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AECL-6479

ATOMIC ENERGY ^ H H u L'ENERGIE ATOMIQUE

OF CANADA UMITED V j j & Y DU CANADA LIMITÉE

ON THE PROPERTIES OFCOLLISION PROBABILITY INTEGRALS INANNULAR GEOMETRY-II EVALUATION

Propriétés des intégralesde probabilité de collision dans

l'évaluation de la géométrie-II annulaire

M.S. MILGRAM and K.W. SLY

Chalk River Nuclear Laboratories Laboratoires nucléaires de Chalk River

Chalk River, Ontario

February 1979 février

ATOMIC ENERGY OF CANADA LIMITED

ON THE PROPERTIES OF COLLISION PROBABILITY INTEGRALSIN ANNULAR GEOMETRY-11 EVALUATION

Michael S. Milgram and Kenneth N. Sly

Chalk River Nuclear Laboratories

Chalk River , Ontario KOJ 1J0

7979 February

AECL-6479

Propriétés des intégrales de probabilité

de collision dans l'évaluation de la géométrie-II annulaire

Michael S. Milgram et Kenneth N. Sly

Résumé

II est nécessaire d'avoir les probabilités de transmissioninternes-externes pi° et externes-externes P°° pour calculer larépartition des flux neutroniques dans les régions annulairesinfiniment longues. Pour 0 _<̂ K _<_ 1 et pour tous les 0 < T. ondonne des algorithmes efficaces permettant de calculer cesprobabilités en fonction de deux variables (le rapport des rayonsinternes/externes K et la section efficace E) et, ce, avec deserreurs fractionnaires inférieures à 2 x 10~5.

L'Energie Atomique du Canada, LimitéeLaboratoires nucléaires de Chalk River

Chalk River, Ontario KOJ 1J0

Février 1979

AECL-6479

ATOMIC ENERGY OF CANADA LIMITED

ON THE PROPERTIES OF COLLISION PROBABILITY INTEGRALSIN ANNULAR GEOMETRY-II EVALUATION

Michael S. Milgram and Kenneth N. Sly

ABSTRACT

To calculate neutron f lux d is t r ibut ions in i n f i n i t e l y long annularregions, the inner-outer and outer-outer transmission probabi l i t iespio and Poo are required. Ef f ic ient algorithms for the computation ofthese probabi l i t ies as functions of two variables (the ra t io of inner/outer rad i i K, and cross-section l) are given fo r 0 < K < 1 and a l l0 < Z, with f ract ional errors less than 2 x 10"5 .

Chalk River Nuclear Laboratories

Chalk River, Ontario KOJ 1J0

1979 February

AECL-6479

TABLE OF CONTENTS

INTRODUCTION 1

EVALUATION OF P00 3

EVALUATION OF P10 7

SUMMARY 15

APPENDIX A 18

TABLE A.I 24

REFERENCES 25

ACKNOWLEDGEMENTS 28

FIGURES 29

TABLES 34

SUBROUTINE LISTINGS 62

ON THE PROPERTIES OF COLLISION PROBABILITY INTEGRALS

IN ANNULAR GEOMETRY-11 EVALUATION

1. INTRODUCTION

In a previous paper [ 1 ] , referred to as I , the analytic evaluation

of two integrals fundamental to neutron transport calculations in annular

geometry was described. The integrals were expressed as an i n f i n i t e

sum of Meijer's G-function, several applications were presented, and

the claim was made that the form of the analytic representations

permits e f f i c ien t numerical evaluation of f i r s t - f l i g h t co l l i s ion

probabi l i t ies. The purpose of this paper is to substantiate that

claim by reporting an e f f i c i en t method for the numerical evaluation

of two of the most commonly calculated probabi l i t ies , P00 and P10.

I t is well-known [2] that of the eight probabi l i t ies in annular

geometry, two are independent; the other six may be obtained from

conservation of probabi l i ty and reciproci ty arguments. In the usual

superscript notation we have:

P i v / (2. P°i .

0 V / (2 x

- Pvi -

2 )pVO

using Figure 1 to define the variables and treating P 0 0 and P 1 0 as

independent probabilities represented in terms of the fundamental

integrals investigated in I. To summarize,

poo . 4 i 1 ' " , * _ i f" IT L 3 , ^ X ) K ; TTJ Ki3(2xcose)cos6de

sin^ic

i- A I I'O A fP10 = - M (X,K) /K = - / Ki3(xR)cos9d6

TT 3 0 0 TT I

These probabilities form the basis of the J method[3] of solving

the transport equation, and certain of them are used extensively in

transport codes such as LATREP[4], HAMMER[5], RABBLE[6], RAHAB[7], and

the THESEUS[8] option of WIMS. Methods for the numerical evaluation

of some have been given by B0NALUMI[9], KENNEDY[4], and MULLER and

LINNARTZpO], all of which are accurate to two or three significant

figures over a range of both variables, but tend to break down[4] in

certain limits. Numerical integration can be used [6,7,11] but in

typical problems where the routines are called several thousand times,

this can be very slow and expensive [7,11].

On the premise[12] that "any approximation that will be

used several millions of times deserves a little tender care and

grooming", the analytic expansions from I have been used to isolate

the singularities of the integrals in both variables, and the remaining

functions - represented by infinite series - were fitted with rational

minimax approximations in one variable, demanding a relative

accuracy in the final answer to better than 2 x 10"5 over all ranges

of both variables. When using the REMES algorithm[12,13] for rational

minimax fitting, calculations in the master routines were usually done

in 60 bit double precision arithmetic, corresponding to about 29

significant digits. The final minimax fits will be accurate to the

significance quoted in the following tables on a 60-bit machine,

although a loss of significance may occur on a computer of smallerword length. All integrations to check the claimed accuracy of thefits were performed using the CADRE algorithm[14], and a Bickley-Naylorminimax fit[15] accurate to 13 significant digits.

To summarize, computational accuracy in P00 and P10 to betterthan 2 parts in 10s is claimed for any combination of radius andcross section in the range 0 < K < 1 and 0 f L The upper bound onE is governed by the underflow characteristics of the computer. Thefollowing sections deal with the evaluation of P00 and P10 respectively.

2. EVALUATION OF P00

It is convenient to employ different representations for P00,according to the value of z and K. Explicitly, for 0 * K < 1 and0 < z < 1.4, using Eq. (4.3) of I

P00 = (1 - K ) (1 + 2w(l - 1(1 - K ) ) + w 1 / 2F_! (K)

+ w 3/2 z w1 [Hn. (K) + F, (K) log w]) (2.1)i=O n

where analytic expressions for F ^ K ) , H^-(K) are given in the Appendix.

If F.J(K) and HI(K) were computed to full machine precision, thenumber of terms I in the series (2.1) required to produce a fractional errorin P00 of less than 1 x 10"5 is conservatively estimated by

I = [3.65852Z + .878]

where [ ] means the greatest integer less than the quantity in thebrackets.

As the value of I increases in Eq, (2.1), the fine t iers get more

d i f f i c u l t to f i t to a constant accuracy, without unduly increasing the

order of the rational polynomials. However, the contribution of the

higher order terms to the accuracy of the f inal answer decreases

rapidly, so we accept lower accuracy f i t s for higher order terms in

the series. This stratagem, which is used throughout this work,

results in a sl ight loss of precision in the f inal answer from the

goal of 5 signif icant d ig i ts , but repays i t se l f by the rapid and

ef f ic ient algorithms that result . I t is worth noting, that i f the

geometrical functions F-(ic) and H.(K) are precomputed and stored,

considerable arithmetic w i l l be saved i f z remains in the range

0 5 z 5 1-4 throughout a burnup or slowing-down calculation. The

coefficients of the rational minimax f i t s to F-(K) and H.(K) are

given in Tables I and I I , using the minimax notation that a function

is f i t t e d to the rat io of the polynomials u /v of degree [m, n+1]

shown in the tables, where

vn i=0

For l . ^ z 57.5 the minimum number of terms in the series is

found by using expansions about X2 = 1 or about X2 = 0. For

0 5 A2 < .696, Eq. (4.3) of I - equivalent to transposing the two

sums in Eq. (2.1) and (A.I) - suggests

P 0 0 = (1 - K) S (]-K)Z A » ( 2 . 2 )5=0 '

/ \o1? / iV

I [i] fx U+m.s ^W y2, 0, 1/2;*, -1 - I,

Rational minimax approximations to A.(w) for two ranges of w are given

in Table I I I , where again acceptable accuracy decreases as Z increases.

Figure 2 i l lustrates how the series in Eq. (2.2) may be truncated, with

a fractional error of less than 1 x 10"s, while Table IV gives conservative

estimates of the equations for the curves in Fig. 2, so that I i may be

easily computed.

In the eventuality that 1.4 < z ^ 7.5 and .696 s A2 < 1, we f ind

from Eq. (4.1) of I

P00 = PCYL(x) - X2-v/i - A2 e ' 2 z ^ 2 (1 - A 2 ) £ B£(z) (z-zo (£))z£ (2.3)P C Y L ( x ) - X - v / i - A e ^ 2 ( 1 - A ) B £ ( z ) ( z - z o ( £

*" 3/2,0,l/2;-l/2J/(z-zo(£))

CYL(x) --W'1(2.4)

is the transmission probability for a cylinder. Table V gives the

coefficients of tne rational minimax approximations for B^(z) and the

values of zo(^), and Figure 3 i l lustrates how I 2 may be ef f ic ient ly

chosen, again to obtain a fractional error of less than 10~5 i f B.(z)

were exact. Conservative estimates for the curves of Figure 3 are given

in Table VI. Extracting the factor exp(-2z)z , the asymptotic

form of Bp(z), improves the efficiency of the minimax f i t s .

Rational f i t s for the function P~y. (x) are given in Table VII

for 1.4 5' x f 7.7. For 7.7 < x, i t is possible to derive an asymptotic

form using properties of G-functions [16]. We f ind

Isr(3/2H)V

3/2 2 Z-* rU+3)ïï x z=o

which, together with Table VIII, permits the accurate determination

of Pryi(x) in the desired range; six significant digits suffice for

the purposes described here.

For z > 7.5, an asymptotic form for P00 may be found from Eq. (5.1)

or (4.3) of I. Since the asymptotic series for P00 coincides with

that for Ppy.(x), it is convenient to use Eq. (4.3) of I and write

p 0 ° =exp(-2w) (2-6)

This form gives the required accuracy for all values of X2 when

7.5 < z.

The algorithms given here have been used within a computer routine

to calculate P00. This routine was tested using 40 000 pseudo-random

values of 0 5 x and 0 < K < 1. The average time required for one

calculation of P00 was .233 x 10"3 s on a CDC 6600 computer, about

76 times faster than straightforv/ard numerical integration to the

same accuracy. Of 40 000 cases, .3% had a fractional error greater

than 1 x 10"5; the largest fractional error observed was 1.66 x 10"5.

3. EVALUATION OF P io

The properties of P10 are very di f ferent from P00 and the

function is more d i f f i c u l t to calculate. Two basic formulae obtained

from I are employed. The f i r s t - Eq. (3.9) of I - is

i)rU+2) (3.2)

The second is Eq, (5.6) of I which gives

n + r(3.5)

2 " Z ^ Y

n rV r j / 2 (3.6)

n+r 2(n+r)-2t Ki3(z/ft)dt (3.7a)

" * \ 4

;2. 3/2-n-r \

1-n-r,3/2,0,1/2;/(3.7b)

e n ( r ) = (")r fT2^r7 r(r(1/2+n-r)

It is convenient to subdivide the segment of the ( K 2 , Z ) plane of

interest into six regions, whose boundaries are indicated by the

heavier curves in Figures 4 and 5. Conservative estimates of these

curves are given in Table IX. In each region, a different numerical

evaluation method is appropriate, as described in the following

sections labelled A - F corresponding to the respective areas in

Figs. 4 and 5. The objective is a fractional error in P10 less

than 1 x 10"5.

CASE A: K 2 % 0, z £ 5.0

The basic formula here is Eq. (3.1), which converges slowly

for increasing values of n. In addition, the functions gAz) are

oscillatory, which makes it difficult to determine the integer I

simply, the power series expansion of gAz) given in Eq. (A.2)

suffers from severe numerical instability as z increases, and the

asymptotic form of g^z) does not result in sufficient accuracy

for values of z in the rcnge of interest. Accordingly, invoke the

recursion formula[16]

t 1/4 z2gr; (z))/(U+l )«>+2)) (3.9)

A similar result holds for h.+,(z) according to Eq.(3.3).

From I, identify

go(z) = 2 Ki2(z) (3.10)

hQ(z) = -4/z Ki3(z) (3.11)

where Ki (z) are the Bickley-Nayler functions, which obey

and negative order functions can be calculated by the (numerically

stable) recursion formula

= ¥ • KlVz) -

So taking I=10in Eq.(3.1), applying Eqs. (3.9)-(3.13) to each of

gç(z) and hj(z) and evaluating the sum algebraically[17], it is possible

to express P10 in the form

Pi0 = f (KiQ(z) P0(z,n) + K1x(z) Pjz.n) + K1z(z) P2(z,n)) (3.14)

where P^(z,n) are polynomials in z, n and /̂ îï.

The analytic (unnested) form of these functions is given in

Table X, factored in such a way that the coefficient of each power

of z is a polynomial in n and J-n which can be precompiled for cases

in which the geometry remains invariant, and only three evaluations

[15] of Bickley-Nayler functions are required for each calculation

of P10 in this region.

The choice 1=10 is a compromise that covers most of the area

of this region of the plane. In a small section, values of I such

that 1=11 and 12 are required. The curves A , illustrated inni} n

Fig. 4, delineate the points at which one more term must be added

to the series (3.1) used to augment (3.14). Conservative estimates

of the equations of these curves are given in Table XI, and rational

minimax approximations for go(z) and h.(z) for £=11 and 12 in the appropriate

range of z are recorded in Table XII, together with the locations of the

zeros of g£(z); each g^(z) approximation is multiplied by the factor

CASE B: 0 < KZ <.5, z £ 5.0

In this range, employ Eq.(A.2) in Eq. (3.1) and transpose

the sums. The result for y = z2/4 is

Pi0 = i + ^ [ ( M n ) + & R_i(n) + y

i=0 (3.15)

where Q and R, functions of the geometry only, may be precomputed

in some applications. Analytic expressions for these functions are

given in Appendix A, and rational minimax approximations (as functions

- 12 -

of the variable K to facilitate fitting) are given in Table XIII.

The curves Bm n which delineate values of I needed in Eq. (3.15)

to achieve the desired accuracy are illustrated in Fig, 4, and

conservatively defined in Table XIV.

CASE C: .5 < K 2 5 1., z £ 5.0

In this range, perform the analytic continuation rr+l/n as

outlined in Eq.(4.5) of I. The result (using y = z2/4) is

pio = ! + ^-( _rJ + ^ S . ^ l / n H ' y Y] y1 [S^l/n) - log y

Rational minimax approximations to the functions S and T

are given in Table XV, conservative estimates of the curves Cm n

used to delineate values of I in Eq. (3.16) are illustrated in

Fig. 4 and given in Table XVI, and analytic expressions for the

functions S(n), T(n) are recorded in the Appendix.

CASE D: K 2 > 0 , z intermediate

Here we discuss the evaluation of P10 for intermediate

values of z; the governing equation is (3.4). As indicated in I,

the function )z vanishes like exp(-z) whereas /i has the

asymptotic exponential behaviour exp(-z/y).

Thus for some area of the plane (region D of fig. 4), /2 does

not contribute, and so use Eq.(2.G) to write (setting v = z/y)

(3.17)

where values of Ii are delimited by the curves Dni,n illustrated in

Figs. 4 and 5, and conservatively estimated in Table XVII. The

identification of the functions °Jn(v) is recorded in Table XVIII,

based on an analysis given in Appendix A.

CASE E: K 2 > 0, z intermediate

In that region of the plane in which /2 contributes, it is

possible to identify

(3.18),?!

using Eq.(3.8) . The t r i p l e sums in Eq. (3.6) are then re-ordered and

the resu l t is

= Y :exp(-z)z~3 /2 £ Y n < H - n ( z ) (3. IS)n=0

together with

pio . p i o . J g } ( 3 . 2 0 )1 C K I 2

because o f Eqs. (3 .17 ) and ( 3 . 4 ) .

The functions H 0(z) and'H' (z) are ident i f ied in Appendix A,n™ x» n

rational minimax approximations t o ^ (z) are given in Table XIX, and

conservative estimates of the boundary curves IL which delimit

values of 12 in Eq.(3.19) are i l lust rated in Figs. 4 and 5 and

recorded in Table XX.

CASE F: K2 > 0, z/y large

In this range of values ( i l lustrated in Fig. 5) the only

contributions come from /1 in Eq.(3.4). Extract the f i r s t few

terms of the asymptotic series for the G-function defined in Eq.(3.7b)

and analytical ly[17] sum the series (3.5) over n and r. The result

is an asymptotic series given by

where P2n(ï) 1S a polynomial in y of degree 2n identified in Table

XXI, and required values of I depend on the value of v = z/y according

to Table XXII as illustrated in Fig. 5 by the curves F

The algorithms described in this section were used in a computer

routine to calculate P10 using pseudo-random values of 0 5 x and

0 * K ^ 1. The average time for one calculation of P10(with no

precalculation of geometrically invariant functions) was .67xlO~3 s

on a CDC 6600 computer, about 40 times faster than simple numerical

integration with the same accuracy. Of 80 000 cases, .15% had a fractional

error greater than 1 x 10"5; the largest fractional error observed

was .19 x 10"11. Most of these errors are clustered about the A-F

boundary.

4. SUMMARY

Algorithms are given which permit the swift and accurate

evaluation of the collision probabilities P10 and P00 as a function

of two variables lying in the range 0 S < s 1 and 0 * z. The method

gives results which are correct to a fractional error of less than

2 x 10"5; six other probabilities may be evaluated simultaneously

- 16 -

with no loss of significant figures, by employing Eq. (1.1) as part

of the algorithms, because of the form of Eq. (3.16) and its

analogues.

The method is designed to take advantage of characteristics

of a large set of problems for which the probabilities are required.

In these cases - nuclear transmutation or neutron slowing-down -

the geometry remains invariant and only the cross section varies.

Precalculation of geometrical quantities thus leads to extremely

efficient evaluations because in a majority of practical cases the

variable z is small and only a few terms in the series are nt_eded.

Although the form of some of the equations appear to have

superficial logarithmic singularities (eg. log w in Eq. 2.1) this is

in fact not so, because in each instance the logarithm is multiplied

by a quantity that approaches zero faster than the logarithm diverges

3/2(eg. w log w in Eq. 2.1). A simple test for w less than the

smallest floating point machine quantity will eliminate all possible

difficulties associated with this phenomenon, because w underflows

more rapidly than does log w.

From the point of view of numerical analysis, the summations

are not the optimal ones, because no attempt has been made at

economization of terms by re-ordering using Chebyshev polynomials.

- 17 -

Thus the error term is a jagged sawtooth rather than a more desirable

ripple. This is particularly evident in region A, where future

effort might be profitably expended someday.

For practical applications however, the method is superior

in both speed and accuracy in comparison to contemporary techniques

of evaluation, and may have potential application to other functions

of two variables (eg. Sievert's integral).

- 18 -

APPENDIX A

In th is Appendix, complicated analyt ic expressions appearing in

the text are summarized.

The functions F^(K) and H.J(K) appear in Eq.(2.1) and may be

iden t i f i ed as follows (i|> is the digamma funct ion) :

2Fi H - 3 / 2 ,

(-1/2, 3/2; 5/2;

H , ( K ) = F. (K) + | r (1 /2+ i ) L T(5/2 +t+i)jii(IL) ( - ) * /l-tcV1 ' r ( i+ l ) r ( i+2) ir(5/2+i-£) r(7/2+i+i) r (^+ i ) t 2 J

The funct ion g5(z) is defined in Eq.(3.1 ) and may be wr i t ten as

a power series

- 19 -

g£(z) = -2ir»y r(3/2+A)/r(3/2-A) + 2

+ „ > r(2+£+i)r(i+l/2)yi

denoting

y = z2/4

- log y

The functions Q(n), R(n) are used in Eq. (3.15). Their analytic

expressions are[18]:

Q-, (n) = 2 ( - n )1 / 2 - | 2Fi(l, - l/2;3/2;n)

- 20 -

2F, (3/2,- l /2;2;n)

Q i ( n ) = P r(2+£+i)

s>=0ru+i) ru+2) r(2-w-i)

-E r(3/2+g,+irU+l/2) r(£+3/2) r(5/2-s,+i)

.1/2(2

- (-ri)i + 5 / 2 V n& (4+&+21)

^ r U + i + 3 )

r(3/2n+ir(£+l/2 r(5/2+i-

2îi -£)

p = r(i/2+i)/(r(i+i)r(i+2)r(3/2+i))

The functions S(n) and T(n) are used in Eq. (3.16) and are

defined analyt ical ly by[18]:

s , ( n ) - o ( i

T i ( n )

= 1. Y" (Vn)* rU+3/2) r(£+1/2)

= log(-Vn)

, 1 ; 5/2;l/n) (A.4)

/1/2,3/2,1

- 22 -

a = X"2r(l/2+i)/(r(2+i)r(3/2+i))

The function / (v) appearing in Eq. (3.17) and (3.7b) may be

obtained analytically. For n=0 and n=l, we have

°(v) = 2E3(v)

(v) = E^v) + E3(v)

These results may be derived from Eq. (3.7(b)), Eq. (6.1) of I, the

identity

K1J(V) = Ev+1(v)

and contiguity relations between G-functions given in reference [16].

Furthermore, i t is easily shown that / , ( v ) obeys

and the relationship [19].

- 23 -

valid for all m>0 allows the integrals to be evaluated.

The exponential integral E (v) obeys the recursion formula

En(v) = — (exp(-v) -v En .(v)). (A.6b)n-1 ""'

so substituting Eqs. (A.6) into (A.5), and doing some algebra'- , allows

the eventual ident i f icat ion of ^ " (v ) in terms of E-,(v), exp(-v) and a1 '

polynomial in (inverse) powers of v. These are given in unnested form

in Table XVIII. Approximations to the function E-j(v) are well-known[20].

The functions Hn(z) appear in Eq. (3.18). Using Eq. (3.8) they

may be calculated from their integral representation

Hn_£(z) = 2 f e"1 ( l+t2 /z)2 n"2 £"7 / 2 k(z+tz)dt (A.7)

;2,2+£-n= ex

where minimax approximations to the functions

k(z) = /i" ezKi3(z)

are given elsewhere[21], and a 136-point Gauss-Hermite algorithm was

used[22]. The funct ions^ (z) are represented as combinations of

H .(z) in Table A . I .

TABLE A.I

Expressions for the functions'^- (z) appearing in Eq. (3.19) in terms of

the basic functions H _9 (z) defined in Eq.(A.7)

= 3/8 G2

= 5/16 G3

= 35/128

1/4 GQ

3/15 G1

+ 5/32

= 63/256 G5 + 35/256

9/64 GQ

+ 15/128

Gl =- H

G2 = H_2 - HQ +

G3 = H_3 - H.!

G4 = H-4 " H-2

H4 " H6

G5 = H_5 - H_3 + H5 - H?

- 25 -

REFERENCES

1. M.S. Milgram, J.Math. Pnys. 18,12,2456 (1977).

2. H. Mark!, Int. Conf. on the Peaceful Uses of Nuclear Energy,

paper P/640, Session 3.1, Geneva (1964). A/C0NF.28/.

3. D. EmendSrfer, "Physics Assumptions and Applications of

Collision Probability Methods", in Proceedings of the 1973

Conference on Mathematical Models and Computational

Techniques for Analysis of Nuclear Systems, Paper VII (5),

VII-136, Ann Arbor (1973).

4. M.S. Milgram, "A Guide to LATREP(1975)", Atomic Energy of

Canada Ltd., report AECL 5036(1975); G.J. Phillips and

0. Griffiths, "LATREP Users Manual", Atomic Energy of Canada

Ltd., report AECL-3857(1970).

5. S. Jabbawy et al, "Water-Moderated Reactor Analysis with

ENDF/B Data" Symposium on Applications of Nuclear Data in

Science and Technology, p.147, conf. IAEA-SM-170/8 Paris

(1973) STI/PUB/343

6. P.H. Kier, A.A. Robba, "RABBLE, A Program for Computation

of Resonance Absorption in Multiregion Reactor Cells,"

Argonne National Laboratory report ANL-7326(1967).

7. H.C. Honeck, "The JOSHUA System", Nodule CREEP, DPSTM-500(1970).

8. A. Jonsson, "THESEUS, A One Group Collision Probability

Routine for Annular Systems", AEEW-R253(1963).

- 26 -

9. R. Bonaiumi, Energia Nucleare, 8, 5, 326 (1961).

10. A. Minier and E. Linnartz, Nukleonik, 5,1,23 (1963).

11. G. Doherty, "Some Methods of Calculating First Flight Collision

Probabilities in Slab and Cylindrical Lattices", AAEC/TM489

(1969).

12. F.S. Acton, "Numerical Methods That Work", Harper S Row,

N.Y. (1970).

13. J.H. Johnson, J.M. Blair, "REMES2: A FORTRAN Program to Calculate

Rational Minimax Approximations to a Given Function", Atomic

Energy of Canada Ltd. report AECL-4210(1973).

14. M.B. Carver, V.J. Jones "An Evaluation of Available

Quadrature Algorithms and Selection for the AECL FORTRAN

Mathematical Library", Atomic Energy of Canada Ltd. report

AECL-5605(1977).

15. J.M. Blair, C.A. Edwards, J.H. Johnson, "Math. Comp.

32,143,876(1978).

16. Y. Luke, "The Special Functions and their Approximations"

Academic Press, New York (1969), Vol. I.

17. The complicated algebra was done with the algebraic

manipulation code SCHOONSCHIP, Computer Phys. Comm. 8, 1,

(1974).

- 27 -

18. The numerical evaluation of some of these functions (which are

very slow to converge in some ranges) can be fac i l i ta ted by

using integer differences between the parameters to lower the

order of hypergeometric series [P.W. Karlsson, J. Math. Phys.

12, 2,270 (1970)] and then using linear transformations given

in ref. 16.

19. J. LeCaine, Nat11. Res. Counc. Can. Rep. NRC-1533(1945),

Eq. (1.1.1).

20. W.J. Cody and H.C. Thacher, J r . , Math. Comp. 22,641(1968).

21. P.A. Christ ie, J.M. Blair , "Rational Chebyshev Approximations

for the Bickley Function Ki3(x) , x > 6", Atomic Energy of

Canada Limited, report AECL-5820(1977).

22. A.H. Stroud, D. Secrest, "Gaussian Quadrature Formulas",

Prentice-Hall (1966), Table 5 p.250-251.

- 28 -

ACKNOWLEDGEMENTS

We are very grateful to Jim Blair for many helpful conversations

and ideas during the course of this work. Special thanks are also

due Bruce Winterbon for his comments on the perversities of asymptotic

series, and of course typists Sharon Baker and Sandra Lusk.

- 29 -

MACROSCOPICCROSS SECTION

R2 = l + x 2 - 2< cos \p

FIG. I Illustration of geometry and notation used in the text.

- 30 -

O CIRCLED NUMBERS REFER TO THE VALUEOF I , IN THE APPROPRIATE REGION

.6 .696

FIG. II Illustration of truncation boundaries for Ii estimated inTable IV for use in Eq. (2.2).

- 31 -

Oc iRCLED NUMBERSL- REFER TO VALUES I 2

IN THE APPROPRIATEREGION

•696 -75 -80 -85 -90 -95 I 0

FIG. I l l I l l u s t r a t i o n of t runcat ion boundaries fo r I 2 estimatedin Table VI fo r use in Eq. (2 .3 ) .

- 32 -

3 15) )

ni)19)1

• 9 1 0

FIG. IV Illustration of truncation boundaries in regions A-E foruse in calculating P10 when z<15.

- 33 -

FIG. V Illustration of truncation boundaries in regions D, E andF for use in calculating Pio when 2>15.

TABLE I

Rational Minimax Approximations to F-U") in Eq. (2,1)

F (K)-1

RANGE 0.0 - 1.0

U0 -.63909 19680 1000E+02Ul -.45713 21674 2000E+02U2 -.62724 63006 0000E+01

V0 .45190 61988 6000E+02VI .22077 10740 3000E+02

F (K)0

0.0 - 1.0

.25960 19131 2000E+02

.24911 98637 3000E+02

.64901 83887 0000E+01

.14685 30710 8100E+03

.23092 96882 7000E+02

0.0 - 1.0

.40004 28803 0000E+00

.52472 94040 0000E+00

.21787 70760 0000E+00

.54311 43026 3700E+02

.81959 41337 0000E+01

F (K)2

0.0 - 1.0

.27911 24240 0000E-02

.48449 24300 0000E-02

.29708 17700 0000E-02

.12125 41967 5150E+02

.52526 52994 0000E+01

ACCURACY(SIG.DIG.)

6.84 6.45 5.45 4.37

F <ic)3

RANGE 0.0 - 1.0

U0 .27920 59000 0000E-04Ul .61729 52000 0000E-04U2 .53758 88000 0000E-04

V0 .58212 20334 4790E+01VI -.40106 78892 0600E+01

F (K)4

0.0 - 1.0

.25412 52100 0000E-06

.69021 40000 0000E-06

.83645 97000 0000E-06

.36314 74934 3410E+01

.33345 90337 0858E+01

F (K)5

0.0 - X.0

.19507 02800 0000E-08

.62875 25000 0000E-08

.10519 48000 0000E-07

.25990 03906 8749E+01

.29152 59896 3052E+01

ACCURACY(SIG.DIG.)

3.69 3.18 2.77

ACCURACY = -ALOG10(MINIMAX ERROR)

TABLE II

Rational Minimax Approximations to H.(K) in Eq. (2.1)

RANGE 0.0 - 1.0

U0 -.87312 22496 1000E+02Ul -.81664 83533 9000E+02U2 -.19671 84841 1000E+02

V0 .14420 76365 3700E+03VI .45579 41387 600BE+02

0.0 - 1.0

.37940 44610 0000E+01

.46842 31404 0090E+01

.17564 22197 0000E+01

.15808 52755 9900E+03

.26658 23378 0000E+01

H (K)2

0.0 - 1.0

,14414 647]0 0000E-01.23576 61880 0030E-01.13169 25840 0000E-01

.17040 05928 260SE+02

.57386 93276 3000E+01

ACCURACY(SIG.DIG.)

6.71 6.07 4.61

H (K)3

RANGE 0.0 - 1.0

U0 -.13477 58450 0000E-03Ul -.28340 28300 0000E-03U2 -.22723 76600 0000E-03

V0 .69243 57232 8480E+01VI -.42603 41917 9500E+01

H (K)4

0.0 - 1.0

.12397 30000 0000E-05

.32333 74000 0300E-05

.36348 87000 0000E-05

.40367 65334 7545E+01

.34708 13999 5870E+01

H (K)5

0.0 - 1.0

.97887 96000 0009E-08

.30570 07000 0000E-07

.47651 16000 0000E-07

.27896 62653 9613E+01

.29973 81779 3758E+01

ACCURACY(SIG.DIG.)

3.85 3.30 2.86

- 36 -

TABLE HI

Rational Minimax Approximations to A,, (w) in Eq. (2.2)

RANGE 1.96 - 39.0

U0 -.84169 23789 0000E+01Ul .23872 25484 5320E+02U2 .34153 77155 7820E+01U3 .37527 53460 5181E+00

V0 -.18170 40625 5000E+02VI .26218 17959 5000E+02V2 .60445 63133 7063E+02V3 .91882 3B277 2P1O8E+01V4

A (w)1

1.96 - 39.0

.78846 29620 0000E+02

.91206 13209 7000E+01

.42443 21758 7700E+00

.37345 90665 0000E-03

.14615 59362 3570E+04

.90910 31648 722OE+03

.20294 31379 2037E+03

.15675 91578 2814E+02

A (w)2

1.96 -• 39.0

.12318 69840 0000E+O5

.12554 27038 8000E+04

.12319 26244 9850E+07

.35573 03278 5220E+06

.96029 41566 9336E+05

.34930 41580 3710E+04

.33892 58415 14902+03

ACCURACY(SIG.DIG.)

5.65 4.70 3.61

V0VIV2

A (w)0

39.0 - 72.7

.37507 87678 0000E+00

.62005 50000 0000E-06

.66559 97000 0000E-02

A (w)1

39.0 -• 72.7

.45884 40989 0000E+00

.47251 42100 0000E-04

.38543 99288 3000E+02

.15269 25801 1740E+O1

A (w)2

39.0 - 72.7

.30068 81660 0000E+01

.16131 35897 083BE+04

.13797 14904 9897E+03

.36495 05719 5490E+01

ACCURACY(SIG.DIG. )

5.86 5.30 4.54

V0VIV2V3

A (W)3

1.96 - 39.0

.47979 45550 0000E+02

.87737 49000 0000E-01

.18776 42412 0500E+05

.18604 08633 7880E+04

.60350 26071 8700E+03

.26033 04938 7670E+01

A (w)4

1.96 - 39.0

.18630 97800 0000E+02

.25483 64595 6300E+05

.54017 42516 4300E+83

.67456 19227 3280E+03

.14500 45468 8329E+02

A [ w )5

1.96 - 39.0

.18194 78200 0000E-01

.31542 48000 0000E-03

.7698? 07313 4000Ev02

.47372 46155 0000E+00

ACCURACY(SIG.DIG.)

2.84 2. 54 1.74

A (w)3

A (w)4

V0VIV2

39.0 - 72.7

.22619 08000 0000E+00

.37930 30375 0160E+05

.29610 27116 2818E+04

.80816 50346 1561E+02

3 9.0 - 72.7

.37362 96900 0000E-02

.32703 43400 0000E-04

.90153 35918 9170E+03

.53510 50385 2453E+02

ACCURACY(SIG.DIG.)

2.96 2.51

* ACCURACY = -ALOG10(MINIHAX ERROR)

- 37 -

TABLE IV

CONSERVATIVE ESTIMATES OF THE BOUNDARY CURVES IN FIG. 2AND EQ. (2.2)

Equation

w = 5.33.333 A2

w = 178.570 A2

w = 155.556 A2

w = 160.000 A2

- 5.996

- 18.400

- 44.889

- 73.000

TABLE V

Rational Minimax Approximations to Bp (z) and z (£) in Eq. (23)

U001U2

1.4 - 7.5

.24730 67990 0000E+00

.1S600 66872 9000E+01

.11280 78289 0300E+01

.10624 86895 0000E+01

.25435 12709 5000E+01

B (Z)1

1.4 - 7.5

.95429 96902 0000E+00

.33540 60010 0000E+00

.75168 17122 0000E+00

.26384 59985 0000E+00

.16018 69973 6000E+01

B (z)2

1.4 - 7.5

.29982 51890 0000E+00

.29743 44675 0000E+00

.17848 50600 0000E-03

.12327 62080 0000E+00

.53177 49825 0000E+00

3.194 93032 09937

ACCURACY(SIG.DIG.)

5.92 4.93 4.37

B (Z)3

RANGE 1.4 - 4.87

U0 -.36992 26951 4200E+00Ul .51800 39405 8400E+00U2 -.85829 41275 4900F-01

V0 .80351 12665 1000E-01VI -.19074 81397 8100E+00V2 .53609 95728 8480E+00

Z0 2.468 73314 67198

B (Z)4

1.4 - 3.85

.10965 37513 3000E+01

.42508 27310 0300E+00

.38846 59265 0000E-01

.90163 63288 4000E+01

.83618 16165 2000E+01

.68677 46458 4200E+01

2.087 37415 92691

B (z)5

1.4 - 3.1

.36324 72420 0000E-01

.19586 05807 0000E-01

.26562 31630 0000E-02

.11811 28946 7000E+01

.19173 81565 1100E+01

1.841 64293 80076

ACCURACY(SIG.DIG.)

5.45 3.62 2.48

- 39 -

TABLE VI

CONSERVATIVE ESTIMATES OF THE BOUNDARY CURVES IN FIG. 3AND EQ. (2.3)

Equation

••= 126.148

= -111.068

= -273.068

= -172.719

= -120.17

- 85.715

+ 520.145

+ 821.134

+ 558.473

+ 354.221

401.933

567.974

414.455

254.571

TABLE VII

Rational Minimax Approximations to Pc (x) in Eg. (2.3)

PCYL(x)

RANGE 1.4 - 7.7

U0UlU2U3U4

V0VIV2V3V4

*ACCURACY

(SIG.DIG.)

.24568

.96032- . 2 9 6 5 9

.17173

.49156

.25813

.13366- . 5 4 6 6 7

.11211- . 2 6 5 3 1

0054651912061005034355169

1862765605071692290745486

3100E+017400E+003770E+004387E+004000E-03

1000E+004353E+027070E+018520E+021511E+01

* ACCURACY = -ALOG10(MINIMAX ERROR)

- 40 -

TABLE VIII

Threshold Values of x Defining the Number of Terms inThe Asymptotic Series for P^y.(x) to give desired

Accuracy Between 5 and 9 Significant Digits

VACCURACY2*

1) I, = no. of terms in asymptotic sum (Eq. 2.5) required to give desired

accuracy for x. < x < x,,

2) ACCURACY = log]0 |i-SUM/PCY|_|

CYL calculated to relative accuracy of 1 x 10 and Ki^(x) accurate

to M 3 s.d.

e.g . f o r accuracy of 9.00 s .d . and 12.45 < x < 12.85 use 9 terms in series

TABLE IX

Equations of the boundary curves defined by the heavy lines in

Figures 4 and 5, which subdivide the quadrant into different regions.

ContiguousRegion Boundaries

(A-D also)

D.I, 9

C_ 1 , 5

E o s - i

Curve Descriptionand Equation

z = 4.21 + 8 . 2 9 K 2 - 1 2 . 2 K "

- D8,7(Table XVII) f o r z <

in te rsec t ion wi th D 9 , 6 ;

- D9 ,8(Table XVII) fo r z <

in te rsec t ion wi th D_i,9 ;

z = . 1 8 3 3 / ( K 2 - .078) + 9.6364

z < E0,_i and z < F5,_i

z/y = 36.0

Da,7(Table XVII) i f K2 < .5

K2 = .5

z = 5.5801 - 6.68K2 + 6.51924K1*

z/y = 36.0

z= -7.358/(/y- 1) - 2.910 + 3.508/f

- 42 -

TAM.F. X>0 = E x p r e s s i o n s fo r t h e p o l y n o m i a l s P . U . i ) . ippcnr in t i in F-<;. ( 1 . 1 * )

+ 1 . 2 7 3 2 3 9 5 4 4 7 3 5 2 * 2 ( 1 1

+ 2 ( 3 )

* ( - ( I .25*SQMET*ETA * 0 . 1 5 6 2 5 * S Q M E T * E T A * » 2 + 6 . 8 3 5 9 3 7 5 E - 2 *SQMFT*ETA»*3 * 1 . 8 4 5 2 1 4B4 1 7 5 E - ? *SQMET*ETA* *4

+ 2 . 4 6 7 3 4 6 1 9 1 4 P 6 2 E - 2*SQMET»F.TA** 5 + 1 . 7 1 83 3d 3 8 3 3 0 0 8 E - 2*SOMET*ETA**6 + 1 . 26 57 344 14 1 2 7 8 E - 2 • S O « E T > E T A * * 7

+ 9 . 7 1 2 7 5 3 8 1 7 4 ? 9 1 E - 3 * S O M E T * E T A * * 8 + 7 . 6 8 9 2 6 3 4 3 8 8 f l 5 9 E - 3 * S O M E T * E T A • • 9 * 6 . 2 3 8 7 B H 7 4 4 6 6 7 S F - 3 * S 0 « F T * E T A " 1»

- 0 . 3 3 9 5 3 0 5 4 5 : 6 2 7 * E T A * * 2 - 4 . 8 5 B 4 3 6 3 6 0 B 9 5 9 E - 2 * E T A * « 1 - 1 . 6 1 6 8 1 2 1 2 P 2 9 8 6 E - 2 * E T A ' « 4 - 7 . ? 4 9 1 4 6 P P . 1 3 5 7 4 E - 3 * E T A * * 5

• 3 . 9 5 7 2 3 2 4 6 2 2 6 9 3 E - 3*ETA*»6 • 2 . 374 3 3 9 4 7 7 3 6 1 6E- <*ETA**7 - 1 . 5 16 Ï 1 7 3 B 8 8 8 I E - 3 *FTA" • 8 - 1 . Pr> 1 1 7B 1 5B7P.B 1 E- 3 * E T A « * 9

- 7 . 5 0 8 4 1 5 4 1 9 3 4 3 4 E - 4 * E T A * * 1 B )

+ Z ( 5 )

* ( 1 . P 4 1 6 6 6 6 6 6 6 6 6 7 E - 2 * S O M E T * E T A « * 2 • 1 . 8 8 8 0 2 P . 8 31 3 3 13E- 2 «PQMET'FTA* • ? . 8 . P I 5 9 5 P 5 2 P B 3 3 3 E - 3 * S Q M E T » E T A * * 4

- 4 . 2 7 5 5 I 2 6 9 5 H 2 5 E - 1*SQMET*ETA**5 • 2 . 6 P 5 5 6 5 388 9 9 7 4 E - 3 >SOMFT*ETA* " 6 • 1 . 7 ' ! 2 6 6 ! 7 1 9 5 4 9 F - 1«S0MET*ETA * * ''

- I . 2 2 6 7 0 U P 8 4 7 2 7 E - 3*SBMET*ETA**8 . 9 . 09 I 34 4 P 9 2 S 6 9 1 E- 4 •POMFT*F.TA " 9 . 6 . 9 8 1 H 1 S 29S 5 •= 2 fiF- 4 * S Q M F 7 ' F T A * * 1 P

+ 2 . 2 6 1 5 3 6 9 6 8 4 1 8 1 E - 2 * E T A * * 3 - 1 . 84 7 - R 5 2 8 H 34 1 3 F - 3 "ETA • * 4 - 2 . 82 «P.P52B94 I t> IF • 3«FTA« * 5 • 2 . ' 7 54 8 1 5 35 792 3E- 3 *ETA* ' 6

- 1 . B 9 P 9 8 5 2 P . 9 2 4 2 9 E - 1*ETA**7 - 1 . 5 0 9 8 36 3 8 5 6 0 4 3E- 3 «PTA»» 8 - ! . 22 3C44 9 9 4 9 1 11 F - 1 *E1V.**9

- 1 . 0 f l 6 7 P 1 0 6 4 2 7 3 6 E - 3 « F T A * * 1 0 )

+ Z ( 7 )

* ( - 2 . 1 7<!1 3 8 8 8 8 8 S 8 9 I T - 4 * S Q M E T * E T A * * 3 + S . 544 92 1 fl 75F.- 4 •PO^ET*ETA • *4 • 2 . f> I f 7 6 9 i 1 i.'Mf; ! K- i *."0MFT*ETA* • 5

+ 3 . 4 6 2 8 9 7 4 0 6 6 8 4 E - 5*5QMET*ETA**6 - 1 . 6 ? 4 4 7 2 8 9 2 3 9 7 9 E - 5*PQMET*F.T^ • * 7 • 3 . T ! 76 H Q 9 ? 9 r i P ( i r E r- *:"0MFT *FT^ * * 8

- 3 . 5 8 P . 7 3 7 3 1 1 7 , 4 F - 5 * S Q M E T * E T A * * 9 - 1 . 4 9 7 1 36 1 2 2 9 6 1 E • 5*P0MFT«FTA* * ! 0 - 6 . f>72 r; 5 79 r j6 7BB! 4 - F T - « • 4

+ 4 . 1 8 3 9 5 8 2 5 2 6 2 4 6 E - 4 * E T A » * 5 t 3 . ] 3 5 7 6 6 1 4 8 2 P 9 3E • 4 *FTA • * 6 » : : . 1 r > 7 P 2 6 ? 5 6 4 8 1 ' ) F 4 * F T « * * ~ * 1 . ': Z 4 2 9 ! 4 346 64 ?E- 4 *ETA* * 8

t 1 . 11 71 3 6 6 5 2 7 9 7 8 E - 4 « E T A « * 9 * 8 . 4 6 2 6 0 8 6 371 18 5E • 5 * F T A « " 1 il i

+ 2 ( 9 )

* ( 2 . 7 1 2 6 7 3 6 1 111 1 1 E - 6 «SQMET«ETA* «4 - 1 . 9 P 5 6 5 i ? l 1 8 0 5 6 E - 5 'SOMET'ETA * • 5 + 2 . 2 2 9 4 7 0 6 2 4 1 3 I IE • fi*S0MET*FTA"6

t 4 . 9 3 6 2 1 P 6 9 2 8 7 5 E - 6 * S O H E T * E T A * * 7 + 4 . 6 1 6 7 2 8 6 1 4 2 4 7 E - 6 » S 0 M E T * E T A * * 8 + 1 . 8 4 8 6 7 2 5 5 3 7 3 9 2 F - 6*SCMET*ETA**9

* 3 . 1 3 4 4 1 6 1 3 4 1 8 B 5 E - 6 * S O M E T * E T A * * 1 0 + 1 . C88 76 2 37(15 7 1 '.F - S 'UTA" « 5 • 1 . 6 ; 7 1 r. 1 M r:l>? 9 1 ! " - ^ * FTA* «6

- B . 361 7 P 8 3 9 2 7 5 4 I E - 6 * E T A * * 7 - 4 . P 2 7 2 1 1 5 3 2 7 7 6 E - 6 « E T A « * 8 - 1 . B84 1 7 7 7 P 3 9 2 7 I E • 6*FTA*« 9 - 7 . 945(182 ? 1 5 2 4 6 E - ^«ETA« * 1 P I

+ Z ( l 1 )

* ( • 2 . 26P .561 1 4 2 5 9 2 6 E - 8*SQMET*ETA** 5 * 2 . 5 P 6 8 P ! P 6 P 2 6 7 9 F . - 7*S0MET"F.TA * • 6 • I . 5H4 68 [ 6 I 4PPr-~<-' 7*S0MFT*ETA** 7 :

- 1 . 4 2 D l ( 1 1 4 9 8 6 2 D 4 E - 7 * S Q M E T * E T A * * f l - 9 . 6 2 7 7 8 3 5 24 ! i l 1 E - 8 "PQMET'FTA* «9 - 6 . 1 99 19 >27 ! 9 0 F - «•.-•0>!F:T*FT/' • • 1 P

- 1 . 12 3 9 3 B 5 1 9B7 1 4 E - 7 * E T A * * 6 + 3 . P 2 P . 3 8 8 9 6 11 6 22E - 7 *ET7>** 7 - 6 . 8 4 ? 9fl ? 8 5 5 5 8 M F - 8"FTA* * R • î . 6 6 Q 2 ' 8 7 6 1 5 3 "^6E- B*ETA * * 9

- 4 . 2 7 6 6 7 6 6 2 5 9 4 9 9 E - 8 * E T A » * 1 B I

* Z I 1 3 )

* ( 1 . 3 4 5 5 7 2 J 2 7 7 3 3 7 E - ] f l * S C > « | - T * E T A * * 6 -• 2 , ] 7 1 5 37 " 2 2 8 8 2 7 E - 9 * S O M F T « F T A * * 7 < ; . 8 4 f 9 ? l « 2 J t l ' 1 5 P 2 F - 9*R0MET*FTA* " 8

+ 1 . 5 4 7 1 8 6 4 2 2 0 2 5 4 E - 9 * S Q M E T * E T A * « 9 ' 5 . 45 19 79 1 221 4 79E- 1 ("*SOMFT*ETA** 1 P •• ~ . '194 0 7 ' 8 ' .""1 1 : "E- 1 P 'ETA • • 7

- 3 . 364 1 9 0 4 11 7781 E- 9 * E T A * * 8 + 4 . B 9 9 9 6 B 1 81 4 9 5 2 E - 1 fl*ETA* *9 + 1 . 2 ?P ' 7 8 9 6 ' 11 8 2'.'- 9 *ET,% * * ! (' ]

* 2 ( 1 5 1

* ( - 6 . P . P 7 B 1 8 8 7 3 8 1 1 1 E - 1 3 * S O M E T * E T A * * 7 + I . 3 ! ? 1 8 1 5 8 1 4 9 4 I E • I 1 « P O M F T ' E T A * « 8 2 . 8 6 9 9 . ) i 9 1 2 2 P 7 F - 1 1 " ? 0 M E T * E T A " " 9

• 6 . 1 6 ( 1 2 B O 3 4 3 7 9 6 7 E - 1 2 * S O H E T * E T A * * 1 I > - 4 . 1 5 6 H 2 1 9 6 5 1 3 5 5 F - 1 2 * E T A * * 8 + 2 . 5 ! 7 6 5.) P 4 2 6 5 6 7 F - 1 1 « E T A * • 9

- 1 . 5 3 8 1 1 6 2 7 3 8 8 1 9 E - 1 1 * E T A * * 1 P )

* Z ( 1 7 )

* ( 2 . P . 8 5 7 7 P 4 4 2 2 9 5 5 E - 1 5 * S Ç M E T * E T A * * B - 6 . f ! 9 2 1 8 7 8 1 3 5 " 0 2 E - - 1 4 » 5 0 M E T » F T A « * 9 • 1 . 9 . ' 6 7 : ? 6 6 ' t 6 1 7 9 F - 1 1 * S O M F T « F T A * * 1 0

+ 1 . 6 4 7 B 7 I 4 0 P . P . 0 4 8 E - 1 4 * E T A * * 9 - 1 . 3 5 7 5 3 7 8 1 P . S 7 6 4 F . - 1 " E T A " * 1 0 I

> 7. I 19)

* ( ' , . 7<iI«(ifi7H4|54.->,r. 18»S<J«ET*ETA**9 * 2 . 1 5 7271 ?B5 ! »9) E- ! 6 *SONET*KT» * * ] P - '••. M . J 1 6 P S IS ' -1 T K - 1 7>F7A • ' IP I

* 7 121 I

* ( 1 . 11 6 7 M P 6 9 I ?6E-_ 2P*r.QMET*ETA** 1 0 )

- 43 -

TAU].!- .\ U-orU'ti)

+ 1 . 2 7 3 2 3 9 5 4 4 7 3 5 2 * 2 [ f l l

+ 2 ( 2 )

* i 0 . 7 5 « S Q M E T * E T A -> P . 1 5 6 2 5 * S O M E T < E T A * * 2 t 6 . « 1 5 9 37 ri?. • / * S O M F . T * E T A * * 3 » 1 . 3 4 5 2 ] 4 8 4 37 5 F - - 2 * S Q M E T * E T A * * 4

+ 2 . 4 6 7 3 4 6 1 9 1 4 0 6 2 E - 2 » S Q M E T * E T A « * 5 + 1 . 7 1 3 3 3 P 1 8 < J 0 H 8 F - 2 * S 0 M F T * E T A « - 6 * 1 . 2 6 5 7 3 4 4 3 4 1 2 7 8 E ~ ? * S O M E T * E T A * * 7

+ 9 . 7 1 2 7 5 3 8 1 7 4 1 9 1 E - 3 * S C M Ë 7 * F . T A * * 8 + 7 . 6 8 9 2 6 14 1B8P. 5 0F.- l * S 0 M F . T * F T A * * 9 t 6 . 2 36 7 8 S 7 4 4 6 6 7 5 E - 3 * S 0 M E T * E T A * * l 0

- 0 . 8 4 8 8 2 6 3 6 3 1 5 6 8 * E T A * 0 . ] 6 ° 7 6 5 2 7 2 6 3 1 4 * F T A « • 2 + 7 . 2 7 5 6 5 4 5 4 I 3 4 3 8 E - 2 * F . T A * « 3 ' 4 . A 4 2 0 3 0 2 C 0 7 4 6 5 E - 2 » E T A * * 4

+ 2 . 5 7 2 2 0 1 ] f ! B 4 7 5 1 E - 2 * E T A » * 5 » 1 . 7 8 0 7 5 4 fi fl 8 B 2 1 2E • 2 «ETA • * 6 * 1 . 3 P 5 8 8 f i 7 ] 2 5 4 8 9 E - 2 « F T A > * 7 ^ 9 . 9 8 6 1 9 2 5 0 7 7 2 6 8 E - 3 * E T A * * 8

+ 7 . 8 8 3 8 3 6 1 9 0 3 1 0 6 E - 3 * E T A « * 9 + 6 . 3 8 2 ] 5 3 1 S 6 4 4 1 9 F - • * F T A « » 1 0 )

+ 2 ( 4 )

* ( - 9 . 3 7 5 E - 2 * S 0 M E T * E T A * " 2 • 1 . 4 9 7 3 9 5 8 3 3 3 3 3 J E - 2 * S 0 M E T * E T . « * * 3 - 3 . 2 1 4 5 ! S ? ? 9 1 6 6 7 F - 3 * S Q . M E T * F T A " 4 :

- 3 . 2 6 5 3 8 0 8 5 9 3 7 5 E - . 1 * S Q M F T * F T A * * 5 + 5 . 1 6 6 3 7 1 6 6 34 1 1 5 1 ' - 4 * S C M E T * E T A * * 6 * 7 . 5 2 5 8 2 4 16 r i ! 6 1 7 F - 4 * S O M E T * E T A * " 7

+ 7 . 8 4 1 8 2 3 8 8 8 6 7 6 5 E - 4 * S Q M E T * E T A * * 8 » 7 . 4 4 8 1 8 9 9 4 361 4 9E • 4 ' Ï Ï O M F T ' F . T A " * 1 i 6 . S 3 2 . H i 8 9 1 1 1 9 9 E - 4 * S Q M E T * E T A * * I B

+ 5 . 6 5 8 P 4 2 4 2 1 0 4 5 2 E - 2 * E T A * * 2 - 5 . 1 7 3 7 9 8 7 8 4 « 5 5 S E - 2 * E T A * * 3 - 2.7947752365162E- 2 « F T A * * 4 - 1 . 6 9 ! 4 7 B 1 1 1 4 2 3 5 E - 2 * E T A * * 5

-- 1 . 1 2 5 2 1 3 0 6 8 8 1 7 E - 2 * E T A * * 6 - 8 . f P 3P 1 8 ? 7 8 2 3 9 8 F - .' * F T A * * 7 - 5 . 9 7 5 9 5 8 1 9!i9t< 3 9 R - .' * Î T A * * fl - 4 . 6 2 9 ? P 6 1 6 5 6 B 4 6 E - 3 * E T A * * 9

- 3 . 6 9 B 2 8 1 5 7 2 2 P B 5 E - 3 * E T A * « 1 E )

+ Z I 6 )

* I 4 . 1 2 3 2 6 3 8 8 8 8 8 a 9 E - 3 * S O H E T * E T A * * l - 9 . 3 5 8 7 2 3 9 5 8 .' ? 1 3F: • 4 " S Q M E T ' L T A * M • 1 . <<ï 7 0 4 R f . 7 6 2 1 5 3 E - 3 * S O H E T * E T A * » 5

- 8 . 6 8 5 2 1 7 9 6 3 3 2 4 7 E - 4 * S Q H E T * E T A * * 6 - 6 . 7 6 9 f i 1 7 . ' 9 6 r . 7 < ) 9 F - 4 - S C M E T ' F T A ' • 7 - 5 . ? -29 i 'P ' I " j 9 9 B 8 3 E - 4 • S 0 M F T * E T A * * 8

- 4 . 2 7 0 5 2 7 3 9 6 R 3 5 7 E - 4 * S O M F T * E T A * * 9 - 3 . 4R4f i 7 9 1 7 2 5 1 1 6F.- 4 « S Q ^ F Ï ' F T A » • 1 !' • 1 . 6 I 6fl 1 ? 1 2 H 2 9 B 6 F . • 3 ' E T A * * 3

+ 4 . 1 3 1 B 5 3 1 9 6 3 1 8 7 E - 3 * E T A " 4 + 1 . 7 2 3 3 5 5 2 9 51 3 3 IF.- >*F.1 A * * 5 + » . 2 4 7 2 9 3 '" 81 4 7 6 6 1 - • 4 " F TA • - n * 4 . 3 9 1 8 ( 1 0 3 3 6 9 1 3 8 E - 4 * E T A » « 7

+ 2 . 5 1 7 3 7 H 6 6 8 5 1 1 2 E - 4 * F T A * * B • ! . 5 1 7 9 2 1 2 6 2 7 B 6 1 E - 4 « E T A * * 9 ^ 9 . 4 6 9 8 5 C 1 1 S I 5 6 4 F - 5 " E T A « « 1 P i

+ Z < 8 )

* ( - 8 . 9 5 1 8 2 2 9 1 6 6 6 6 7 E - 5 « S O H F T * E T A » * 4 + 8 . 3 6 1 8 I 6 4 B 6 2 5 E - 5 * S 0 H E T * E T A * « 5 + 5 . 9("d 9 1 2 8 1 4 6 7 6 1 E - 5 * S 0 M E T * E T A " * 6

+ 3 . 7 6 5 3 6 7 3 2 6 2 8 2 3 E - 5 * S 0 M E T * E T A * * 7 -*• 2 . 4 5 6 6 6 0 4 2 6 7 9 8 3 E - r, *SQMET * F T A * * 8 ^ 1 . 6 6 P 6 7 1 ? 1 6 7 1 8 F - 5 * S Q M E T * E T A • * 9

+ 1 . ] 6 D 6 9 1 8 2 3 0 2 6 8 E - 5 1 S Q M E T > E T A * M B * 2 . 5 6 6 3 6 8 4 1 4 91 8 4 E - 5 " E T A • • 4 - 1 . 3 2 2 H 6 B 5 9 2 8 3 6 8 E - 4 * E T A * * 5

- 1 . 8 2 4 d 3 a 4 6 4 9 8 3 4 E - 5 * E T A * * 6 + 9 . 3 9 7 5 P 9 3 7 2 3 6 5 1 E - 6 * E T A * * 7 + 1 . 5 5 3 7 6 1 C 2 C 8 3 6 1 E - 5 * E T A * * 8 ^ 1 . 5 7 8 5 1 ' C 3 8 8 4 6 6 E - 5 * E T A * * 9

+ 1 . 4 4 1 8 7 3 4 9 7 7 4 E - 5 * E T A * * 1 B )

+ ZUQ)

* ( 1.1528862847222E-6*SQMET«ETA**5 - 2.265PB173C9441E-6*50MET-ETA«* 6 - 9.741 (11184B8664E-7*SOMET*ETA**7

- 3.B927934479(Î11E-7*SOHEÏ*ETA*«8 - 1 . 6H98869244 IC7E-8«SQMET*ETA**9 ^ 1 . P94 Î84 3P48 1 31E-7*S0MET«ETA»»l lî

- 2.5922913585035E-7«ETA*»5 + 2 . 2605 5B57626 1 4E- 6*ETA** 6 - 6 . 78 1 ? ? 3 ?92 362 8F. - 7-FTA ' * 7 - 9 . 329 1 1 54687888E-7'ETA"8

- 7.7797813777751E-7*ETA*»9 -• 5 . 9836S 1 4545234E-7*ETA»*1 n i

* ZU2)

* ( - 9.8226772624559E-9«SQMET*ETA*»6 • 3 . 26271 2 3B i 3 ! H5E- S *S0MFT*ETA* *7 -i 3 . P59 I ("ifi48l"449E-9*SQKET*ETA* *8 :

- 6.39170612B2867E-9*SOMET*ETA*«9 - 8.2636885793°44E-9*S0MET*ETA*«1B » 1.812791]597926E-9*ETA**f

- 2.4058992418171E-8«ETA>"7 + 2 . 1 64 ] 39C691 4 52E • 8 «FTA * « fl * '. . 6594971 327 3] 9K- B^ETA** 9 * 9 . 472 4932B7 2 ? 1 5E- 9*ETA*« 1 0 )

+ Z(14)

* ( 5.946948685073E-il*KÇ)MET*ETA'"7 - 2 . 97 7666741 ] 2 55F.- 1 (I*SQMFT*ETA ** 8 • 9 . 78 5 78 3 1 11 9874E- 1 1 *SQMET*ETA**9 :

+ 1.4866467424537E-le*S0PET*r:TA"l(l - 9 . 296 364 92 2P 1 ' 5F- 1 2-ETA" * 7 i 1 . 748S 1 I<.1953Î94E- 1 B«ETA*«8

- 2.9069873631677E-lfl*ETA*-9 - 1 . 2258 93 70 1 366 7E-1 C-ETA* * 1 0 )

+ Z(16)

* ( - 2.69H6438705612E-13'SOMET«ETA**8 + 1.8894472994145E-12«SQMFT*ETA«»9 - I . 'S 517 I75J6855E- 12•50MET*ETA " 1fl

- 3.6456333027504E- 14*ETA**8 - 9 . 2393666 lf43PE-l ,1*ETA"9 * 2 . 39229 1783 I 39E- 1 2*FTA** 1 P I

+ Z(18)

* ( 9.4439B50581714E-16*SOMET*I:TA««9 - fl.84726 !469CB77E- 15*30MET«ETA** 1P - 1 . ]28679P4IP992E- ] 6«FTA**9

+ 3.709931234B892F-1 5*ETA«*1(1 I

+ ZI20)

* ( - 2.6467162809432E- 1 8»SQMET*ETA«*10 • 2 .8287695265644E--1 9>ETA •• 1 0 )

- 44 -

TABLE X (cont 'd)

• 2.«SQMET

+ 2 ( 1 )

* ( - 1 . 2 7 3 2 3 9 5 4 4 7 3 5 2 + 0 .8488263631568«ETA + 0 .1697652726314*ETA** 2 + 7 .2756545413438E-2*ETA**3

+ 4 .O420303007465E-2«ETA»M + 2 .5722011004751E-2*ETA*«5 + 1 .7807546P80212E-2«ETA**6 + 1 .3P58867125489E-2«ETA**7

• 9 .9B61925e77268E-3*ETA**8 * 7 .8B383619031P6E-3*ETA**9 • 6 .38215310644]9E-3*ETA**1P. )

+ 2 ( 3 )

* ( - 9.375E-2*SQMET«ETA**2 - 1 . 4973958 33 3333E-2»SQMET*ETA** 3 - 3 . 21 45 1 82 291 667E--3*SOMET«ETA*M :

- 3.265380859375E-4*SQMET*ETA**5 + 5 . 166371 663411 5E»4*SQMET*ETA"6 • 7 . 525824 365 1 6 1 7E-4 •SQMET«ETA**7

+ 7 .8J!8238886765E-4*SQMET*ETA*"8 + 7.448]B99436]49E-1*SQMET*ETA**9 + 6.B323318911199E-4«SOMET*ETA**10

+ 5 .6588424210452E-2*ETA**2 - 5 .17379878495->6E-2*ETA**3 - 2 .7947752365162E-2«ETA**4 - 1 . 6 9 1 4 7 0 1 ] ] 4 2 3 5 E - 2 * E T A * * 5

•• 1 .125213068817E-2*ETA**6 - 8 .0030183782398E-3*ETA**7 - 5 . 97595B1 989P39E-3*rTA**8 -• 4 .6293P.61 656046E-3*ETA**9

- 3.690281 5722PO5E-3'ETA**1P )

' Z(5)

* ( 4.1232638888889E-3«SOMET*ETA**3 - 9.3587239583333E-4*SQMET*ETA**4 - 1.0779486762]53E-3•SQMET*ETA*»5

- 8.6852179633247E-4*SÇMET*ETA**6 -- 6 . 76961 739f)5799F.-4*SQMET*ETA**7 - 5 . ?29083P599883E-4*SOMET*ETA»*8

- 4.27fl527396B357E-4*SQMET*ETA**9 - 3 . 4846791 72591 6t:-4*SOMET«ETA**l (I • 1 .61 681 212H2986E-3*ETA"3

» 4.1318531963!87E-3*ETA*M + 1.7233552951331E-3*ETA**5 * 8 . 247J93.1814766E-4'ETA««6 * 4 . 39] 8P.P.3369 1 38E-4*ErA«* 7

+ 2.51737B6685112E- 4*ETA**8 + 1 . 51 792 1 2627B8 1 E-4*ETA**9 <• 9 . 46985(111 8 1 564E- 5»ETA** 1 0 )

+ Z(7)

* ( - 8.951822916fi667E-5«SQMET*ETA**4 * 8.36lfl1«406?5E-5*S0"ET>ETA" 5 * 5.9P09]2B146701E-5*SOHET*ETA«*6

+ 3.765367.1262823E-5*SQ«FT*ETA**7 + 2. 4566604267983E-5"SOHET*ETA*«8 * 1. 66F67] 31 6718E--5*S0MET*ETA«*9

+ 1.1606918238268E-5*SQMET*ETA**m + 2.5663684449184E-5*FTA««4 - 1.322PS85928366E-4*ETA**5

- 1.82403H4649B34E-5«ETA**6 + 9 . 397509 J 72365 IE-6*ETA*« 7 -• 1 . 55 376 1 0208?61E-5'ETA** 8 * 1 . 578 501" 3884B6E-5*ETA**9

+ 1.44187349774E- 5'ETA»*!? )

+ Z(9)

* ( 1.1528862847?22E-6*S(3MET*ETA**5 - 2 . 26500] 7 3<!944lE-6*SOMET*ETA**6 - 9 . 741 011 84H8664E- 7*SQMET*ET/i**7

- 3.B927934479H1]E-7*SQMET«ETA*«8 - 1.6098869244107E-8*S0HET*ETA**9 * 1.0945843048131E-7*EQMET*ETA*''1P

- 2.5922913585035E-7*ETA**5 + 2.2605505762614E-6*ETA*«6 - 6.781233392362BE-7*ETA** 7 - 9.32911546B7888E-7«ETA«'8

- 7.7797813777751E-7*ETA**9 - 5.9836814545234E-7*ETA»•)0 )

+ Z(ll)

* ( - 9.8226772624559E-9*SQHET*ETA««6 * 3.2627]23013105E-8*SOWET*ETA**7 + 3.0591 (156480449E-9*S0MET*ETA*»8:

- 6.3917061202B67E-9*SOMET«ETA»*9 - 8.2636885791944E- 9*S0MET*ETA*M0 * 1.812791 : 597926E-9«ETA*«6

- 2.4058992418171E-8*ETA««7 + 2. I 641 !9fl6<U452E- S»ETA»*8 * 1 . 6594171 3273Î9E- 8*ETA**9 » 9 . 4724 93207231 5E-9*ETA**!'0 )

+ Z ( 1 3 )

* ( 5.94694B685073E-1]•E0MFT»ETA»*7 - 2.97766674]1255E-10»SOMFT*ETA*«8 + 9.7857831119874E-11*SQMET*ETA*•9:

+ 1 .4866467424537C-10*S3HET»ETA**10 - 9 . 296 3649220 ] 35E-1 2«ETA* • 7 t 1 . 7488 1 •12953294E- ] G*ETA"8

- 2.9069873631677E-!0»ETA««9 - 1.225898701366 7E-1fl*ETA«*10 )

+ Z(15)

* ( - 2.6906438705612E-13*SQMET*ETA**8 + 1.8894472994145E-12*PQMET*ETA**9 • ].59c1717546B55E-12*S0MET»ETA**1P

• 3.6456333027504E-14*ETA»*8 - 9.239366630438E-13»ETA**9 * 2.3922937B1139E-12*ETA*«]0 I

+ 2(17)

* ( 9.44390505B1714E-16*SQMET*ETA»«9 - 8.847261469B877E-15«S0MFT*ETA*»]p - l.]2867904]0992E-16*ETA«*9

+ 3.7099312340892E-15*ETA**]0 )

+ 2(19)

* ( - 2.6467162809432E-1B*SOMET*ETA**1<> * 2 . 828 769526 564 4E-] 9»ETA »*11» ) tl>.

NOTE: 7.{\.\ = ?.**(I.-t )ETA » ", _SIJMET = >'—,

- 45 -

TABLE XI

Equations of the curves Am in Fig. 4 and Fig. 5 for use inm » ri

Eq. (3.1)

Curve Equation

A z = .0065454/ (K 2 - .285) + 4.8181811 0 , 1 1

A z = . 0 3 3 3 3 3 / ( K 2 - .29) + 4.333331 1 , 1 2

T A B L E X I I

R a t i o n a l M i n i m a x A p p r o x i m a t i o n s t o g £ ( z ) a n d h £ ( z ) i n E q . ( 3 . 1 ) f o r 8 = 1 1 , 1 2

U0UlU2U3

5.54 - 6.46

.33248 19800 0000E-01

.43037 26000 0000E-02

.13945 78870 0000E+02

.55922 67828 0236E+01

8 1 2 ( z )

5.56 - 6 .0

.29568 03000 0000E-02

.35036 57000 0000E-03

.55856 44142 3291E+01

5.54 - 6.46

.46077 34403 0000E-01

.23280 76668 6000E-01

.37245 26778 9000E-02

.19184 86643 8000E-03

h12(z)

5.56 6.0

.86500 43800 0000E-02

.38150 50820 0000E-02

.37155 27390 0000E-03

ACCURACY(SIG.DIG.)

2.80 2.62 4.11 3.31

* ACCURACY = -ALOG10(MINIMAX RELATIVE ERROR)

TABLE XIII

Rational Minimax Approximations for Q. and R. in Eq. (3.15)

Q (K)0

0.0 - 0.707106781186

U0UlU2U3U4U5

V0VIV2V3

ACCURACY(SIG.DIG. )

U0UlU2U3U4U5

V0VIV2

*ACCURACY(SIG.DIG.)

-.30780.56370.20834.43084

.36261

.78798

.52334

-.11806,43339

-.79917,82085

-.43520.98359

,13219.65684,12360

00333 5050E+0189496 1260E+0046699 3519E+0162860 2140E+00

83710 3854E+0190601 6027E+0187556 8108E+01

0.707106781186

49225 5460E-0550802 5570E-0532706 1710E-0b90365 3000E-0537547 5900E-0513614 6000E-06

28414 7881E+0066861 6439E+0060261 3194E+01

Q (K)1

0-0 - 0.707106781186

.14164 50475 8827E+00

.17897 42533 4204E+00

.21211 44170 5500E-01

.16118 57523 9000E-01

.16687 16430 8248E+01

.44445 39100 1617E+01

.37976 98260 1539E+01

Q (K)5

0.0 - 0.707106781186

. 12121 63398 9500E-07

.50035 07193 7700E-07

. 10817 80844 9330E-06

.12902 15454 7000E-06

.79364 50563 1200E-07

.20680 95946 7100E-07

.58812 29579 9420E-01

.35781 53417 7503E+00

.86074 59034 3422E+00

0.0 - 0.707106781186

.34360 54298 7670E-01

.70634 99455 2240E-01

.54616 39306 3510E-01

.18341 21031 2370E-01

.56672 08382 0431E+01

.19507 05265 6796E+02

.23688 19828 8328E+02

.11128 93477 5143E+02

Q (K)6

0.0 - 0.69

.10747 47075 5300E-09

.48580 81044 7500E-09

.11907 73821 0780E-08

.15895 07276 0250E-08

. 10898 51063 2900E-08

.31483 65198 8000E-09

.29864 59349 9360E-01

.21697 68818 5810E+00

.64032 00287 6808E+00

Q («)3

0.0 - 0.707106781186

.86804 96860 6630E-04

.26368 66128 6822E-03

.37713 38719 0148E-03

.29741 65933 4010E-03

.11609 13960 7230E-03

.18942 28035 7000E-04

.30924 84020 3259E+00

.12465 55402 2212E+01

.18048 88920 7880E+01

Q (K)7

3.0 - 0.66

.20823 52534 0000E-12

.49013 94841 00B0E-12

.91678 03730 0000E-12

.66474 50960 0000E-12

.42334 20379 6128E-02

.47590 75516 0266E-01

.19977 59374 5477E+00

A' rij|,Ary -ALOG10(MINIMAX RELATIVE ERROR)

TABLE X I I I(CONTINUED)

R ( K )

RANGE 0 .0 - 0 . 7 0 7 1 0 6 7 8 1 1 8 6

U0 - . 6 6 9 7 4 18439 2314E+01Ul . 47220 46779 7420E+01U2 . 9 6 2 1 2 61250 0060E+01U3 - . 9 2 8 5 5 46576 2760E+01U4 . 1 6 3 5 5 17913 5040E+01

V0 . 3 1 4 0 7 13027 7046E+01VI . 3 2 2 1 9 95667 0980E+01V2 - . 3 3 5 6 1 30905 5810E+01V3 - . 2 2 5 6 0 15422 9610E+01V4

R ( < )

0.0 - 0.707106781186

.13416 88052 5320E+01

.31432 14715 8766E+01

.24724 57804 4850E+01

.72408 79145 4130E+00

.53104 74207 7300E-01

.65525 90331 2904E+01

.61140 98940 5578E+01

.43919 93671 7660E+01

.35157 71755 3893E+01

0.0 - 0.707106781186

.18798 49010 7814E+00

.39223 24944 2718E+00

.28916 17953 2575E+00

.84901 45199 6130E-01

.10823 44940 1160E+02

.30986 77893 1341E+02

.28498 29360 0303E+02

.64936 89652 1387E+01

.25148 97110 1491E+01

0.0 - 0.707106781186

.15896 49837 4788E-02

.39714 32374 0040E-02

.44562 58979 6750E-02

.24799 32862 4410E-02

.40512 38044 6100E-03

.17317 84258 2332E+01

.70829 20893 4274E+01

.11113 95154 6674E+02

.82502 52181 4108E+01

.28610 72668 2414E+01

ACCURACY(SIG.DIG.)

U0UlU2U3U4

V0VIV2V3V4

R (K)4

0.0 - 0.707106781186

.27637 55067 1560E-05

.67776 06393 5300E-05

.87850 16748 6600E-05

.47704 94722 4300E-05

.85570 73949 5473E-01

.49227 84294 3897E+00

.12060 62167 8681E+01

.16632 24774 7201E+01

.13460 53777 0380E+01

R (*. )5

0.0 - 0.707106781186

.21583 90744 6900E-07

.64032 42909 7800E-07

.10894 57262 6750E-06

.85323 43245 6900E-07

.18942 72901 0000E-07

.26752 11460 7976E-01

.18300 81354 3377E+00

.54338 69433 1260E+00

.87572 51763 7214E+00

0.0 - 0.69

.22742 34773 6300E-09

.64188 88496 8400E-09

.12283 31995 9200E-08

.81023 13867 2000E-09

.15134 81209 8793E-01

.12985 22835 5216E+00

.50226 14284 2159E+00

.11072 40389 9646E+01

.17934 46171 2576E+01

R (<!7

0.0 - 0.66

.10379 29671 0000E-11

.24709 50667 0000E-11

.44733 83889 0000E-11

.31796 00517 0000E-11

.47881 83590 5315E-02

.51870 21296 4481E-01

.21199 36103 7845E+00

ACCURACY(SIG. DIG. )

6.16 5. 33 5.08 3.06

ACCURACY = -ALOG10(MINIMAX RELATIVE ERROR)

TABLE X I I I(CONTINUED)

U0UlU2U3U4

V0VIV2V3

0.0 - 0 . 6 2

,32888 27200 0000E-14

.12922 47652 7902E-02,24096 93818 4644E-01.42830 32326 2377E-01

R (K)—1

0.0 - 0.707106781186

.15743 40048

.30309 35311

.14718 19743

.97772 13849

.11142 50379

2127E+030542E+034258E+035100E+010770E+02

,39358 50135 8633E+02,13949 21985 0581E+02,37593 11174 5460E+02.16079 19005 1590E+02

ACCURACY(SIG.DIG.)

1.63 8.41

TABLE XIII(CONTINUED)

U0UlU2U3U4

V0VIV2V3

0.0 - 0.62

15044 65530 0000E-14.36636 50054 0000E-14.78511 11102 0000E-14.60726 57430 0000E-14

27890 39147 9852E-02,35696 44993 4384E-01.16807 58616 2404E+00

Q (K)-1

0.0 - 0.707106781186

,80132 06735 2530E+02.10787 60073 7832E+03.31665 41295 4000E+01.38111 56411 1020E+02.65910 70790 6400E+01

.31467 78951 2250E+02

.17648 10951 6660E+02

.25592 20148 5640E+02

.13090 25219 4020E+02

ACCURACY(SIG.DIG. )

2.96 8.10

- 51 -

TABLE XIV

Equations of the curves B in F ig , 4, fo r use in Eq.(3.15)

Curve Equation

Bi.o z = .20 + . 2 7 3 3 2 K 2 - . 1 3 3 3 3 4 K "

B 2 , I z = .55 + 1 .3064K 2 - 1 . 0 2 1 4 1 K "

B3 ,2 z = 1.0 + 2 . 1 9 7 2 K 2 - 1 . 7 6 8 7 1 K "

BI. ,3 z = 1.5 + 3 . 0 9 2 3 K 2 - 2 . 7 5 2 2 9 K "

Bs,., z = 2.0 + 3 . 8 3 9 9 K 2 - 3 . 3 5 5 3 K "

B6,5 z = 2.55 + 4 . 8 0 6 2 K 2 - 4 . 7 6 5 7 K "

B7 s 6 z = 3.08 + 6 . 4 4 6 0 K 2 - 8 . 7 0 9 2 2 K "

B8,7 z = 3.62 + 7 . 8 5 4 2 K 2 - 1 2 . 0 5 7 2 K "

TABLE XV

Rational Minimax Approximations for S. and T- in Eq. (3.16)

U001U2U3U405

V0VIV2V3

0.0 - 0.414213562373

.89793

.11609

.19228

.25264

.10804

.16704

119615364362857379289463582282

3500E+009100E+000260E+016000E+007000E+010000E+00

.10767 50405 6210E+01

s </7)1

0.0 - 0.414213562373

-.23722 19481 2000E-01-.11931 00197 0000E-01-.13371 12790 0000E+00.19698 10700 0000E-02

-.17848 89931 0003E+00

.17442 95021 7150E+01

.87710 72396 1000E+00

S (/7)2

0.0 - 0.414213562373

.12370 55678 8000E-03

.17316 56975 0000E-03

.27075 16188 0000E-03

.26331 76227 3714E+00

.36868 05439 0460E+00

.59378 24363 7410E+00

.16235 91313 6140E+01

S (/7)3

0.0 - 0.414213562373

.12776 15142 0000E-04

.13631 00800 0000E-04

.53748 29220 0000E-04

.54311 28760 0000E-04

.92434 46800 0000E-04

-.93765 95532 9584E+00

ACCURACY(SIG.DIG.)

U001U2U3U4U5

V0VIV2V3V4

S (/ï)4

0.0- 0.414213562373

.25935 28884 0000E-06

.30431 37550 000JE-06

.10559 21428 0000E-05

.10867 97830 0000E-05

.13402 25870 0000E-05

-.85102 11959 2699E+00

0.0 - 0.414213562373

.12016 82200 0000E-08

.22730 06734 7397E+00

.13144 97629 5315E-01

.11243 68386 3736E+01

S (-Y)-1

0.0 - 0.414213562373

.12666

.22845

.24857

.56827

.20697

.13690

000007954064186670508046890410

0000E-110000E-078200E-039729E-014398E+010542E+02

.83179 41482 6000E-05

. 27266 03799 J009E-02

.14241 11688 8828E+00

.16281 36181 0842E+01

.31571 37070 8841E+01

ACCURACY(SIG.DIG. )

5.06 3.20 6.82

* ACCURACY** ACCURACY

-ALOG10(MINIMAX RELATIVE ERROR)-ALOG10(MINIMAX ABSOLUTE ERROR)

TABLE XV(CONTINUED)

uaUlU2

V0VIV2V3

T ( > • )

0.707106781186 - 1.0

.17517 41609 9000E+02

.20783 29030 5900E+02

.47877 14197 2700E+02

.33525 68142 800BE+02

T (<)1

0.707106781186 - 1.0

.38354 49162 4100E-01

.18957 93809 8500E-01

.44193 04218 0250E+00

.16071 66774 3045E+01

.17496 18511 5548E+01

T (<)2

0.707106781186 - 1.0

.45672 18845 Z100E-03

.46798 64023 2900E-03

.23838 29227 7800E-03

.40789 98510 6133E-01

.54679 16828 5271E+00

.98093 91810 6611E-01

T (<)3

0.707106781186 - 1.0

.46012 65210 0000E-05

.62377 00572 2623E+00

.31303 57842 6379E+01

.65359 47141 4575E+01

.37671 57824 9320E+01

6.82 1.1 7.29 5.29

U0UlU2U3

0.707106781186 - 1.0

.75790 90940 0000E-07

.43610 47000 0000E-09

.59461 53330 3707E-0]

.58014 72480 0141E-0]

0.707106781186 - 1.0

.12501 75700 0000E-08

.38304 89659 0931E-01

.39106 78557 8933E-01

0.0 - 0.414213562373

.16756 53858 0590E+02

.13975 37603 7180E+02

.17905 38875 6006E+01

.45526 52779 3000E+01

.83782 69156 6400E+01

. 12402 88681 1000E+00

4.59 4.42 7.80

- 54 -

TABLE XVI

Equations of the curves C in Fig. 4, for use in Eq. (3.16)m j fi

Curve Equation

Ci.o z = 1.892 - 4.9754K2 + 3.9650K"

C2,I z = 3.4391 - 8.24781K 2 + 6.69611K1*

C3,2 z = 4.6303 - 9.9724K2 + 8.2120K"

U,3 z = 5.6776 - 11.1567K2 + 9.2954K"

Cs,* z = 4.9197 - 6.8699K2 + 6.50998K"

- 55 -

TABLE XVII

Equations of the curves Dmjn i n Figs. 4 and 5, fo r use i n Eq. (3.17)

Curve Equation

D z = . 0 8 9 8 2 6 / ( K 2 - .8) + 42.2295 - 4 2 . 6 5 5 K 2

D.,,3 z = .52086/(K 2 - .5} + 8.2057 - 7.9405K 2

Ds.i, z = .27933/(K 2 - .35) + 10.4571 - 10.7462K 2

D6,5 2 = .16109/(K 2 - .26) + 10.8867 - 11.5360K 2

D 7,G z = .29945/(K 2 - .17) +8.9529 - 9.4990K

**De,7 z = , 4 8 2 3 / ( K 2 - .123) + 5.20801 - 5.38447K2 - 1.5799K14

D9,s z = . 2255 / (K 2 - .095) + 7.453

* second arm of hyperbola exists in region D

** second arm of hyperbola exists in region B

- 56 -

TABLE XVIIIIdentification of ̂ "(v) in Eq. (3.17J

CRLI1(0)=1 +EXPMV*(l.-V)

1 +E1V*(V**2)

CRLI1(1)=1 +EXPMV*(5.E-1-5.E-1*V)

1 +E1V*(1.+5.E-1*V**2)

C R L I 1 ( 2 ) =1 + E X P M V * ( 0 . 3 7 5 - 0 . 3 7 5 * V + 0 . 7 5 * V * * ( - 2 ) + 0 . 7 5 * V * * ( - 1 ) )

1 + E 1 V * ( 5 . E - 1 + 0 . 3 7 5 * V * * 2 )

C R L I 1 ( 3 ) =1 + E X P M V * ( 0 . 3 1 2 5 - 0 . 3 1 2 5 * V + 3 . 7 5 * V * * ( ~ 4 ) + 3 . 7 5 * V * * ( - 2 ) + 2 . 2 5 * V * * ( - 2 ) + V

1 * * ( - l ) )

1 + E 1 V * ( 0 . 3 7 5 + 0 . 3 1 2 5 * V * * 2 )

C R L I 1 ( 4 ) =1 +EXPMV* ( 0 . 2 7 3 4 3 7 5 - 0 . 2 7 3 4 3 7 5*V + 6 . 5 6 2 5 E 1 * V * * ( - 6 ) + 6 . 5 6 2 5 E 1 * V * * ( - 5 ) +1 3 . 4 6 8 7 5 E 1 * V * * ( - 4 ) + l . 2 8 1 2 5 E 1 * V * * ( - 3 ) + 3 . 9 5 3 1 2 5 * V * * ( - 2 ) + 1 . 1 4 0 6 2 5 * V *1 * ( - U )

1 +E1V*(0.3125+0.2734375*V**2)

CRLI1(5)=2 +EXPMV* (0.246 0 93 75-0.246P93 7 5*V + 2.48062 5E3*V**(-8)+2.48B62 5E3*V*1 *(-7)+l.273125E3*V**(-6)+4.4625E2*V**(-5)+l.21172875E2*V**(-4)+2

*V**(-3J+5.75*V**(-2)+l.234375*V**(-1))

1 +E1V*(0.27 3 4 375+0.2460 9 37 5*V**2)

CRLI1(6)=1 +EXPMV*(0.225 58 59375-0.22 558 593 7 5*V+1.6 37 212 5E5*V**(~10)+1.637211 2 5E5*V**(-9)+8.310 093 75E4*V**(-8)+2.85271875E4*V**(-7)+7.46648431 7 5E3*V**(-6)+1.595671875E3*V**(-5)+2.92546875E2*V**(-4)+4.8093751 E1*V**(-3)+7.5996093 75*V**(~2)+l.302734375*V**(-]))

1 +EIV*(0.24609375+0.2255859375*V**2)

CRLI1(7)=+EXPMV*(0.2094 7265625-0.2C94 7 26 56 2 5*V+1.67 2 29 56 25E7*V**(-12)+1.672 2956 2 5E7*V**(-11)+8.4433 387 5E6*V**(-10)+2 869020.*V**(-9)+7.38 65 0 39062 5E5*V**(-8)+l.53931640625E5*V**(-7)+2.71228125E4*V**(-6)+4.175 76 562 5E3*V**(-5)+5.784 9 02 34 375E2*V**(-4)+7.4521484 375E1*V**(-3)+9.4824 21875*V**(-2)+l.35546875*V**(-1))

- 57 -

TABLE XVIII (CONTINUED)

CRLI1(8)=1 +EXPMV*(0.196380615234375-0.196380615234375*V+2.4457323515625E9*IV** (~14)+2.445 7323515625E9*V**(-13)+1. 23.1 22765390625E9*V** (-12) +1 4.1598353671875E8*V**(-11)+1.061476491796875E8*V**(-10(+2.183 6071 808 593 7 5E7*V**(-9)+•3.77671833984375E6*V**(-8)+5.659 5111328125E5*1 V**(-7)+7.5234 89 50195 3125E4*V**(-6)+9.0575 78613 28125E3*V**(-5)+l1 . 008822509765625E3*V**(~4)+1.068679199218 75E2*V**(-3)+1.138775631 4 765625El*V**(-2)+l.39776611328125*V**(-1))

1 +E1V*( 0.2094 7265625+0.196 38 06.152 343 75*V**2)

CRLI1(9)=1 +EXPMV*( 0.18 54 70 58.10 54 688-0.1854 70 581054 688 *V+4. 8 50 7024 9726 562 5E1 11*V**(-16)+4.8 50 7024 972656 25E11*V**(-15)+2.437579910390625E11*V1 **(-14)+8.2067907796875E10*V**(-13)+2.08289646017578IE 10*V**(-121 )+4.252 33421894 5312E9*V**(-ll)+7.278 481546875E8*V**(-10)+1.075 301 9962890625E8*V**(-9)+1.401649332275391E7*V**(-8)+1.64082043212891 06E6*V**(-7)+1.751983374023437E5*V**(~6)+1.7321748046875E4*V**(-1 5)+1.613434753417969E3*V**(-4)+1.451559448242187E2*V**(-3)+1.3301 92 0410156 2 5E1*V**(-2)+l.43267822265625*V** (-1))

1 +EIV*(0.19638 0 61523437 5+0.185 4705810 54688*V**2)

NOTE: V=Z*SQRT(GAMMA)

EXPMV=EXP(-V)

E1V=E (V)

CRLI1 (N)= Q (V)

TABLE XIX

Rational Minimax Approximations for'H"" (z) in Eq. (3.19)

U0UlU2

V0VIV2

4.0 - 30.0

.14951 37850 0000E+02

.63740 23508 0000E+01

.99822 14300 0000E+01

.78785 35644 0000E+01

4.0 - 30.0

.10212 20030 0000E+02

.63824 67553 0000E+01

.63589 07300 0000E+01

.71922 81360 0000E+01

4.0 - 30.0

.44273 02122 0000E+02

.92706 58849 0000E+01

.63735 49521 6900E+01

.20560 75136 0000E+02

.21850 63364 8000E+02

.69506 10297 7300E+01

4.0 - 30.0

.15711 52237 0000E+03

.51874 26400 0000E+00

.63189 48176 0000Ë+01

.44686 70710 0000E+02

.31917 61411 7000E+02

.47673 37536 9000E+01

ACCURACY( S I C DIG. )

4 . 3 5 4 . 2 6 4 . 9 4 3 . 9 1

U0UlU2U3U4

V0VIV2V3

4.0 - 30.0

.23199 26488 8000E+03

.25765 53936 5800E+02

.52920 47591 3900E+01

.11786 89726 3500E-01

.21774 58630 0000E+01

.80574 70811 2000E+01

.51245 25340 2900E+01

3 .4 - 3 0 . 0

. 1 9 6 8 4 60491 0130E+03. 3 3 3 7 2 05190 712 3E+0 3. 3 9 4 0 5 50134 6508E+02. 4 8 3 5 1 16161 3037E+01. 1 5 4 8 7 05159 5628E-01

. 7 9 6 3 1 40664 0300E+22

. 9 9 6 2 7 06538 1859E+02

. 4 8 5 3 6 79247 8110E+02

. 1 1 0 2 9 84516 1352E+02

ACCURACY(SIG.DIG.)

3 .65 3.66

ACCURACY = -ALOG10(MINIMAX RELATIVE ERROR)

- 59 -

TABLE XX

Equations of the curves E in Figs. 4 and 5, for use in Eq.(3.19)

Curve Equation

Ei,o z = -6.32732/(/r - 1) - 4.6525 + 8.1524/y

E2, i z = -6.89926/(/y - 1) -6.6153 + 5.72377/y

E3,2 z = -6.9201/(/y - 1) - 7.1176+ 3.71219/y7

Eu,3 z = -7.5967/(/y - 1) - 6.5243- 2.20893/y

Es,.» z = -6.7982/(/y - 1) - 5.6831- 2.00732/y

TABLE XXI

Explicit expressions for the polynomial P_(Y) for use in Eq. (3.21)

P (Y)nx ' '

Coefficient Y° yi ï! YJ

45/128

-315/256

- 2784

-15312

example: P 2(Y) = 3/8(32-52Y2 + 15y"

TABLE XXII

Equations of the curves F in Fig. 5 for use in Eq.(3.21)ni j n

Curve Eouation

F 2 , 3

Fl»,5

F2 » S

F"ts 5

z/y = 191

z/y = 80.2

z/y = 48.1

K2 = .04

- 62 -

*DECK CPROBS

SUBROUTINE CPROBS(X,OK,PP,SPACE,NU)C£ * * * * * * + * * * * * * * * * * * * * * * * * * * * * * * * * * * *CC THIS SUBROUTINE CALCULATES 8 ANNULAR COLLISION PROBABILITIESC (CF. AECL-6479) BY M.S.MILGRAM AND K.N.SLY; ALSO SHORTER NOTEC PUBLISHED IN JOURNAL OF COMPUTATIONAL PHYSICSCC MACHINE - CDC 6600/CYBER 175CC INPUT VARIABLES -C X = SIGMA * BC QK = KAPPA = R(INNER)/R(OUTER)C SPACE = WORKING SPACE ARRAYC DIMENSIONED AT MOST 34 * NC (N = NUMBER OF REGIONS)CC OUTPUT -C PP(8) WHEREC PP(1) = PVIC PP(2) = PVOC PP(3) = PVVC PP(4) = PIVC PP(5) = POVC PP(6) = PIOC PP(7) = POOC PP(8) = POIC ACCURATE TO 2.E-5 FRACTIONAL ERROR FOR ALL X AND QKC NU = NUMBER OF LOCATIONS OF ARRAY "SPACE" USED ON EACHC CALL TO SETPRBCC ENTRY POINTS -C SETPRBC CPROBSCC CALLING SEQUENCE -C 1) CALCULATE PP FOR EACH RING (N=NO. OF RINGS)C DIMENSION SPACE{34),PP(S)C DO 1 1=1,NC X = . . .C QK = . . .C CALL SETPRB(X,QK,PP (1) ,SPACE(1) ,NU)C CALL CPROBS(X,QK,PP(1),SPACE(1),NU)CCCC USE PP IN CALCULATION

CCC 1 CONTINUECC 2) PRECALCULATION OF INVARIANT QUANTITIESC DIMENSION NS(N),PP(8,N)C COMMON SPACE(1)

- 63 -

C NUS = 1C DO 1 1=1,NC QK = ...C CALL, SETPRB (X,QK,PP, SPACE (NUS) , NO)C NS(I) = NUSC NUS = NUS + NUC 1 CONTINUECC DO 2 1=1,NC X = . . .C QK = . . .C CALL CPROBS(X,QK,PP(1,I) ,SPACE (NS (I)) ,C NU)CCCC USE PP IN CALCULATIONCCCC 2 CONTINUEC

DIMENSION PP(8) ,SPACE (1)

QKSQ=QK*QKQLSQ=1.-QKSQZ=X*SQRT(QLSQ)GAMMA=(1.-QK)/(1.+QK)SQG=SQRT(GAMMA)ZRTG=Z*SQGIF (QKSQ.LE..5) GO Tû 50BOUND=5.5801+QKSQ*(6.5192 4*QKSQ-6.680 0)IF (Z.LT.BOUND) GO TO 10

9 CONTINUEIF (ZRTG.GE.36.) GO TO 54

PID=SUMG(Z,GAMMA,QK,QKSQ,SQG,ZRTG,PP)GO TO 500

12 PID=SUMG8(Z,GAMMA,QK,QKSQ,SQG,ZRTG,PP)GO TO 500

13 PID=SUMG9(Z,GAMMA,QK.QKSQ,SQG,ZRTG,PP)GO TO 50 0

10 CONTINUEETI=-QLSQ/QKSQ

PID=D3(Z,ETI,QLSQ,QKSQ,QK,SQG,SPACE(14),NU,PP)

GO TO 500

50 CURVE=4.21 + QKSQ*(8.29-12.2*QKSQ)

- 64 -

CINF=QKSQ-.123BOUND=.4823+CINF*(5.20801-QKSQ*(5.38447+1.5799*QKSQ))IF (Z.GT.CURVE) GO TO 51

IF((CINF.LE.0.).OR.(Z*CINF.LT.BOUND)) GO TO 60GO TO 9

51 IF (Z*CINF.GE.BOUND) GO TO 9

IF (ZRTG.LT.36.0) GO TO 55

SQG1=SQG-1.BOUND=-7.358-SQG1*(2.810-3.508 *SQG)

IF (Z*SQG1.GT.BOUND) GO TO 57

54 PID=G4ASY(GAMMA,QK,ZRTG,PP)GO TO 500

55 CINF=QKSQ-.095IF (CINF.LE.0.) GO TO 56

BOUND=.2255/CINF+7.453IF (Z.GE.BOUND) GO TO 12

56 CINF=QKSQ-.078IF (CINF.LE.0.) GO TO 57

BOW7D=.1833/CINF+9.6364IF (Z.GE.BOUND) GO TO 13

57 CONTINUEETA=-QKSQ/QLSQPin=SUMGL(Z,ETA,QKSQ,QK,PP)GO TO 50fl

60 PID=E3(Z,QKSQ,QK,SPACE(14),NU,PP)

500 CALL POOA(X,Z,QK,QKSQfQLSQ,SPACE(l),NU,PP)

PP{5)=QK*PP(4)-PP{5)TXLSQ=2.+X*QLSQ?P(2)=PP(5)/TXLSQPP(1)=QK*PP(4)/TXLSQP?(3)=1.0-PP(1)-PP(2)RETURN

ENTRY SETPRB

NU=0QKSQ=QK*QKCALL POOSET(X,Z,QK,QKSQ,QLSQ,SPACE(1),NU)IF(QKSQ.LE.0.5) GO TO 150SOG=SQRT((1•-QK)/(1.+QK))?ID=5SS ET (Z.ETI,QLSQ,QKSQ,QK,SQG, SPACE (NU+1),NU)

- 65 -

GO TO 700150 PID=QQSET(Z,Q.KSQ,Q.K, SPACE (NU+1) ,NU)700 RETURN

175END

- 66 -

*DECK E3FUNCTION E3(Z2,QKSQ,QK,QQ,NU,PP)

DIMENSION PP(8)

DIMENSION QQ(20)

DIMENSION Dl(9), D2(9), D3(9)DATA Dl /.2, .55, 1.0, 1.5, 2.0, 2.55, 3.08, 3.62, 4.21DATA D2 / .27332, 1.3064, 2.1972, 3.0923, 3.8399, 4.8062,1 6.4460, 7.8540 , 8.29 /DATA D3 / -.13334, -1.02141, -1.76871, -2.75229, -3.3553,1 -4.7657, -8.70922, -12.0572, -12.20 /P5Z=.5*ZZY=P5Z*P5ZE3 = 0.IF (Y.LE.0.) GO TO 26YI=1.ALY=ALOG(Y)DO 10 11=1,9

YI=YI*YE3=E3+(QQ(II+10)-ALY*QQ(II))*YI

BOUND=D1(II)+QKSQ*(D2(II)+QKSQ*D3(II))IF (ZZ.LT.BOUND) GO TO 25

10 CONTINUE25 CONTINUE

E3=E3+QQ(10)+P5Z*QQ(20)

26 PP(4)=-P5Z*E3PP(6)=1.0-PP(4)PP(8)=QK*PP(6)

RETURN

ENTRY QQSET

DO 110 11=1,9QQ(II)=Q(II,QK)

110 QQ(II+10)=R(II,QK)QQ(10)=QM1(QK)QQ(20)=RM1 (QK)NU=NU+2 0RETURN

- 67 -

*DECK D3FUNCTION D3(Z,ETI,QL,SQ,QKSQ,QKrSQG,SS,NU,PP)

DIMENSION PP(8)

DIMENSION SS(14)

DIMENSION El (6), E2(6), E3(6)DATA El / 1.892, 3.4391, 4.6303, 5.6776, 4.9197, 5.5801 /DATA E2 /~4.9754, -8.24781, -9.9724, -11.1567, -6.8699, -6.68/DATA E3/ 3.965, 6.69611, 8.212, 9.2954, 6.50998, 6.51924 /P5Z=.5*ZY=P5Z*P5ZD3 = 0.IF (Y.LE.0.) GO TO 20ALY=ALOG(Y)YI = 1.DO 18 11=1,6

Y::=YI*YD3=D3+(SS(II)~ALY*SS(II+7))*YI

BOUND=E1(II)+QKSQ*(E2(II)+QKSQ*E3(II))IF (Z.LT.BOUND) GO TO 19

18 CONTINUE

19 D3=P5Z*(SS(14)*SQRT(~ETI)+P5Z*SS(7)-D3*QLSQ)

20 PP(4)=-D3PP(6)=1.0-PP(4)PP{8)=QK*PP(6)RETURN

ENTRY SSSET

DO 118 11=1,6SS(II)=S(II,SQG)

118 SS(II+7)=T(II,QK)SS(7)=SM1(SQG)SS(14)=TM1(SQG)NU=NU+14RETURN

- 68 -

*DECK POOASUBROUTINE POOA(X,Z,QK,QKSQQ,QLSQ,FFP,NU,PP)

DIMENSION PP(8)

DIMENSION FFF(13)

CCOMMON /QKCS/ QKMK,ZSQ,QKPK,QKSQDIMENSION F F ( 3 5 ) , HH(30)

C SQPI2 = S Q R T ( P I ) / 2 .DATA SQPI2 /.88622692545276/DATA FO3 /I.3333333333333/

CCCC ARRAY FF CONTAINS THE COEFFICIENTS FOR THE RATIONAL MINIMAXC APPROXIMATION OF FUNCTION FC

DATA FF / -.63909 19680 1E+02 , -.45713 21674 2E+02 ,C-.62724 63006E+01 , .45190 61988 6E+02 , .22877 10740 3E+02 ,C-.25960 19131 2E+02 , -.24911 98637 3E+02 , -.64961 8J387E+01 ,C-.146R5 30710 81E+03 , -.23092 96882 7E+02 , .40004 28803 ,C.52472 9404 , .21787 7076 , .54311 43026 37E+02 ,C-.B1959 41337E+01 , .27911 2424E-02 , .48449 243E-02 ,C.29708 177E-W2 , .12125 41967 515E+02 , -.52526 52994E+01 ,C.27920 59E-04 , .61729 52E-04 , .53758 88E-04 , .58212 20334 479E+C01 , -.40106 78892 06E+01 , .25412 521E-06 . .69021 40E-06 ,C.83645 97E-06 , .36314 74934 341E+01 . -.33345 90337 0858E+01 ,C.19507 028E-08 , .62875 25E-08 , .10519 48E-07 ,C.25090 03906 8749E+01 , -.29152 59896 3052E+01 /

CCC ARRAY HH CONTAINS THE COEFFICIENTS FOR THE RATIONAL MINIMAXC APPROXIMATION OF FUNCTION HC

DATA HH / -.87312 22496 1E+02 » -.81664 83533 9E+f,2 ,C-.19671 84841 1E+02 , .14420 76365 37E+03 , .45579 41387 6E+02 ,C-.37940 44610E+01 , -.46842 31404E+01 , -.17564 22197E+01 ,C.15808 52755 99E+03 , -.26658 23378E+01 , -.14414 6471E-01 ,C-.23576 6188E-01 , -.13169 2584E-01 , .17040 05928 26E+02 ,C-.57386 93276 3E+01 , -.13477 5845E-03 , -.28340 283E-03 ,C-.22723 766E-03 , .69243 57232 848E+01 , -.42603 41917 95E+01 ,C-.12397 3E-95 , -.32333 74E-05 , -.36348 87E-05 , .40367 65334 754C5E+B1 , -.34708 13999 587E+01 , -.97887 96E-08 , -.30570 07E-07 ,C-.47651 16E-07 , .27896 62653 9613E+01 , -.29973 81779 3758E+01 /

CcC RATIONAL HINIMAX APPROXIMATION FOR FUNCTION FC

C (FF(II+3) - L QK*(FF(II+4)+QK))CCC RATIONAL MINIMAX APPROXIMATION FOR FUNCTION Hr

- 69 -

H(II,QK)=(HH(II)+QK*(HH(II+l)+QK*HH(II+2)))C (HH(II+3)+QK*(HH(II+4)+QK))

CQKSQ=QKSQQQKPK=1.+QKQKMK=1.~QKZSQ=Z*ZW=2.*ZSQ/QKPKIF (Z .GE. I .4 ) GO TO 398QK2=.5*QKMKSQW=SQRT(W)ALW=0.IF (W.NE.0.) ALW=ALOG(W)

PP(5)=W*(2.0-FO3*QK2)+SQW*FFF{1)SUM1=FFF(8)SUM2=FFF(2)

CIM=3.6585*Z+2.878

CIF(IM.LT.3) GO TO 207Wl=l.DO 206 11=3,IMWI=WI*W

TERM1=FFF(II+6)*WITERM2=FFF(II)*WI

SUM1=SUM1+TERM1SUM2=SUM2+TERM2

206 CONTINUE

207 PP(5)=PP(5)+W*SQW*(SUMl+SUM2*ALW)PP(5)=QKMK*PP{5)PP(7)=QKMK+PP(5)GO TO 201

C398 IF(Z.GE.7.5)GO TO 399

PP(7)=POOSUM(QK,Z,QLSQ,X,W)GO TO 200

C339 CONTINUE

SQW=SQRT(W)ALW=ALOG(W)FASYS=QLSQ*SQPI2*EXP(-SQW-SQW-.75*ALW)/QKPKCALS=PCYLDR(X)PP(7)=CALS-FASYS

200 PP(5^=PP(7)-QKMK

201 CONTINUERETURN

ENTRY POOSET

FFF(1)=F(1,QK)FFF(2)=F(6,QK)

- 70 -

FFF(8)=H(1,QK)111=6DO 106 11=3,7FFF(II+6)=H(III,QK)111=111+5

106 FFF(II)=F(III,QK)NU=NU+13RETURN

- 71 -

* D E C K QF U N C T I O N Q ( I I . Q K )D I M E N S I O N QCfô ( 7 ) , Q C 1 ( 7 ) , Q C 2 ( 8 ) , Q C 3 ( 9 ) , Q C 4 ( 9 ) , Q C 5 ( 9 ) , Q C 6 ( 9 ) , Q C 7 ( 7 ) ,

? Q C 8 { 7 )DATA QC0 /$ -.30780 00333 50501E1 , .56370 89496 126$ .43084 62860 214 , .36261 83710 38544E1 ,S .52334 87556 8108E1 /DATA QCl /$ -.14164 50475 8827

.20834 4669S 3519E1.78798 90601 6027E1 ,

.17897 42533 4204 .21211 44170 55E-1.16687 16430 82483E1 , .44445 39100 16169E1

.70634 99455 224E-1.56672 08382 04310 5E1.23688 19823 83279 6E2

- . 5 4 6 1 6 3 9 3 0 6 3 5 1 E - 1

. - . 7 9 9 1 7 3 2 7 0 6 1 7 . 1 E - 5

. . 9 8 3 5 9 1 3 6 1 4 6 E - 6 ,3 9 2 ,

$ -.16118 57523 90E-1 , .$ , .37976 98260 1539E1DATA QC2 /$ -.34360 54298 767E-1 ,$ , .18341 21031 237E-1 ,$ .19507 05265 67962 0E2$ .11128 93477 51429 8E2 /DATA QC3 /$ -.86804 96860 663E-4 , .26368 66128 6822E-3 , -.37713 38719 0148E$-3 , .29741 65933 401E-3 , -.11609 13960 723E-3 , .18942 28035 70E$-4 , .30924 84020 32594 44 , .12465 55402 22122 521E1 ,$ .18048 88920 78804 00E1 /DATA QC4 /$ -.11806 49225 546E-5 , .43339 50802 557E-5 ,$ , .82085 90365 30E-5 , -.43520 37547 59E-5 ,$ .13219 28414 78808 105 , .65684 66861 64390$ .12360 60261 31940 797E1 /DATA QC5 /$ -.12121 63398 95E-7 , .50035 07193 77E-7 , -.10817 80844 933E-6 ,$ .12902 15454 700E-6 , -.79364 50563 12E-7 , .20680 95946 71E-7 ,$ .58812 29579 94199 857E-1 , .35781 53417 75027 1677 ,$ .86074 59034 34223 8976 /DATA QC6 /$ -.10747 47075 50E-9 , .48580 81044 75E-9 , -.11907 73821 078E-8 ,$ .15895- 07276 025E-8 , -.10898 51063 29E-8 , .31483 65198 8E-9 ,$ .29864 59349 93595 58271E-1 , .21697 68818 58096 83898 6 ,$ .64032 00287 68075 74192 4 /DATA QC7 /$ -.20823 52534E-12 , .49013 94841E-12 , -.91678 0373E-12 ,$ .66474 5096E-12 , .42334 20879 61279 97713E-2 ,$ .47590 75516 02662 49019 0E-1 , .19977 59374 54772 39210 4 /DATA QC8 /$ -.15044 65530E-14 , .36636 50054E-14 , -.78511 11102E-14 ,$ .60726 5743E-14 , .27890 39147 98521 94810 11E-2 ,$ .35696 44993 43843 56597 441E-1 , .16807 58616 24042 16307 2726 /GO TO (20,21,22,23,24,25,26,27,28) IIQ=(QC0(1)+QK*(QC0(2)+QK*(QC0(3)+QK*QC0(4)))) /$ (QC0 (5)+QK*(QC0(6)+QK*(QC0(7)+QK)))RETURN

Q=(QC1(1)+QK*(QC1(2)+QK*(QC1(3)+QK*QC1(4)))) /$ (QC1(5)+QK*(QC1(6)+QK*{QC1(7)+QK)))RETURN

Q=(QC2(1)+QK*(QC2(2)+QK*(QC2(3)+QK*QC2(4)))) /$ (QC2(5)+QK*(QC2(6)+QK*(QC2(7)+QK*(QC2(8)+QK))))DPTHDM

- 72 -

C23 Q=(QC3{1)+QK*(QC3(2)+QK*(QC3(3)+QK*(QC3(4)+QK*(QC3(5)+QK*QC3(6'»)) )

$)) / (QC3(7)+QK*{QC3(8)+QK*(CC3(9)+QK)))RETURN

C24 Q=(QC4(1)+QK*(QC4(2)+QK*(QC4(3)+QK*(QC4(4)+QK*(QC4(5)+QK*QC4(6))))

$)) /(QC4(7)+QK*(QC4(8)+QK*(QC4(9)+QK))) .RETURN

C25 Q=(QC5(1)+QK*(QC5(2)+QK*(QC5(3)+QK*(QC5(4)+QK*(QC5(5)+QK*QC5(6))))

$)) / (QC5(7)+QK*(QC5(8)+QK*(QC5(9)+QK)))RETURN

C26 Q=(QC6(l)+QK*(QC6(2)+QK*(Qt6(3)+QK*(QC6(4)+QK*(QC6(5)+QK*QC6(6))))

$)) / (QC6(7)+QK*(QC6(8)+QK*(QC6(9)+QK)))RETURN

C27 Q=(QC7(1)+QK*(QC7{2)+QK*(QC7(3)+QK*QC7(4)))) /

$ <QC7(5)+QK*(QC7(6)+QK*(QC7(7)+QK)))RETURN

C28 Q=(QC8(1)+QK*(QC8(2)+QK*(QC8(3)+QK*QC8(4))) ) /

$ (QC8 (5)+QK*(QC8(6)+QK*(QC8(7)+QK)))RETURN

- 73 -

*DECK QM1FUNCTION QMl(QK)

CDIMENSION QM1C(9)DATA QM1C /$ -.80132 06735 253E2 , .10787 60073 7832E3 , .31665 41295 40E1 ,S -.38111 56411 102E2 , .65910 70790 64Ë1 , .31467 78951 225E2 ,$ -.17648 10951 666E2 , -.25592 20148 564E2 , .13090 25219 402E2

CQM1= (QM1C(1)+QK*(QM1C(2)+QK*(QM1C(3)+QK*(QM1C(4)+QK*QM1C(5)) ) ) )$ /(QM1C(6)+QK*(QM1C(7)+QK*(QM1C(8)+QK*(QM1C(9)+QK))))RETURNEND

- 74 -

•DECK RFUNCTION R(II,QK)DIMENSION RC0(9),RC1(9),P.C2'9),RC3(10),RC4(9),RC5(9),PC6(9) ,$ RC7(7),RC8(4)DATA RC0 / -.66974 18439 2314E1 , .47220 46779 742E1 ,$ .96212 61250 006E1 , -.92855 46576 276E1 , .16355 17913 504E1 ,S .31407 13027 7046E1 , .32219 95667 098E1 , -.33561 30905 581E1 ,$ -.22560 15422 961E1 /DATA RC1 / -.13416 88052 53199E1 , .3143214715 8766E1 ,S -.24724 57804 4850E1 , .72408 79145 413 , -.53104 74207 73E-1 ,$ .65525 90331 29038E1 , .61140 98940 5578E1 , -.43919 93671 7660E1$ , -.35157 71755 3893E1 /DATA RC2 / .18798 49010 7814 , -.39223 24944 2718 . .28916 17953 2$575 , -.84901 45199 613E-1 , -.10823 44940 11603 5E2 ,$ -.30986 77893 13407 2E2 , -.28498 2936U 03034 0E2 ,S -.64936 89652 13866E1 , .25148 97110 14912E1 /DATA RC3 / -.15896 49837 4788E-2 , .39714 32374 004E-2 ,$ -.44562 58979 675E-2 , .24799 32862 441E-2 , -.40512 38044 61E-3,$ .17317 84258 23320 55E1 , .70829 2B893 42741 52E1 ,$ .11113 95154 66741 02E2 , .82502 52181 41081 1E1 .$ .28610 72668 24144 0E1 /DATA RC4 / -.27637 55067 156E-5 , .67776 06393 53E-5 ,$ -.87850 16748 66E-5 , .47704 94722 43E-5 , .85570 73949 54730 18E$-1 , .49227 84294 38970 041 , .12060 62167 86805 483E1 ,$ .16632 24774 72009 535E1 , .13460 53777 63803 956E1 /DATA RC5 / -.21583 90744 69E-7 , .64032 42909 78E-7 ,S -.10894 57262 675E-6 , .85328 43245 69E-7 , -.18942 72901 0E-7 ,$ .26752 11460 79760 245E-1 , .18300 81354 33771 B496 ,S .54338 69433 12605 8907 , .87572 51763 72140 4431 /DATA RC6 / -.22742 34773 63E-9 , .64188 88496 84E-9 ,$ -.12283 31995 92E-8 , .81023 13867 2E-9 ,$ .15134 81209 87927 50129E-1 , . 129SS }k><^ 52157 82974 9 ,$ .50226 14284 21585 47324 . .11072 40^d9 96457 29895E1 ,S .17934 46171 25763 25978 7E1 /DATA RC7 / -.10379 29671E-11 , .24709 50667E-11 , -.44733 83889E-1$1 , .31796 00517E-11 , .47881 83590 53145 5828E-2 ,$ .51870 21296 44809 10292E-1 , .21199 36103 78447 08653 4 /DATA RC8 / -.32888 272E-14 . .12922 47652 79019 18136E-2 ,$ .24096 93818 46435 12026 3E-1 , .42830 32326 23770 09763 4E-1 /GO TO (20,21,22,23,24,25,26,27,28) II

20 R=(RC0(1)+QK*(RC0(2)+QK*(RC0(3)+QK*(RC0(4)+QK*RC0(5))1 ) )$ / (RC0(6)+QK*(RC0(7)+QK*(RC0(8)+QK*{RC0{9)+QK))))RETURN

C21 R=(RC1(1)+QK*(RC1(2)+QK*(RC1(3)+QK*(RC1(4)+QK*RC1(5)))))

$ / (RC1(6)+QK*(RC1(7)+QK*(RC1(8)+QK*{RC1(9)+QK))))RETURN

C22 R=(RC2(1)+QK*(RC2(2)+QK*(RC2(3)+QK*RC2(4)))) /

S (RC2(5)+QK*(RC2(6)+QK*(RC2(7)+QK*(RC2(8)+QK*(RC2(9)+QK)))))RETURN

C23 R=(RC3(1)+QK*(RC3(2)+QK*(RC3(3)+QK*(RC3(4)+QK*RC3(5))))) /

S (RC3(6)+QK*(RC3(7)+QK*(RC3(8)+QK*(RC3(9)+QK*(RC3(10)+QK)))))RETURN

- 75 -

$ (RC4(5)+QK*(RC4(6)+QK*(RC4(7)+QK*(RC4(8)+QK*(RC4(9)+QK)))))RETURN

C25 R=(RC5(1)+QK*(RC5^)+QK*(RC5(3)+QK*(RC5(4)+QK*RC5(5))) ) )

$ / (RC5(6)+QK*(RC5(7)+QK*(RC5(8)+QK*(RC5(9)+QK))))RETURN

$ (RC6{5)+QK*(RC6(6)+QK*(RC6(7)+QK*(RC6(8)+QK*(RC6(9)+QK)))))RETURN

C27 R=(RC7(1)+QK*(RC7(2)+QK*(RC7(3)+QK*RC7(4))) ) /

$ (RC7(5)+QK*(RC7(6)+QK*(RC7(7)+QK)))RETURN

C28 R=RC8(1) / (RC8 (2)+QK*(RC8(3)+QK*(RC8(4)+QK)))

RETURNC

- 76 -

*DECK RM1FUNCTION RMl(QK)

CDIMENSION RM1C(9)DATA RM1C /$ .15743 40048 21267E3 , -.30309 35311 05422E3 , .14718 19743 4258E$3 , .97772 13849 51E1 , -.11142 50379 077E2 , .39358 50135 8633E2,$ -.13949 21985 0581E2 , -.37593 11174 546E2 , .16079 19005 159E2 /

CRM1=(RM1C(1)+QK*(RM1C(2)+QK*(RM1C(3)+QK*(RM1C(4)+QK*RM1C(5)))))$ /(RM1C(6)+QK*{RM1C(7)+QK*(RM1C(8)+QK*(RM1C(9)+QK))))RETURNEND

- 77 -

*DECK SFUNCTION S (II,RHO)DIMENSION S0(7),S1(7),S2(7),S3(6),S4(6),S5(4)DATA S0 / .89793 11961 35 , .11609 53643 91 , .19228 62857 026E+1$ , .25264 37928 6 , .10804 94635 70E+1 , .16704 82282 ,$ .10767 50405 621E+1 /DATA SI / -.23722 19481 2E-1 , -.1.1931 00197E-1 , -.13371 12790 0,$ .19698 107E-2 , -.17848 89931 , -.17442 95021 715E+1 ,$ -.87710 72396 1 /DATA S2 / -.12370 55678 8E-3 , .17316 56975E-3 , -.27075 16188E-3$ , -.26331 76227 3714 , .36868 05439 046 , .59378 24363 741 ,$ -.16235 91313 614E+1 /DATA S3 / -.12776 15142E-4 , .13631 0080E-4 , -.53748 2922E-4 ,$ .54311 2876E-4 , -.92434 468E-4 , -.93765 95532 9584 /DATA S4 / -.25935 28884E-6 , .30431 3755E-6 , -.10559 21428E-5 ,$ .10867 9783E-5 , -.13402 2587E-5 , -.85102 11959 26987 /DATA S5 / .12016 822E-8 , .22730 06734 73974 0 ,5 .13144 97629 53148E-1 , -.11243 68386 37361 3E+1 /

GO TO (10,11,12,13,14,15) II10 S =(S0(1)+RHO*(S0(2)+RHO*(S0(3)+RHO*(S0(4)+RHO*(S0(5)+RHO*S0(6))))

$)) / (S0(7)+RHO)RETURN

11 S =(S1(1)+RHO*(S1(2)+RHO*(S1(3)+RHO*(S1(4)+RHO*S1 (5)) ) )•) /$ (S1(6)+RHO*(S1(7)+RHO))RETURN

12 S =(S2(1)+RHO*(S2(2)+RHO*S2(3))) / (S2(4)+RHO*(S2(5)+RHO*(S2(6)+S RHO*(S2(7)+RHO))))RETURN

13 S =(S3(1)+RHO*(S3(2)+RHO*(S3(3)+RHO*(S3(4)+RHO*S3(5))))) /$ (S3(6)+RHO)RETURN

14 S =(S4(1)+RHO*(S4(2)+RHO*(S4(3)+RHO*(S4(4)+RHO*S4(5))))) /$ (S4(6)+RHO)RETURN

15 S =S5(1) / (S5(2)+RHO*(S5(3)+RHO*(S5(4)+RHO)) )RETURNEND

- 78 -

*DECK SMIFUNCTION SMI(RHO)DIMENSION SM1C{11)DATA SMIC / .I2666E-11 , -.22845 7954E-7 , -.24857 64186 82E-3 ,S -.56827 67050 9729E-1 , -.20697 80468 43980 7E1 ,

. $ -.13690 90410 05422 0E2 , -.83179 41482 6E-5 , -.27266 03799 80095E-2 , -.14241 11688 88283 , -.16281 36181 08416 3E1 ,$ -.31571 3" '0 88409 4E1 /SMI = (SMIC(1)+RHO*(SMIC(2)+RHO*(SMIC(3)+RHO*(SMIC(4)+RHO*$ (SM1C(5)+RHO*SM1C(6)))))) / (SMIC(7)+RHO*(SMIC(8)+RHO*(SMIC(9)+$ RHO*(SM1C(10)+RHO*(SM1C(11)+RHO)))))RETURNEND

- 79 -

*DECK TFUNCTION T(II,QK)DIMENSION TC0(4),TCI(5),TC2(6),TC3(5),TC4(4),TC5(3)DATA TC0 /.17517 41609 9E+2 , .20783 29030 59E+2 , .47877 14197 27$E+2 , .33525 68142 80E+2 /DATA TCI / .38354 49162 41E-1 , -.18957 93809 85E-1 , .44193 04218$025 , .16071 66774 3045E+1 , .17496 18511 5548E+1 /DATA TC2 / .45672 18845 11E-3 , -.46798 64023 29E-3 ,$ .23838 29227 78E-3 , .40789 98510 6133E-1 , .54679 16828 52711 ,$ .98093 91810 6611E-1 /DATA TC3 / .46012 6521E-5 , .62377 00572 26233 . -.31303 57842 637$94E+1 , ,65359 47141 45752 3E+1 , -.37671 57824 93196 0E+1 /DATA TC4 / .75790 9094E-7 , .43610 47E-9 , -.59461 53330 37065E-1$ , .58014 72480 01413E-1 /DAT* TC5 / .12501 7570E-R , -.38304 89659 09308 1E-1 ,$ .39106 78557 89331 5E-1 /GO TO (10,11,12,13,14,15) II

10 T=TC0(1)/(TC0(2)+QK*(TC0(3)+QK*(TC0(4)+QK)))RETURN

C11 T=(TC1(1)+QK*TC1(2))/(TCI(3)+QK*(TCI(4)+QK*(TCI(5)+QK)))

RETURNC12 T=(TC2(1)+QK*(TC2(2)+QK*TC2(3)))/(TC2(4)+QK*(TC2(5)+QK*(TC2(6)+QK)

$))RETURN

C13 T=TC3(1)/(TC3(2)+QK*(TC3(3)+QK*(TC3(4)+QK*(TC3(5)+QK)))>

RETURNC14 T=(TC4(1)+QK*TC4(2))/(TC4(3)+QK*(TC4(4)+QK))

RETURNC15 T=TC5(1)/(TC5(2)+QK*(TC5(3)+QK))

RETURNC

- 80 -

*DECK TM1FUNCTION TMl(RHO)DIMENSION TM1C(6)DATA TM1C / -.16756 53858 059E2 , .13975 37603 718E2 ,$ -.17905 38875 6E1 , .45526 52779 3E1 , .83782 69156 64E1$ .12402 88681 1 /

5 TMl =(TM1C(1)+RHO*(TM1C(2)+RHO*(TM1C(3)+RHO*TM.1C(4) )) ) /S (TM1C(5)+RHO*(TM1C(6)+RHO))RETURNEND

*DECK SUMG

FUNCTION SUMG(Z,GAMMA,QK,QKSQ,SQG,Y,PP)

DIMENSION PP(8)

DIMENSION A(5), B{5), C(5), D(5)DIMENSION Al (6), Bi{6), Cl(6)DATA A/ .089826, .52086, .27933, .16109, .29945 /DATA B / 42.2295 , 8.2057 , 10.4571 , 10.8867 , 8.9529 /DATA C /~42.655,-7.9405,-10.7462,-11.536,-9.4990 /DATA D / .80, .50, .35, .26, .17 /DATA Al /~7.358, -6.32732, -6.89926, -6.9201, -7.5967, -6.7982 /DATA Bl /-2.910 , -4.6525 , -6.6153 , -7.1176 , -6.5243 , -5.6831/DATA Cl/ 3.508, 8.1524, 5.72377, 3.71219, -2.20893, -2.00732 /CURVE(N)=A1(N)/SQG1+B1(N)+C1(N)*SQG

NNN=71 CONTINUE

SQG1=SQG-1.GMSQ=GAMMA*GAMMAGN=1.FTR=.5 *(1.+QK)N=0FTRZ=0.SUMGZ=0.GMP=1.DO 2 N=l,6IF (Z.GE.CURVE(N)) GO TO 14SUMGZ=SUMGZ+CH(Z,N)*GMPGMP=GMP*GAMMA

2 CONTINUE14 IF (N.EQ.l) GO TO 15

FTRZ=GAMMA*EXP(-Z)/(Z*SQRT(Z))

15 SUMG=G402 4E[Y,0)IF (NNN.GT.7) GO TO 17NB=1CINF=QKSQ-D(1)IF (CINP.LT.0.) NB=2DO 16 NN=NB,5CINF=QKSQ-D(NN)BOUND=A(NN)+CINF*(B(NN)+C(NN!*QKSQ)IF (Z+CINF.LE.BOUND) GO TO 16NNN=NN+1GO TO 17

16 CONTINUENNN=7

17 CONTINUEGO=GNDO 18 NN=1,NNNN=NN-1GOO=GOGO=GNGN=GN*(.5+N)/(N+l.)*GMSQGF=GN

- 82 -

IF (NN.GT.l) GF=GF-GMSQ*GOOG1=G4024E(Y,NN)SUMG=SUMG+G1*GF

18 CONTINUEPP(8)=(SUMG-SUMGZ*FTRZ)*FTRPP(6)=PP(8)/QKPP(4)=1.0-PP(6)RETURN

ENTRY SUMG8NNN=8GO TO 1

ENTRY SUMG9NNN=9GO TO 1

- 83 -

*DECK CHFUNCTION CH(Z,NN)DIMENSION CH0C(4),CH1C(4),CH2C(6),CH3C(6),CH4C(7),CH5C(9)DATA CH0C/ -.14951 3785E2 , -.63740 235<38El , .99822 143E1 ,$ .78785 35644E1 /DATA CHIC/ -.10212 2003E2 . -.63824 67553E1 , -.63589 073E1 ,$ .71922 81360E1 /DATA CH2C/ -.44273 02122E2 , -.92706 5S849E1 , -.63735 49521 69E1$ , .20560 75136E2 , -.21850 63364 8E2 , .69506 10297 73E1 /DATA CH3C/ -.15711 52237E3 , .51874 2640 , -.63189 48176 0E1 ,$ .44686 7071E2 , -.31917 61411 7E2 , .47673 37536 9E1 /DATA CH4C / -.23199 26488 8E3 , .25765 53936 58E2 ,$ -.52920 47591 39E1 , -.11786 89726 35E-1 , -.21774 5863E1 ,$ .80574 70811 2E1 , -.51245 25340 29E1 /DATA CH5C / .19684 60491 013E3 , -.33372 05190 71233E3 ,$ .39405 50134 65082E2 , -.48351 16161 30372 9E1 ,$ -.15487 05159 56277E--1 , .79631 40664 03E2 , -.99627 06538 1859E2$ , .48536 79247 81099E2 , -.11029 84516 13519 55E2 /GO TO (10,11,12,13,14,15) NN

C1 0 C H = ( C H 0 C ( 1 ) + Z + C H 0 C ( 2 ) ) / ( C H 0 C ( 3 ) + Z * ( C H 0 C ( 4 ) + Z ) )

R E T U R NC

1 1 C H = ( C H 1 C ( 1 ) + Z * C H 1 C { 2 ) ) / ( C H I C ( 3 ) + Z * ( C H 1 C ( 4 ) + Z ) )R E T U R N

C1 2 C H = ( C H 2 C ( 1 ) + Z * ( C H 2 C ( 2 ) + Z * C H 2 C ( 3 ) ) ) /

$ ( C H 2 C ( 4 ) + Z * ( C H 2 C ( 5 ) + Z * ( C H 2 C ( 6 ) + Z ) ) )R E T U R N

C1 3 CH= ( C H 3 C ( 1 ) + Z * ( C H 3 C ( 2 ) + Z * C H 3 C ( 3 ) ) ) /

$ ( C H 3 C ( 4 ' + Z * ( C H 3 C ( 5 ) + Z * ( C H 3 C ( 6 ) + 2 ! ) )

R E T U R N

C1 4 C H = ( C H 4 C ( l ) + Z * ( C H 4 C ( 2 ) + Z * ( C H 4 C m + Z + C H 4 C ( 4 ) ) ) ) /

$ ( C H 4 C ( 5 ) + Z * ( C H 4 C ( 6 ) + 2 * ( C H 4 C ( 7 ) + Z ) ) )R E T U R N

C1 5 C H = ( C H 5 C ( 1 ) + Z * ( C H 5 C ( 2 ) + Z * ( C H 5 C ( 3 ) + Z * ( C H 5 C ( 4 ) + Z * C H 5 C ( 5 ) ) ) ) ) /

$ ( C H 5 C ( 6 ) + Z * ( C H 5 C ( 7 ) + Z * ( C H 5 C ( 8 ) + Z * ( C H 5 C ( 9 ) + Z ) ) ) )R E T U R N

- 84 -

*DECK G4024EFUNCTION G4024E(V,N)DIMENSION CIV(71)DATA (CIV(I),I=1,51)

.37500000000000E+00,

.22500000000000E+01,

.11406250000000E+01,

.34687500000000E+02,

.12343750000000E+01,

.12117187500000E+03,

.24806250000000E+04,

.75996093750000E+01,

.15956718750000E+04,

.83100937500000E+05,

.13554687500000E+01,

.57849023437500E+03,

.15393164062500E+06,

.84433387500000E+07,

.13977661132813E+01,

.10088225097656E+04,-56595111328125E+06,

DATA (CIV(I),1=52,71)

.31250000000000E+00,

.37500000000000E+01,

.39531250000000E+01,

.65625000000000E+02,

.57500000000000E+01,

.44625000000000E+03,

.22558593750000E+0 0,

.48093750000000E+02,

.74664843750000E+04,

.16372125000000E+06,

.94824218750000E+01,

.41757656250000E+04,

.73865039062500E+06,

.16722956250000E+08,

.11387756347656E+02,•90575786132813E+04,.37767183398438E+07,

10614764917969E+09,24457323515625E+10,13309204101563E+02,17321748046875E+05,14016493322754E+08,42523342189453E+10,24375799103906E+12,

NP1=N+1GO TO (10,11,12,13,14,15,16,17,18,19)

.41598353671875E+09,

.18547058105469E+00,

.14515594482422E+03,

.17519833740234E+06,

.10753099628906E+09.

.20828964601758E+11,

.48507024972656E+12/

.10000000000000E+01

.27343750000000E+00

.12812500000000E+02

.24609375000000E+00

.27546875000000E+02

.12731250000000E+04

.13027343750000E+01

.29254687500000E+03-28527187500000E+05.20947265625000E+00.74521484375000E+02.27122812500000E+05.286902000E0000E+07•19638061523438E+00.10686791992188E+03.75234895019531E+05.21836078085938E+0Ê

.12312276539063E+1È

.14326782226563E+0]

.16134347534180E+04

.16408204321289E+0:

.72784815468750E+0S

.82067907796875E+1]

VSQ=V*V S VI=1./V S OMV=1.-VEXPMV=EXP(-V)E1V=EN(V,1)G4024E=EXPMV*OMV+E1V*VSQGO TO 20

VI2=VI*(1.+VI)

G4024E=.5*EXPMV*OMV+E1V*(1.+.5*VSQ)GO TO 20

G4024E=EXPMV*(CIV(1)*OMV+.75*VI2)+E1V*(.5+CIV(1)*VSQ)GO TO 20

G4024E=EXPMV*(CIV(2)*OMV+ VI*(CIV(3)+VI*(CIV(4)+CIV(5)*VI2)))1 +E1V*(CIV(1)+CIV(2)*VSQ)

GO TO 20

- 85 -

C14 G4024E=EXPMV*(CIV(6)*OMV+VI*(CIV(7)+VI*{CIV(8)+VI*{CIV(9)+VI*(

1 C I V ( 1 0 ) + C I V ( 1 1 ) * V I 2 ) ) ) ) )2 +E1V*(CIV(2)+CIV(6)*VSQ)

GO TO 20CC N=5C15 G40 24E=r:'XPMV* (CIV (12) *OMV+VI * (CIV (13)+VI * (CIV (14 )+VI* (CIV (15 ) +

1 V I * ( C I V ( 1 6 ) + V I * ( C I V ( 1 7 ) + V I * ( C I V ( 1 8 ) + C I V ( 1 9 ) * V I 2 ) ) ) ) ) ) )2 +E1V*(CIV(6)+CIV(12)*VSQ)

GO TO 20CC N=6C16 G4 0 2 4E=EXPMV*{CIV(20)*OMV+VI*(CIV(21)+VI*(CIV(22)+VI*(CIV(23)+

1 V I * ( C I V ( 2 4 ) + V I * ( C I V ( 2 5 ) + V I * ( C I V ( 2 6 ) + V I * ( C I V ( 2 7 ) + V I + ( C I V ( 2 8 ) +2 C I V ( 2 9 ) * V I 2 ) ) ) ) ) ) ) ) )3 +E1V*(CIV(12)+CIV(20)*VSQ)

GO TO 20CC N=7C17 G4024E=EXPMV*;CIV(30)*OMV+VI*(CIV(31)+VI*(CIV(32)+VI*(CIV(33)+

1 V I * ( C I V ( 3 4 ) + V I * ( C I V ( 3 5 ) + V I * ( C I V ( 3 6 ) + V I * ( C I V ( 3 7 ) + V I * ( Ç I V ( 3 8 ) +2 V I * ( C I V ( 3 9 ) + V I * ( C I V ( 4 0 ) + C I V ( 4 1 ) * V I 2 ) ) ) ) ) ) ) ) ) ) )3 +E1V*(CIV(20)+CIV(30)*VSQ)

GO TO 20CC N=8C18 G4024E=EXPMV*(CIV(4 2)*OMV+VI*(CIV(4 3 ) + V I * ( C I V ( 4 4 ) + V I * ( C I V ( 4 5 )+

1 V I * ( C I V ( 4 6 ) + V I * ( C I V ( 4 7 ) + V I * ( C I V ( 4 8 ) + V I * ( C I V ( 4 9 ) + V I * ( C I V ( 5 0 ) + V I *2 ( C I V ( 5 1 ) + V I * ( C I V ( 5 2 ) + V I * ( C I V ( 5 3 ) + V I * ( C I V ( 5 4 ) + C I V ( 5 5 ) * V I 2 ) ) » ) ) ) ) ) ) )3 ) ) )4 +E1V*(CIV(30)+CIV(42)*VSQ)

GO TO 20CC N=9C19 G4024E=EXPMV*(CIV(56)*OMV+VI*(CIV(5 7 ) + V I * ( C I V ( 5 8 ) + V I * ( C I V ( 5 9 ) +

1 V I * ( C I V ( 6 0 ) + V I * ( C I V ( 6 1 ) + V I * ( C I V ( 6 2 ) + V I * ( C I V ( 6 3 ) + V I * ( C I V ( 6 4 ) + V I * (2 C I V ( 6 5 ) + V I * ( C I V ( 6 6 ) + V I * ( C I V ( 6 7 ) + V I * ( C I V ( 6 8 ) + V I * ( C I V ( 6 9 ) + V I * ( C I V ( 7 03 ) + C I V ( 7 1 ) * V l 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) )5 +E1V*(CIV(42)+CIV(56)*VSQ)

20 CONTINUERETURNEND

- 86 -

*DECK SUMGL

FUNCTION SUMGL(ZZ,ETA,QKSQ,QK,PP)

DIMENSION PP(8)

DIMENSION Al(2), Bl(2), Cl(2)DIMENSION P0(11), Pl(10), P2(10), ETAP(10)DIMENSION AKIN(10)EQUIVALENCE (AKIN,ETAP)

EQUIVALENCE (ETAP(l),ETA1),(ETAP(2),ETA2),(ETAP(3),ETA3),1(ETAP(4),ETA4),(ETAP(5),ETA5),(ETAP(6),ETA6),(ETAP(7),ETA7),2(ETAP(8),ETA8),(ETAP( 9 ) ,ETA9),(ETAP(10),ETA10)

DATA Al /.0065454, .033333 /DATA Bl / 4.81818, 4.33333 /DATA Cl / .285, .29 /

DATA P0(1)/ 1.2732 39544 7352 /

ETA1=ETADO 87 LL=2,10

87 ETAP(LL)=ETAP(LL-1)*ETAZSQ=ZZ*ZZFZ=.5*ZZSQMET=SQRT(-ETA)

1 P0(2)=1 +ETA1*(-0.25*SQMET)1 +ETA2*(-0.3395305452627+0.15625*SQMET)1 +ETA3*(-4.8504363608959E-2+6.8359375E-2*SQMET)1 +ETA4*(-1.6168121202986E-2+3.84521484375E-2*SQMET)1 +ETA5*(-7.3491460013574E-3+2.4673461914062E-2*SQMET)1 +ETA6*(-3.9572324622693E-3+1.7183303833008E-2*SQMET)1 +ETA7*(-2.3743394773616E-3+1.2657344341278E-2*SQMET)1 +ETA8*(-1.536337308881E-3+9.7127538174391E-3*SQMET)1 +ETA9*(-1.0511781587081E-3+7.6892634388059E-3*SQMET)1 +ETA10*(-7.5084154193434E-4+6.2387887446675E-3*SQMET)

1 P0(3)=1 +ETA2* (1.0416666666667E-2*SQMET)1 +ETA3*(2.2635369684181E-2-1.8880208333333E-2*SQMET)1 +ETA4+(-1.8477852803413E-3-8.0159505208333E-3*SQMET)1 +ETA5*(-2.8230052894103E-3-4.2755126953125E-3*SQMET)1 +ETA6*(-2.3754815357923E-3-2.6055653889974E-3*SQMET)1 +ETA7*(-1.3909852092429E-3-1.7332661719549E-3*SQMET)1 +ETA8*(-1.5098363856043E-3-1.2267011084727E-3*SQMET)1 +ETA9*(-1.2230449949131E-3-9.0913440928691E-4*SQMET)1 +ETA1B*(-1.0067010642736E-3-6.9818152955526E-4+SQMET)

1 P0(4)=1 +ETA3*(-2.1701388888889E-4*SQMET)1 +ETA4*(-6.672557956788E-4+8.544921875E-4*SQMET)1 +ETA5*(4.1839582526246E-4+2.0107693142361E-4*SQMET)1 +ETA6*(3.1357661482093E-4+3.462897406684E-5*SQMET)1 +ETA7*(2.1570262564899E-4-1.6344728923979E-5*SQMET)

- 87 -

1 +ETA8*(1.5242914346643E-4-3.2176899295005E-5*SQMET)1 +ETA9*{1.1171366527978E-4-3.580737331754E-5*SQMET)1 +ETA10*(8.46260863.71185E 5 3 . 497136122961E-5*SQMET)

1 P0{5)=1 +ETA4*(2.7126736111111E-6*SQMET)1 +ETA5*(1.0887623705715E-5--1.9056532118056E-5*SQMET)1 +ETA6*(-1.6271613450299E-5+2.2294786241319E-6*SQMET)1 +ETA7*(-8.3617083927543E-6+4.936210692875E-6*SQMET)1 +ETA8*(-4.027211532776E-6+4.616728614247E-6*SQMET)1 +ETA9*(-1.8841777039271E-6+3.8486725537392E-6*SQHET)1 +ETA10*(~7.945082335246E~7+3.1344161341805E-6*SQMET)

1 P0(6)=1 +ETA5*(-2.26056i3425926E-8*SQMET)1 +ETA6*(-1.1239305190714E-7+2.5068010602679E-7*SQMET)1 +ETA7*{3.0203889631622E-7-1.5846816i40057E-7*SQMET)1 +ETA8*(6.8439838555864E-8-1.4201014986204E-7*SQMET)1 +ETA9*(-1. 6692387615356E-8-9.6277835241311E-8*SQMET)1 +ETA10*(-4.2766766259499E-8-6.19939327194E-8*SQMET)+0.

1 P0(7)=1 +ETA6*(1.3455722277337E-10*SQMET)1 +ETA7*(7.9948738329317E-10-2.1715373228827E-9*SQMET)1 +ETA8*(-3.3641904117781E-9+2.8469202209502E-9*SQMET)1 +ETA9*(4.8999681814952E-10+1.5473864220254E-9*SQMET)1 +ETAl0*(1.2303789631182E-9+5.4539791223479E-10*SQHET)

1 P0(8)=1 +ETA7*(-6.0070188738111E-13*SQMET)1 +ETA8*(-4.1560219651355E-12+1.3321315814941E-11*SQMET)1 +ETA9*(2.5176540426567E-11-2.869941912207Ë-11*SQMET)1 +ETA10*(-1.5381162738819E-11-6.1602003437967E-12*SQMET)

1 P0(9)=1 +ETA8*(2.0857704422955E-15*SQMET)1 +ETA9*U-6478714000048E-14-6.0921878335382E-14*SQMET)1 +ETA10*(-1.3575378108764E-13+1.9267126696179E-13*SQMET)

1 P0{10)=1 +ETA9*(-5.7938067841542E-18*SQMET)1 +ETA10*(-5.1483605383472E-17+2.157?.712851091E-16*SQMET)

1 P0(11)=+ETA10*(1.316774269126E-20*SQMET)

1 Pl(l)=1 +ETA1*(-0.8488263631568+0.75*SQMET)1 +ETA2*(0.1697652726314+0.15625*SQHET)1 +ETA3*(7.2756545413438E-2+6.8359375E-2*SQMET)1 +ETA4*(4.0420 303007465E-2+3.84521484375E-2*SQMET)1 +ETA5*(2.5722011004751E-2+2.4673461914062E-2*SQMET)1 +ETA6*(1.7807546080212E-2+1.7183303833008E-2*SQHET)1 +ETA7*(1.3058867125489E-2+1.2657344341278E-2*SQMET)1 +ETA8*(9.9861925077268E-3+9.712753817439lE-3*SQMET)1 +ETA9*(7.8838361903106E-3+7.6892634388059E-3*SQMET)1 +ETA10*(6. 3821531064419E-3+6.2387887446675E-3*SQMET)

- 88 -

1 Pl(2)=1 +ETA2*(5.6588424210452E-2-9.375E~2*SQMET)1 +ETA3*(-5.1737987849556E-2-1.497 3 9583 33333E~2*SQMET)1 +ETA4*(-2.7947752365162E-2-3.2145182291667E~3*SQMET)1 +ETA5*(-1.6914701114235E-2-3.265380859375E-4*SQMET)1 +ETA6*(-1.125213068817E-2+5.1663716634115E-4*SQMET)1 +ETA7*(-8.0030183782398E-3+7.5258243651617E-4*SQMET)1 +ETA8*(-5.9759 581989039E-3+7.84182 38886765E-4*SQMET)1 +ETA9*(-4.6293061656046E-3+7.4481899436149E-4*SQMET)1 +EÏA10*(-3.6902815722005E-3+6.8323318911199E-4*SQMET)

1 PI (3) =1 +ETA3*(--1.616 812120 2986E~3+4.123 26 3 888888 9E-3*SQMET)1 +ETA4*(4.1318 531963187E-3-9.3587239583333E-4*SQMET)1 +ETA5*(1.7233552951331E-3-1.0779486762153E-3*SQMET)1 +ETA6*(8.2472933814766E-4-8.6852179633247E~4*SQMET)1 +ETA7*(4.3 918003369138E-4-6.769617 398 5799E-4*SQMET)1 +ETA8*(2.5173706685112E-4-5.3290830599883E~4*SQMET)1 +ETA9*(1.5179212627081E-4-4.2705273960357E-4*SQMET)1 +ETA10*(9.4698501181564E-5-3.4846791725916E-4*SQMET)

1 Pl(4)=1 +ETA4*(2.5663684449184E-5-8.9518229166667E-5*SQMET)1 +ETA5*(-1.3 220 6 8 5928 368E-4+8.36181640625E-5*SQMET)1 +ETA6*{-1.8240304649834E-5+5.9009128146701E-5*SQMET)1 +ETA7*(9.397 50 9372 36 51E-6+3.765 36 7 32628 2 3E~5*SQMET)1 +ETA8*(1.5537610208361E-5+2.4566604267983E-5*SQMET)1 +ETA9*(1.578500388486E-5+1.6606 71316718E-5*SQMET)1 +ETA10*(1.44187349774E-5+1.1606918230268E-5*SQMET)

1 Pl(5)=1 +ETA5*(-2.5922913585035E-7+1.1528862847222E-6*SQMET)1 +ETA6*(2.2605505762614E-6-2.2650017309441E-6*SQMET)1 +ETA7*(-6.7812333923628E-7-9.7410118408664E-7*SQMET)1 +ETA8*(-9.3291154687888E-7-3.0927934479011E-7*SQMET)1 +ETA9*(-7.7797813777751E-7-1.6098869244107E-8*SQMET)1 +ETA10*(-5.9836814545234E-7+1.094 584 304 8131E-7*SQMET)

1 Pl(6)=1 +ETA6*(1.8127911597926E-9-9.8226 772624559E-9*SQMET)1 +ETA7*(-2.4 05 8 992 418171E-8+3.262712301310 5E-8*SQMET)1 +ETA8*(2.1641390693452E-8+3.05910564 80449E-9*SQMET)1 +ETA9*(1.6 594 9713 2 7319E-8-6.3917 06120 28 6 7E-9*SQMET)1 +ETA10*(9.4 724 9 3 20 72 315E-9-8.2636885793944E-9*SQMET)

1 Pl(7}=1 +ETA7*(-9.2963649220135E-12+5.946948685073E-11*SQMET)1 +ETA8*(1.7488102953294E-10-2.9776667411255E-10*SQMET)1 +ETA9*(-2.9069873631677E~10+9.7 857 8 311198 7 4E~11*SQMET)1 +ETA10*(-1.2258987013667E-10+1.486 646 7424537E~10*SQMET)

1 Pl(8)=1 +ETA8*(3.64563 3 30 2 7 504E~14-2.690 64 3 870 5612E-13*SQMET)1 +ETA9*(-9.2 39 36 6 6304 38E-13+1.8894 4 7299414 5E-12*SQMET)1 +ETA10*(2.3 92 293 78 3139'E-12-1.595171754 68 55E-12*SQMET)

- 89 -

1 +ETA9*(-1.1286790410992E-16+9.4439050581714E-16*SQMET)1 +ETA10*(3.7099312340892E-15-8.8472614690877E-15*SQMET)

1 P1(10)=+ETA10*(2.8287695265644E-19~2.6467162809432E-18*SQMET)

1 P2(1)=-P0(1)+2.*SQMET1 +0.8488263631568*ETA11 +0.1697652726314*ETA21 +7.2756545413438E-2*ETA31 +4.0420303007465E~2*ETA41 +2.5722011004751E-2*ETA51 +1.7807546080212E~2*ETA61 +1.3058867125489E~2*ETA71 +9.9861925077268E~3*ETA81 +7.8838361903106E-3*ETA9I +6.3821531064419E-3*ETA10

1 P2(2)=1 +ETA2*(5.6588424210452E-2-9.375E-2*SQMET>1 +ETA3*(-5.1737987849556E-2-1.4973958333333E-2*SQMET)1 +ETA4*(-2.7947752365162E-2-3.2145182291667E-3*SQMET)1 +ETA5*(-1.6914701114235E-2-3.265380859375E-4*SQMET)1 +ETA6*(-1.125213068817E-2+5.1663716634115E-4*SQMET)1 +ETA7*(-8.0030183782398E-3+7.5258243651617E-4*SQMET)1 +ETA8*(-5.9759581989039E-3+7.8418238886765E-4*SQMET)1 +ETA9*(-4.6293061656046E-3+7.4481899436149E-4*SQMET)1 +ETA10*(-3.6902815722005E-3+6.8323318911199E-4*SQMET)

1 P2(3)=1 +ETA3*(-1.6168121202986E-3+4.1232638888889E-3*SQMET)1 +ETA4*(4.1318531963187E-3-9.3587239583333E-4*SQMET)2 +ETA5*a.7233552951331E-3-1.0779486762153E~3*SQMET)1 +ETA6*(8.2472933814766E-4-8.6852179633247E-4*SQHET)1 +ETA7*U.3918003369138E-4-6.7696173985799E~4*SQMET)1 +ETA8*(2.5173706685112E-4-5.3290830599883E-4*SQHET)1 +ETA9*(1.5179212627081E-4-4.2705273960357E-4*SQMET)1 +ETA10*(9.4698501181564E-5-3.4846791725916E-4*SQMET)

1 P2(4)=1 +ETA4*(2.5663684449184E-5-8.9518229166667E-5*SQMET)1 +ETA5*(-1.3220685928368E-4+8.36181640625E-5*SQMET)1 +ETA6*(-1.8240304649834E-5+5.9009128146701E-5*SQMET)1 +ETA7*(9.3975093723551E-6+3.7653673262823E-5*SQMET)1 +ETA8*(1.5537610208361E-5+2.4566604267983E-5*SQMET)1 +ETA9*(1.578500388486E-5+1.660671316718E-5*SQMET)1 +ETA10*(1.44187349774E-5+1.1606918230268E-5*SQMET)

1 P2(5)=1 +ETA5*(-2.5922913585035E-7+1.1528862847222E-6*SQHET)1 +ETA6*(2.2605505762614E-6-2.2650017309441E-6*SQMET)1 +ETA7*(-6.7812333923628E-7-9.7410118408664E-7*SQMET)1 +ETA8*(-9.3291154687888E-7-3.0927934479011E-7*SQHET)1 +ETA9*(-7.7797813777751E-7-1.6098869244107E-8*SQMET)1 +ETA10*(-5.9836814545234E-7+1.0945843048131E-7*SQMET)

1 P2(6)=1 +ETA6*(1.8127911597926E-9-9.8226772624559E-9*SQMET)

- 90 -

1 +ETA7*(-2.4058992418171E-8+3.262712301310 5E~8*SQMET)1 +ETA8*(2.1641390693452E~8+3.0591056480449E-9*SQMET)1 +ETA9*(1.6594971327319E-8-6.3917061202867E-9*SQMET)1 +ETA10*(9.472 49 3 20 72315E-9-8.26368R579394 4E-9*SQMET)

1 P2(7)=1 +ETA7*("9.296 3649220135E-12+5.946948685073E-11*SQMET)1 +ETA8*(1.7488102953294E-10-2.9776667411255E-10*SQMET)1 +ETA9*(-2.906 98 73631677E-10+9.7857831119874E-11*SQMET)1 +ETA10*(-1.2 25 898701366 7E-10+1.4 866467424537E-10*SQMET)

1 P2(8)=1 +ETA8*(3.64 5 63330 27 50 4E-14-2.690 6 4 387 0 5612E-13*SQMET)1 +ETA9*(-9.239366630438E-13+1.8894472994145E-12*SQMET)1 +ETA10*(2.392 293783139E-12-1.5951717546855E-12*SQMET)

1 P2(9)=1 +ETA9*(-1.128 6 790410992E-16+9.44 390 50 581714E-16*SQMET)1 +ETA10*(3.7099312340892E-15~8.8472 6146 90877E-15*SQMET)

1 P 2 ( 1 0 ) =1 + E T A 1 0 * ( 2 . 8 2 8 7 6 9 5 2 6 5 6 4 4 E - 1 9 - 2 . 6 4 6 7 1 6 2 8 0 9 4 3 2 E - 1 8 * S Q M E T )

P 0 Z = P 0 ( 1 ) + Z S Q * ( P 0 ( 2 ) + Z S Q * ( P 0 ( 3 ) + Z S Q * ( P 0 ( 4 ) + Z S Q * ( P 0 ( 5 ) + Z S Q * ( P 0 ( 6 ) +1 Z S Q * ( P 0 ( 7 ) + Z S Q * ( P 0 ( 8 ) + Z S Q * ( P 0 ( 9 ) + Z S Q * ( P 0 ( 1 0 ) + Z S Q * P 0 ( 1 1 ) ) ) ) ) ) ) ) ) )

P 1 Z = P 0 ( 1 ) / Z Z + Z Z * ( P 1 ( 1 ) + Z S Q * ( P 1 ( 2 ) + Z S Q * ( P l ( 3 ) + Z S Q * ( P l ( 4 ) + Z S Q * ( P l ( 5 )1 + Z S r M P I ( 6 ) + Z S Q * ( P i ( 7 ) + Z S Q * ( P I ( 8 ) + Z S Q * ( P l ( 9 ) + Z S Q * P l ( 1 0 ) ) ) ) ) ) ) ) ) )

P 2 Z = P 2 ( 1 ) + Z S Q * ( P 2 ( 2 ) + Z S Q * ( P 2 ( 3 ) + Z S Q * ( P 2 ( 4 ) + Z S Q * ( P 2 ( 5 ) + Z S Q * ( P 2 ( 6 ) +1 Z S Q * ( P 2 ( 7 ) + Z S Q * ( P 2 ( 8 ) + Z S Q * ( P 2 ( 9 ) + Z S Q * P 2 ( 1 0 ) ) ) ) ) ) ) ) )

E T A 1 1 = E T A 1 0C A L L K I N ( Z Z , A K I N )S U M G L = A K 0 B E S ( Z Z ) * P 0 Z + A K I N ( 1 ) * P 1 Z + A K I N ( 2 ) * P 2 Z

DO 4 LL=1,2CINF=QKSQ-C1(LL)IF (ZZ*CINF.LE. (Al (LL)+B1 (LL)*CINF)) GO TO 5ETAll=-ETA*ETAll

4 SUMGL=SUMGL+ETA11*(SQMET*GL1(ZZ,LL)-GL2(ZZ , LL) )C5 CONTINUE

PP(6)=PZ*SUMGLPP(8)=QK*PP(6)PP(4)=1.0-PP(6)RETURNEND

- 91 -

*DECK GL1FUNCTION GL1(Z,L)DIMENSION GLlll(3), GL112(2)DATA Zlll /.5592 2678 2802 36E+1/DATA Z112 /.5585 6441 4232 91E+1/DATA GLlll /$ .33248 198E-1 , -.43037 26E-2 , -.13945 78870 0E2 /DATA GL112 /$ .29568 03E-2 , -.35836 57E-3 /IP (L-l) 11,11,12

11 GL1=(GL111(1)+Z*GL111(2)) / (GLlll (3)+Z)*(Z-Z111)RETURN

C12 GL1=(GL112(1)+Z*GL112(2))*(Z-Z112)

RETURNC

- 92 -

•DECK GL2FUNCTION GL2{Z,L)DIMENSION GL211(4), GL212(3)DATA GL211 / -.46077 34403E-1 , .23280 76668 6E-1 ,$ -.37245 26778 9E-2 , .19184 86643 8E-3 /DATA GL212 / .86500 438E--2 , -.38150 5082E-2 , .37155 2739E-3 /IF (L-l) 11,11,12

C11 GL2=(GL211(1)+Z*(GL211(2)+Z*(GL211(3)+Z*GL211 (4) )) )

RETURNC12 GL2=(GL212(1)+Z*(GL212(2)+Z*GL212(3)))

RETURNC

- 93 -

*DECK G4ASYFUNCTION G4ASY(GAMMA,QK,ZRTG,PP)

DIMENSION PP(8)

G4ASYC=1.+QKGAMSQ=GAMMA*GAMMAG4ASY=G4ASYC*EXP(-ZRTG)/ZRTG*SQRT(1.-GAMSQ)ZRGI=1./(ZRTG*(1.-GAMSQ))CORRA=1. -1. 5* (2 . --GAMSQ) *ZRGIZRGP=ZRGI*ZRGICORRA=CORRA+.375*ZRGP*((15.*GAMSQ-52.)*GAMSQ+32,)IF((ZRTG.GT.191.).AND.(QK.GT.0.20)) GO TO 75ZRGP=ZRGP+ZRGICORRA=CORRA-.312 5*ZRGP*(192.-GAMSQ*(504.-GAMSQ*(420.-87.*GAMSQ)))IF((ZRTG.GT.80.2).AND.(QK.GT.0.20)) GO TO 75ZRGP=ZRGP*ZRGICORRA=CORRA+.3515625*ZRGP*(1024.+GAMSQ*(-3 712.+GAMSQ*(4912.+GAMSQ$* (-2 784.+4 5 5.*GAMSQ))))IF((ZRTG.GT.48.1).AND.(QK.GT.0.20)) GO TO 75ZRGP=ZRGP*ZRGICORRA=CORRA-1.23046875*ZRGP*(2048.+GAMSQ*(-94 72.+GAMSQ*(17248.+$GAMSQ*(-15312.+GAMSQ*(6618.-899.*GAMSQ)))))

75 PP(8)=G4ASY*CORRAPP(6)=PP(8)/QKPP(4)=1.0-PP(6)RETURNEND

- 94 -

*DECK PCYLDRFUNCTION PCYLDR(X)DIMENSION R(10),A(8)

CC ARRAY R CONTAINS THE COEFFICIENTS FOR THE RATIONAL MINIMAXC APPROXIMATION OF FUNCTION PCYLDRC

DATA R /.24568 00546 31E+01 , .96032 51912 74 , -.29659 06100 377,C .17173 50343 4387 , .49156 55169 40E-03 , .25813 18627 1 ,C .13366 65605 4353E+02 , -.54667 07169 7070E+01 ,C .11211 22907 8520E+02 , -.26531 45486 1511E+01 /

CDATA A /.1875 , .1171875 , .384 521484375 ,C .30281066894531E+01 , .43718290328980E+02 , . 10047399938107E+04 ,C .33674488855060E+05 , .15504295910351E+07 /

CIF(X.LE.7.7) GO TO 10

CC CALCULATE ASYMPTOTIC FORM OF PCYLDR TO 6 SIG. DIG.C

X2S=1./(X*X)PCYLDR=(A(2)*X2S+A(1))*X2SIF(X.GE.38.) GO TO 83XIS=X2S*X2S*X2SPCYLDR=PCYLDR+A(3)*XISIF(X.GE.16.1) GO TO 83XIS=XIS*X2SPCYLDR=PCYLDR+A(4)*XISI F ( X . G E . 1 1 . 4 5 ) GO TO 83X I S = X I S * X 2 SPCYLDR=PCYLDR+A(5)*XISI F ( X . G E . 9 . 9 ) GO TO 8 3XTS=XIS*X2SPCYLDR=PCYLDR+A(6)*XISI F ( X . G E . 9 . 2 ) GO TO 83X I S = X I S * X 2 SPCYLDR=PCYLDR+A(7)*XISI F ( X . G E . 8 . 6 5 ) GO TO 8 3PCYLDR=PCYLDR+A(8)*XIS*X2S

8 3 RETURNCC RATIONAL MINIMAX F I T TO PCYLDR WHEN 1.4@X@7.7C (ACCURACY 6 . 1 S I G . D I G . )C10 P C Y L D R = ( R ( 1 ) + X * ( R ( 2 ) + X * ( R ( 3 ) + X * ( R ( 4 ) + X * R ( 5 ) ) ) ) )

C / ( R ( 6 ) + X * ( R ( 7 ) + X * ( R ( 8 ) + X * ( R ( 9 ) + X * ( R ( 1 0 ) + X ) ) ) ) )RETURNEND

- 95 -

*DECK POOSUMFUNCTION POOSUM(QK,Z,QLSQ,X,W)COMMON /QKCS/ QKMK,ZSQ,QKPK,QKSQDIMENSION S01(8),S02(3),5iilû),S12(4),S21(7),a22(4)fS31(6),S32(4)f

C S41(5),S42(4),S51 (4)DIMENSION R0(5),R1(5),R2(5),R3(6),K4(6),R5(5)

C ZI IS THE LOCATION OF THE ZERO IN THE FUNCTION G3124 WHEN L = IDATA Z2 /3.1949303209937/DATA Z3 /2.4687331467198/DATA Z4 /2.0873741592691/DATA Z5 /I.8416429380076/

CC ARRAYS R0,R1,R2,R3,R4,AND R5 CONTAIN THE COEFFICIENTS FOR THEC RATIONAL MINIMAX APPROXIMATIONS OF FUNCTION G3124C RI IS FOR G3124 WHEN L = I

DATA R0 /.24730 6799 , .19600 66872 9E+01 , .11280 78289 03E+01 ,C .10624 86895E+01 , .25435 12709 5E+01 /DATA Rl /.95429 96902 , .33540 60010 , -.75168 17122 ,C .26384 59985 , .16018 69973 6E+01 /DATA R2 ,/-.29982 5189 , .29743 44675 , .17848 506E-03 ,

C -.12327 6208 , .53177 49825 /DATA R3 / -.36992 26951 42 , .51800 39405 84 , -.85829 41275 49E-1

C , .80351 12665 1E-01 , -.19074 81897 81 , .53639 95728 848 /DATA R4 / .10965 37513 3E+01 , -.42508 27310 , .38846 59265E-01 ,C .90163 63288 4E+01 , -.83618 16165 2E+01 , .68677 46458 42E+01 /DATA R5 / .36324 7242E-01 , -.19586 05807E-01 , .26562 3163E-02 ,C .11811 28946 7E+01 , -.19173 81565 11E+01 /

CC ARRAYS S01,S02,S11,S12,S21,S22,S31,S32,S41,S42,AND S51 CONTAIN THEC COEFFICIENTS FOR THE RATIONAL MINIMAX APPROXIMATIONS OF FUNCTION G3235CC FUNCTION G32 35 IS SPLIT INTO TWO RANGESC 1) 1.96 TO 39.0 , ANDC 2) 39.0 TO 72.7C S U IS FOR G3235 WHEN I = L AND J = RANGE

DATA S01 / -.84169 23789E+01 , .23872 25484 532E+02 ,C .34153 77155 782E+0.1 , .37527 53460 5181E+00 , -.18170 40625 5E+0C2 , .26218 17959 500E+02 , .60445 63133 7063E+02 ,C .91882 30277 2008E+01 /DATA S02 / .37507 87678 , -.62005 5E-06 , .66559 970E-02 /DATA Sll /.78846 29620E+02 , .91206 13209 7E+01 , .42443 21758 77,C .37345 90665E-03 , .14615 59362 357E+04 , .90910 31648 722E+03 ,C .20294 31379 20371E+03 , .15675 91578 28141E+02 /DATA S12 / .45884 40989E+00 , .47251 421E-04 , .38543 99288 3E+02C , -.15269 25801 174E+01 /DATA S21 / .12318 69840E+05 , .12554 27038 8E+04 ,C .12319 26244 985E+07 , .35573 03278 522E+06 ,C .96029 41566 9336E+05 , .34930 41580 3710E+04 ,C ,33892 58415 1490E+03 /DATA S22 / .30068 8166E+01 , .18131 35897 083E+04 , .13797 14904 9

C897E+03 , -.36495 05719 549E+01 /DATA S31 / .47979 4555E+02 , .87737 49E-01 , .18776 42412 05E+05 ,

C .18604 08633 788E+04 , .60350 26071 870E+03 , -.26033 04938 767EC+01 /DATA S32 / .22619 080 r -.37930 30375 016E+05 , .29610 27116 2818E

C+04 , -.80816 50346 1561E+02 /DATA S41 / .18630 978E+02 , .25483 64595 63E+05 , .54017 42516 43E

- 96 -

C+03 , .67456 19227 328E+03 , --.14500 45468 8329E+02 /DATA S42 / .37362 969E-02 , -.32703 434E-04 , .90153 35918 917E+03

C , -.53510 50385 2453E+02 /DATA S51 / .18194 782E-01 , -.31542 48E-03 , .76983 07313 4E+02 ,

C -.47372 46155 /CC BAR AND CBAR ARE BOUNDARIES THAT DELIMIT THE NUMBER OF TEPMS IN THEC SUM, DEPENDING ON THE VALUES OF Z AND QLSQC

BAR(EM,B)=EM*QLSQ-BC

CBAR(A,B,C)=A+B*QLSQ+C*QL4

I F ( Q L S Q . G E . . 6 9 6 ) GO TO 20I F ( W . L T . 3 9 . 0 ) GO TO 31

CCC RATIONAL MINIMAX APPROXIMATIONS FOR FUNCTION G3235 WHEN 39.0@W@72.7C21 G3235=(S02(1)+W*S02(2)) / (S02(3)+W)22 G3235=G3235+QKMK*(S12(1)+W*S12(2)) / (S12(3)+W*(S12(4)+W))

IF(W.GE.BAR(533.333,5.996))GO TO 28QK2=QKMK*QKMK

2 3 G3235=G3235+QK2*(S22(1)/(S22(2)+W*(S22(3)+W*(S22(4)+W)))}IF(W.GE.BAR(178.57,18.4))GO TO 28QK2=QK2*QKMK

24 G32 35=G3 235+QK2*(S32(1)/(S32(2)+W*(S32(3)+W*(S32(4)+W))))IF(W.GE.BAR(155.556,44.889))GO TO 28QK2=QK2*QKMK

25 G32 35=G3 2 3 5+QK2*(S42(1)+W*S42(2))/(S42(3)+W*(S4 2(4)+W))GO TO 28

CC RATIONAL MINIMAX APPROXIMATIONS FOR FUNCTION G3235 WHEN 1.96@W<39.0C31 G32 35=(S01(1)+W*(S01(2)+W*(S01(3)+W*S01 (4))))/

C (S01(5)+W*(S01(6)+W*(S01(7)+W*(S01(8)+W))))32 G3235=G3235+QKMK*(S11(1)+W*(S11(2)+W*(S11(3)+W*S11(4))))/

C (Sll(5)+W*(Sll (6)+W*(Sll(7)+W*(Sll(8)+W))))IF(W.GE.BAR{533.333,5.996))GO TO 28QK2=QKMK*QKMK

33 G32 35=G3 235+QK2*(S21(1)+W*S21(2))/(S21(3)+W*(S21(4)+W*(S21 (5)+W*C (S21(6)+W*(S21(7)+W) ) )) )IF(W.GE.BAR(178.57,18.4))GO TO 28QK2=QK2*QKMK

34 G3235=G3235+QK2*(S31(1)+W*S31(2))/(S31(3)+W*(S31(4)+W*(S31(5)+W*C (S31(6)+W))))

I F ( W . G E . B A R ( 1 5 5 . 5 5 6 , 4 4 . 8 8 9 ) ) G O TO 2 8QK2=QK2*QKMK

3 5 G32 35=G3 2 3 5 + Q K 2 * ( S 4 1 ( 1 ) / ( S 4 1 ( 2 ) + W * ( S 4 1 ( 3 ) + W * ( S 4 1 ( 4 ) + W * ( S 4 1 ( 5 ) + W ) ) )C))

I F ( W . G E . B A R ( 1 6 0 . 0 , 7 3 . 0 ) ) G O TO 28QK2=QK2*QKMK

36 G 3 2 3 3 - G 3 2 3 5 + Q K 2 * ( S 5 1 ( 1 ) + W * S 5 1 ( 2 ) ) / { S 5 1 ( 3 ) + W * ( S 5 1 ( 4 ) + W ) )28 POOSUM=QKMK*G3235

RETURNC

- 97 -

20 CONTINUEPCY=PCYLDR(X)FTR=QLSQ*QKQL4=QLSQ*QLSQFZ=EXP(-Z-Z)/SQRT(Z)QK2=Z*QKSQ

CC RATIONAL MINIMAX APPROXIMATIONS FOR FUNCTION G3124C

G3124=(R0(1)+Z*(R0(2)+Z*R0(3)))/(R0(4)+Z*(R0(5)+Z))*FZIF(ZSQ.GT.BAR(~85.715,-126.148)) GO TO 39FZ=FZ*QK2G3124=G3124+(R1(1)+Z*(R1(2)+Z*R1(3)))/(R1(4)+Z*(RI(5)+Z))*FZIF(ZSQ.GT.CBAR(-111.068,520.145,~401.933)) GO TO 39FZ=FZ*QK2G3124=G3124+(R2(1)+Z*(R2(2)+Z*R2(3)))/(R2(4)+Z*(R2(5)+Z))*(Z-Z2)C*FZIF(ZSQ.GT.CBAR(~273.068,821.134,~567.974)) GO TO 39FZ=FZ*QK2G3124=G3124+(R3(1)+Z*(R3(2)+Z*R3(3)))/(R3(4)+Z*(R3(5)+Z*(R3(6)+Z))C)*(Z-Z3)*FZIF(ZSQ.GT.CBAR(-172.719,558.473,-414.455)) GO TO 39FZ=FZ*QK2G3124=G3124+(R4(1)+Z*(R4(2)+Z*R4(3)))/(R4(4)+Z*(R4(5)+Z*(R4(6)+Z))C)*(Z-Z4)*FZIF(Z.GT.CBAR(-120.17,354.221,~254.571)) GO TO 39FZ=FZ*QK2G3124=G3124+(R5(1)+Z*(R5(2)+Z*R5(3)))/(R5(4)+Z*(R5(5)+Z))*(Z-Z5)C*FZ

39 POOSUM=PCY-FTR*G3124RETURNEND

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