n")*"

In the last sua, the change of variables i' • k - k - i-,

ijfil ,, and then the use of the inequality (« • "i)'* * °î*,î«

.. B !, lead to the factorisation of the q-1 BUBS over i,£ I , n x q —•

6 into 4-1 equal terms, each being upper bounded by e . The

resulting sua over i is bounded separately for the cases A • 0,

1, 2, and the equations (6.1$) and (6.16), respectively, are

obtained, Q.E.D.

Identifying in Lemma 6.1, the function C of (6.13)

with (6.9d) and then vith (6.9c), ve get

CORQLLAKV 6.1, If a., b€c[x ,x_], then the following 1 * O B'

bounds are uniformly valid on (x , ,x 1 for the local n~1 n*

discretization errors (6.7),

( 6 . 2 1 ) | t j ( j ) ( x ) | £ [ (x - x n _ 1 ) V p / v p s ] l U k > B ( J ) H •*IIA*"(n)||||«k JpP

* ( I i l | A a Bn l | | | y ( i ) i | + | |ob"'| | , vp - k - j • pk 0 ,

n « 1 , 2 , . . . , l , j • 0 , 1 , . . . . x - 1 , p • 1 , 2 ,

To get«bounds to the propagators (6.6), Lemma 6.1 has to

be supplemented with the following result. LEMMA 6.2. If' 7 * (I ;s ) is the function defined in

q,n o o

(6.1b); then an uppe** bound to the sum

< 6 ' 2 2 ) S P,n<Vo> " ÎJ.1 F q.n^o'-o) • ».6 K-1'*J' * ' 0 ' 1

is given by

< 6 ' 2 3 > I 8 P , n < V o > l * « • V A ^ v A I ) I U k # n < i 0 ) H I W . » m ( n ) H H « j ! U A < p ) .

- 29 -

where

XA(p>l) - XA(1)exp[«(e6HAa"(n)ll H e ^ H ) *°) ,

aad the meanings of the quantities arising in the r.h. sides of

these equations are those established in Leana 6.1.

Proof. Taking into account the bound (6.. 15), we get froi

(6.22),

I S P.h<V86>l * «^X.n^o»1 ,,Aa,,(n,,, <«**<')

If p«l , the sum over q reduces to (vx' "* » w ni^ e ' o r P*1 it is

bounded by (VA'.) exp<A ), with A giren by (6.16), Q.E.D.

CORvUARK 6.2. Let F (0;I ;s ) denote the function q,D. o o

defined by equation (6.8).

If 0 £ i < k-X0, then

'6'2h»> lEqVq.n'^VoH i l«k"j/U-J)!]»«k#B(J)H HA."(n)H r

x He i+1nllexp*l2 • (Vlloa"(n)i1 » « k j n « ) ' ^ j ,

* * x - *„_!» * E (xn-1 ,3sJ * J " °» ' •••••• k"' • I f k - k e S i $ k - 1, then

(6-2>b> | i ; . . \ . .<°*v'o>l * [6 "J 0/(k - j > ko> sK. f .<^M

x ||A.B(n)ime.+1>nl|exp«[u(Sp8||AaB(n)|| lle^H* °l.

4 - x - x n _ r *€(«„_,.«„] . J " 0, 1 *"1-

Proof. The r.h..side of equation (6.2l»a) i» nothin»ţ but the bound (6.23), provided tha function G(0>i it^) of (6.8) is identified with the auxiliary function 0 „("Li*») in (6.22).

- 30 -

When it i» 0 1(i„;s„) which is taken for G(0;i ;s ) in (6.22),

then the bound (6.2Ub) is obtained from Lemma 6.2, «i.E.D.

From Corollaries 5.1, 5.2, and 6.2, bounds are easily

established for the expression (6.6) of the propagator D i + 1 n

(x ,x) at x - x and at x C (x ,x ) and thus the proof of % n—1 n n

Theorem 3.3 is immediate.

7. Discretisation er-rors of finite algorithms in p-th order

perturbation theory

7.1. Algorithms, propagators of the accumulated discretization

errors, and local discretization errors

The algorithm (6.6), as well as that of reference

show the conceptual drawback that they are not finite. Indeed,

according to Lemma 2.3, complete information about the

perturbations (1.*) ia obtained only provided an infinity of data

(the coefficients of the Legendre serti.es developments (2.7) of

m m' Aa^ n and Ab ) is stored for each of them. As the computer

memory is finite, we have to solve the problem pf implementing an

algorithm vhi,ch uses a finite number only of these coefficients.

We shall suppose a finite,cut off, M > m, for Aa" , i • 0, 1, .. x ,n

., k-k0» "> • 1. 2, ..., H, and allow for a cut off M' 2 m' for

AbB , n * 1, 2, ..., n, with 11' not necessarily equal to M. Then

the information to be stored taw the approximate representation

of the coefficients of equation (1.1a) will fit within (k-k +1)

(M+l)*(M'+l) memory locations, of the computer. the teehnique of deriving a finite algorithm is similar

to that developed in section 6. Let «* , p ( j ,(x] f x £ (x ,,x ] , n • , . n n*i nJ

• 1# 2, ..., 4f, i • 0» 1, ..., fc-1, denote the solution obtained

in the pHh order approximation of perturbation .theory with

- 31

finite cuts for the Legendre series expansions of the .perturbations (l.a) and let TM,p(j'(x) be its associated n discretization error.

Equation (6.1) is replaced by a. new "dressed" zeroth order approximation on (x ,,x ),

n~ i n (7.la) z»>°<*Ux) - Li- C«>."'0(i><«> • b"'(x), n '•l i(n n n

(7.1b) • ; * 0 < i , ( » ; . 1 ) - » ; : ţ ( J ) < « ; . , ) . • • « . 3 *,

«t,0<d,(**> • «j» J - 0, 1. .... k-i,

whence «^(xl^C^'^.^xJOc"^.,,!,,).

Similar to (6.2), we get

(7.2) zj|»0j>(x) • *Jt'J(J)(«) • l\ «• bţj'(s)(aJ/3xj)dk n(s.x),

where

( T . 3 ) « ; ; n( j ) (x ) . i j : i « ; : t ( i , ( « ; . 1 ) ( » , / » . J ) * l + J f B ( » l l - i - « ) .

Let the reminder (2.7) be written

(7.u) Agjtx) - '^ . .^«^(t tx) ) • ?r-ll+1.;p1(t(«)) - Ag"*M(x) + Ag"(x), o n

m M and, as diseusaed above, o>nly terms of the type Ag ' (x) be allowed in the algorithm. Then, from (5.9) we have the splittings (7.5a) vj(iqis,t) - vJ»M(iqis,t) • Yj(iqis,t),

(7.5b) «J'(iqis,t) - «•'»M'(iq;s,t) • «J'(iqis,t),

with obvious meanings for the r.b.s. terms, and equation ($.8) can be written

(7.*) y<J>(x) - v<"(x) • /^i*.I[lliTj-*(i4,.1.«)y<i»>(.1)

- 32 -

i

*5'*"'u.,.,.»)]. where

•(T.T) r^Jhx) - T « > ( . ) • ^ ^ « . J l i , ^ , * - , . » ) » ^ (•;>

• •rev,.»)]. Similar to (6.1»), we have

(T.8) ^»(x) - ,;.<><*>(,> + i V ^ ^ ^ - ^ ' ^ K + i j * ^

Mow, p times iteration of (7-6) and uae of (T.T) and

(T.8) yield the algorithm

(T.9) . ;•»«>(,) - i ; . 0 ' ; , n ( - n i o i . o ) • i ; . 1 »;: 1 . B (Mo*- . ) .

*€ [*„_!.»„]. n « 1 , 2 , ...,H, j • 0, 1, .... k-1, p • 1, 2...

and an equation of propagation of the type (3.2) for the

accumulated discretization errors T " ' P ( J ' ( X ) , with the

propagators

"."> « t a i î V l - » ) " Dîii!n<«n-1'") * lî-l'Î.^O»1.*-»). and the loctal diacretization errors

(T.1U •£•»"><,) - fJf.(a*iDi.0) • (3»i 0». 0)

The quantity D ? ^ ( x „ ,,x) of (T.10) was defined in

(5.10), whereas in (7.9)-<7.11),

<7'12> 'q,n<«'Vo> * I l p V 8 / 1 ' ^ " ^ with the variables I , S , q • 0, 1, ..., defined in equations

(6.10) and (6.11), respectively. Furtber,

- 33 -

(7.i3) r;;;(i0.80) . 1 .

and

(T.U.) CMC-Ui,î.4) - »;, 0 ( i 4 >(» q).

(T.J*«) GM(o;iqi.q) « o ^ / » - * X * i . a K - i ' V »

(7.tad) G»(2;iq;.q) -J. J ^ i f l V t J V l ' V ' ^ V

y - •, M ,

(7.He) «M(3;iq;.q) •/*^ id. q t 16»(l q S« 4 + |.. q). p - « \ M«

Ae M,M' —*- •, the present results go into those of the

prarioua section a. follows: r j j - 'JiB«.'J.nM) - ^ . n * " 1 ) 1

Fî B<°> — '„ B( 0 )J Fî »(1) ^ '« m(l>* 'î « ( 2 ) — °» Fo'n(3)^°V q,n q,n q,n q,n q,n q,n

F" .(2) — F ., n(2)i F"'(3) -» F n(1). Therefore, «"•9<j) — P»n P*««n * p,n p,n ^ n

,P(J). D»«-PU) _ D?(J) . tM,p(j) _^ tp(j) „ d TM,p(j) Tp(j) n i+1,D i+1,n n n n n

7.2. Proof of Theorem 3.»

With the'trivial aodifieation *•*—•• r"*« in the r.h.

side of equation (6.lb), Laaaaa 6.1 and 6.2 remain valid and they

oaa •• uaad to derive hounds to the quantities 0-!, „ and

t«*.p(j) „ w , u # n

^ha bounds to the propagators l>.X? £ » aquation (7.10),

ara similar to those obtained to D?i, in section 6. • jti,n

- 3k -

More interesting is the derivation of hounds to the local

discretisation errors (7.11). Mere, ve distinguish three groups

of teras:(i) -the first two taras; (ii) the sua over q; (iii) the

last tvo teras. Bounds are separately derived for each group.

(i) For the first group of teras, use of Leaaa 6.1 yields

the following unifora bound at xe(i ,,i 1. 1 o~l a-*

(7.15) |»;,n<2;i0;.0)| > |03;i0;.0>| * C*^'V>«*k.»Cj,«

n « i, a i, i * o, l, .... t-i, p » i,2, ...

Here and in the following»

vp - k - j • pk0 , p • 0, 1, 2, ...

Further, ||Aa*' (n)|| and ||daB*N(n)H denote obvious extensions to

the.finite ease of equations (3.H) and (3.8a) respectively,

ii; vhile the other quantities nave the aeanings given in Leaaa 6.1.

(ii) For the second group of teras, a unifora hound at

x£(x .»s ) is obtained by coabined use of Leaaas 6.1 and 6.2.

We get,

- | |aaB»M(n) | | « • k , B l l< ! i l l»»? f B l l l|yCA>ll • | |AoJ'| | x % ) ,

a • I , 2 , . . . , » , j • 0 , 1, . . . , B - 1 , p • 2 , 3 , . . . ,

where th>e fo l lowing a d d i t i o n a l q u a n t i t i e s have been introduced

v i t h r e s p e c t to ( 7 . 1 5 ) *

XM(2) « e \

xH(p>2) - x M (2)exp[*(e*HA«'' M (n) | | H«k > n | | ) °J .

- 35 -

(iii) Bounds to the tbird «roup of terms, vhieh are

uniformly Valid at X£(* B-1,*BJ • •*• fo«Bd using Leaaa 5.1. He

S«t

(T.'tT.) |»;tfJtS*I0*.0)| S (*V°/vo'.)H«1[tn(J)HlilK.n« »

yli)"-

(T'1Tb) l0 3*V»o>| * ( /VNle milllAb;;'»,

n • 1. 2, ..., I, j • 0, 1, .... k-1.

To ensure tbe validity of tbe bounds (7.15)-(7.IT), it is

sufficient to require a., bt C[x ,x_J in (1.1a). If aore

stringent differentiability conditions are iapoeedon these

coefficients, tban Learns $.2 yields sharpened error bounde at x •

• x_ inatead of (7.17a-b).

Let ^ C ^ ^ . x J , i - 0, 1 k-k0, bCCBfx0,xj], R -

• aaxCl.M-k^). If If 2 k-2, tban for all values i • 0, 1, ....

k-1.

<7.»8«) | 0 2 i v * n > l * [hr2/<M • *y]i t-o2"1**!2)

• ^iM6*i,BH ,,wk! O»1)». * • *-**i*** B » 1, 2, ...,•.

If, however, X < k-2, then the sharpened bound (7.18a) bolda for

i » k-M-1, k-M, .... k-1, while for j • 0, 1, ..., k-M-2, it is

tha unifora beaad (7.17a) which baa to be uaed at x • xfl.

If N' < M 2 k-2,'let bCC*'[x0,xM], R' • aax(l ,M»-ke+2),

while other viae, b » e [i .i,] aa above. Then for all values j •

• 0, 1, .... k-1,

(T.it.) K . o i w i . [I.;,,2/(M. • «)t]il!J0,ri(V8>

• IIAbJ'll | |a k f B ( j ) l l , M» - M»-k*j*2, • ' • 1, 2 , . . . . t.

for K f W < k-2 , tba s p l i t t i n g : snarpened/unifora bounds at

- 36 -

x • x gets similar to tbat for |pM (?;I ;X ) | . n I o,n o n I ?o make a Meaningful comparison of tbe bovnds (7.15)-

(T.19), va ah all .impose the simplifying condiţiona • • •*, M « « N' > k-2 (the aaae reault is obtained imposing H » •• • M'>k-2) and ,Q ahall require aj, b€ C y[x o,x f], Q > aax(M+1, M-ko+2), which ensure tba validity ft Lemma 2.k. With-conveniently defined constants K, to Ka, Vt get at every n - 1, 2, ..., N, and I

j • 0, 1 k-l. vtlpi+1 .

(T.i9) |';, n<2;i o i. 0)| • |^, n(3ii 0;. 0)| i i,.„l . P - i.2.. J

v,(p) • k - j - 1 • (••I)(P*1) +*pk0. «0 - xe(x n_ 1,xj

<7'20> ll-\Knl2ilo''oA * K , n < 3 ' V o > l > * KahnVa(M> + ' .

va(M) - k - o * a * 1 * M * k o , a 0 « x € . ( x n _ 1 , x n J , p « 2 , 3 , .

i \(M) » 2M • 2.

The proof of Theorem 3.*» is thereby easily achieved.

Acknowledgements. The author is deeply indebted to Dr. M. D. Smookc for sending hia results before publication. He gratefully acknowledges the kind support pf Prof. V.K. Fedyanin, Head of the Group of Theory of Condensed Matter, and of Prof. V. G. Soloviev, Deputy Director of the Laboratory of Theoretical Physics, during tbe period of preparation of the present work at JIIR-Dubna.

37

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