C O M I T E T U L DE STAT PENTRU E N E R G I A N U C L E A R A

INSTITUTUL CENTRAL DE FIZICA

BUC UMŞTI - M AOUfULI

R O M A N I A

CENTRAL INSTITUTE OF PHYSICS

INSTITUTE FOR PHYSICS AND NUCLEAR ENGINEERING

Bucharest, P.0.8.5206, ROMANIA

department oi Fundamentai Phytic*

MC- 3-1 9 7 % March

A step function perturbative numerical method for the solution of

coupled differential equations arising from the Schrodinger equation

II, Computational Aspects

L.Gr.Jxaru

Abstract: The me.th.od developed in the previous pa.pe.1 [preprint, C.l.Ph. [Bucharest] ,PiC-2-7S, J97S) is here investigated irom computational point oi view.Special emphasis is paid to the. two basic descriptors oi the. eiiiciencyt the. volume. oi memory required and the com­putational eiiort [timing]. Next, two experimental cases are ie.poA.ted, They [i] confirm the theoretical estimates ior the late oi convergence oi each version 0& the. present method and [ii] show that the present method is substantially iaster than the othe.it>, Spe -ciiically, it it iound that ior typical physical pro­blems it is iaster by a faeton, oi ten up to twenty than the methods' commonly used, viz, Humerov and de Vogelaere. The data spotted also allow an indirect comparison with the method oi Gordon,It is shown that, while this exhibits the samz rate as our basic,lowest order version, the computational eiiort ior the lat­ter is, in case oi systems with nine equations, only hali than icr the method oi Gordon. At the end oi the paper some types oi physical problems are suggested which should be the most benefitting ii sol­ved numerically with the present method,

1. Introduction

I» the provioue paper £ |J , which wi l l be hereafter quo tad an I , we presented

a perturbative method for Aolving the nystena or second order differential equations

arl ning from the SchroVUn^or equation.

The reference potential i a in each interval a matrix of constants which i e

determined in a suitable way from the original potential matrix and the major point

of tho whole approach i s that the oxact perturbative corrections can be evaluated

analytically in any order of the perturbation theory. In ref. I we presented the

expressions of the f i r s t ard second order corrections and delimited five versions of

the method, in accordance with the nunber of corrections actually included into the

algorithm. In particular» we have shown that the most accurate version i s of neventb.

order. Tb our knowledge, th i s ia the hi chest order ever reached by the perturbaţi ve

methods for solving the systems*

l a this paper we discuss the computational aspects of this aethod and i t s aain

version*, f i r s t the program l a discussed shortly. Ye actually i n s i s t in some detai l

on same aspects such as the computational ef fort , i . e. t iming for which we give

soae theoretical estimates,) and the memory requirementa. The next two secţiona are

devoted to the experioental evidence aimed to compara the present aethod with the

other ones. Ve mention two numerical examples. The first(Section 3 ) i s a system which

admits exact solutions which ara then taken as reference for the investigation of the

convergence. Iha second(Section 4 ) i s a system of interest in molecular physics and

which was investigated frequently, see L2, 3J and references therein.

f inal ly , we try to answer the question of which kinds of physical problems

should V the moat benefitt ing i f solved with the present aethod.

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im The PrOjTra»

A prO£raa was written which uses only s ingls precision arithmetics to solve

the following system of K coupled equations:

Y " » (J(x) - B I ) Y . »£«. h}. C?.l)

uor«V(x) i s the potential matrix, i . e . a K by K rea] eyusetric aatrix the tleaonte

of which, V. ( x ) , srr known functions of x; the energy S i s a constant and X i e the

h by N unit oatrix.

As compared vith eq . (2 . l ) of I the only difference i s that the solution I i s

no longer a coluna vector tut a K by K matrix. This i e so say that the procrea actal ly

nolvoo simultaneously the systea (2.1) of [ t ] with If s e t s of i n i t i a l conditions T (a)

and "¥ ( a ) . Such an approach became necessary because in physical applications the

usurl quantities of interest as , for instance, the s matrix or the eicenenerjies,

need K linear independent sets cf solutions of the system to be determined. Cloarly

a l l the theory developed in the preceding paper remain the aaae except for eq.(3.1)

which presorves i t s fora but with y(J). j (<f ) , y(o) and y (o) thought now as N

by 1 matrices.

The program has two main sections. The f i r s t deals with the finding of the

partit ion of the integration interval that i s consistant with the preset accuracy

in the resul ts . The Becond section consists of the very algorithm to propagate the

solution throucn eq . (3 . l ) of [ t ] .

The f i r s t section has the following input parameters : the number of equations

in the systea, N; the i n i t i a l and final ends of the domain, a and b; the- lowor M.<X upjvr

limit* of the energies at which the oyatca(2.t) w i l l be colved. S and \i ; the ain - m

required accuracy in the results , T0L7; the i n i t i a l increment of the step .ii=e, .[.

I t alno need» the expressions Of the functions V. ( x ) , x*[a, bl for any pair

i» j • 1» 2, . . . , X. These are tfvon by the user as a soparate RI1SCTI0» of the .ir<;uo-

cnto 1 , j and x.

In outwit i t i^voo tho number of the intervals of the part i t ion,n, ;».o l.-n,:th

oi' each intorvnl, and a l l tho relevant quantities in oach interval which v.. 11 l,- urcd.

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«action* Theas are the Dira* Batrlces vhlrh duterejiaje Uw Mpprotiantin*

pnrabollo potential l e Uw working basis , to which on» sud* the traanforaation ont r i t

TitM the working tenia of tha prav loua Internal to Uts oe*» tn Um mrrvnt tnU>rwtl.

Tho three satf-lces which deurnein» too parabolic pot «tit inl in UM> current int­

erval labelled by 9 »re in order V , nao eq.(2.t?a) of I . atilrh In a diacon»l m i i P

aatr ix , «ai V and « which ara tha wei^hte of tha linear and quadratic t*»raa in tha P 9

perturbation, ooo eqs . (2 . M) and (2.12b) of I . îhey *xn eyaaetric real aaU-icea. *a

t for tho tranaforaation aatrix tala i a T. - U, and T - U ,U for p - 2 , 3, . . . , a,

1 1 p p-l p

«bora U i a tho unitar/ diagonalisation aatrix in tha p-th interval. Ooo aora

transforsstion aatrix, denoted aa T . , auat bo addad to thia a»t, which rotates tho

aolutiona obtainad at tho r .h.a . of tha la s t intarval in tha vor* AC baaia of i t , to

thoir valuee into tha original baaia of tha aquation {2.1), 2 2

AII together, tha eterate of thoaa quanti tiaa requires n(2» • 2K • l ) « H

«ord locations. ( for tho diagonal and tho syanetric natricoo tho coapreooed etorage

•oda i a ueed.) In addition, aa aany aa seven square satrlcee ( i . o. 7W locations)

ara necessary to atora interaodisry results in tha inner procedura aa for instance

the amtricra of tha partial error» t . , t » t , fc and £_, ae» eqe.(6.1 - 4) of I .

for nystesa which we encounter frequently in physics aa aany an twenty or

thirty intervals are usually suff ic ient to guarantee at least three exact figurée in

the f ina l résulta. As a se t t er of fact, on our ccaputer, an 131 37o/l35 for which

the as in aaaory i s of only 192 C, no storage problem occurred for eysteae of up to

2$ equations. ( Mots that the object progrès takes approximately 58K.)

The procedure to find the appropriate partition i e the one presented in Section

6 of I . lbs f i r s t b. i s taken as H for the f i r s t interval and as h for the p-th

interval . Ine length of the la s t interval l a fixed according to tbe condition that i t s

r.h. end I s exactly b . I t i s a l so worthy to be noted that we use our in hose subroutine

for the aatrix diagonali ration ( thia i e baaed on the standard QR procedure) and also

that the detailed approach to evaluate the appropriate step eiae i s applied when the

reUUve difference between two successive h t • « ie î ^ e r than .3 while the rapid

_ « -

on* otherwie*, • • • tte dlecusaioa 1» Sees of I .

Th* tia* Mûwnid with the execution of tala section of th* prognm depends

on the expreeeion* of tho potential functions, ibr tnatan— i t i s eubeteatlally lerg-*r

i f the** ezpresaiooe involve exponenţiala than when they con tola only arlthaetic

opération». Ia fact, i t i s gust this (preliminary) section too on* which i« th- aost

tise consuaing. fortunately this i s run only once, at tho beginning of tho «nolo procédai •

Consequently, for problema vhich need rtpoated integrations of th* eystsa, e. £« at

different ener^l**, tho tiao spent with tho preliminary section représenta only a aaall

fraction froa the total effort. This euc£*st* that the probleaa which roooi.ro repeated

integration are among the onea the coat benefitting- froa this method.

To end the preeentati.M-. of the first section we note again that the partition

resulted free It ia obviously consistent only with the hlghoet accurate version of our

aethod, the version i i i . In fact, there i s so problem to develop accurate procedures for

generating partitions «uitable forth* other, weaker versions. However, w* did not put

thie possibility in practice aaialy because of two practical reasons» first, the weaker

versions require certainly finer partitions, that i s to say aore intervals than i l i , so

tht.t for finer preset accuracy this night lead to the exceeding of the neaory area

(at leant for our computer). The second reason i s that the very ooaparisoa of the results

ci van by various versions at the same partition of the integration domain i s sufficient

enough for drawing* eoae interesting- conclusion» on the relative aerits of each version.

The second cection of the prograa has the following input parameters» n, the

number of the intervals of the partition; the five quantities vhich are relevant for

each interval.naaely the matrices V , V , v and T , p • 1, 2, . . . , n/ and T ) . P P P p * n+1

nnd *he interval sise h J (All these were supplied by tho previous cection of the prograa.)

E, th* energy at which the eystes (?. l) i s to be Integrated; the initial coodiUons

i . e . th« two N by X aatrices Y (a) and ¥ * ( * ) .

In return i t yields the nuaerlcal solution at the outer end b, i . e . the R by

N aatrices Y(b) and Y ^ b ) .

To this aia it coaja»tes the initial valu* probi** by eq(3.l) ot I tor the first

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interval, then £0es to the second interval vhore the aaae ia repeated, and co on.

Clearly, the central part of the procedure coacists, i a each interval, in the evaluation

of the propagation aatrieee u, v, u t>nd v .

lïie following five individual versions ware actually ispleaented i n the

prograa for the evaluation of the propegatorsi

version i . The eq.(2.1o) of I i s approximated by the se t of dia, ,cnal equations

yIi + (E " <?ii + V L ( / - Ş - ^ J ^ ^5» ^ i • °' A - 1 » 2 N- <*.*)

Tha off-iia£onal contributions

^ - ^ . ^ ^ ( ^ - ^ . i . J . (2.3)

are t isply disregarded. The propajatora are then diagonal matrices which are computed

according to the forau Las fciven in tho Appendix of I . Xote that this version aakes 1 2 5

are of only V. , V. and IT. . Ito order ot convergence i s of h . i i i i i i

verşi jn i i * Sie diagonal elements of the propaga tors are evaluated as i a the previous

version* 2ie off-diagonal elesents are given by the f i r s t order contribution» from the

approximate perturbation

^ a p p r o * _ ( y ^ + b v 2 _ ) ( j - _ h} ^ . # .# {M)

Thus i t uses V,, , v l . ,V7. and y1 a PP r o x « v1 * yf. . I t s order of convergence ia i i i i i i i j i j i j

of h and there i e an energy dependence of the results which i s diaphra^ed by the

deviation of the original potential from i t s representation by the piecewise linear

functions*

version i i i , . 3his i e the same as the previous one plus the second order corrections

fron V a p p r o x . i t s order of convergence i s again of h but the numerical results

should be better since now sooe parts of the higher order contributions in the error

are retained in the aleorithst,

vorwlon i i . Die diagonal éléments of the propagators are computed ae in version i while

the off-diaconal ones ere civen by the f i r s t order perturbaţive corrections fro» the

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original perturbation, e q . ( 2 . 3 ) . The rate of convergence i o aGain of h . "cwcver, the

retraite ar» expected to be equally accurate for any energy.

•.vrr'.oii 111. i M i la exactly the verolon i i piua tho uoco.-.d order correction») from

7 . I t s order i s of h and tee results are affected by an er-er^y dependence

••fhicîi i s diaphrajTiei by the d«viatica of the pieco- iaj p~nii.oiic potential fros the

exact one»

As compared with the nethods which l i e upon the piccewise l intor rt-ffrt.-.ce

potential, the veimoa i should yield results which are as accurate aa with the -othoa

of Cordon [4] while the results obtained with the method of Rosenthal and Cordon f5j

l i e soaevhera between the ones given by the versions i i ar.d i i i .

Chir pro^rai was r.̂ n on only or.e computer, an I DM 37o/lJ5 located at the

Icstxtute of Physic? and Nuclear tii^ineorico ir. Bucharest, and eoae words are now

in ordor with respect with the timing. l a fact , vo iuiov from our personal experience

chat when tisin^s ore reported froi sooe computer, tr.ese data are fuiiy rtiuva&t

only for the readers who have already cone experience witn enca a cc-: - tor. Vo

accordingly feel that our reader wi l l be helped bettor if , besides reporting the

axnariaental timings from our computer, we also try to describe the computational

effort in teraa of soae general uni.ta that are aore or l e s s independent of computer.

This wil l provide the reader with the poss ib i l i ty of estimating the equivalent tiaing

tor this program on h i s computer, then compare i t with the one for the program he i s

currently using for solving systems of coupled equations.

In so doing, we counted the basic operations oî our program in terms of soae

elementary tine unit ( e . t . u . ) defined as the t iae to compute one floating point

single precision multiplication + onr addition + one reference main storage, «'or our

computer these arm In order X ~ 3 4 / 8 , X .. ~ 14/*a and IT ~ 3 > s so that mult. / odd / r.ta.s. '

1 e. t .u. «» 5o^ta.

Our working assumption ia that the multiplication and Addition tuken AS much

to two thirds and one forth from one e . t . u . , respectively. To our kuovlrdfl» such

(mettone m a i n ewtiafaciory valid tor many computers. rt>r inrtnnco, on nn I!*i 360/63

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(thia « M the ooaputer « M l a [A~\ ) oaa baa t * . t .u .~7 /»a # f roa «nick Uw three

iodlvidoal coatrleutioaa talc* in outer 4 . 5 / a . 1.7^a and about ly ia . Aa .'or DM

other aiitfcajetio opaimtiooa we eaeuee that tha auetraotioe and tha diviaioa take aa

audi aa the ad i i t ioa and tha Multiplication» respectively.

Va ocmeaBtiata only on tha eecond aaotloa or the procrea beeaueo thia i a

thu one cal led repeatedly «ad alas because, aa explained above, i t i a entirely

indepaaoV.-.t of tha exprea&iona of tha potential funcţiona- The h u m ad r ader can paaa

ovor tha cooaidaratlona which follow to conclud* i n eqs.(2.9 - 13).

there ara thraa aain eequancea in tha computation of tha oecorui «action.

Tfce f i r s t cooaieta of tha preparation of tha i n i t i a l condition» for tha current

intorval and of tha computation of tha sat of the working functions ţ - X . Toe • •

aacood sequence coaputaa tha aatrlx propaţr.tora u(h), v(h) t u (h) aed r (h) while in

tha third acqueace tha nuser-ical values ara computed of the solution and i t » f i r s t

derivativa at tha r .b . e . of tha currant interval. Clearly, tha firsthand tha. third

sequences ara identical for a l l versions while the second i s version dependent.

Aa for tha f i re t sequence, supposa that ve have the nuaerical values of y

and y at tha f inal and of tha previous Interval ( p - l ) . Denote then by / ? * * * and

y vrmwm sLnce they refer to the working basis of that interval, tiiey can ba used as

In i t ia l data for tha integration inside tha current interval (p) only after preault-

lpl icat ion by tha rotation matrix T :

y(o) - if™, y'(o) - y p r w . (2.5)

This i s tha f i r s t task of this sequence and i t takos 2 N e . t . u . As for the effort

associated with the conputaUon of the seven functions ^ , n , ? , j> , y ,

r , , a n d ^ l £ , l c - 1 , 2, . . . . N,(this i s the second task) this takes as ouch as

5o K e . t . u .

In the third sequence one f irs t has to compute tho four oatrix products

A m u(h) y(o) , B - v(t) y (o) , C - u (h) y(o) , 0 - v (h) y (o) , («VO

nnd f ina l ly odd each pair Vo yield tho r .h .o . ronulto,

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jrOO - A • B . J*00 - C * O. (2-7)

As for the products, eqs . (2 .6 ) , titer* are tM C U N : «boa UM version, i i s »»s i . 2

the propagators ar* diagonal aatricea so that the computational effort i s o f 4 H a . t . u .

voi le for a l l tha r w i n i n g versions tha propagator» ara central I by B aatricta and

thua eqs.(2.6) require 4 !r e . t . u . In tha both casa* tha maul t i n - A, B, C an* J) are

General aatricee. thus tho additions (2 .7) and tha associated storages taic* a T ^ x i a a t a l j

2/5 II2 e . t . u .

In conclusion, tha execution of tha f i r s t and third sequences takes tot -ier

a ooaputatioaal affort of

( 2 l 3 • 4.67 K2 • 5o D) e . t . u ,

tor tha version i and of

( f i r * .67 X • 5o X) a. t .u.

for tha other versions.

Now evaluata the computational effort for the record sec,i.mce.

Version i . In this case only the diagonal elements of the propagators ara

computed by the fortaulas gi.ven in the Appendix of I and this requires about 7o N e . t . n .

Wrrpion i i . The din/pnal eleaenta are coaputed as for the previous version. I approi 1 2 For the off-diagonal elements f i r s t coapute the matrix 7 « V -t- V h and this

takes 3(M +• l ) / 2 e . t . u . Xext, the formulas (3 .6 , 0, 11, M) of I for X ţ l . 8 and

(3.8, 1o, 12, K) of I for X > ; . 3 are used, «ote that only tha partial contributions

that iff froa the linear terra of the por turbat ton, a re taxen. For filed i and j the

cooputaUoa ot the EOt of the four numbers u .(h), v. (h) , u- (h) and v.- (n) require o

in averse* 5o e . t . u . If ona further aakes ase of the fact that the propagators obey

none syoaetry relationships,

\,(h) - v w - u!'i(h) -- u!ICh)- i w - -* . '>>• \'^ - - u!i(n). va.*») .-.<» full offort to oompiito tha a'.l N(K-1 ) off-dia(7>jial jleaentii of tr.o four propQ^.j/s

in of fibout 25 N(N- 1) B. t .u .

It in r< r*» urn/ul to r>"inll :at "...Ilur h'rr i=?iy rciflilof "'.'<;. rh.T'-i'orr',

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for higher precât accuracies, the cosputr : .on of the fir.-;t order corrections o.

•he off-diagonal elements cf the propagators =ates use la in iy of their Taylor

counterparts. Then, i f the c o a p t a t i o n proeeeds with f : v a i ( er.d no I t ia ir. the

prograa) and takes successively ; s i t I , . t 2, . . , Ï , tne cocl"i'icents of the

powers of A ( which depend oc i Alone l ) should be c=puted only once aed tina takes

about 3o e . t . u . They are thea stored and usee for the correspondis multipli.cations

with tiii» powers of the j dépendent A ' s . l a th i s -ay thu total effort becoees o f abc-

( 3o W • 12.5 KO»-!)) e . t . u .

Version i i i f . The zeroth and f i r s t order corrections are co^put^d as i t tr.e

previous two versions. 7o cospute tr.e second order contri.--ti.ons use i s =ade of t.-.e

forculag (4.1oa - d) of I . A." explained there, th i s implies as .zasy as 4 ï individual

a u l t i p l i c a t i o n s . "Bie '.unber of additions i s however larjor so that, a i l t o g e t h e r ,

these foraulas need as such as 5.5 X e . t . u . Cnce coaputed, tr.ese corrections are adacd

2

to the previous ones and t h i s operation needs 2 X e . t . u . Gone contributions proporti­

onal to V also e x i s t but they are irrelevant for th i s approxiaate estimation of the

t i a i n c

Vorolon i i . The diagonal aatrix elesento of tho propica>o r» , r o calculated

s e i n version 1* Jbr the off-diagonal elements the procedure *,i»en for the version 1 \ I aaproXx

i i i s repeated [ tor true V rather than V ; to which the f i ro t order correct­

ions fro* T are added, The very addition takes a» such no 4 / j P ( M - 1 ) e . t . u . while

the c o a p t a t i o n of the new correction requires about Jo x(H - l) e . t . u . lor Uie

general case and (20 N -i 1o N(N - l ) ) e . t . u . for the l i a i t of swi l l h'n. Kote that

no sysMtry rrlationnliip occurs for these corrections.

Vorsion i i i . I t l s p l i e s the seas addltlor.*' effort over i i as i t were with

i i i over i i . .

This discussion nhcvn that when each of the five versions I s taken independ­

ent ly Uie overall emitting U s e / s t e p i n given in e . t . un i . s by the following

approiisate fonwlas i

t — 2 M5* 4.67 M2 • 12o X (.-..9)

f 6 î:5 • 26 y2 + 75 S lor c^^ra.1 casa (2-loa)

X l l l 6 li3 • 13 »i2 • 137 K for mail h , (?-1ob)

r H .5 » + 29 X • 95 X for cenoral cas* (".*!•)

m i d l . 5 I 3 • 16 K2 • 177 X *>r r=.-.ll h, ( ? - " » )

f 6 IS3 • 56 * t u ) » for c«"«rai- C M * (?•'*»)

'ii ^ 1 * 6 a 3 • 25 S 2 + I<8 K for euail h , (?.U'b)

,' H.5 K • 59 K • 65 K for con-ml «i™ (r.13n)

l 11.5 fc • 21 K • 140 X for t m l l I» - V?-»3b)

Obviously thés* forau ian £iv« only trio lovant l i a i t * of tr.e computational effort» ID fact,

*>.• prOfcTaa also contniiM other typical operations ac, for in.ntanc*, IF ntbtoaentn,

30 loop«, repeated CALL'a of vnrioua subroutines, and thope ara aloo Une coomau.n£. Th«

2 3

tiMfi associated to thia ara proportional to X> K and tt . then, to reproduc* the

ejperiaental t ia incs , the coeff ic ients of th* power» of H in these foraulaa should bo

multiplied by BOM factor» 1 , 1 - , and f_ yhics ara ^reetar than the unity and whoa*

values depend of each coapute.-. Ibr our computer they ranc» batwean 1.4 ana t.7 for

the a l l f iv* versions. Aa for otltar coaputara, our personal feel ing i s that they should

not be too different of t'nese values,e»y, up to two or so.

Thaoratically thane factors « i l l coa* closer to unity i f th* whole procedura

would be procraaaed in ASSiKHLT. Th<3 ia however a formidable task. Instead, ve have

proM u—ad ia AS3BUJLT only th* subroutines which coatributa ir the coefficient* of K .

( All th* othara ara l e f t in KHRIUX.} lhaaa ara the satrix Multiplication and the calcul­

ation ot th* eacond order correction** With t h i s iapleaant, the coeff ic ients of • i a

oc,a. (2 .9 - ' 3 ) raaain pracUcally unchanged but M M f a e t o n in UM coefficient» of » and

of H which, M eentiooed, jangs batman 1.4. and 1.7.for our computer, ar* a t l l l

noceacary to reproducă tha CPU tiain^a.

A» an i l lustrat ion of th* pract ical i ty of this theoretical estimation for lining»

w« npply i t to prodict tho tlain<; to be oxpeci<>rl for th* coaputatioi.nl efi'art/intorva'l

lu ca»e of un lui 3°o/â5 ( 1 o . t . u . ~ 7 / ' » ) lor the v.irmon i and 91 » 9 . TÎ>o forau in ( ? . ) )

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gives 26 as. for * >• f » 1.5 and 31 ms. for the extrane case f • f - 2. Theue

estimates compare quite favourably *ith the experimental value of about 5o as.

which corresponds to the method of Gordon, see fig. 1 of ref. [4].( Remember that

the both methods are of the sane order, h ̂

3. 4 Teat System

The resul ts from a tent system are quite instruct ive at loant for two ronoona.

Firet , they produce experimental confirmation for tne theoretical entimet^s of the

order of convergence, as evaluated in the f i f th Suction of I . r^cond, they are of help

in the understanding of nome spec i f io computational aepecto.

The t e s t system reads

'1

'2 "

3 - 2 x - S - x 1 +• x

-x - 1 - 2 x - B 1 - x

1 + x 1 - X 1 - 2 x - S

and i s solved on xe[n, \o\ for E - o, with the i n i t i a l condiţiona

t • t y,(o) - y2(o) - y3(o) - 1, VjCo) - 2 , y?(o) - y2(o) - o.

(3.1)

(3.2)

The exact solution reads

y , (» ) - (1 + x)e*p(*)# y 2 («) - ( l -x )exp(x ) , y ? (x) - x exp(x) . (3 .3)

Obviously, th i s system meets the banic requirements for this progi-air. In fact ,

the potential i s a roal symmetric matrix. Moreover, each matrix element of the potential

i s a l inear funotion. This fact has two consequences, i l r o t , the deviation of the

computed resu l t s from the exact resu l t s must be unsigned only to the algorithme themselves

and not to the preliminary section of the program. Second, the pairs ( i i , i i ) und

( i i i 1 , i i i ) should give ident i c answers, and t h i s was confirmed in the experimental

runs.

In table 1 relative deviations, z m ( z° M C t(Jo) - ZCOapUt'(lo))/zexact(lc>),

- 12 -

"• - yj t y ? • y, , y. , y 2 • y , ire pronontod, in unit» of In , for Uio »h r*-"

riUevnnt verrions i , i i , niiil n i . , «t lèverai values of TOLV. i.oto in panning thnt

for t.hin nyetcti i t happnnn thnt tli» intnrvala of the purtition are nearly eijuidintunt

for each T3LV; of couroo, the lnn^th of M.« last interval i c always taken, an tho corroop-

oiuiing fraction that en.iuros Uw correct b - 1o,

Th» lnrgeat orror actually reached for each TOI.V i s plotted on f i j . I .for tho

three versions. As the deviations for i i i . converge to a rol&tivo background of 5.1 o ,

the data at email TOLV'a are displaced by this quantity through the indicated arrows.

These data car. U» interpreted an follows, First, the value of TOLV rcpresontn th

predetermined accuracy for the version i i i , which for this case gives identical re oui t s

with i i i . 'lljen, ideally, the data cooputwl by this version or.ould l i e alon« none of

the two bisectors of the right handed quadrants. However the experimental data regain

parallel to one of thorn but are displaced by a factor of about l / 5 . This in quite natural

since the global error, as defined by eq.(b.5) of I , represents an upper «stitoation

of the real error, llie f i g , 1 also shows that the data for the versions i ajid i i l i e

along some l ines with the slopes of 1/3 j-id 2/3, respectively. This confiras the error

analysis as well . In fact, the version i i s a third order method in each interval so

that, the deviations of the rosul ta computed for the r .h . s . of the whole doaain Ghould

exhibit a quadratic decrease; for the versions i i . and i i i . the final results behave accor-

5 - 1 4 7-1 6

ding to the h =h and h =n rules,respectively.These data also suytrsst taat, while

the experimental deviations exhibit fairly .soLotonous decreasing '"or s t a l l TOLV's, tr.ia

decrease might exhibit soae osci l lat ions for larger tolerances, sea i i for TÛLV between

.5 and ,25. This tendency was observed frequently in our experimental tes ts with other

nynteos and i t i o closely related with the largeness of the steps involved, vhuo, the

uoer ohould be aware that, in general, the partition deteraincd by tho f i r s t section

of this prochain can not identify the exact position of the error to be expected for i i i .

It only guarantees that thiB error la not_ outvarrtg the anple AOB of the two

bisectors, ao ficurûted in the c irc le .

Ae for '•.ho back^Tound deviation (of 5.1o for thio example) vtc source i s iho

accuTiultition of the inherent ."iur.d off errors. Cur experience on t.hin ai>d :cany other

- 13 -

teat cystous ehovs that the highest accuracy actually reached by th i s program i e of

at least four exact decimal figures.

Ste tabla 1 also presents the aorreopondlns timings. T.ie timings In tho laet

row refer to the computational effort to obtain throe sets of individual solutions,

(tfe took the sane i n i t i a l values for oach se t . ) Note in passing that the experimental

timings agree pri t ty well with their theoretical estimates. eqe.(2.9 — " ) with some

multiplying factor which, as explained, rangea beetven 1.4 and 1.7.

An easy task wax for us to compare the present prograa with the usual euoroutir.ee

as supplied by the Subroutine Scientif ic Package for the VS/l machines [ 7 ] . These are

the subroutines KKGS (Runs» - Kutta method) and HPCI. (Hamming predictor - corrector

method for linear systems), ^ r these subroutines only one set of i n i t i a l conditions

ia taken so that the reported effort tshould be multiplied by three and then compared

with the one for the present orogram. Note in addition that the cystem (3.1) actually

favours the claesic&l methods. In fact , the expressions of the potential functions

are so simple that the computational effort to evaluate them i e practically negligible

ai.d, in addition, the solution of the system i s well behaved ( i . e . i t does not exhibit

narrow osc i l la t ions as usually happens in physics), BO that only a decent amount of

steps i s needed.

We found that for the both subroutines the aaxiu.ua relative accuracy i s very

oitnilar as for our program. To reach it ,each of the two c lass ical methods needs as many

as two hundred intervale and about four seconds of computation. Thus a global effort

of qbout twelve seconds i s necessary for three aete of solutions at some given energy

i . e . of about s ix times larger than with the present verakon i i i . .

4. Humori col Rosul te, for jn Hiynlcal Problem

A prob lea which con be described in terns of coupled differential equations

i s tho one of the rotational excitation of a diatomic molecule by neutral particle

impact. It was often investigated in the l i terature, uoe, for instance, (.2, 3 J.

following AUimn,we denote the entrance channel by the quantum numbers J and 1, the

- 14 -

exit channols by J' and 1* while J « j • 1 - j' + 1" ropresont» the total ancilar

nom en tua of the system. One arrivas at the system

<-dr J J r J J k bT j-,1» J

(4.0

where

« i s the kinetic energy of the incident particle in the center of mass system, I ia th>i

momentum of inert ia of the rotator and A i s the reduced ones of the system of particles .

Thm coupling matrix element rende

tyv; J | ï l j " l " i 4 > - ^ j . j . ^ i . i - V r ) • ^ O ' l S J " ! - : J) V2C r ) , (4.2)

where the coefficient f c«n be evaluated by tho specif ic formulas &iven by Bernstein

et a l l . [ s ]or by the funeral formulae given by Luster j r . , [ ? ] .

The number of equations of the eystem (4.1) i s fixed by either tha corresponding

nelectlon rules fumichod by the conservation of the a n u l a r moaentum or by the vory

convenience in terms of the computational ef fort . Suppose for moment that i t i s K. Then

K linear independent se ts of solutions of eq . (4 . l ) should be computed, which are used

j ii(m) next for the evaluation of tha S matrix* The partial sots arc donoted by y . , . , »

m • 1, 2, . . . . N. Then the syaten can bo written in •: suitable jiatrix fora:

M

k

where.

V l a Z < ^ > " ^ ^ U.3)

E a ^ e - ^ j ( j • i ) ^ » y^i B ) ( r ) , i « Ù M ' ) , k . (JM- ) r

Vii(r) - Vl» j ' l - -^ ' (J* + ° + V ( 1 V 1? + %o ( r ) + t2^"Lt'i'1,i J)%*Vr>» ' r h h*-

v i k ( r ) " v j » i ' , j - i - " ^ 5 f2

( j , 1 , * j M 1 " ; J ) V r ) ' ^

It ount be ooiphacizod that the whole entrance channel dopendenco has boon absorbed into S,

- 15 -

Thin i o of ce r t a in advantage for o w program s ince , once an approximate parabol ic f i t

of the po t en t i a l a t a a iven angular oooantua J was performed by the f i r a t s ec t ion , fur ther

ca lcula t ion» fordiflferant values of both the energy e and of tho entrance channel quantum

numbers by means of the second sec t ion ara performed very quickly with the appropr ia te

•alue for E.

TLor» a r e seve ra l numerical r e s u l t s reported for the p a r t i c u l a r case

J •- 6, J - 1 - o , —— - looo.o , £m 2.351, a • 1.1, h* L

Vo(r) - r - 1 2 - 2T6, V2(r) - .2283 VQ(r) . (4.5)

This descr ibes the exc i t a t i on of the r o t a t o r from the j =» o s t a t e to l eve l s up to J ' » 2, 4

and 6, giving r i s e to s e t s of four, nine and sixteen coupled equat ions . In p a r t i c u l a r ,

All ison considered t h i s problea to compare the eff iciency of several numerical method o

nemely matrix h'uinerov, i t e r a t i v e Nuxerov, da Vo^sluere, ar.d Cordon. His i n t e r e s t was mainly

x'ocuood on the numerical valuea x'or the so called transition satrix oleaenta which are

the square modulus of the corresponding 3 matrix d e f e a t s , Ho found that the i t e r a t i v e

liuaaro» i s the f a s t e s t , a fac tor of two over tho matrix h'uaerov and s l i j a t l y l e s s over

the de Vogelaere nethod. With respect to the method of Gordon, Allison shows t ha t t h i s

ehould be prefer red only when lover prec is ion io required in t.-.e r e s u l t s , lie a l so observed

that the convercence of the diagonal elements i s elow ar.d, moreover, they appear to

converge to values t h a t d i f f e r sy up to three on the th i rd decimal p lace .

The slow r a t e of convergence- for thin netnod cur. bo understood a t onca. In i._»ct

2 4

the t h e o r e t i c a l p red ic t ions show t h a t i t behaves liice h whilo the h ru le i a appropriate

for the o ther methods inves t ica tod by th i s author.(Thane eot.iT.Atoo rofor , obviounly, to

tne convergence of the r e s u l t s a t the very and of the in tegra t ion doaaln and t h i s in the

otily relevant in uuch ca l cu la t ions , Tho order of -his global convergence can bo derived

d i r ec t ly from the rer .u l ts of the e r ro r ana lys i s on individual i n t e r v a l s . Une i s nade of

tho following r u l e : Cor liotnodo which y ie ld simultaneously tho solut ion ana i t s dur iva t ive

in each in te rva l tho global ortisr i n r.v.alle- \vj >ue tr.an tho omor on t. s ingle in te rva l ;

i t i o lioallor by two for the iiiot.jodr, which compute only t.io solut ion, ".'no usthodn of

Jo Vogelaore and Gordon boloi.g to the fir:if cU.i:; while Ino '...^.••;ov l ike .".oihodn to tho

- 16 -

r^oond c l a n o . )

I t i s d i f f i c u l t to t , ive n d m i n i t e ox.jl.-ina'.ion f o r t.ho n«»cond o b s e r v a t i o n . Wo f o o l ,

howover, that a major coure* for tho r.onconvor,,once o f the computed data to the o i a c t onoa

c m bo anclfjied to the uiuiccurate eva luat ion of the s t e p s i z e s o f the p a r t i t i o n . Ono oenaa

that i n t e r v a l a ara çeuoin tod >)iich a r s , an a r u l e , too Lar^e to be conoi s t e n t with the

prenet accuracy and ouch a conc lus ion i s a l s o supported by the t imingc reported i n Table I I I

o f [2\.

A computer CDC 64oo was used in a l l c a l c u l a t i o n s of A l l i o o n . To f ind the p lace o f

the present method ve took the sums p a r t i c u l a r case and a l e o progranraod SODS o f the method H

uaed by A l i i non for our computer. They are the matrix Nuaerov and the de Vocel»ere xethodn.

The way to obta in the S - latr ix elemente frota the s o l u ţ i o n a ca l cu la ted a t the o u t e r

end o f the demain i s s i m i l a r with that used by A l l i s o n and i t connis to mainly in the une

o f the R matrix a» an in terned 'ary s t a ^ e . Tue H matrix i s obta ined by matching the oeymptotic

condiţ iona with the computed s o l u t i o n s , ibr the "uaerov method the va lues o f j are taken

from two v a l u e s o f r, lar^o onouj^i so tr.at the t eras V., aro n e g l i g i b l e , itor the present

and the de Vogelaere methods, which produce both T and only the f i n a l po int r , i s

s u f f i c i e n t . In f a c t , the asymptotic condi t ions read:

^? - A i / A } + Vi'V' ^-6ai

^\jrJ ' VW + V l ^ P ' i. » - 1. 2, . . . . K, (4.6b)

which lead t o the numerical va lues of A. and 3 . Here AS ua

'•i(Vs ^iVi.(kxiV' :vrr>3 "uViAiV' -̂7)

whore J and n are tho uriual 3enaei -»nd Neumann funct ions which can be projra.ime'i a i r u c t l y

t o o t h e r with t h e i r d e r i v a t i v e s , the a n a l y t i c formulas of whons are:

• i j . ^ " ă^Jj^^x) -t- ^ v * ) - M 1 + 1 ^ ' ) ^ » V-<««a)

n j U ) - j ( r . 1 M ( x ) + - n ^ x ) - n U ) ) . (4 .«b)

Thon, 'ho H l o t r i » elf.Tiinta ;ire ;p.ver. by tho o'i'intton

- 17 -

S. - (11. A ) r C3A _ 1 ) . i4.9) ua u» 11 i a

from v hi cil the S matrix recuits as

s - (r + ia)(i - IR)"1. U.io)

This one point procedure ia sore direct and much faster than that used by Lentcr f î j în

connection vlth t.ie de Voselaere aethod. la that paper, Lester calculates tne S matrix

via phat"» ehifts and amplitudes. The latter are determined Dy the data at so-e innor points.

As mentioned above, we compare the following aethods:

1. The oatrix Niuaerov method in which we uso the )i by ÎI catricea 0 for y(r , , ) i n i t i a l ' and I for y(r. . . . , + h) to start the steo by step inte/^ration out to the * i n i t i a l . j r i**

aatching points which are rr — h and rc. "ote that the step _ize i 3 once doubled

as explained below.

2. The aasrir do Vo^elaere aethod in which the i n i t i a l conditions are y(r. . _ ,) • 0 i n i t i a l '

and y'(r , • t i a l ) — I» Note that this aleoritha, when proera-.aed in i t s raatrix

form, proves more ef f ic ient than when one uses thu repeated integration ove* the

whole dooain which i s to be porformed N t^moa, for each individual set of i n i t i a l

conditions, as used by Allison and Lester.

3 . Our present single precision program which embeds the five different versions

to generate the propaga tore. The i n i t i a l conditions are the ease ae for the de

Vogelser» method.

For a l l threo methods, the actual forts of the second i n i t i a l matrix condition , for which

ve uee the unit matrix, i^ema to be irrelevant from computational point of view. In fact,we

also triod other nonsingular matrices to obtain identical S matrices,

following Allison, the i n i t i a l point was fixed at o,75.

For the methods 1 and 2 we used threes different partitions:

(a) 6o steps a t b - o.o15 followed by 14o stepe at h - o.o3, which result in r f -5 .85 ,

(b) loo steps at h » o.oo7 followed by 35o steps at h - o.o14, 1 . e. r m 6.35,

as used by All ison, and

(c) 14o «taps at h • o.oo5 followed by 49o steps at o .o l , i . e . r f - 6.35,

«'or our aothod the final point was always r . - 6.35*

- 18 -

lu j-.oào preliminary chocks wa havn found tha t the r.injlo precision procramn for

tho ;-4.iod8 1 cmd 2 are ty far unsa t infac tory since they yield d r t j a t i c accumulation

DC tr.o rouiii off e r rors which affoct even the f i r s t f i ,>re in th-3 ronul ta . l'oie phunoaenon

i s cor.nocted with the r e l a t i ve ly short word length of our computer, wnich i s about of

coven aecinal places i a tr.e aar.tic:ia. (.-'or co=parir.on, we r.ole that the saae i s i h i i i n of

atout eleven plaças on tr.e CJC cosputar uaed by All ison.) Then we had to re—write the

ooth algorithms in double precision arithmetic:;* All tr.o data presented below refer to

th i s double procioion computation, '.'o t;et a honest info.-nation concerning the efficiency

of thet:e -ethodG we ware quite careful ir. tr .eir proi,ra.-r.am,i» '.i'o mention, however, tha t

the doubla precision n a t i i x mult ipl icat ion subroutines which thoy often une, wore writ t e i

ii. rCT.'lAî;; the sa,-e ie true for the .subroutine for the doubla precision, a a t r i x inversion

required by the Eothod 1.

The prograas for ir.e three methods were rue for a l l the three caoes of coupled

equations i . o . with four, nine, and s ixteen equations. We i n s i s t however on. the case !» — ~

nince t h i s i s the one which a lso received aore a t t e n t i o n in [2\m

A aajor point ia the inves t igat ion ia that we do not restr ic t the comparison only

2

to the 1st matrix eleratnts as Allison did, but a le» include I t s real and imaginary parte,

Re S and Ia S. In oo doing we meet tocetner both the imediate purpose of having a sore

reliable compari eon of these methods as well as the practical necess i t i e s . As a matter of

fact , there are often cases, for instance when consider polarization phenomena, for which

the knowledge of Re S and In S i s essent ia l . For the problem at hand th is inclusion i s

mainly important since i t unveils some surprising conclusions which are simply masked i s

tho truncated analysis roported by All ison. Bio table 2 presents a synthesis of tho results in which the modulus of the largest

2 —4

deviation of the computed Re S, la S and | S | are civen in units of 1o tocether with

the total nuabor of steps required and the CPU t iao , i n seconds. The data froo the version

i i i of our method with TOLV » o.oooo75 vere taken a s reference*

Tho mont «tricking obcervation in connection with the methods 1 and 2 i s that th — •->

in a lar^e diocrepancy between the accuracies reached by Ro S and l a S on one hand, and 2

|5t ,on the other. In foot , the former ones a r e vorse by about two orders of magnitude th*

- 19 -

the lat ter . TO explain i t , wo recall that a reliable critorion which guarantees accurate

solutions from the méthode baaed on the truncated Taylor ser ies , and this i s the caso

for tho methods 1 and 2, i e that, in the ro;-.ion in which tho notations exhibit ooci l lat lone,

the etep aise muet be substantially osallor than the wavolonjth of the nolUion. Unfortun­

ately , none of the partitiona (a ) , (b) und (c) full"il3 thin critorion.(ibr lnotance, tho

o top Discs of the partition {a) aro about throe tlnao l^r^r than tho corresponding

wavelengths.) This causae dramatic loas of tho accuracy soinly for the quantities which

are the most close dependent of the wavelength. Tho oat; cos Re S and l a S are of this

type while \s\ dépende mainly of the amplitudes of the wavefur.cti.ons. This both explains

the aboveaentioned discrepancy and suggests that the saae phenomenon happens for any other

numerical method which l ien upon truncated Taylor serios for the :-- utions. In particular,

i t impliea that the basic conclusion of Allison, according to which the i terat ive Nuaerov

2 method i s so e f f i f i ent , proves ful ly valid only for 131 aatrix eieaent3.

In contrast, for the methods based on perturbative expansions the throe matrices

exhibit comparable accuracies. This theoretical statement i s confirmed in ful l by the data

presented i i table 2 for our aethod. ibr each of the five versions c? i t one additional

column, A . . t *»• Inserted, which 'gives the .'.argent fro* the 2N quantities

JL 2 JL 2 1 / | S | - 11 for i - 1, 2, . . . , îr, and | £ _ lS| - 1 I for j - 1, 2 ». j=r J i-1 J

These data are of use to compare the influence of each core tora in the pertuxbative oerie»

when constructing bettor and better approximations for the solution of the working cystem.

A short discussion io necessary to clarify this point. To f ix the id.-as, tako the

version i . Take f i r s t some coaree partit ion. ?,'/ the very construction of this version the

original syetca I s replaced by a set of independent equations in each interval, "ha ex.vt

S matrix for this working system i s obviously unitary. The deviation of the COT mi tod S

matrix for this ey^te-! from uni.tarity has therefore two sources. The one i s the truncation

of the very algorithm which i e used to integrate each of tno independent equations which

fora this working system. The second source i s tht. contribution of the round off errors

which, when accumulated, generate soae ground contribution.

Connidor now a finer partit ion. Once a^ain, tr,o ujrï.lr.c syate.-a i s n cet of indepen­

dent equations in oach interval . The exact. S matrix of the newer working syatcm i s

- 2C -

obviously different from the previous one. Tot i t in equally u ' tory. Li for the doviation

froo the unitarity of the computed S aatrix , th i s i s now oxpectod U, bo i-aalicr tfcar

in the previous caae. In fact , the background should bo «oro or l cos t i e s a o ' i~ U10

number of intervals i s not too different) but the truncation-error duo to the very t : .prit/ci

i s Burely smaller simply because no are the intervals of the part i t ion, ror . f iner , ^nd

f iner'part i t ions the deviation becomes smaller and saal lor unt i l i t i-oaches tho b.->ck^round.

îhe l a t t e r case con be therefore interpreted as indicating that the truncation err ir ia

deeply svaaped into the round off background. In other words, the connu tod S aatrix now

represents jus t the exact S aatrix of the corresponding working system, except for n»1

tvackground, Thus, the doviaUon of the computed S aatrix froa the exact S aatr ix of '•v <

o r i g n a l system should be assigned only to the very deviation of the working system f-xn

the original one.

This discussion provides us with an appropriate tool in the analysis of the retriita

given in table 2. In fac t , the inspection of the column / A . Tor the version i indica'-*1*

2 a background behaviour for a l l part i t ions . Therefore, the deviations i n He S, lia S and \3\

must be assigned only to the approximation of the original potential aatrix by se ts of

diagonal potent ials . Fbr the other versions, the data for A . • + sh<w a monotonous decreet* unit

ing for i i . and i i , but the background-like behaviour for i i i . and i i i , at loast for

TOLV ^ o . o l . This can be understood ao an experimental proof of the fact that, to generate

the exact solutions for the corresponding vorlcing system, the use of the f i r s t order

corrections i s not suff ic ient while the addition of the second ordor correction in .

The cross investigation of the rr.jults given by various versions i s a lso fru i t fu l

to draw soiae conclusions about the piecovise polynomial representation of the potent ia l s .

For instance, i f tho vernions i , i i i . and i i i are coraparcd for the name partit ion, one

real izes that the deviations result ing from the f i r s t version are not much vorae than for

the other versions and this ind ica tes that the eystes i s weakly coupled. In particular,

this helps us in understanding that i t was just th i s fcaturn the one which ennurad the

roouirkable Accuracy of tho method of Cordon, as pointed out by Allinon. The corapnrinon

of tho vorniono l i t , a f |d i i i also fluggeeta that the approximation of Mie off dinj:on#il

matrix elemer.tn of the o r i g n a l potential i e only s l ight ly b«U»r 'tirou/h p-irabolan Umn

through iitrai^ht lii.or,. Ve c;>. thus conclude that this ex^apl» in by no se&na tho jent

one to benefit in fu l l frwa tho advantages of tho vereiono of higher order o f our method.

Yot, the correi»pontlin£ tiaa.n,;n or the nuaber of otepa required, which nre snibotanUaiily

roallcr than for the c la s s i ca l nethodn (an order of ttafcnitude or no ) provide impressive

evidence in tho favour of this net of «ethodo»

5. Concluziona

Ve developed here a perturbat' ve method for the numerical solution of nystcms of coupled

equations ariRinç from tho Schrfldingor equation. ?ivo different verrions, quoted an i , i i ,

i i i , i i , and i i i , wore dencribed and tne neglected terns aro in each of then of tho

o-Jor of h , h , h , h , and h . "3ie versio«3 i and i i are enorgy independent. Tho er.crcy

dependence of i i and i i i , 10 of the fon» (v. . - 2) h t^d i t ia diaphra^ned by the second

order deriva tivea of the original potential matrix elements in each interval. Tho energy

dependence of the version i i i reads (V . — ïi) h ar.d i3 diaphrngnod by tho third derivative

of the original potent ial . Thus, for higher energies, a l l these versions prove by far more

e f f ic ient than the c lass ica l methods .( Recall that for the c lass ical raethods, the onerey

dependence hao the form (V,. - i) h for any potential . Exaopleo of such .aothono aro

Kuaerov, Runge - Kutta, and de Vogelaere. ) A3 compared with other perturbaţi ve methods,

for a fiven partition, the version i i s aiout as eff icient as the method of Jor^n

while tbe method of Rosenthal and Cordon should yield results which are a- -•curate

as those given by the versions i i . or i i i , , according to the shape of the potential.

Yet, the coaputational effort/step seems to be substantially smaller for our verniona.

In practice, the version of choice depends on tho otructuro of the potential funct­

ions. Tor the nearly uncoupled systems the version i should be preferred. 'Jeer, tho coût ions

are coupled closely» either the version i i i . or i i i should be used, and th is depends on both

the nhapo of the off diagonal elements of the potential a&trix and the ran0-e of the energies,

i'or instance, whon the o f f diagonal potontial matrix elemente are represented accurately

cnou^i by piecewice l inear functions and the energica aro not too h i j i , wo would recomseud

tho veroiori i i i . ţ the vorr.ion i l i io the only ur.oful for complex prob Icon at higher

^ior^ic* and for -.-.ach the potential z^ti lx elements arc ruprerented acoirately c::.y

!.y p i e c v i s o parabolic functions. Al i the venions ^uvo t.io co'.utio:: in. analytic fam

at •-'.y -.oir.t of thi» domain. , r.n; n:, ly at tho =es'n points, and this fact i s aico useful

in practicul applications.

Tho problème in which repeated ir.tojration at different energies 13 noce-rary

are porhaps tho =oat benefitting when use i s soda of then* versions, ouch problems of

txaodia" e interest are tha following:

(a) Inve3tijption of the energy dependence of the S rj.trix elegants for co.-.plex

rystcas and higher oner^cs;

(b) Computation of tr.o er.cr^' rpoctr-a ann of the corrospondi-j ei^or.fu.-.etionc

for coaplex cystr.rts. In particular, the correspORdir.^; integrals for tha normalization

can be evaluated analytically as shown in [j\;

(c) Construction of r ea l i s t i c 'oases for f^rtccr calculations such as for the

usual riartree i\>c< procedure;

(d) Treatrent of problems which involve periodic potentials and cyclic boundary

conditions, Such probl&as occur (,uite often in tho solid state physics, j'or thei, or.ly

the numerical integration or. a sin^lo c e l l would bo cufficient to produce the coi-rospor.d-

ins one c e l l transfer oatrix. Ihen the total transfer matrix rosult3 at or.ee an tr.e n'th

power of i t where n i s the number of c e l l s encountered;

Note also that the formalisa developed in Cection 2 of I indicates that t'r.a

piecewiae constant potential enables analytic evaluation? of th,: corrections frca perturb­

ations of rather general form, not only polynomials, as used in this ::ot of ; in^;;-,. This

fact i s of real importance because i t civea the poss ib i l i ty to develop pjrturhativo

alcoritnas of high order of accuracy for oystcas cf aore general fora. l-'or instance,

a type of perturbation of laaedlate in te rost i e

ac

where p(J) l a none aatrix of polynomials. Thore la no d i f f icu l ty to ovaluate analytic

crprwsnion» for tho corrections fro» this perturbation. Ouch corrections would be useful

for deriving the Algorithm for the syatoa

- 23

- y ( B i - ( \7s(x) • ffJCx) + ^ ( , ) L )) Y - o. dx2

where \ J / ( ^ and \ 7 2 ) **• *y«"»«tric / anti=y=actric aa tr icos . 3ucU sysîroo occur

when the Bom Oppcnheiaor theory i s applied to the iar.y body problcas, soc for instance,

ţg"] and ţ lo ] .

Acknowledgements

Bio author l a deeply indebted to Or. V. A« Lcsior ;r. ,who sent to h u his

program based on the de Vocolaere scheme. He also t i r . e s to tr.arjc ?r. S. Cocstantiaescu,

Dr. D. Eazilu, and Dr. K. Koldovan for tiieir kindly help in the pro£rair-.ipa i a the

ASSSfflLT language.

- 24 -

Hrrerrs.c<ţft

1. I~Cr.Ixant, preprint C.I.Ph.(r*H-h«r—t). C3-;--TC, 1178.

? . A. C A l l i s o n , J . Coaput. Ptjya. iŞ ( l J7o) , 573.

3 . V. A. Las tcr J r . , ir. • Xelhoda o f Coaputatior.il phys ics", ( « . J . i ld»-r r S. FemfctrJi,

and K. Botenberc, 3 t !a . ) , Val . I o , pp. 221 - 241 , Acodcaic ? i » n s , .Vow Xorc, 1711, *r«&

references there in .

4 . R. C Cordon, in. "i."ntboàs o f Cozputatiocal Pijr.Lcs", ^£. J . Aluar, 3 . .-'ernbacn, «.-id

X. itotrnbarft, 2 d s . ) , Vol. 'o, pp. o l — 11o, Acideeic ?r«un, .Wi» Toric, 1771.

5 . A. D o w n ^ l And a. C. Conior., J . Cfcca. Hr.ys. €i_ ( î 'J . 'w, »»îil.

6 . U S r . I x a r u , X.T.Crm'a, and « . - . t 'opa, preprint C.I.Fh. ;i»iohare3t) .1T-J-J6-77, l j W .

7 . • • • If* ^ s t c » / 3oo S c i e n t i f i c a^broutir.» ? & « • * * , V«raioo I I I , QI 2o - o2o5 - 14»

Aucunt i<jr/©.

0 . R. 3 . B r n o U i r , , A* >»l{?»rno, ••• S. V» Kiacey, a o i I . C. ? e r c i v a i , Proc. «toy. Coc.

S e r . A ţ 7 £ C t 9 6 3 ) , 4Zi*.

9 . I~ Gr» Ixaru. 2»v. itou». i'i,yn. _ţ£, ( t ' j67) . 5ci9.

l o . A. ? . rtilîa&n, L. X. itonooarvv, and 0» I . Vi£Ăta<y. J . ?«V'i. 3 - Atoa. :-tol«c p h y s .

Vol. 9 , Ko. 13 ( 1 9 7 6 ) , 2255.

- 25 -

Table 1

Experimental results for the eyBtes (3.1) fron three version»of the prosent aothod.

The relative deviation* are eiven la units of 1o .

TOLV 1 0.5 Tiding for the

preliminary sta^e (seconds) 1.65 1.77 Number of intervals required 6 7

Exact solution Version-* i i i , i i i , i i i , i i i J- 1 1 1 !

y (lo) » 242 291.1 176 26 11 o5 41 57 152 96 4 32 23 3o

y (Io) - - 185 238.1 185 11 i l 15 42 36- 15o 95 3 39 26 55

y (Io) - 22o 264.6 13o 24 11 1o 41 12 152 06 4 13 27 51

y^ lo ) - 264 317.6 155 33 12 69 47 77 167 6j 2 21 22 47

y^O0) - - 22o 264.6 214 o9 8 84 34 69 13o 93 7 14 35 91

y*(1o) - 242 291.1 182 o4 11 o5 41 63 15o 99 4 46 23 53

Timing for the actual integration (seconds) .16 .Jo .41 .2o .35 .5o

Table 1 (continued)

0.25 °«1 ° * ° 5

2.11 2.21 2.58 7 8 9

i i l 1 i i i , i • i i , i i i , i i * ţ * * * ,

117 96 47 18 49 87 3 7 - 1 9 9 9 99 71 56 - 2 72 5 86

137 28 - 42 16 23 1o7 21 - 2 65 8 26 86 45 - 2 Qo 5 48

ţ26 65 7 17 47 96 32 - 2 20 9 21 78 26 - 2 76 5 60

125 43 2 32 22 48 98 38 - 83 12 45 54 36 - 1 58 8 62

127 o7 - 2 48 11 67 92 32 - 3 93 5 81 11o o7 - 4 13 2 36

126 17 14 17 77 95 62 - 2 24 9 43 79 79 - 2 74 5 77

.2o .35 .48 .26 .45 .61 .29 .5o .64

26

Table 1 (continued)

o.o25 o . o l O.OO'J

i

57 49

69 28

£2 8o

51 97

77 12

63 4o

2.94 11

U 1

- 2 57

- 2 74

- 2 7o

- 2 3o

- 3 H

- 2 63

i U 1

3 5o

3 14

3 34

4 45

2 17

3 42

i

4£ bo

45 77

46 34

«ic 16

55 io

46 95

3.3o 12

U .

•• 2 19

- 2 10

- 2 19

- 2 za

- 2 06

- 2 18

1 1 1 1

1 Û2

1 â4

1 63

1 53

1 79

1 i 7

i

34 C6

33 So

37 o9

3o o1

46 6o

37 55

3.51 14

U 1

- 1 73

- 1 72

- 1 73

- 1 73

- 1 74

- 1 73

i i i . i

36

C--9

93

1 11

77

95

.33 .56 .76 .35 .60 .82 .43 .7o .45

Table 1 (continued)

O.OO25 <*.301 O.OOOŞ

i

29 39

29 72

29 54

25 87

34 67

29 87

3.87 15

U l

- 1 32

- 1 32

- 1 32

- 1 36

- 1 3o

- 1 33

i i i ,

53

52

52

51

56

53

i

2o

23

21

17

L

94

19

95

60

27 84

22 26

4.66 ' 18

U 1

- 8 6

- 0 5

- (36

- ao

- 8 4

- 8 6

i i i ,

27

25

26

29

24

26

i

16 62

18 41

17 43

14 5o

21 37

17 62

5.11 2o

" 1

- 61

- 59

- 60

- 62

- 60

- 61

i i i ,

15

H

15

15

H

15

.45 .75 1.o2 .53 .09 1.22 .61 l.oo I.36

- 27

Tabla t (continued)

i

o.ooo25

5.63 22

l i 1 i i i , i

o.oool

6.75 26

U l i i i , i

0.00005

7.34 29

U 1 i i i , i

0.000025

8.12 32

i i , i i i ,

13 47 _ 41 io 9 9o - 2 4 6.4 7 8 3 - 1 5 5.9 6 77 - 0 . 7 5.2

14 36 - 4o 1o 1o 62 - 23 6.4 8 47 - 15 5.8 6 64 - 0.4 5.2

13 87 - 4o 1o 1o 22 - 24 6.4 8 15 - 15 5.9 6 49 - 8.6 5.2

12 24 - 41 1o 7 75 - 25 6.2 6 27 - 16 5.6 5 14 - 8.3 4.0

16 09 - 42 1o 13 59 - 24 5.7 1o 7o - 15 5.5 8 35 - 9.o 4.8

13 99 - 41 1o 1o 41 - 25 6.0 8 28 - 15 5.6 6 60 - 9.1 4.8

.76 l . lo 1.49 .79 1.28 1.77 .87 1.45 1.98 .96 1.6o 2.18

- 2fl -

Table 2

Experimental reculte from two c lass ical nothods (l.uaorov and de Vo^door»)

and f ive versions of the prenant aothod, for tho ryntom (4.3 - 5) with N • 9 .

Clnnfiical Method n

2 * Partition *rJT*_ a i x | ^ R o S | a a i | A I a 3 | aaxjA/SI | CPU t iae Tbtal

number )f steps ( lo units) (to units) ( lo units) (seconds)

Numerov

de Vorelaere

M (b)

(c)

M

W (c)

2oo

45o

63o

2oo

45o

63o

4495

228

5o

57o2

364

1o3

4437

224

51

5351

559

1o1

5o

2

l

13O

3

1

2o9

442

624

142

3o5

414

Perturbaţi va methods

I . Preliminary calculation ( f ire t section of our program)

ÏDIV

o . l

o.ol

o.ool

O.0O01

number ot steps required

15

21

3o

44

CPU tiae (seconds)

57.2

79.8

1o7.4

156.2

- 29 -

Tabla 2 (continued)

II. Second section

TOW '

Version 1 — — — o . l

o.ol

o.ool

o.oool

Version 1 1 , 0.1

o.ol

o.ool

o.ocol

Version H i o. l

o.ol

o.ool

0.0001

Var .on i i o . l

o.ol

o.ool

o.oool

Version 1 U o.1

o.ol

o.ool

o.oool

a a x | û R » s| ,A

( lo units)

257

1o4

83

38

97

8o

18

4

17

25

2

2

79

72

• * i

2

32

2o

3

-

•aSxl&lB S |

( l o units)

528

8a

7o

46

151

87

15

4

26

' 36

2

1

134

74

16

2

42

23

3

-

max\ 0 |S | 2 |

l lo unite

235

138

38

22

8o

1o8

H

5

8

13

1

2

54

92

13

3

31

2

2

-

A u n i t

i) ( lo~5units)

1.8

1.5

1.3

1.7

176

78

18

14.9

3 .8

1.5

2.o

0.9

174

7ù

17.9

14.9

12.1

1.5

2.1

1.4

C?U time

(seconds)

2.8

3.6

5.4

7.9

7.6

lo.a

14.4

21.1

11.2

16.o

21.1

31-o

11.5

15.6

21.3

31.1

15.2

2o.3

21). 3

41.3

- 30

g 0.5 ^025 ^ 0.1 ^ •-

Û Version / o Vers/on iïf

o Version iiit

I *:0.01 - A L *

A»

aooj

o.oooi— _ •

A 1 i , . i . , i i L-J I L

-0.0001 —

0.001 -

-0.01 -

F i g . l . Experimental evidence for the convergence of the present irethod.

CENTRAL INSTITUTE OF PHYSICS Documentation Office

Buchareff, P.O.B. 5206

ROMANIA