THE UNIVERSITY OF MICHIGAN 7133-1-T AFCRL-65-283(I) Tables of Prolate Spheroidal Functions for m = 0: Volume I H. E. Hunter, D. B. Kirk, T. B. A. Senior and E. R. Wittenberg Contract No. AF19(628)-4328 Project 5635 Task 563502 Scientific Report No. 1, Volume I April 1965 Prepared for Air Force Cambridge Pesearch Laboratories Office of Aerospace Research United States Air Force Laurence G. Hanscom Field Bedford Massachusetts 01731

THE UNIVERSITY OF MICHIGAN 7133-1-T SUMMARY The eigenvalues Xon(c) and normalization constants N on(c) are tabulated for c=O.1(0.1)10.0 and n ranging from 0 up to a maximum of 20. The radial functions of the first and second kinds, and their first derivatives, are given for the same c and n for -=1- 0000500, 1 0050378, 1. 0206207 and 1- 1547005, corresponding to prolate spheroids of length-to-width ratios 100:1, 10:1, 5:1 and 2:1 respectively. Values of the angular functions are contained in a companion report (7133-2-T). All computations were carried out on The University of Michigan IBM 7090 computer. The program, which can also be used for the determination of the functions of orders 1 and 2, is described in some detail.

THE UNIVERSITY OF MICHIGAN 7133-1-T I. INTRODUCTION Prolate spheroidal functions occur naturally in solutions of the wave equation in regions bounded internally or externally by a prolate spheroid and, over the years, have been used in the analysis of such problems as the scattering of an acoustic wave by a rigid spheroid (Spence and Granger, 1951), the scattering of an electromagnetic wave by a perfectly conducting body of this shape (Siegel et al, 1956), and the radiation from an excited slot cut in the surface of a metallic spheroid (Chu and Stratton, 1941). A survey of these (and other) applications of spheroidal functions has been given by Flammer (1957), and since the angular functions Smn(c, rT) for m =0 are also the eigenfunctions of the finite Fourier trans - form (see, for example, Slepian and Pollak, 1961), they also play a prominent part in studies of the radiation patterns of planar slots (Rhodes, 1963). In the course of these investigations, many spheroidal functions computations were performed, and most of the data in existence prior to 1957 was incorporated into the tables presented by Flammer. Additional data has become available since that time (a list of the more important sources is contained in Sleator, 1964), but even so the resulting tabulations are far from complete. Not unnaturally, individual authors have confined themselves to values of the separation parameters and arguments of most interest for the particular application being considered, and the coverage provided by these piecemeal attacks is spotty at best. This fact was impressed on us during a recent study (Sleator, 1964) of the scattering and radiating properties of prolate spheroidal bodies. Even for the more simple problems of the scattering of a plane acoustic wave by a hard or soft prolate spheroid at end-on incidence, or the radiation of electromagnetic energy by a uniformly excited azimuthal slot on a perfectly conducting spheroid, all of which are cases where the solution can be expressed as an infinite sum involving only the ________________________ 1 ________________1_______

THE UNIVERSITY OF MICHIGAN 7133-1-T spheroidal functions for m = 0, the existing tabulations are not sufficient for numerical work at other than the smaller values of c. From time to time we have endeavored to remedy this situation, at least in part, by hand computations of the spheroidal functions for selected c and n, but this process gets progressively more tedious as c increases. With the advent of digital computers of large storage capacity, however, such computations have become both feasible and economical. Prompted finally by the desire for information about the scattering behavior of a spheroid, with data of sufficient scope and accuracy to serve as a standard against which to measure the effectiveness of various approximate methods for the determination of the scattering, a comprehensive computation of the spheroidal functions was undertaken. Although attention was concentrated on the functions of order zero (i. e. m=0), the program was so arranged that by change of the input conditions only, the functions of orders 1 and 2 could also be computed. A detailed description of the program, along with specimens of the various print-outs, is given in Section III. In Table 1 the eigenvalues on((c) and normalization constants Non(c) are given for c=0. 1(0. 1)10. 0 and n=0(l)nl, where n1 ranges from 14 for c=0. 1 to 20 for c 0. 8. The radial functions of the first and second kinds, and their first derivatives with respect to A, have been computed for the same c, for (substantially) the same n, and for 1 = 1- 0000500, 1. 0050378, 1- 0206207 and 1- 1547005, corresponding to prolate spheroids whose major and minor axes are in the ratios 100:1, 10:1, 5:1 and 2:1 respectively. The results are given in Table 2. Values for the angular functions are contained in a companion report (7133-2-T). 2

THE UNIVERSITY OF MICHIGAN 7133-1-T II. THE FUNCTIONS The notation and definitions which we have employed are identical to those of Flammer (1957). For the sake of completeness, we list here the formulae used and give a brief description of the methods by which the functions were computed. We shall not, however, attempt to justify these formulae, and for details and proofs the reader is referred to the above-mentioned work of Flammer. The angular function Smn(c, rl) satisfies the differential equation d rl 12m72) d Smhn(c, r] + mn-c2 r2- 2_ Sm(C, r)=0 Similarly, for the radial function Rmn(c, l), d- [2-1)d Rmn(C,)- Ximn-c2~2+ ]Rmn (c' )=0 where the separation constants Xmn and m are the same as in the preceding equation. Xmn is also the eigenvalue, and is approximated by either a power series or an asymptotic series, after which an iteration scheme is applied until a sufficiently accurate value is obtained. The angular and radial functions are expanded in terms of Legendre functions. The expansion coefficients drmnc) satisfy the recursion formula (2m+r+2)(2m+r+l)c2 n, c)m+r)(m+r+l)-kC) (2m+2r+3)(2m+2r+5) r+2 [ r)(m+r) () 2(m+r)(m+r+1)-2m2 -1 2 dmn(c (2m+2r-1)(2m+2r+3) c r r(r-1)c2 I> + r-r-1)c d (c) =O ~ (r 0). (1) (2m+2r-3)(2m+2r-1) r-2 3

THE UNIVERSITY OF MICHIGAN 7133-1-T Let m= (m+r)(m+r+l)+ 1c2 - 4m -1 (r0) r 2 (2m+2r-1)(2m+2r+3)] (r) tim r(r-1)(2m+r)(2m+r-l)c4 (r 2) Or (2m+2r-1)2 (2m+2r-3)(2m+2r+l) mn Nm (2m+r)(2m+r-l)c2 dr (r2) r (2m+2r-1)(2m+2r+l) dmn r-2 (note the omission of the negative sign from the last of these equations in Flammer's book). We have then 1m m r (r )2) (2) m m = mn r+ r+2= r mn mm N r m= TMr n In N = 7-X and N -X 2 o mn N3=7 1 mn and using equation (2), the continued fraction m m m nm r+2 r+4 r+6 N = *r (4) r+2 m m m r+2 mn r+4 mn r+6 mn is obtained. From equation (3), however, m m Nm mI r rr-2 N =7 -- (5) r+2 r n- mn m r-2 mn Yr-4 mn and these two expressions for N lead to an equation whose roots are the desired r+2 eigenvalues. __________________________ 4

THE UNIVERSITY OF MICHIGAN 7133-1-T The power series with which to approximate X (c) is mn(c) =,mn 2k (6) rnn k2k where'mn n(n+) rnmn 1 1 (2m-1)(2m+l)] n(n+l) 2 2 (2n-1)(2n+3) i rmn 1 (n-m-l)(n-m)(n+m-l) (n+m) (n-m+l) (n-m+2)(n+m+l)(n+m+2) 4 2 L (2n-3)(2n-l)32n+1) (2n(2n+3)3(2n+5) I mn= (4m2 1) (n-m+l)(n-m+2)(n+m+l)(n+m+2) 6 L (2n-1)(2n+1)(2n+3) (2n+5)(2n+7) (n-m-l)(n-m)(n+m-1)(n+m) (2n-5)(2n-3)(2n-l) (2n+1)(2n+3)J,mn= 2 (4m 2 -1 (n-m-1) (n- m) (n+m-l) (n+m) 8'L ( 5L(2n-5) (2n-3)(2n-1) (2n+1)(2n+3) (n-m+l) (n-m+2)(n+m+l) (n+m+2) 1 (2n-)2 (2n+l)(2n+3) (2n+5)(2n+7)2 1 (n-m-3)(n-m-2) (n-m-l)(n-m) (n+m- 3)(n+m-2)(n+m-l)(n+m) 16 L(2n-7)(2n-5) (2n-3) 3(2n-1) (2n+l) (n-m+l) (n-m+2) (n-m+(n-3)n- m+4) (n+m+l) (n+m+2) (n+m+3) (n+m+4) (2n+l)(2n+3)4(2n+5) 3(2n+7)2(2n+9) r 92 2 2 2 2 2 2 21 1 (n-m+l) (n-m+2) (n+m+1) (n+m+2) (n-m-1) (n-m) (n+m-1) (n+m)2 (2n+)82 2n+ 3) (2n+5)2 (2n-3)2(2n-l)7(2n+l)2 1 (n-m-) ) (n-m+l) (n-mn+2) (n+m-1) (n+m) (n+m+l)(n+m+2) 2 L (2n-3)(2n-1) 4(2n+)2 (2n+3) 4(2n+5)

THE UNIVERSITY OF MICHIGAN 7133-1-T The power series (6) converges rapidly for small values of c. For larger c, however, we use the asymptotic series (National Bureau of Standards, 1964, p. 756): 2_ (c)= cq+m (q + 5)- (q2+11 - 32 m2) mn 8 64c 1 2 4 5 (q2+26q2+21)-384m2 (q2+1)] 1024c2 m -1 (33q5+1594q 3+5621q)- 128 (37q3+167q)+ 8-q] - 1 2 (63 q6+ 4943 q4+43327q2 +22470) c 2562 2 4 3m 2 - 5(115 q4+1310 q2+735)+ 3m (q+ 1) e- - (527q7 61529 q5+1043961q3+2241599q) c" _1024 2 32-1024 (5739q +127550q3+298951q) m4 3m6q +512 (355 q+1505q)- +.. where q = 2(n-m)+l. The machine program described in Section III shows how the choice between the two methods of approximating X (c) was made. -— 6 —--------------- 6 ------------

THE UNIVERSITY OF MICHIGAN 7133-1-T Having found a first approximation to Xmn(c), Bouwkamp's iteration technique is then applied. From the continued fraction forms (4) and (5) m m Ui( Imn)-m - n-m n-m-2 Ul( ) n-m mn m m*' Tn-m-2 mn n-m-4 mn m m U ( )= - n-m+2 n-m+4 2 mn m Tm -X r,o-x rn-m+4 mn Tn-m+2 nm +4mn leading to the correction 1( )+ U2("mn) 6X = mn DI Im m 2 Im 2 m.2 - n-m n-m n-m-2 nm+2 nm+ n-m+ 1-+ ---- m m - - +'' + /~m ~2 / m2 \/ \2 DIm DDm L(Nm N m L +2, n —m+2 n -m+4 Iteration is continued until a preassigned accuracy is attained, or until a specified count is met. inn, ran mn The quantity Nr is proportional to d /dr 2 and to determine the individual r r r-2 values of the expansion coefficients, the normalization that is used is S (c, O)=Pm(O)= (-l (n+m) (n-m) even, nm n n n-mi, n+m, 2 ( 2 )'( )' 2 2 S' (c, )=Pm (o) 1(nm-1 )2(n+m+l) (n-m) odd. mnn n n n-m-l n-+m+1 2 (-2 )! ( 2 7

THE UNIVERSITY OF MICHIGAN 7133-1-T The resulting normalizing relations are O (-1) d/ (r+2m)! dmn (_1 m/ (n+m) 2 - -r(r,(r+2m), r?-m(n)-(n+m) (n-m) even, 0 (1) (r+2m)! - ( n+- m mn(- (r+2 ml)! (2r+2m = - (n-m) odd, r r-l(r+am+l)! r n n-m-n ml) nm r 1 2 ( 2 2 2 2 )! ( 2 where the prime attached to the summation symbol indicates that the summation is is then evaluated, and the X (c) and N (c) are listed in Table 1. The angular functions are given by co S (c,r)= d m (r), (n-m) even, (7) man 2r d am+2r r=o O0 Smn(c, r Z) = dmn +2r+l() (n-m) odd, (8) r=o where -1 r < 1, but since S (c, r) is even or odd according as (n-m) is even or odd respectively, it is sufficient to restrict the computation to 0< rl7 1. Thus,for example S 0(c,-r) = -So(c, r7); S 2(c, -t7)=S02(, r)

THE UNIVERSITY OF MICHIGAN 7133-1-T The Legendre function is defined as mn (r)=(_12)/2 dm P (ri) n d m n dr and formulae for the differentiated functions S' (c, r;) can be obtained from (7) and mn (8) by substituting p' in place of P. The angular functions were computed for r = 0(0.05)1.0 and the results are presented in the companion report. To evaluate the radial functions it is convenient to rewrite the recursion formula (1) in the form A d M+B nd +C d =0 r+2 r+2 r r r-2 r-2 where r > 0 and Amc) (2m+r)(2m+r-1) 2 Ar (2m+2r-1)(2m+2r+l) Bmrc)= (m+r)(m+r+l) (M+ 2(m+r)(m+r+l)-2m2-1 2] r [ Xmn (2m+2r-l) (2m+2r+3) cj C m(c) (r+l)(r+2) r (2m+2r+l)(2m+2r+3) If we now extend the range of r to include negative values, we have dmn _Am d CM A -k+2 mn -k-2 -k -k mn -k-2 and though the dn for r -2m vanish, we can define mn lim -r+p p/r p -> p ______________________________,9 -______________

THE UNIVERSITY OF MICHIGAN 7133-1-T and for r > 2m these values exist and are non-zero. If, for example, we put -k= -2+ p, then for m = O on on oA p/2 = lim -2+p _ lim P P don p- 0O on p-O oC0 o o 0O on -4 -2 -2 Bon -4 and lim p lim (p-l)c 2 p-tO p p->O (2p-1)(2p+l) Similarly, for -k= -1 + p and m = 0, don on A o p/l - lim -l+p lim p 1+p on p -* O on Pp-0 C0 0 on -3 -1 -3 and A0 2 lim l+p _ lim (p+I) c 1 2 p-O p p*O (2p+-1)(2-p+3) 3 More generally, it can be shown that, for any m and n Am 2 lim -r+p - c P' o 0 p (-2m+l)(-2m+3) (n-m) odd 2 (2m+l)(-2m+l) (n- even Other ratios of the form dp /+2 / dp/, may be evaluated using the continued fractions. ________________________ 10

THE UNIVERSITY OF MICHIGAN 7133-1-T The radial functions depend on e as well as on c and n. A, in turn, is related to the major and minor axes of the spheroid, a and b respectively, by the equation 1 -hi and in the tables the radial variable ~ is printed as XI. Since ~ > 1, the Legendre function P m() is defined as n pm()=(2_l)m/2 d P n dCdm n For the radial function of the first kind, (1) 1 ) R (c,)- (c, mn k(1) (c) mn mn where 00 co Sn(c )= d P+2r( (n-m) even, Smn L-) 1m 2r m+2r r=o co S (C,)=Y. mn pm Smn (c,= r+l m+2r+l() (n-) odd r=o and the joining factors are defined by (2m+l)(n+m)! 7 d7 mc) (2 r) /,v1)~ ^- r r;,(c)-. r=o k( (c)= -— +-, (n-m) even n 2 +mdn(c)c m!( n-m)! (nm 0 2 2 _______________- 11. —-------------— ~~~~~~~~'

THE UNIVERSITY OF MICHIGAN - 7133-1-T O I.,'mn,.(2rm+r)! (2m+3)(n+m+l)! ) dr (c)2r! ~~~r=(1) r__ k( ) (c)= -r1 (n-m) odd rnn n+rn rnn nm+i, n-m-l n+m+l, 2 d n(c)c m+m!( )!( For the radial function of the second kind, R (C' (2) (c) r n 2k dr M)rnrn k (c) r= -2m,2m+l + 2md (c)PM r=2m 2m+1l P/r r-m-l n where the Q (g) is the associated Legendre function of the second kind, and n-rn,nr 1 (n+mr m 2n-m (2m) (n-n 2 )!n+m d (c) oo' k(2)(c) 2 2 -2m g drn ( +r) mn r (2m-i)m! (n+m)! c r=o (n-m) even, n-m,n-m- 2n+m+l,mn,, 2 (2m);( 2 Cn (2), 2 2 -2W.( —) ( — n 0 l(2m+r)! mn (c)=-m-2 (2m-3)(2m-l)m! (n+m+l)' c r=l (n-m) odd. rn' Here again, the derivative functions are obtained by substituting P (r) and Mt n Qm() in place of Pn() and Qm(). n n n 12

THE UNIVERSITY OF MICHIGAN 7133-1-T III. THE COMPUTER PROGRAM This section of the report will supply more detailed information about the computer program used to prepare the tables of the prolate radial and angular wave functions. A general description of the program, including some documentation of computational difficulties, followed by a brief description of the subroutines, input requirements for the program, illustrations of various output options, the program listing, and some examples will be presented. 3. 1 General Description The program was written in FORTRAN II and UMAP for use on the IBM 7090 at The University of Michigan. The system configuration uses tape 7 (B1) for input, tape 6 (A3) for output and (for this particular program) tapes 2 (B2), 3(B3), 4(A4), and 9 (B5) for intermediate storage. With the modification of such statements as WOT (for WRITE OUTPUT TAPE), RIT (for READ INPUT TAPE), ELOG (LOGe) and SIN, it may be adapted to any standard system. FTRAP is a system subroutine which simply sets underflow traps to zero. The program was written as a group of subroutines for simplicity, ease of check-out and speed of attaining the final product. It has not been'condensed' since the expense of reprogramming would probably not be justified by a material saving in execution time. Some'debris' may exist where tests and variables have been changed or deleted. The initial attempt to calculate both the radial and angular functions over the entire range in one pass failed because of various convergence and overflow difficulties. Rather than ignore the knowledge that we have gained from this experience, the authors feel that some documentation may be of value in assisting those who might wish to use the program in the future. The initial run, using a single tolerance of 1 x 10-8 for all convergence tests, failed in the eigenvalue calculations for the following cases, listed as (c, n): (7.4, 0), (8.5, 0), (8.6, 0), (8.7, 0), (8.9, 0), (9.3, 0), (9.8, 0), (9.8, 4), (9.9, 4) and (10.0, 4). _________________________ 13 _____,__

THE UNIVERSITY OF MICHIGAN 7133-1-T It was found that the tolerance criterion was too severe for the first seven of these cases. In the case of (8.6, 0), for example, both U1 and U2 in Bouwkamp's correction formula printed as 16.829322 (maximum machine capacity), yet the difference oscillated between + and - (1.4060035x10-6). The tolerance was eased to 1x10-6 for the calculation of these seven values. A test which has proved valuable in hand calculations (though it has no mathematical basis) is that the values of dr must be positive for r, n, negative for r=n+2 (and thereafter alternate in sign). This test is in the program. Its purpose is to select the other eigenvalue approximation if the one first chosen fails (this test may well have been applied during the course of the calculation). It did not hold for (9. 9, 4) and (10. 0, 4) and was bypassed (by binary patch) for these cases. m The correction routine for (9.8, 4) resulted in a very low Nr value which caused the process to converge on the eigenvalue for n=6 rather than for n=4. This may be the break-off point for the test cited above, and is the one isolated case where a hand approximation was inserted into the program. The correction routine adjusted it properly. If an eigenvalue failure occurs, the program will set a dump parameter and repeat the calculation giving intermediate printouts of the routines involved, after which the computation will continue normally. The tables of eigenvalues, normalization coefficients and S values were therefore run with all tolerances set to lxl0 except in the ten cases listed above. When calculating the R values, it became apparent from the Wronskian check that a tolerance of lx108 resulted in the loss of too many terms in the d and dp r pk series. Thus separate tolerances were used - one for the eigenvalues and one for the series. These are designated as EGCV and CVT respectively in the input parameters. EGCV was set at lx106 and CVT at xl10-20 for the R run calculation. Simultaneous S calculations on isolated values were also made to determine whether 14

. THE UNIVERSITY OF MICHIGAN - 7133-1-T the easing of the eigenvalue tolerance would have any effect on the S values. No differences were found in the S values to six significant figures. As anticipated, three cases failed. The appropriate changes were then made in the program, and the calculations completed. Overflow problems were encountered in both the S and R runs. This necessitated putting a test on maximum values in the program since these overflows appeared not only in DCAL but in the continued fraction routine and the joining factor routine as well. When overflow difficulties were encountered in the continued fraction routine (CF2) which evaluates'from the top down', an alternate routine (CF1) which evaluates'from the bottom up' was inserted. There are currently two potential overflow error signals which are set in the program: (1) if the absolute value of the product of the d ratios exceeds Ix1030, and (2), if an R2 or R2' values exceeds lx10. If either of these conditions is reached, subsequent calculations for higher n are deleted and the next value of c is chosen. This accounts for the deletion of some of the S tables for high n at low values of c. Storage allocation for most of the arrays was set at 20 (permitting subscripts to 38 or 39) and 40 values of the Legendre tables were computed. Large values of e cause the Legendre values to increase rapidly and for large n, 20 terms of decreasing dr and dpr values may have been insufficient to offset the increasing Legendre functions in a few instances. As a specific example,a value of 1.6666667 for g gave a Wronskian check of.63x10-7 at c=.l, n=0 but only.42x10-1 at c=9.6, n=0. For higher n the check improved. This merely emphasizes the fact that for certain ranges of arguments the program should probably be expanded. Timing estimates are variable due to repeated use of Legendre tables in a large run but only one use in a single case. With the valuable hindsight gained from the experience just described, the set of tables would take about one hour running time on the 7090. +Henceforth, we shall frequently denote the radial functions of the first and second kinds by R1 and R2 respectively. A prime will be added to denote the derivative. ______________________15 _______________________________

THE UNIVERSITY OF MICHIGAN 7133-1-T The program may be broken down into the following components. 1 Main program 7 DCAL 13 HYPER (subroutines) 2 PQCAL 8 SCAL 14 GAM 3 APROX 9 DNCAL 15 BETA 4 ASYM 10 FJOIN 16 FACT 5 CORR 11 R1CAL 17 CF1 6 FNORM 12 R2CAL 18 CF2 A brief description of the input, output and purpose of each component will now be given. 1. Main Program This program reads the input data, controls the flow of the problem and, with the exception of the printing of the Legendre tables, controls the printout. The eigenvalue calculation is also imbedded in this phase of the program. This is done since the convergence upon the correct eigenvalue is not checked until after the d values are computed. A simple editing of gross errors in input data is also made. A good idea of the overall program may be quickly obtained from the Flow Chart (Section 3. 4). 2. PQCAL This subroutine calculates 40 values of the necessary Legendre tables for all values of e and ri. It will calculate 40 values of P, P', Q, and Q' and negative values, if required, for m = 1 and m = 2. INPUT OUTPUT X - (g or r;) PP P table MM- m QQ Q table STEST (1 for P only, 0 for P and Q) QPR Q Prime table IPQT (Print Option) QNEG Q Negative for m / 0 QNEGP Q Negative Prime for m / 0 PPR P Prime. 16

THE UNIVERSITY OF MICHIGAN 7133-1-T The methods by which the values are calculated are outlined below. The functions of the first kind were obtained from the recurrence relation (2v+l)zP(z)-(v+p)Pp (z) (z)= v - () v -v+l and the derivatives from d pAL (v-=+l)P1 (z)-(v+l)zP~(z) d-pz ) +1 V (10) dz v 2 z -1 The first two values are calculated explicitly. The tables for z = 1 are calculated directly. The Legendre functions of the second kind for 1. 00 < z I| 1. 01 and 1A = 0 are found using ( 1 )=P (z) - Pkl P z)P-k(). (11) 1 2 v z- k Qk=l When p is 1 or 2 and 1. 00 < |Iz < 1.01, a recurrence relationship is used, once for p = 1 and twice for,p = 2, after the Q values for the E = 0 case have been found by equation (11). The recurrence formula is (v-p)z Q(Z_)-(v +M)Q& (z) +l ( -v v-1 Q +l(z) 2 (12) V (z 2_ 1) ( 1/2 When z| > 1.01, the program uses 1A ie-Pv-1'6 r y ~'(v++) -v - fl1 Pzl)2/2 Q (z)=eTM 2-v-1 7r1/2 F(v++3 ) Vl(z2l/2 (13) F(2, l 2;v+3/2; ) - 17 z 17

THE UNIVERSITY OF MICHIGAN 7133-1-T or v(v+ -V-3-I 52_1)/F2 24v+p l+v+tC 3 1 GP (z)=(_ Z v 1-3-5 —-(2v l) 2' 2'2 z2'2 since v and j are positive integers and z is real. The subroutine to evaluate this hypergeometric series is HYPER. After having calculated the Q values, the derivatives are obtained from d (zv —P+l)Qv+l (z)-(v+l)zQe(z) d ) -z V+)1 2 (14) dz v 2 The values of Q for v < p which are required in the R2 calculation for p > 0 are computed by using (10) with Q substituted for P in the recurrence relation. 3. APROX This is an eigenvalue approximation routine using the polynomial approximation. INPUT OUTPUT FN - (Floating) n EIGEN - Eigenvalue approximation FM - (Floating) m C -c PTEST - Optional dump This approximation is used first whenever n > 6, or whenever n~ 6 and c < 5. If the incorrect eigenvalue is obtained ASYM is tried. 4. ASYM This is an eigenvalue approximation routine using the asymptotic series. Input and output are the same as APROX. The routine is first used whenever n A 6 and c > 5. If the incorrect eigenvalue is obtained, APROX is then tried. If both fail, an appropriate remark is printed and a snapshot dump is taken. 5. CORR This routine is used to obtain the correction to the eigenvalue approximation. ___________________________ 18

THE UNIVERSITY OF MICHIGAN. 7133-1-T INPUT OUTPUT n DELAM - Correction factor m FLAM - Eigenvalue c RTEST - Convergence test PTEST - Optional print. 6. FNORM This routine calculates the normalization factor for the sum of the d values. INPUT OUTPUT n ANS - Normalization factor m EOTEST (Even/odd test) PTEST ERR (Error indication) 7. DCAL This routine calculates the d values, N array, and the normalization factor of the Legendre function Nmn. INPUT OUTPUT m FN1 - Final N array (in common) c D - d array FNANS - Eigenvalue ITRM - Number of d values DTEST FNMN - Normalization Nmn PTEST ERR - Error flag NUM - Number of items in N array (in common) 8. SCAL This routine calculates the function Smn(c, ri). INPUT OUTPUT D - d array S - Smn(c, 7) ND - Number of d's SPRIME - Snn(c, r) P - Array of P values DERP - Array of P' values PTEST ~~_____ ______.... 19 ___________

THE UNIVERSITY OF MICHIGAN - 7133-1-T 9. DNCAL This is a d negative calculation routine. This also calculates the d values. p/r INPUT OUTPUT m DNEG - There will be m of these c DRHO - d/ array p/r FLAM - Eigenvalue NDRAT- Number of di, DSTART - Starting D value DNTST - Convergence test (eliminated) PTEST (1) (2) 10. FJOIN This routine calculates the joining factors k and k Inn Inn INPUT OUTPUT m FK1 - k (1) joining factor mn n FK2 - k2 joining factor c D - d array DNEG - DNEG array NOD - Number of d's PTEST (1) 11. R1CAL This routine calculates R( (c, z) by effectively computing mn 1 OD d mn Pm (z) k(1) U 2r m+2r k (c) r=o INPUT OUTPUT D - d array R - R( ND R1PR - Rm) FK1 - Joining factor P - P array for R values DERP - P' array for R PTEST 20

THE UNIVERSITY OF MICHIGAN 7133-1-T 12. R2CAL This routine calculates n)(c, z) and R(m)n(, z) INPUT OUTPUT m R2 - Rl D R2DER-R(2)1 mn ND DNEG Q - Q array P - P array PDER - P' array QDER - Q' array QNEG - Q negative array QDRNEG - Q' negative array DRHO NDRHO FK2 - Joining factor PTEST The remaining subroutines are used by the other routines listed above. 13. HYPER This routine evaluates the hypergeometric series oo (A) K(B)K K L=0 K! (c) z where (A)K=A(A+i). *(A+K-1), (A) =1. K=0 K 0 An answer will be returned if the relative difference of two successive terms -< x10-8,or at the end of 2000 terms, whichever occurs first. INPUT OUTPUT A ANS B ERR C Z This routine is used to evaluate the tables of the Legendre functions. 14 - 15. GAM and BETA These are external FUNCTION statements which evaluate r and S1m in the CORR subroutine during the calculation of the Nrm evarrat, 7 r array. ________________________ 21 ________________________

THE UNIVERSITY OF MICHIGAN - 7133-1-T INPUT FM - (Floating m) r c 16. FACT This external function prepares a table of N! where N < 33. INPUT N 17 and 18. CF1 and CF2 These are continued fraction functions written in FAP for the evaluation of dpr values in DNCAL. CF1 evaluates the expansion'from p/r the bottom', while CF2 evaluates'from the top'. Expansion is of the form al F=b + a b0 bl+a2 b2 + a3 etc. INPUT OUTPUT B ZERO - bo m - Number of iterations performed A - Array (al, a2,... ) F - Value of the fraction B - Array (bl, b2, b3,... ) N - Number of elements The subroutines used are modifications of the University of Michigan Institute of Science and Technology Computation Department Program WR-58, written by R. Gould. 22....

THE UNIVERSITY OF MICHIGAN 7133-1-T 3.2 Input Specifications for Main Program The input for one'case' consists of one card if NZ (the number of ~ values) is 0, or two cards if NZ / 0 ( < 5). Much of the data read in consists of options controlling the printed output. The data description and card columns are as follows. CARD 1 DATA DESCRIPTION COLUMN ID Case number 1 - 5 IST S test 6 IRT R test 7 IEGD Print eigenvalues and d's 8 ISP Print P values for S 9 IRPQ Print P and Q values for R 10 IRHO Print DNEG and RHO for R 11 PTST Intermediate dump 12 CVT Series convergence tolerance 13 - 18 M m 19 CS c start 20 - 24 CIN c increment 25 - 29 NOC Number of c's 30 - 32 NS n start 33 - 34 NIN n increment 35 - 36 NON Number of n's 37 - 38 ETAS ETA start 39 - 45 ETA IN ETA increment 44 - 48 NOE Number of ETA's 49 - 50 NZ Number of Z values ( 5) 55 NIT Number of iterations 56 - 58 IPAGE Starting page number 59 - 61 IEGFN Print eigenvalues and normalization 62 LN CNT Line count for S print (If 0, new page for each n) 63 - 64 EGCV Eigenvalue convergence tolerance 65 - 69 ___________________________ 23 _____

THE UNIVERSITY OF MICHIGAN. 7133-1-T CARD 2 (Must not be present if NZ, Column 55, Card 1, is zero ). DATA COLUMN Zl 1- 10 Z2 11 - 20 Z3 21 - 30 Z4 31 - 40 Z5 41 - 50 (Decimals should be punched.) Although the data descriptions are rather self-explanatory, a few additional comments might be helpful. Columns 6, 7, 8, 9, 10, 11, 12 and 62 are individually 0 or 1 choices - 0 to bypass the option, 1 to take it. If both IST and IRT are 0, an error statement will be printed and the case deleted. LNCNT controls the number of lines per page when the S table is printed. If this value is 0, a new page will be started for each n. The convergence tests or tolerance limits are read in from Columns 13 - 18 and 65 - 69. It is suggested that a form such as 1.E-8 be used. The answers, of course, may vary according to the magnitude of this number. The number of c values (NOC) and the number of ETA values (NOE) are used as input to minimize rounding problems which could result by continuous addition of the respective increments. Double precision arithmetic was not used in this program.'Cases' may be run consecutively. 3. 3 Output Description Since the output is primarily in the form of tables, it is also largely selfexplanatory (see exhibits). One feature which will be of assistance to the analyst is the printout of the relativized error in the Wronskian computed during the generation of P and Q values (for m = 0) in the R calculation. 24

THE UNIVERSITY OF MICHIGAN 7133-1-T It might be noted that the input is designed such that an entire array of values or one single case may be computed. If any difficulty is encountered for one value, it may be handled individually with a T' punch in Column 12 to give sufficient data to isolate the problem. Since PTST (or PTEST as it is sometimes called) gives a rather massive printout, it should be set to zero except for isolated cases. c is printed out to one, and Ye to two decimal places. If intermediate values are required, the input provisions already exist. The output formats, however, must be modified to print the number of decimal places desired. 3. 4 Exhibits In this section, the flow charts of the entire program, typical examples of the input and output, and the 7090 program itself are presented. The following listing indicates where each can be found. Page Flow Chart No. 1 26 Flow Chart No. 2 27 Input Example 28 One-page Specimen Output (optional and tabular) Heading Page 29 Legendre Functions for r7 Value 30 Legendre Functions for e Value, m=0 31 Legendre Functions for 9 Value, m=2 32 S Table 33 R Table 34 Eigenvalues and d Values 35 Eigenvalues and Normalization 36 Joining Factors, d Negative and d Values 37 Detailed Printout Example r 38 7090 Program 39 - 65 25 -

START FLOW CHART 1 Read and print inA^ - ~put data. Initialize Edit input for obvious errors Print ( Er^ror-? Yes. error v message No 1".ST.EST on? No Initialize Initialize Compute and Compute and store tables of store tables of P, P and P' for r P, -Q, Q' for Yes Yes Save SP on? Save RP on?'-o No Write tables on Write tables on C;| l l t ] oubutTape 6 output Tape 6 output Tape 6 All r1done? es All g done?. No No Increment Choose next 7r e I I I~ value 26

FLOW CHART 2 Initialize c and n Calculate X, d' Nmn Yes Write c, n, X, Save ds d on Tape 2 N o'N o - Non Yee Wrt Save Nmn? STEST on? N on Tape Initialize No --- r1 -- I Calculate kl, k |Calculate SS' * DNEG and and write on DRHO * Output Tape 6 avDWrite k Dk No Iave DHROses _ DNEG and DRHO on Tape Increment o t Initialize r7 N —.Finished? ------- Yes Calculate R1,Ri, (RTEST on? JR R2, _ No Increment n and c fin.?- (Both tests on?. V Yes No 0 l ____X No L________ o_____ No R- - Yes Radtape 9 Choose next Read Tape _Yes Write R's on Write R's on Writ e omBsts on_ Output Tape 6 Tape 4 No eW RTEST ones Yes Save d's? ) finished? No _ y — R ad tape 9 RaChoose next S ve Nmn? Write on 6 value RTEST on? I Yes No Yes Read tape 3 (^)< —- -— ^Save DRHO' ^^e on 6 -- A 27

050031111001.E-200 0.1.5 20000102.3.3 03 402o0001111.E-6 II I I I II II I I IolIoooooloooooooIloooooooooolllloloooooooooooooooooIIlllooooooooooooooooooo I 2 3 45 I 7? I 1011 H2I3 S 1 7 a 1t 20212223242526272 3123313 41422427343345 7 401 3323334353 I37M3S4SIN42 4U4447#4S4sS O7U0 7 173747576 7 7 7m4o0 2222222222222222|2222222222222|222222|222222222222222222|22222222222222222222222 3333|333333331|3333333|1333|3333333333333||333||333|333333333333333|33333333333333 111 I I l l 1111 I I I 11 11 I I I111 I1 I 1111 111 I I I I I I I I I I III111 I I 111 1I II 3 2 3 4 5 3 7 8 3 10 1 1213 14 1511 3 1 20 21 22324 2526 221 30 31 3233 313I36 373S3404143 4344454647 43 4 5051 535455356515533 162 33 465666 6 370 71 12731 4576 3 3 3 37 444*444444444444444444444444444444444444444444444444444|4444444444444444444444444 CARD 1 1.00005 1.0050378 1.0206207 1. 1547005 I I I I 5|555555555555|55555555555|55555555555555555555555555555555555|555|5555555555555 oollloooolllooooololooloooooolo66666666666666666666666666666666666666666666666666666666666666|66666666666 1 3 45 I 7 I 110n 21 M H 17~1~2~2222324 2521 2/23293s31 g4.,423233341sc u s s s 42s4s#47#4$uMMasso gmsnm74 9999 999 9999999 999 99 99999 99999 9999999999999999999999999 9 999999 99999999 1 2 3 4 S I 7 I I 1011 12 13 14 1516 17 18 " 20 2127 23 24 25 26 27 212930 31 32 33 34 336 37 3839 40 4142 43 44 45 46 47 414915 51 52S354 5556 759/55I616263665 66676869 70 71 7273747576 T 7 787980 5081 CARD 1 1.00005 1.0050378 1.0206207 1.1547005 111111111111111111111111111111 111111111111111111111111111111111111111111111111 22222222222222222222222|22|22222222222222222222222222222222222222222222222222222 33|33333333|3333|3333|333333333|333333333333333333333333333333333333333333333333 4444444444444444444444444444444444|444444444444444444444444444444444444444444444 5555555|555555|555555555555555555|5555|55555555555555555555555555555555555555555 666666666666666666666666|66666666666666666666666666666666666666666666666666666 7777777777777777I7|777777777|777777|77777777777777777777777777777777777777777777 8818888888818s888881888888888831888s888888838888818888188888888888888888888888B8 9 99 99999999999999999999999999999999999999999999 999999999999999999999399 1 2 3 4 S e 7 e S 112 13 14 15 17 20 21 22 23 t 2 52 2677 n28 29 30 31 32 33 343 373I83940 41 4243 4445 4647 4749 53SI S4 5 X 57X"5 " 1 2 65 66676869 70n12,4757677879a m 5061 CARD 2 28

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SCOMPILE FORTRAN,PRINTOBJECPUC EC,PUNCHOJECEXECUTEDUMP,I/ODUMP GMNOOO DIMENSION PS(40,42),PPS(40,42),PR(40,5),RPR(40,5),RQ(40,5),RPQ(40, 15),RNQ(4,5),RPNQ(4,5),Z(5),ETARR(42),P(40),Q(40),QPR(40), 2QNEG(4),QNEGP(4),FNF( 20)20),0(20),DNEG(2),DRHO(20),DJRAT(20) 3 PPRIME(40),DMRHO(22) COMMON FNF,NFNFEOTST,ID, IPAGE CALL FTRAP ND = 1 1000 NRK = 0 NREC=0 NRR=O REWIND 2 REWIND 3 REWIND 4 REWIND 9 LINES =1 LIN2 = 1 RIT 7,1, IDISTIRTIEGDISP,IRPQ,IRHO,PTEST,CVT,M,CS,CIN,NOC, INS,NIN,NON,ETAS,ETAINNOE,NZNITPAGE, IEGFN,LNCNT, EGCV 1 FORMAT(I5,6I1,F1.0,F6.0,11,2F5.0,I3,312,2F5.0,12,4X11,213,1,12, 1F5.0) IF (NZ) 2,3,2 2 RIT 7,4, (Z(I),I=1,NZ) 4 FORMAT (5F10.0) 3 W0T6t5I,I ST,IST,IRT,IEG FN,I ISP,IRPQ,IRHO,PTEST,CVT,EGCV, 1 M,CSCINNOCNS,NINNON,ETAS,ETAIN,NOE,NIT,NZ 5 FORMAT(lH1,50X15HINPUT FOR CASE I5/IH0O,30X, 9HS TEST = I1,30X 9HR 1TEST = II/ 81H02X7HIEGD = I1,3X8HIEGFN = I1,3X7H ISP = 11,3X7HIRPQ = I1,5X7HIR 9HO = Ii, 2 5X,8HPTEST = F1.0,5X,5HCVT = E10.2,5X7HEGCV = E10.2//60X 5H 3M = I/1iHOlOXlOHC START = F8.3,10X14HC INCREMENT = F8.3,1OX18HNO 4. OF C VALUES = 14,/lHO1OXO1HN START = I2,10X14HN INCREMENT = 12,1 50X18HNO. OF N VALUES = I2,/lHOlOX12HETA START = F8.3,1OX16HETA INC 6REMENT = F8.3,10X20HNO. OF ETA VALUES = 12,/lHO,20X23HNUMBER OF IT 7ERATIONS = I3,3X22HNUMBER OF XI VALUES = I1,/) IF(NZ) 6,7,6 6 WOT 6,8,(Z(I),I=1,NZ) 8 FORMAT(1HO,1OX,5E20.8) 7 IRST =IST+IRT WOT 6,520,LNCNT 520 FORMAT(1H015X46HS FUNC. PAGE CONTROL (0 IF CONTROLLED BY N) = 12) IF(IRST) 15,15,530 530 IF (NS - M) 15,9,9 9 IF(NZ-5) 14,14,15 14 IF(M-2) 10,10,15 10 IF (NOE-41) 11,11,15 15 WOT 6, 13 13 FORMAT(1HO,50X21HAN INPUT ERROR EXISTS) GO TO 1000 39

11 WCT 6, 12 12 FORMAT (1IO,50X24HCOMPUTATION WILL PROCEED) FM=M IF(IST) 16,21,16 16 STEST=l. DC 18 IETA=l,NOE ETA=ETAS+FLOATF(IETA-1)*ETAIN 20 CALL PQCAL(ETA,M,P,QQPR,QEGEGQNEGP,PPRIME,STESTISP) DO 17 l=1,40 PS(I,IETA) =P(I) 17 PPS(I,IETA)=PPRIME(I) 18 ETARR(IETA)=ETA 21 IF (IRT) 22,28,22 22 STEST=O. 00251= 1,NZ CALL PQCAL(Z(I),MP,Q,QPRQNEGQNEGP,PPRIME,STEST,IRPQ) DC23J= 1,40 PR(J,I) =P(J) RPR (J,I) =PPRIME(J) RQIJ,I) = Q(J) 23 RPQ(J,I) = QPR(J) IF (M) 26,25, 26 26 DC 27 J=l,4 RNQ(J,I)= QNEG(J) 27 RPNQ(J,I) =QNEGP(J) 25 CCNTINUE 28 DC 400 IIC=1,NOC C=CS+FLOATF(IIC-1)*CIN DERR=O. IERR = 0 DC 55 IIN=1,NON N=NS+( IIN-1)NIN IRTEST = C 300 EITST=O. NPOS = (N-M)/2 + 1 NFNEG =NPCS+1 FN=N IC =C IF(N-6) 130, 130,131 130 IF(IC-5) 131,131,132 131 CALL APROX (FNFM,C,FLOW,PTEST) EINM =0. GC TO 129 132 CALL ASYM (FN,FM,CFLCW,PTEST) EINM =1. 129 CALL CORR(N,M,FLCW,C,EGCV,PTEST,APREV) FLAM =FLOW +APREV IF(ABSF((FLAM - FLOW)/FLCW) - EGCV) 106,106,124 124 DC110 I= 1,NIT CALL CORR(N,M,FLAM,C,EGCVPTEST,ANEW) IF (ABSF (ANEW)-ABSF(APREV)) 113,113,103 103 IF(ABSF(ANEW) - EGCV) 113,113,121 40

113 FLOW =FLAM +ANEW 102 IF(ABSF((FLAM - FLOW)/FLOW) - EGCV) 106,106,108 108 FLAM =FLOW 110 CCNTINUE GC TO 121 106 CALL FNORM (N,M,EOTST,ANS,PTEST,ERR) CALL CCAL(MC,FNF,ANS,D,CVT,PTEST,ND,FNMN,DERR) DC 120 I=1,NPOS IF (D(I)) 121,120,120 120 CCNTINUE IF(D(NFNEG)) 30,30,121 121 IF (EITST) 123,122,123 122 EITST =1. IF (EINM) 131,132,131 123 WCT 6,125, N,M,CFLAM,(D(I ),I=1,ND) 125 FCRMAT (28HO BOTH APPROXIMATIONS FAILED,216,2E16.8/(8E16.8)) IF(IRTEST) 1351,1352,1351 1352 IRTEST = 1 PSAV = PTEST PTEST = 1. GC TO 300 1351 PTEST = PSAV IRTEST = 0 30 NREC = NREC + 1 IF(IEGD) 31,32,31 31 WRITE TAPE 2, C,N,FLAM,ND,(D(I),I=1,ND) 32 IF(IEGFN) 201,200,201 201 WRITE TAPE 9, C,NFLAM,FNMN 200 IF(IST) 33,3070,33 33 DC 39 I = 1,NCE ISUB = 40*(I-1)+1 CALL SCAL(D,NDPS(ISUB),PPS(ISUB),SSPRIMEPTEST) IF (LNCNT) 21C,211,210 210 IF (LINES-1) 212,213,212 213 LINES = LINES+1 214 WOT 6,37,ID,MIPAGE 37 FCRMAT(lHLlOX5HCASE I5,25X17HS VALUES FOR M = I2,32X6H PAGE I4// 1OXtHC, 5X, HN,4X,3HETA, 1X,1HS,13X,7HS PRIME) LIN2 =1 IPAGE =IPAGE +1 WCT 6,215,C,N,ETARR(I),S,SPRIME 215 FORMAT(1X,Fll. I,3XI2,F7.2,El7.6,El7.6) NPREV =N GC TO 39 212 IF (N-NPREV) 216,217,216 216 LINES =LINES+2 WCT 6,218,C,N,ETARR(I),S,SPRIME 218 FCRMAT( HOF1l.1,3XI2,F7.2,E17.6,E17.6) NPREV = N LIN2 =1 41

219 IF (LINES - LNCNT) 39,221,221 221 LINES =1 GC TO 39 217 IF(LIN2 - 5) 222,223,222 222 LINES =LINES1+ LIN2 =LIN2 +1 WCT 6,224,ETARR(I),SSPRIME 224 FCRMAT(17X,F7.2,2E17.6) 226 IF (LNCNT) 219,39,219 223 LINES =LINES +2 LIN2 =1 WCT 6, 225,ETARR(I),S,SPRIME 225 FCRMAT (lHO,16X,F7.2,2E17.6) GC TO 226 211 IF (N-NPREV) 214,217,214 39 CONTINUE IF(IRST - 2) 1235,307C,1235 3070 IF(IERR) 400,45,400 45 CALL CNCAL(M,C,FLAMD(l),DNEG,DRHRC,DRHtCVTPTEST) CALL FJOIN (t,N,C,D,DNEG,ND FKlFK2,PTEST) IF( IRHO) 41,42,41 41 IFIM-1) 513,511,512 511 DMRHO(1) = DNEG(l) GO TC 513 512 DMRHO(1) = DNEG(2) CMRHC(2) = CNEG(1) 513 NMCRO = NCRHO + M CO 514 L = 1,NDRHO LL = L +M 514 CMRHO(LL) = DRHO(L) WRITE TAPE 3,C,N,FK1,FK2,NMDRO,(DtRHC(J),J=1,NMCRC) NRK = NRK + 1 42 DC 54 I=1,NZ ISUB= 40*( I-)+1 ISUB2= 4*(I-1)+1 CALL RlCAL(D,ND,FK1,PR( I SB),RPR( ISLB),R1,RPR,PTEST) CALL R2CAL(M,C,NCDNEGR( I SUB),PR(I SUB),RPR I SUB)RPQ( I SUB),RN (I ISUtB2)tRPNQ( ISUB2),DR D NDC,RHO,FK2,R2,R2PRPTEST) ZZ=Z(I) IF(R2-1.E25) 3010,3011,3011 3010 IF(R2PR - 1.E25) 3012,3011,3011 3011 IERR=1 3012 CCNTINUE CHECK1 = R1 * R2PR - R2*R1PR CHECK2 = 1. /(C( ZZ**2 - 1. )) RERR = ABSF'(CHECK1 - CHECK2)/CHECK2) IF (IRST-2) 43,44,43 44 WRITE TAPE 4,C,N,Z Z,R1,R1PR,R2,R2PR,RERR NRR = NRR + 1 GC TO 54 43 ASSIGN 54 TO NAP 62 IF (LINES -1) 46,47,46 42

47 WCT 6,48, IDM,IPAGE 48 FCRMAT(1HI,IOX5HCASE 15,24X17HR VALLES FOR M = 12,33X6H PAGE I4// 14XIHC,6XIHN 110X2HXI13X2HR112X8HR1 PRIMElOX2HR212X8HR2 PRIME,5X9HWR. CHECK) IPAGE =IPAGE +1 NPREVR = N WCT 6,49,C,NtZZ,Rl,RlPR,R2,R2PR,RERR 49 FCRMAT( XF5.1,6,F16. 7,E16.6,E16.6,E16.6,E16.6,E13.3) LINES =LINES +1 GO TO 313 46 IF (N-NPREVR) 310,311,310 310 WCT6, 312, C,N, ZZ R1,R 1PR,R2, R2PR RERR 312 FORMAT (1HO,F5.1,16,F16.7,4E16.6,E13.3) NPREVR = N LINES =LINES +2 GC TO 313 311 WCT 6,314, ZZ,R1,R1PR,R2,R2PR,RERR 314 FORMAT (12X,F16.7,4E16.6,E13.3) LINES =LINES +1 313 IF(LINES-50) 50,51,51 51 LINES =1 50 GC TO NAP,(5CC,54) 54 CCNTINUE 1235 IF(DERR) 1234,55,1234 1234 LINES=1 GC TO 400 55 CCNTINUE 400 CCNTINUE 56 REWIND 2 REWIND 3 REWIND 9 REWINC 4 IF( IRST -2) 60,61,6C 61 LINES =1 ASSIGN 500 TO NAP DC 500 I =l,NRR READ TAPE 4,C,tNZZR 1,RPR,R2,R2PRRERR GC TO 62 500 CCNTINUE 60 IF(IEGD) 63,64,63 63 LINES = 1 DC 75 1= 1,NREC READ TAPE 2, C, N, FLAM,ND,(D(J),J=1,ND) IF (LINES -1) 65,66,65 66 WCT 6,67, ID,M, IPAGE 67 FCRMAT(1HlO1X5HCASE 15,15X33HEIGENVALUES AND D VALUES FOR M = 12, 126X6H PAGE 14// 14XIHC3X1HN 5XO0HEIGENVALLE48X 8HD VALUES) IPAGE =IPAGE +1 65 IF (ND - 7) 6S,69,9S8 43

69 WCT 6,92,C,N,FLAM, (D(J),J1,ND) 92 FORMAT( 1XF4.2, 14, E 15. 6, 7E 15.6) LINES = LINES + 1 GO TO 74 98 WCT 6,68,CN,FLAM, (D(J),J=1,ND) 68 FORMAT(IXF4.2,14,E15.6,7E15.6/(24X,7E15.6)) 71 IF (ND -14) 72,72,73 72 LINES = LINES +2 GC TO 74 73 LINES = LINES +3 74 IF (LINES -51) 75,76,76 76 LINES =1 75 CCNTINUE 64 IF (IEGFN ) 410,664,410 410 LINES =1 CO 440 I = 1,NREC READ TAPE 9,CN,FLAM,FNMN IF(LINES - 1) 413,412,413 412 wOT 6, 414,ID,M,IPAGE CPREV=C 414 FORMAT(Hl, OX5HCASE I5, 15X45HEIGENVALUES AND NCRMALIZATICN FACTOR 1 FOR M = I2,14X6F PAGE I4// 24XlHC3XLHN5X CHE IGENVALUE5X13HNORMAL I ZAT IN) IPAGE =IPAGE+1 WCT6,419,C,N,FLAM,FNMN 419 FORMAT( lXF4. 1,I4E,E15.6,E18.6) GC ro 411 413 IF(C-CPREV) 415,416,415 415 WCT 6,417, C,N,FLAM,FNMN 417 FORMAT ( HO, F4.1,I4,E15.6,E18.6) LINES = LINES + I CPREV =C GC TO 411 416 WCT 6,418,N,FLAM,FNMN 418 FORMAT (5X,14,E15.6,E18.6) 411 LINES = LINES + 1 IF(LINES -51) 440,437,437 437 LINES = 1 44C CONTINUE 664 IF (IRT) 80, 000,80 80 IF (IRHO) 81, 10C0,81 81 LINES =1 DC 88 I = 1,NRK READ TAPE 3,C,N,FK1,FK2,NMDRC, (DPRHO(J),J=1,NMCRO) IF(LINES -1) 82,83,82 83 WCT 6,84,1D, M,IPAGE 84 FCRMAT(1H1,lOX,51CASE I5,1X,53HJCINING FACTORS, D NEGATIVE AND D R IHC VALUES FOR M = 12, 2OX,6H PAGE I4// 24X1HC3X1HN,14X14 Kl,14X2HK2,10X,25H (M) C NEGATIVE ANO D RHO) IPACE =IPAGE +1 82 IF(NMCRC -6) 94,94,95 94 WCT 6,96, C,N,FK2, (DMRHO' JJ),J=I,hMDRO) 44

96 FORMAT (lX,F4.2,14,2E16.6,6E14.6) LINES = LINES + 1 GO TC 93 95 WCT 6,85,CtN,FK1,FK2,(OMRHO(J),J=1,NMDRO) 85 FCRMAT(lX,F4.2,14,2E16.6,6E14.6/(41X,6E14.6)) IF (NDRHO -12) 86,86,87 86 LINES =LINES +2 GO TO 93 87 LINES = LINES + 4 93 IF(LINES- 51) 88,99,99 99 LINES = 1 88 CCNTINUE GC TO 100OC END SCOMPILE FORTRAN,PRINT OBJECT,PUNCH OBJECT,EXECUTE,DUMP LEGENOO0 SUBROUTINE PQCAL (Xt,M,PP,QQQPRQNGEGQNEGPPPR,STESTIPT) DIMENSION P(4C),C(40),QPR(40),QNEG(4),QNEGP(4),PPR(40)GAM(40) DIMENSION DUMMY(21),PP(40),QQ(40) DIMENSION PTEM (40) COMMON DUMMY,EOTESTIDIPG C C SUBROUTINE TO COMPUTE LEGENDRE FUNCTIONS C M=MM IPQTST=IPQT FM=M X2=X*X IF (ABSF(X-1.)-1.E-8) 1,1,2 C X IS CNE 1 IF (M-l) 3,301,3C0 C M IS ZERO 3 PP(1)=1. PPR(1 )=0. FI=1. DC 302 1=2,40 PP( I)=1. PPR( I )=PPR( I-)+FI 302 FI=FI+1. IF (STEST-1.) 307,10,307 C M IS TWO 300 DC 303 I=1,40 PP( )=0. FI=I-1 303 PPR( I )=6.+24.*FI+18.*FI*(FI-1. )+4.*FI*(FI-1. )*(FI-2. )+.25*Fl*(FI-1 1.)*(FI-2.)*(FI-3.) IF (STEST-1.) 307,10,307 45

C M IS ONE 301 DC 305 I=1,40 PP(I )=0. 305 PPR(I)=0. IF (IPQTST-l) 308,309,308 C X IS NOT CNE 2 Y=ABSF(1.-X2) IF (M-l) 4,5,6 C M IS ZERO 4 P(1)=l. P(2)=X GC TO 7 C M IS CNE 5 P(1)=SQRT(Y) P(2)=3.*XP( 1) GC TO 7 C M IS TWO 6 P(1)=3.*Y P(2)=5.*X P( 1) C CCMPUTE P 7 DC 8 1=3,40 V=I+M-2 8 P(I)=(( 2.*V+1. )* XP( I-1)-(V+FM )P(I-2) ) / V-FM+1. ) DC 80 I=1,40 80 PP(I)=P(I) C CCMPUTE P PRIME DC 9 1=1,39 V=M+I 9 PPR I )=( (V-FM)*P (1+1 )-V*X*P(I) )/( X2-1.) PPR(4C)=((39.+FM)*X*P(40)-(39.+2.*FM)*P(39))/(X2-1.) C TEST P AND P PRIME ONLY IF (STEST-1.)1 1,10,11 C TEST P P PRIME PRINT CUT 10 IF (IPQTST-l) 26,25,26 25 WRITE OUTPUT TAPE 6,1C1,ID,X,M,IPG 1C1 FCRMAT( H1,1OX5HCASE,I5,9X5HETA=,F8.3,5X3HM=,I2,45X5HPAGE,14,/ 144XI18LEGENDRE FLNCTICNS/ 21F,3XlHN,15XlHP,18X7FP PRIME.,/) IPG=IPG+l NN=M DC 50 1=1,40 WRITE OUTPUT TAPE 6,1C4,NN,PP(I),PPR(I) 104 FCRMAT( 2XI3, 6XE16.6,5XE16.6) 50 NN=NN+1 26 RETURN 11 DC 30 I=1,4 QNEG(I)=0. 30 QNEGP(I)=C. IF (X-1.01) 6C,6C,71 60 PTEM(1)=1. PTEM(2)=X DC 61 1=3,40 46

V=I-2 61 PTEM( Il =((2.*V+1. )X*PTE ( I-1)-V*PTE ( 1-2) ) / (V1. ) Z=ELOG((X+l. )/(X-l.)) Q(1)=.5*Z Q(2)=XQ( 1)-l. DC 62 1=3,40 Q( I ).5*PTEM( I )*Z SLM=O. 1=1-1 DC 63 J=l1ll FJ=J K=I I-J+ 63 SUM=SUM+PTEM(J)*PTEM(K)/FJ 62 Q(I)=C( I)-SUM IF (M) 66,67,66 66 YYY=1./SQRT(Y) DC 68 1=1,39 V=I 68 Q( I )=YYY*(V*X*Q( +1)-V*Q() ) Q(40)=(79.*XeC(35)-40.*Q(38))/39. IF (M-1) 67,67,69 69 DC 70 1=1,39 V-=I+1 70 Q(I)=YYY*((V-1.)*X*Q(I+1)-(V+l.)*C(I)) Q(4C)= (81.*XQ( 39)-42.*Q(38) )/39 GC TO 67 71 IF (M-l) 12,13,14 C M IS ZERO 12 A=1. B=.5 GAM( 1)=2. GC TO 15 C M IS CNE 13 A=1.5 8=1. GAM( 1 )=2.6666667 GC TO 15 14 A=2. B=1.5 GAM( 1)=12.8 15 DC 16 I=2,40 FI=I 16 GA( I )=(2.*(2.*FM+FI-l1.)/(2.*(FM+FI)-.) )*GAM(I-1) C CCMPUTE Q XXX=Y**(FP/2.) Z=1./X2 ERR=O. DC 17 1=1,40 V=M+I-I CALL hYPER(A+V/2.,B+V/2.,V+1.5,Z,C(I),ERR) 47

C TEST ERROR IN SUBR. HYPER IF (ERR-1.) 1718, 18 17 Q(I)=(-1.)**M*(2.**(-V-1.)*Q(I)*X**(-V-FM-l1.)*GAM(I)*XXX) 67 DO 81 I=1,40 81 QQ(I)=Q(I) C COMPUTE Q PRIME DO 19 I=1,39 V=M+I 19 QPR( I )=( (V-FM)Q( I +1)-V*X*Q( I ) /Y QPR(40)= ( (39.+FM )*X*Q( 40 ) —( 39.+2.*FM)*Q( 39 ))/Y C TEST P,P PRIME,Q,Q PRIME. PRINT OUT IF (IPQTST-1) 28,27,28 27 IF (M) 40,41.40 41 ZAN=(-1.)/Y WRITE OUTPUT TAPE 6, 192, I,X,M,ZAN, IPG 192 FORMAT(1H110X5HCASE 5,10 X4HXI= E16.8,X3HM= I2,7X lWRONSK IAN= El 16.6,7X5HPAGE I4,/47X18HLEGENDRE FUNCTIONS/ 21H 3XIHN15X1HP16X7HP PRIME17XlHQ17X7HQ PRIME 1X10HREL. ERROR,/) IPG=IPG+1 NN=M DO 42 1=1,40 CHECK=ABSF( ( (P( I )*QPR( I )-( I )PPR(I) )-ZAN)/ZAN) WRITE OUTPUT TAPE 6,205,NN,PP(I),PPR(I),Q(I),QPR(I),CHECK 205 FORMAT (2XI3,5(5XE16.66) 42 NN=NN+1 GO TO 28 40 WRITE OUTPUT TAPE 6,92,II,X,M,IPG 92 FORMAT(1HI,10X5HCASE,15,10X4HXI= E16.8,5X3HM=,I2,37X5HPAGE,I4,/ 144X18HLEGENDRE FUNCTIONS/ 21H,3XlHN,15XlHP,16X7hP PRIME,17X1HQ,17X7HQ PRIME,/) IPG=IPG+1'NN=F DO 51 1=1,40 WRITE OUTPUT TAPE 6,105,NN,PP(I),PPR(I),Q(I),QPR(I) 105 FORMAT(2X3, 4(5XF16.6)) 51 NN=NN+1 28 IF (M-1) 24,20,21 C M IS ONE 20 QNEG(1)=(3.*X*Q(I)-Q(2))/2. QNEG(2)=X*QNEG(1) QNEGP( 1)=-X*QNEG(1)/Y QNECP(2)=-QNEG(1)/Y JJ=2 C TEST WNEG AND QNEGP PRINT OUT IF (IPQTST-1) 24,29,24 C M IS TWO 21 QNEG(1)=(5.*X*Q(1)-Q(2) )/4. QNEC(2)=XQNEG( 1) QNEG(3)=(X*QNEG(2)+QNEG( 1) )/2. QNEG(4)=(-X*QNEG(3)+2.*NrMEG(2) ) QNEGP( 1 )=(-2.X*QNEG( 1) )/Y QNEGP(2) = (-CNEG()-XNEG(2) )/Y 48

QNEGP(3)=(-2.*QNEG(2))/Y QNEGP(4)=(X*QNEG(4)-3.*QNEG(3))/Y JJ=4 C TEST CNEG ANC QNEGP PRINT OUT IF (IPQTST-1) 24,29,24 29 WRITE OUTPUT TAPE 6,103 103 FCRMAT(1HOC3XlHN,14X5HQ NEG,12XllHQ NEG PRIME,/) NN=M-1 DC 52 1=1,JJ WRITE OUTPUT TAPE 6,1C6,NN,QNEG(I),CNEGP(I) 106 FCRMAT( 2XI3,6XE16.6,5XE16.6) 52 NN=NN-1 RETURN C ERROR P PRIMES UNDEFINED 309 WRITE OUTPUT TAPE 6,310,IPG 310 FORMAT (11,97X5HPAGE,I4,////) IPG= I PG+1 308 WRITE OUTPUT TAPE 6,311 311 FORMAT (1CX84HP PRIMES OF 1. FOR M=l ARE UNDEFINED. THEY ARE SET T 10 ZERO *** COMPUTATION CONTINUES) IF (STEST-1.) 315,26,315 C Q AND Q PRIMES UNDEFINED 307 IF (IPQTST-1) 315,314,315 314 WRITE OUTPUT TAPE 6.310, IPG IPG=IPG+1 315 WRITE OUTPUT TAPE 6,316 316 FCRMAT(10X62FQ AND Q PRIMES OF 1. ARE SET TO ZERO *** COMPUTATION 1CCNT INUES) GC TO 304 C FAILURE IN HYPER SUBRCUTINE 18 ERR=O. IF (IPQTST-1) 401,400,401 400 WRITE OUTPUT TAPE 6,310,IPG IPG= IPG+ 4C1 WRITE OUTPUT TAPE 6,4C2 402 FCRMAT(lOX81HFAILURE IN HYPER SUBROUTINE, Q AND Q PRIMES SET TO ZE 1RC *** COMPUTATICN CONTINUES) C Q AND Q PRIMES SET TO ZERO 304 CC 313 I=1,40 CC(I)=0. 313 QPR(I)=0. 24 RETURN END S CCMP ILEFORTRAN,PR INTCBJECT, PNCHCBJECT, EXECUTE,DUMP APROXOOO SUBROUTINE APROX (FN,FM,C,EIGEN,PTEST) APROX C PCWER SERIES EXPANSION (APPRCXIMATICN) FOR EIGENVALUE APROX 49

DIMENSION FL( 5 ),CK( 5)APROX FN2=2.*FN APROX FM2=2.*FM APROX DIF=FN-FM APROX SLM=FN+FM APROX FL( 1)=FN*(FN+1.) APROX FL(2)=.5*( 1.-( (FM2-1. )*(FM22+1. )/( (FN2-1. )*(FN2+3. ) ) ) APROX TA=(DIF+1. )(CIF+2.)*(SU+1*(M.)(SU+2.) APROX DA=(FN2+3. )*3*(FN2+1.)*(FN2+5.) APROX TB=( IF- 1). )* C I F* (SUM- 1.)SUF' APROX DB=(FN2- 1. )3 ( FN2-3.) (FN2+1.) APROX FL ( 3) =. 5* ( Tb/CB- TA/DA ) APROX CCEF=4.*FM*FM- 1. APROX PA=(FN2+3.)**2*(FN2-1.FN-1. FN2+7.) APROX PB= (FN2-1.)**2*(FN2-5. )(FN2+3.) APROX DC=CA*PA APROX CC=DBePB APROX FL(4)=COEF*(TA/DC-TB/DD) APROX A=TB/(DDCPB)-TA/(DC*PA) APROX DE= (FN2-5. )m**2*( FN2-3. (FN3.)2(FN2-7.) (FN2-1.) *DB APROX DF=(FN2+5.)**2*(FN2+7.)**2*(FN2+9.)*(FN2+3.)*DA APROX B=( CIF-3. )*( CIF-2. )*( SUM-3. )( SUM-2. )*TB/DE-( (DIF3. )*(CIF+4. ) * APROX 1(SUM+3.)*(SUM+4. )TA/CF) APROX D=TA*TA/(CA*DA (FN2+3.))-(TBHTB/(CB*CB*(FN2-1.))) APROX E=TA*TB/(CA*CB*(FN2-1.)*(FN2+3.)) APROX FL( 5)COEF**2*2.*A+.0625*B+.125*D+.5*E APROX CK( 1)=1. APROX CK(2)=C*C APROX CK( 3)-CK(2)*CK(2) APROX CK 4 )=CK( 2 )*CK (3) APROX CK( 5)=CK( 3)CK(3) APROX EIGEN=FL( 1)+FL(2 )CK( 2)+FL( 3-K*CK (3) +FL ( 4)CK(4 )+FL ( 5 ) CK ( 5 ) APROX IF(PTEST) 2,1, 2 APROX 1 RETURN APROX 2 WRITE OUTPUT TAPE 6,1CO,FN,FM,SUt,DIF,C,EIGENA,B,D,E,TA, H,PAPB, APROX IDA, DB,DC,CD, DEDF APROX 100 FCRMAT( 11HAPROX DUMP,/2X19HN,M,N+M,N-M,C,EIGEN,/6E20.8 /2X16HIA,B APROX 1D,E,TA,TB,PA,/7E18.8,/2XllHPB,DA TO DF,/7E18.8//3XlHI,12X4HL( I),17 APROX 2X4HC(I)) APROX WRITE OUTPUT TAPE 6,1Cl,(I,FL(ICK(,CK(I),=1,5) APROX 1C1 FCRMAT (IX,I3,2E20.8) APROX RETURN APROX END APROX SCOMPILE FORTRAN,PRINT OBJECT,PLNCH OBJECT,EXECUTE,DUIP ASYMOCOO SUBROUTINE ASYM(FN,FM,C,EIGENPTEST) C ** ASYMPTCTIC EXPANSICN FCR APPROXIMATE LAMBDA DIMENSION TERM(7) Q=2. (FN-FM) 1. 50

FM2=FM*FM C1=C C2=Cl*C1 C3=C2*C1 Q2=Q*Q Q3=Q2*Q TERM( )=Cl*Q TERM(2)=FM2-.125 (Q2+5.) TERM(3)=Q*(32.*FM2-11.-Q2)/(64**C1) TERM(4)=(384.~FM2*(Q2+1*)-5.*(Q2*Q2+26.*Q2+21.))/(1024.*C2) TERM(5)=Q*(FM2*(37.*Q2+167.-16.*FM2)-(33.*Q2*Q2+1594.*Q2+5621.)/ 1128.)/ 128.*C3) TERM(6)=(.5*FM2*(115.*Q2*Q2+1310.*Q2+735.-192.*FM2*(Q2+l.))-(63.* 1Q3*Q3+494C.*Q2*Q2+43327**Q2+22470.)/256.)/(256.*C2*C2) TERM(7)=Q*(FM2*(5739.*Q2*Q2+127550.*Q2+298951.-64.*FM2*(355.*Q2+ 11505.-32.*FM2))/32.-( 527.*Q3*Q3+61529.Q2*Q2+1043961.*Q2+2241599.) 2/1024. )/( 1024.*C 3C2) IF(PTEST) 1.2,1 1 WRITE OUTPUT TAPE 6,100 100 FORMAT (lH1,1OX1OHASYMO DUMP,/3XlHIlOX7HTERM(I),15X6HLAMBDA,/) 2 A=O. 00 3 1=1,7 A=A+TERM( I) IFIPTEST) 4,3,4 4 WRITE OUTPUT TAPE 6,lC01,ITERM(I),A 101 FORMAT (I5,2E20.e) 3 CONTINUE EIGEN=A RETURN END COMPILEFORTRAN,PRINTOBJECT,PUNCHOBJECT,EXECUTE,DUMP CORR 000 SUBROUTINE CORR (NM,FLAM,CRTEST,PTEST,DELAM) DIMENSION FN1(20),FN2(20),G(20),B(20),D2(20),D1(20) CIMENSION FNF(20) COMMON FNFNUM,EOTST DO 40 I = 1,20 FN1(I) = -0. FN2(I) = -0. C2(I) -0. 0 C 1I(I) =-0. FM =M I =N-M 12 =1/2 IM = 2*12 IF(I-IM) 3,4,3 4 ECTST =0. 51

GC rO 5 3 ECTST =l. C FIRST TIME DEVELOP BETA AND GAMMA 5 IFIEOTST) 6,7,6 C N-M IS EVEN, EOTST IS ZERO 7 DC 8 In 1,20 R = 2*1 - 2 G(I)= GAM(FM,R,C) 8 B(I)= BETA(FM,R,C) GC TO 2 C N-M IS ODD 6 00 9 1= 1,20 R = 2*I -1 G(I)= GAM(FM~R,C) 9 B(I)= BETA (FM,R,C) C DEVELOP AND COUNT N VALUES 2 FNIl()=G(l)-FLAM DO 11 1=2,20 FNl(I)=G(I)-FLAM -B(I)/FN1(1-1) IF(ABSF((FN1(I) - FNl(I-1))/FNl(I-i))-RTEST) 12,12,11 12 NUM =1 GC TO 13 11 CONTINUE NUM = 20 C FINC LOCATION OF N SUBSCRIPT N-M+2 13 IF(EOTST) 14,15,14 C N-M+2 MUST BE EVEN 15 J=(N-M+2)/2 GO TO 16 C N-M+2 MUST BE ODD 14 J= (N-M+1)/2 16 Ul = FN(J) C DEVELOP INVERSE ARRAY STARTING WITH 0 WITH NUM-J+1 TERMS FN2(1)=0 K=NUM-J+1 DC17 L=2,K IFN1 = NUM - L + 2 17 FN2(L)= B(IFNI) / (IFNI)- FLAM -FN2(L-1)) U2 =FN2(K) C STORE TO J-1 FROM FN1 AND FROM J TO NUM(O) FROM FN2 JJ=J-1 DC 50 I=1,JJ 50 FNF(l) = FN1(I) CO 51 I = J,NUM 12 = NUM - I + 1 51 FNF(I) = FN2(12) IF (ABSF((U2 - U1)/Ul) - RTEST) 33,33,34 33 DELAM = 0. GO TO 37 C CALCULATE DENOMINATOR C FIRST TERM 34 ISUB = J 52

SUM1=1. Dl( 1)=l DC 22L=2,20 IF(ISUB-1) 20,20,21 21 D1IL)= (B(ISUB)/FN1(ISUB-1)**2)*D1(L-1) SUMI =SUM1 +D1(L) ISUB=ISUB-1 22 CCNTINUE FSUB ISUB WCT 6,24 24 FORMAT (1HO,12X,25HERROR IN FIRST TERM DENC GO TO 26 C SECCND TERM DENOMINATOR 20 ICNT = NUM D2(1)=FN2(K)~*2/B(J+1) SUM2 =D2(1) D025 L=2,ICNT KK=K-L+ 1 ISUB =L+J-1 D2(L)=(FN2(KK)**2/B(ISUB+l))*D2(L-l) SUM2 = SUM2 + D2(L) IF (SUM2 - 1.E5) 25,25,35 25 CONTINUE 35 DELAM = (U1-U2)/(SLMI+SUM2) 37 IF(PTEST) 26,27,26 26 WCT 6,28,G,B,FN1,FN2,D1,D2,UI,U2,SUMI,SUM2,DELAMEOTST,FLAM,CRTES IT,FNF 28 FORMAT{(H0,26 H CUMP FROM CORR SUEROUTINE/(8E16.8)) 27 RETURN END $ COMPILE FORTRAN,PRINTOBJECT,PUNCHOBJECT,EXECUTE,DUMP FNORMOO SUBROUTINE FNORM(NM,EOTEST,ANS,PTEST,ERR) IF (EOTEST) 1,2,1 C N-M IS OD0 1 IF(N-1) 36,40,40 40 IXP=(N-M-1)/2 LNUM=(N+M+1) LCN2= LNUM/2 LCEN= IXP GC TO 3 C N-M IS EVEN 2 IF (N) 36,21,41 21 ANS= 1. ERR=O. GC TO 35 41 IXP= (N-M)/2 53

LNUM=(N+M) LCN2 = LNUM/2 LCEN= IXP 3 IF (LON2 -20) 2C,20,4 4 WOT 6,5,N,M, IXPLNUMLDEN 5 FORMAT(22HO IN FNORM,LNUM.G. 20 1016) ERR=1. GC TO 36 20 ERR=O. ANS 2 = FACT(LNUM)/(2.**N*FACT(LDEN)*FACT(LDN2)) ANS = (-1.)**IXP *ANS2*2.**M 35 IF (PTEST) 36,37,36 36 WOT 6,38,NMIXP,LNUM,LDENtEOTEST,ANSANS2 38 FCRMAT(13HO FNORM DUMP 5110/(8E16.8)) 37 RETURN END $ COMPILE FORTRAN,PRINTOBJECT, PUNCHOBJECT,EXECUTE,DUMP DCALOOO SUBROUTINE DCAL(M,CFN1,FNANS,D,CTEST,PTEST,ITRM,FNMN,ERR) DIMENSION FN1(20),CF(20),PDR(20),DR(20),D(20),COEX(20) CCMMON FN1,NUMECTST C CALCULATE COEFFICIENTS OF D RATIO FMP=M DC 15 I = 1,NUM IF(EOTST) 1,2,1 1 R=2*I+1 IR=2*I IRX=2*I-1 GC TO 3 2 R=2*I IR=2*I-2 IRX=IR 3 CF(I) = -(2.*FM+R)*(2.*FM+R-1*)*C**2/((2.*FM+2.*R-1.)*(2.*FM+2.* 1R +1.)) DR( I)=FN(I)/CF(I) IF (IR+2*M-34) 30,31,31 31 ILIM=(IR+2*M)/2 SUM2 = 1. DC 32 J=1,ILIM RRR=ILIM+J 32 SUM2 = SUM2 *RRR CCEX(I)=((-1.)**(I-1)*SUM2)/(2.**IRX*FACT(I-1)) GC TO 33 30 COEX(I)=((-l.)**(I-1)*FACT(IR+2*M)/(2.**IRX*FACT(I-l)))/FACT((IR+2 1*M)/2) C CALCULATE PRODUCTS CF D RATIOS 33 IFII-1) 20,21,20 21 PCR(I)=1. GO TO 7 54

20 PCR( I)=PDR(I-1)*CR(I-1) IF(ABSF(PCR( I)) - 1.E30) 7,50,50 7 IF (ABSF(PDR(I))-DTEST) 5,5,15 50 ERR= 1. 15 CCNTINUE ITRM = NUM GO TO 13 5 ITRM=I 13 SUM=O. FTRM=ITRM DO 8 Il1,ITRM 8 SUM=SUM+PCR ( I) CCEX(I) 0(1) = FNANS/SUM DO 9 I =2, ITRM 9 D(I)=D( 1 ) PDR(I) SUM5 = 0. CO 47 I = 1,ITRM IF(EOTST) 41,42,41 41 FIR = 2 * I - 1 GO TO 43 42 FIR = 2 * I - 2 43 IF(M - 1) 44,45,46 44 CCER = (1./(2.*FIR + 1.)) GO TO 47 45 COER = (FIR + 2.) * (FIR + 1.)/(2.*FIR + 3.) GO TO 47 46 COER= (FIR+4. )*(FIR+3.)*(FIR+2.)*(FIR+l. ) /(2.FIR+5.) 47 SUSUM = SUM5 + COER * D(I))*2 FNMN = 2. * SUM5 IF (PTEST) 11,10,11 11 WOT 6,12,FM,C,SUM,DTEST,FNANS,FNlDR,PCR,CF,D,EOTSTCOEX,FTRM, 1 SUM2,FNMN 12 FORMAT (11HO CCAL DUMP/(8E16.8)) 10 RETURN END COMPILE FORTRAN,PRINTOBJECTPUNCHOBJECT,EXECUTE,DUMP SCOOO SUBROUTINE SCAL(CND,P,DERP,SSPRIME,PTEST) DIMENSION D(20),P(40),DERP(40),DUMMY(21) CCMMON DUMMYEOTEST SUM1=0. SUM2=0. DC 4 I=1,ND IF(EOTEST)1,2,1 1 ISUB=2*I GC TO 3 2 ISUB=2*I-1 55

3 SUM1=SUM+0( I )P( I SUB) 4 SUM2=SUM2+D( I) *DERP( ISUB) S=SUM1 SPRIME=SUM2 IF(PTEST) 56,5 5 WCT6,7,DPDERPEOTEST,SUM1, SUM2 7 FCRMAT(1HO,1 O SCAL DLMP/(8E16.8)) 6 RETURN END SCOMPILEFORTRAN,PRINTOBJECT,PUNCHOBJECT,EXECUTE,DUMP DNCALOOO SUBROUTINE CNCAL(M,C,FLAM,DSTART,DNEG,DRHO,NDRAT,DNTST,PTEST) DIMENSION A(24)tB(24),CC(24),DNEG(2),DJRAT(20),DRHO(20), 1FND(20),DUMMY(21),CFB (20),CFA(21),CA(24) CCMMON DUMMY,EOTEST C CALCULATE ALL A,B,C COEF FOR M UP TO 2 (DR = -4) FM =M IF(EOTEST) 22,2C,22 22 ALIM = C**2 /( (-2.*FM+1.)*(-2.*FM+3.)) GC TO 24 20 ALIM = C**2 /( (2.*FM+1.)*(-2.*FM +1.)) 24 DC 5 J = 1,24 IF(EOTEST) 1,2,1 1 RP = -2*J+3 GC TO 4 2 RP = -2*J+2 4 FRP =FM+RP FRPP1 = FRP + 1. FRPM1 =FRP-1. TI = 2.*FRP -1. T2 = T1+2. T3 = T2 +2. A(J) = (C*C*(2.*FM+RP)*(2.*FM+RP-1.))/(T1*T2) B(J) =FRP*FRPP1-FLAM +C*C*(2.*FRP*FRPPI-2.*FM*FM-1.)/(T1*T3) CC(J) =(RP+1.)* (RP+2.)*C*C /(T2*T3) IF(J-1) 6,5,6 6 CA(J) = -CC(J)* A(J-1) 5 CCNTINUE C IF M.NE. 0 CALCULATE 2 0 NEGATIVE VALUES IF (M) 11,10,11 11 CFA(1) =-A(1) DC 7 J = 2,21 K=J-1 CFB(K) =B(J) 7 CFA(J) =CA(J+1) DR1 = CF2(0.,CFA,CFB,20,ND) DNEG (1)= DSTART * DR1 CFA (1) =-A(2) DC 8 J = 2,21 56

CFB (J-1) =B(J+1) CFA(J) = CA(J+2) DR2 = CF2(0.,CFA,CFB,20,KD) 8 DNEG (2) =DR2 *CNEG(1) C CALCULATE D RHO VALUES AND NUMBER OF 0 RHO'S 10 CFA (1) =-ALIM DO 9 J = 2,21 K = J + M CFA(J) = CA(K+1) 9 CFB(J-1) = B(K) DC 12 J = 1,2C NLIMT = 21 - J 38 OJRAT(J) = CF1(0.,CFA(J),CFB(J),NLIMT,ND) IF(ND) 36,37,36 37 NLIMT = NLIMT - 1 GC TO 38 36 FND(J) = ND K = J + M + 1 CFA(J+1) =- A(K) 12 CCNTINUE IF(M-1) 27,28,29 27 DSTAR1 = DSTART GO TO 15 28 DSTAR1 = DNEG(1) GO TO 15 29 DSTAR1 = CNEG(2) 15 DRHO(1) = OSTAR1 * DJRAT(1) DC 16 J = 2,20 DRHO(J)= CRHO(J-1)*DJRAT(J) 16 CCNTINUE NCRAT = 20 21 FNDRAT=NDRAT IF (PTEST) 17,18,17 17 WCT 6,19,FM,A,B,CC,CA,CFA,CFB,ALIM,DJRATDRHOFNDRAT, 1 EOTEST,DR1,DR2,CNEG,FND,DSTART,DNTST,DSTAR1,FLAM,C 19 FCRMAT (12HO CNCAL DUMP/ ( 8E16.8)) 18 RETURN END SCOMPILE FORTRAN,PRINTCBJECT,PUNCHOBJECTEXECUTE,DUMP FJOINOOO SUBROUTINEFJOIN(M,NC,DDNEGNOODFK1,FK2,PTEST) CIMENSION DUMMY(21),C(20),DNEG(2) CCMMON DUMMYEOTEST FP=M IF(EOTEST) 1,2,1 1 ANUM=(2.*FM+3.)*FACT(N+M+1) SIGN = -1. 57

ADEN=(2.oFM-3. )*(2.FM-1.)*FACT(M)*FACT(N+M+l)*C**(M-2) IARG=(N-M- ) /2 JARG=(N+M+1)/2 IPOW=M+1 GC TO 3 2 ANUM=(2.FM+1. )*FACT(N+M) SIGN = 1. ACEN= (2.*FM-1. )*FACT(M)*FACT(N+M)*C**(M-1) IARG=(N-M)/2 JARG=(N+M)/2 IPOW=M 3 ANS=0. DC 4 I=1,NOD IF(M) 5,6,5 5 II=2*(I-1) IF(EOTEST) 7,8,7 7 II=I1+1 8 FRAC=FACT(2*M+II)/FACT(II) GO TO 4 6 FRAC=1. 4 ANS=ANS+D( I )FRAC FK1=ANS/(2.**(N+M)*D(1)*FACT(M))*ANUI/(C**IPOW*FACT(IARG)*FACT(JAR 1G)) IF(M-1) 20,21,21 20 CNN = 0(1) GO TC 23 21 DNN = DNEG(M) 23 FK2=SIGN*(2. )**(N-M)*FAT(t2*M)*FACT IARG)/ADEN*FACT(JARG)*DNN*ANS IF(PTEST) 9 l1C,9 9 WCT6, 100,M,N,NOD IARGRG,JARG, IPOWECTEST,CANUM,ANSFKLFK2,DNN,AEN 100 FORMAT( 1HOFJOIN DUMP,/6I5,/(8E16.8)) 10 RETURN END S COMPILE FORTRANPRINT OBJECT,PUNCH OBJECT,EXECUTE, DUMP R1COO0 StBROUTINE R1CAL(DNDFKl,P,DERPRl,RlPRPTEST) DIMENSION D(20),P(40),DERP(40),DUDMMY(21) CCMMON DUMMY,EOTEST SUM 1=0. SUM2=0. DC 4 I=1,ND IF(EOTEST) 1,21 1 ISUB = 2 I GC TO 3 2 ISUB = 2*1-1 3 SUM1=SUM1+D(I )P(ISUB) 4 SUM2=SUM2+D( I) DERP(ISUB) R1 = SUM1/FK1 R1PR=SUM2/FK 1 58

IF(PTEST)5,6,5 5 WCT 6,7,D,PCERP,SUM1,SUM2,Rl,RIPRFK1 7 FCRMAT(LHO,l1H R1CAL DUMP/(8E16.8)) 6 RETURN END $ COMPILE FORTRAN,PRINT OBJECT,PUNCH OBJECT,EXECUTE, DUMP R2CO0 SUBROUTINE R2CAL(MD,NDONEG,QPPDERQDERQNEGQORNEGDRHONORHO, lFK2,R2,R20ER,PTEST) DIMENSION DNEG(2),D(20),Q(4040),P40),DRHO(20),DUMY(21), LDC(22),PP(20 ),QQ(22),QNEG(4) DIMENSION QD(20),QDRNEG(4),QDER(40),PDER(40),PD(20) CCMMON DUMMY,EOTEST IF(M-1) 1,2,3 2 DO(1)=DNEG(l) IF(EOTEST) 30,31,30 31 QQ(1) = QNEG(2) QD(1) = QDRNEG(2) GO TO 1 30 QQ(a ) = QNEG(1) QD(l) = QDRNEG(1) GO TO 1 3 DC(1)=DNEG(2) DC( 2)=DNEG( 1 ) IF(EOTEST) 32,33,32 33 QQ(1) = QNEG(4) CQ(2) = QNEG(2) QD(1) = QDRNEG(4) QD(2) = QDRNEG(2) GO TO 1 32 QQ(1) = QNEG(3) CQ(2)= QNEG(1) QD(1) = CDRNEG(3) DC(2) = QDRNEG(1) I DC 5 I=1,ND J=I+M IF(EOTEST) 6,7,6 6 K1=2*I GC TO 8 7 K1=2*I-1 8 CC(J)=Q(K1) QC(J)=QDER(K1) 5 OC(J)=D(I) DC 10 I=1,NDRHO IF(EOTEST) 11,12,11 11 Kl=2*I-1 GC TO 9 59

12 K1=2*I 9 PC(I)=PDER(KI) 10 PP(I)=P(KI) NC:=NC+M SUM1=0. SUM2=0. SUM 3=0. SUM4=0. DC 13 I=1,NDQ SUM3=SUM3+DD(I)*CD(I) 13 SUM1=SUMl+DD(I)*CQ(I) DC 14 I=l,NDRHO SUM4=SUM4+DRH( I )PD( I) 14 SUM2=SUM2+DRC( I )PP( I) R2=(SUM1+SUM2)/FK2 R2DER=( SUM3+SUM4)/FK2 IF(PTEST) 20,21,20 20 WCT 6,22,CDPPQCOQDPDSUMlSUM2,SLP3,SUP4,R2,R2DERFK2,Q,P,GDER, 1PCER,CNEG,QDRNEG 22 FCRMAT(12hO R2CAL DUMP/(8E16.8)) 21 RETURN END $COMPILE FORTRAN,PRINT OBJECT,PUNCH OBJECT,EXECUTE,DUMP HYPFINOO SUBROUTINE HYPER (A,BCZt,ANS,ERR) SUM=O. PROC=(A*B*Z)/C IF (PROD) 5,4,5 5 SUM=PROD DC 1 I=2,2000 FI=I-1 PROC=PROD*((A+FI)/(C+FI)*(B+FI)*Z/(FI+1.)) SUM=SUM+PROC IF(ABSF(SLM)-l.E30)2,3,3 2 IF (ABSF(PROD/SUM)-1.E-8) 44,,1 1 CCNTINUE ANS=SUM+1. WRITE OUTPUT TAPE 6,1C0OA,B,C,ZPROD,SUM 100 FCRMAT (21HlHYPER LIMIT EXCEEDED,/6E20.8) ERR=1. RETURN 4 ANS=SUM+1. ERR=O. RETURN 3 ANS=SUM+1. WRITE OUTPUT TAPE 6,lC1,I,A,BC,ZPRCCSUM,ANS 101 FCRMAT (15H1HYPER OVERFLOW,/2XI3,7E18.8) ERR=1. RETURN 60

END S COMPILEFORTRAN,PRINTOBJECTPUNBJEOBJECT,EXECUTE,DUMP GAM 000 FUNCTION GAM(FM,RC) TERMI=(FM+R)*(FM+R+l.) FRACT =(4.*FM*FM-1.)/ ((2.*FM+2.*R-1.)*(2.*FM+2.*R+3.)) TERM2 = (C*C/2.)*(1.-FRACT) GAM =TERM1 +TERM2 RETURN END $ COMPILEFORTRAN,PRINTCBJECT,PUNCHOBJECT,EXECUTE,DUMP BETA 000 FUNCTION BETA(FM,RC) FNUM =R*(R- 1. ) 2.*FM+R)*(2.*FM+R-1.)*C**4 DENOM = (2.*FM+2.*R-1.)**2*(2.*FM+2.*R-3.)*(2.*FM+2.*R+1.) BETA =FNUM/DENOM RETURN END $ COMPILE FCRTRAN,PRINTOBJECTPUNCHOBJECT,EXECUTE,DUMP FACTOOO FUNCTION FACT (N) DIMENSION ANS(33) IF (ISET -5) 1,2,1 1 ISET =5 ANS(1)= 1. DC3I= 2,33 FI= I 3 ANS(I)=FI*ANS(I-1) 2 IF (N-33) 4,4,5 5 WCT 6,7,N 7 FCRMAT (1H0O31H YOU ARE LOOKING FCR FACTORIAL 14) RETURN 4 IF( N) 5,9,8 9 FACT =1. RETURN 8 FACT =ANS(N) RETURN END 61

SASSEMBLEPUNCH OBJECT CF1 000 TITLE CF1......EVALUATION OF CONTINUED FRACTION - METHOD I CF1 001 ENTRY CF1 CF1 002 ~~~~~~~~* Cfl CF1 003 * o.. s.................. PROLOG CF1 034 METHOD I... CFI 005 ~* ~F(K+i) = A(N-K)/(B(N-K)+F(K)), CF1 006 * NWHERE F(O) = O, AND CF1 037 * f~FOR K = O, 1,..., N-. CF1 008 ~* ~RESULT F = FIN) + 8(0). CF1 009 CALLING STATEMENT... CF1 010 ~* ~F = CF1(BZEROtA,B,N,M)t WHERE CF1 011 * BZERO = B(O), CF1 012 e~~* ~A = DIMENSIONED ARRAY CONTAINING A(1)...A(N), CFl 013 e~*~~ B = DIMENSIONED ARRAY CONTAINING B(1)...B(N), CF1 014 ~* ~N = ORDER OF CONTINUED FRACTION CF1 015 ~~~* ~= NUMBER OF ELEMENTS IN ARRAYS A AND B. CF1 016 m*~ ~M = N TO INDICATE NORMAL RETURN, OR CF1 017 * =0= 0 IF DIVIDE CHECK RETURN. CF1 018 CF1 019 CF1 SXA XR1,1 CFL 020 SXA XR2,2 CF1 021 SXA XR4,4 CF1 022 DCT CF1 023 TRA *-1 CF1 024 e CF1 025 *........*..*..O......*LINKAGE CF1 026 CLA 2,4 CF1 027 ADD Al CF1 028 STA LOCA CF1 029 CLA 3,4 CF1 030 ADD Al CF1 031 STA LOCB CF1 032 CLA* 4,4 CF1 033 STO* 5,4 CF1 034 PDX 0,1 CF1 035 TXH *+3,1,0 CF1 036 CLA* 1,4 N = O. CF1 037 TRA XR1-1 CF1 038 CLA* LOCB CF1 039 TRA T2 CF1 040 * CF1 "041 * ITEO.........*....*...ITERATIO CFI 042 Ti BSS 0 AC...F(K) CF1 043 LOCB FAD *, 1 B(N-K)+F(K) CF1 044 T2 STO TEMP CF1 045 LOCA CLA **,1 A(N-K)/(B(N-K)+F(K)) CF1 046 FDP TEMP CF1 047 XCA CF1 048 TIX Tl,,1 DO FOR K = O 1,..., N-1 CF1 049 FAD* 1,4 B(O)+F(N) CF1 050 FRN 62

CTr CF1 051 STZ* 5,4 CFL 052.*~~~~~~~~~ ~~~~CF1 053 ~......................EXIT CF1 054 XR1 AXT *, 1 CF1 055 XR2 AXT **,2 CF1 056 XR4 AXT 4*,4 CF1 057 TRA 6,4 RETURN WITH RESULT IN AC. CF1 058.*~~~~~~~~~ ~~~~CF1 059.......................TEMPORARY STORAGE CF1 060 TEMP PZE CF1 061 * 000*.................. CONSTANTS CF1 062 Al OCT 1 CF1 063,*~~~~~~~~~ ~~~~CFL 064 END CFL 065 SASSEMBLE,PUNCH OBJECT CF2 000 TITLE CF2..... EVALUATION OF CONTINUED FRACTION - METHOD II CF2 001 ENTRY CF2 CF2 002,* CF2 003 *000P...........**... PROLOG CF2 004 METHOD II... CF2 005 ~* ~X(K) = B(K)*X(K-1)+A(K)*X(K-2) CF2 006 -*~~ Y(K) = B(K)*Y(K-1)+A(K)*Y(K-2) CF2 037 -~* ~WHERE X(-1) = 1, X(O) = B(O), AND CF2 008 * YlY(-1) = 0, Y(O) = 1, CF2 009 FOR K = 1, 2,..., N. CF2 010 -* RESULT F = X(N)/Y(N). CF2 011 CALLING STATEMENT... CF2 012 * ~~F = CF2(BZERO,A,B,N,M), WHERE CF2 013 BZERO = B(O), CF2 014 * A = DIMENSIONED ARRAY CONTAINING A(1)...A(N), CF2 015:'~~* ~B = DIMENSIONED ARRAY CONTAINING B(1)...B(N), CF2 016 ~* ~N = ORDER OF CONTINUED FRACTION CF2 017 m*~ -~= MAXIMUM NUMBER OF ITERATIONS PERMITTED, CF2 018 m*~~~ = NUMBER OF ELEMENTS IN ARRAYS A AND B. CF2 019 -~* M~M = ACTUAL NUMBER OF ITERATIONS PERFORMED, OR CF2 020 ~* -= 0 IF DIVIDE CHECK RETURN. CF2 021,,m*~~~~ Cf2 ~CF2 022 CF2 SXA XR1,1 CF2 023 SXA XR2,2 CF2 024 SXA XR4,4 CF2 025 OCT CF2 026 TRA -1 CF2 027, CF2.328 *...................... LINKAGE CF2 029 CLA 2,4 CF2 030 ADD Al CF2 031 63

STA LOCA CF2 032 CLA 3,4 CF2 033 ADD Al CF2 034 STA LOCB CF2 035 CLA* 4,4 CF2 036 PDX 0,1 CF2 037 TXH *+3,1,0 CF2 038 CLA 1,4 N = 0. CF2 039 TRA XR1-1 CF2 040 STD N CF2 041 ~~~~~~~~~~* Cf~~CF2 042 ~*..*. *.*.**** *0000*...INITIALIZATION CF2 043 STZ KMIN2-,2 Y(-1) = 0. CF2 044 CLA* 1,4 CF2 045 STO KMIN1-1 X(O) = 8(0). CF2 046 SLW KMINI-3 CF2 047 CLA Fl CF2 048 STO KMIN1-2 Y(O) = 1. CF2 049 STO KMIN2-1 X(-l) = 1. CF2 050 AXT 1,2 CF2 051 ~~~~~~~~~~~~* ~~CF2 052 *...*e..................RECURSIONS CF2 053 Tl AXT 2,1 CF2 054 T2 LDQ KMIN1,1 CF2 055 LOCB FMP **,2 COMP B(K)*Y(K-1)... CF2 056 STO TEMP CF2 057 LDQ KMIN2,1 COMP A(K)*Y(K-2)... CF2 058 LOCA FMP **,2 CF2 059 FAD TEMP COMP B(K)*Y(K-1) + A(K)*Y(K-2)... CF2 060 FRN CF2 061 LDQ KMIN,11 CF2 062 STQ KMIN2,1 CF2 063 STO KMIN1 1 CF2 064 TIX T2,1,1 SWITCH. CF2 065 CLA KMIN1-1 A(K)/B(K)... CF2 066 FDP KMIN1-2 CF2 067 XCA CF2 068 STO QUO CF2 069 SSP CF2 070 FSB KMIN1-3 SSP CAS TEST TRA ABC TRA N+l TRA N+l ABC CLA QUO TXI N-12,t SLW KMIN1-3 N TXL T1,2,** CF2 076 PXD 0,2 CF2 0 77 STO* 5,4 CF2 078 CLA QUO CF2 079 64

DCT CF2 080 STZ 5,4 CF2 081 IF~~~~~~~~~ ~~~~CF2 082 6..................**...EXIT CF2 083 (Rl AXT **t CF2 084 (R2 AXT **,2 CF2 085 CR4 AXT **,4 CF2 086 TRA 6.4 RETURN WITH RESULT IN AC. CF2 087 CF2 088.S S...o.oee..*o..... TEMPORARY STORAGE CF2 089 )UO PZE CF2 090 PZE CF2 091 PZE Y(K-1) AND YIN). CF2 092 PZE X(K-1). CF2 093 >MIN1 PZE Y(K-2). CF2 094 PZE X(K-2). CF2 095'MIN2 PZE TEMP. CF2 096 EMP SYN KMIN2 CF2 097..................... CONSTANTS CF2 098 il OCT 1 CF2 099:1 OCT 201400000000 CF2 100 EST DEC 1.E-10 CF2 101 END CF2 102 65

THE UNIVERSITY OF MICHIGAN 7133-1-T REFERENCES Chu, L. J. and J. A. Stratton (1941), "Steady-State Solutions of Electromagnetic Field Problems, III:Forced Oscillations of a Prolate Spheroid, " J. Appl. Phys., 12, pp. 241-248. Flammer, C. (1957), Spheroidal Wave Functions, Stanford University Press, Stanford, California. Magnus and Oberhettinger (1954), Formulas and Theorems for the Functions of Mathematical Physics, Chelsea Publishing Company National Bureau of Standards (1964), Handbook of Mathematical Functions; Applied Mathematics Series No. 55., U. S. Government Printing Office. National Bureau of Standards (1945), Tables of Associated Legendre Functions, Columbia Press. Rhodes, D. R (1963), "The Optimum Line Source for the Best Mean-Square Approxi mation to a Given Radiation Pattern," IEEE Trans. PGAP, AP-11, pp. 440-446 Siegel, K. M., F. V. Schultz, B. H. Gere and F. B. Sleator (1956), "The Theoretical and Numerical Determination of the Radar Cross Section of a Prolate Spheroid," IRE Trans. PGAP, AP-4, pp. 261-275. Sleator, F. B. (1964), "Diffraction and Scattering by Regular Bodies-III: The Prolate Spheroid, " The University of Michigan Radiation Laboratory Report No. 3648-6-T. Slepian, D. and H. O. Pollak (1961), "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-I, " Bell Sys. Tech. J., 40, pp. 43-64. Spence, R. D. and S. Granger (1951), "The Scattering of Sound from a Prolate Spheroid, " J. Acoust. Soc.Amer., 23, pp. 701-706. Weeks, W. L. (1958), "Dielectric Coated Spheroidal Radiators, " University of Illinois Antenna Laboratory Report No. 34....________________________ 66

THE UNIVERSITY OF MICHIGAN 7133-1-T PREVIEW OF TABLES The decision to print the tabular entries to six significant digits was based primarily on the belief that six digit precision was obtained for the greater part of the tables. No absolute guarantee, of course, can be made that this is true for every entry but comprehensive checks with various tolerance limits on isolated cases have shown this to be the case. The radial functions are more vulnerable to error due to series limitations as outlined in Section III. Since the scope of these tables is so much broader than that of any other tabulation of the spheroidal functions, it was not possible to make a substantial number of comparisons of data for large c and n. For the range covered by other tables, however, a number of comparisons were carried out with generally favorable results, and for selected values outside these ranges the data was checked by hand computation. Furthermore, several test runs were made with tolerances adjusted to provide for the inclusion of more terms in the summations, and the S and S' functions found thereby differed from the previous data in very few instances, and never by more than one in the sixth significant digit. Since the possibility of losing accuracy in computation is greater for the radial functions than for the others, an extra column was included in the radial function table to show a Wronskian check. This is found from the relation (1) d (2) (2) d (1) _ 1 R (c, ) R(2) (c, )-R( (c, 9)gR (c, 2)mn d mn mn dmn 1) If the left and right hand members are denoted by L and R respectively, the check value listed is 1 —. This can be used to indicate any case in which the accuracy of one or more of the functions may be less than six significant digits. [ ~~~~~~~~~67

THE UNIVERSITY OF MICHIGAN 7133-1-T In the past, various methods have been used for determining the eigenvalues. Consideration was given to the method described by Weeks (1958) in which the eigenvalues are found from those of a symmetric (infinite) matrix. The eventual decision to use the technique explained by Flammer (1957) was due mainly to the availability (in the Radiation Laboratory library) of a large number of check cases which could be used for comparison. Relatively few complications were encountered in the actual computation, and a description of these was given in Section III. It will be noted that, for the smaller values of c, the largest n value computed was less than 20. This was the result of inserting into the program a precautionary test designed to avoid'overflow' in the machine. The loss of information is insignificant for most practical purposes. Since all computer letters are capitalized, the parameters c and n appear in the tables as C and N respectively. 68

THE UNIVERSITY OF MICHIGAN 7133-1-T EIGENVALUES AND NORMALIZATION CONSTANTS 69

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THE UNIVERSITY OF MICHIGAN 7133-1-T RADIAL FUNCTIONS 85

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li ~ r- n ~ O0 ~ ~ ~0 ~ ~ ~ n m 0 ~ n n ~ r-~ m n ~ i ~ n ~ ~ 0 ~ ~ n ~ m L ~ U U V U N m IN 00000 0000 o000 0000 0000 0000 0000 0000 0000 0000 r P.-o oo r0o o N (M r o 10oo o00 r>? C DU-ICoo i-ir Io — m o- i 4 LC% oj < r-4 wwww w wwJ Jw w wJ r ww u Ljww www w ww U.w w l w ww J LU LJ ju J wW u LUW w OOOO a000 0000 0000 000 OO-4O O000'4000 OO O4OOO O-4as0 0NC all0 N fc'j N.tO C' O O 0UL - T-N0 P -Nr- 1A N y r-4N SU-1 r-4 — a r- O 0000 0000 0000 0000 0000 0000 0000 0000:l 0000 - 400 UiwwwJ Www ui w LJ w W ww Ui LL J LW L J L Ww U UJUW UJ W W WW WWWJJ W W ULJW LU LWWW WWI WW W UJ LLI OCM ~O ^O *-'tu??ooi Ne("oO N~r'~o o o rooLA 1Lr oWN co m oot-^Orsi (flQ^O-o M- otU 00 1rO Ln N Itro 6C\Jo Ln C4o oDl 10000 1 O r t-U< CNn* Cf Cr N 0 o L WOOO I COD0000 Co 0000 u Cy M0000 I 0000 0000 s 0000 00-00 0I, 40000 0 000 Ln0 U 0 0 ODNN m M -40oO m I G WO Oo O 4 N 00U) r CO O ~ r 0 LnLaO O-DOc 00uLC-4 ON Go - rOD -<N ir NO O-t- I NO 4r fQ.tML TG 0 1-t 0nO N - O 000 9 rt0?n M 0 N -4 N Coa N ooN m n Co Ln.-4L(M I t a N C'Ln M f -Q CV M 0 - e tO N 9 -4 - " I" I', I I I I I I I I I I* * ~* I* r I* 0 0 0 -<oc-i-<r0 CY - v-4 C4 N N< -4 m mrrtn Nu a quN -r Itn i Ln Win mn 0 %o r~4 ~ h r-pU) u 00o0 0000 000 000 0000 000 0000 0000 00 000000 liiJ U) u l liiii LU L J LU Lii Ui LU LLJ L uiii U L iii wluii LJ LU i ui ui l uiJ ii.J IT ao -. Co N OD-4 N C u N - m o'o rfi cy N N -4 N at -t iu 1 m9 1 (A r-O - It Or aIt- O,-O (4 *4 4 -4 V0c N NN.41r - Q% 0 -4 N P -4oN Ln -4 - ( o (V N Ln C'4 r CV m rCo I I I 1 )1 111 1 1111 1111 111 1 1 1 1 1 1 1 I I I I II L 0 m Un Itn 0 Lfl C- 0 Lf?-OD Co n0 Co N LA pCJ N m rU IcflC%40A N t L1 4 r- 0 0 m 0 N oooi' - o-4 ooooo<-f- (30 NrU oo<Mf- oo o-r Co o oD 0 ooO-r m oNo D oo tr — o -L a, mN oO 0LAOIt OLAO C'.?4 L O 04 U 0 0 7 r- O N OLAO' r.? 0 0U D0 0 0 0L(C0'U O 0 a L A rO. Ir (Xr'.j ga 00 N) ( On NO Jn 1 00 OJD~ <\ OOC'I0A0~ m 0 ( C ONt4IA 0(Ln ITM Ln O-O CLU 0%a0l'.J 00C 000 0 0D000Y0000 0000-4 0000 0000 0004 0000X C- J UQ Ln U.)JL C U Lin C) UJlUU U N WA U r n U UJN LUU U LU Ln U J NL W) U N WI IC N "1 N0NN~o o c r o mo~asfo<r f m fo0'-4 N'.? L LU LA LA Li LA Ln LA LA LA LA 0......"^~tto (ofr* ^f<-o~oooo ~ sjro -~o o ro^r* ^o~? ~ON ~ro ~r ~ ~<if~-m ^^o ~ cr-os crro^ ooo o O^WO~^^iN~orNm *t-^nr -rc i> <~ m.< oo om ~ o O^ ~(<o o'i^-O*y oo m O"^it-< ^''~jN m~ oo~0 ~ r~ ~ ~M( ~<-~ ~0^-^- m o-sr o-fM ^N *(j ~ 0rj ~m >t(0 ~-< -(T0 -(\ oor~n o o~~ ~ocp~n oo -m oo-i cor- o oiin co -i oof^i Oco ~i, f^ o o~ o~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~o~ ~ ~ ~ ~ ~ ir~ro(M ~ mf fM ~ mcQ(M ~ rorM ~ mf~t o ir~r<^<\o fcf~fQ(\0~f ~M~ ~<~io ~ ~c ~ ooo^ oo~or oo~~ ooor^ oo~~ oo~of Oo~^ O~of- 010^ oo'f^ -omO o O4 o o<4 o o < omoO Oomo < o~OO O~O o o<< OmO 4 OinO d O~OO3 000OO00O 000O 000-000-1000O000-O000- 000-O000200 oO ~ o-* rsOfn Of O O ~ O ~ OO O ~O~ ~ fO O~~ ~O O4 - | O O4 ~ ~O O4 f ~O OO -- OO -- O O- O O- OO-nOO- OO -- OO - u ~ * * * ~ *****~~~~~09 r-' r-y-' f-r-p-f'-.^.f~~~~~~~~~~~~~~~~~~~~~~~~~~~r O 9

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he in LA It UN Mn It (n n 10 m,% >o9o 10 rnn 1o N 1 tnW n 0 oo tn ~o >o 0 un 1 10 0 Len 1 U0000 0000 0000 0000 0000 0000 0000 0000 0000 40000 I I I X U.J W i W JUJ WWUJW LU WLUW WWWLUJ ui u LU WL W LULULlJW WUJ WJ UWW L WU U UJ LU LAj W U UJWLJW CuCO n PO4 % 0 U C7U I 7 m a c c-L r4 n m 0 o o r U c-Lr~- o 0 r \j0 o.lt C co r- mO CO 0 -f c co a cO'000M-40'4I N N -4'-"O "'-4-00 00 c00o - 0 N -co <o 00(It 00 00 0 j a, 4It -4 4 rN0 -4o L4I 000O 0000 0000 0000 0000 0000 0000 0000 0000 0000 W W W W WW W W W W W LU W W W WUJ WW UW W WW uW u W U JWW LUWWL UW UJ ui W WWWJ J N 0 It C I 9 co r~~ <y> m G% r- r- m i) a )- Ln -r Q o Ln m - co c5o (n No m Cq 0 as VN c \- NOooo -C M Cli n OD i i~ rm c i o oN e f L a j m o oir 4 o - 00(0 rM -mooI r00 - 00 0 0- 00-0 C 0000 000- 0 0000 N04 -4 0000 N - -iO(N-4 U0000 0 000 0000 00 00 0000 000 0 000 0 0000 0000 UW LUWJ WWW W WWW W uWuW WWWW LU L J i J W WWW WW LU UI WWWW W LU UWW WJ WWU) C 4 flu CY, I r.- cI 0 A a, Cc LA 0 C, N Vt) ODOi0 N c It rOD 0 0r C.-4N 0( a \JL tACOD 4 m r - rC (4?40 0 W L 0 L UW4M-Ua 0 c c n cx ICus m 1 4 a a0% tl 0 o N P- 0 r CID 01U c c O c O r I 6 0 4 fU M Nr j0 LtrNe0J0 UiUr"J0 u% r eONU L rc 0 LA- v eJ0 Lr CJo L U n 0 ODl C, C rC It Nc Lc f or- I 0 -orfa N 9ooco r- rt ooI roo r N - oIt cor oot- o ai o, o iM a,04J OD oo or o0 o-r e OLt o-4a4iN 0'0-4-tI t^ 0tC (n) 0L ( -4 01 0 P. I0m m-o r — f4 -4m 0 0 0L OLAO. 0 0 0 iL 0 CT 0 0n - ) 4<000 0 0004LC 4 000~~t -4 0000 00 L001 0000 00C JU1 00 -li OOJ-N fC —l 4f 4 C^A 0000 0000 0000 0000 0000 0000 0000 00000 000 0000 4-(4)- - 0 -4 - - (I - OD 4 -4 -4 -4 -4 -4 -4 - -4+ -4 -4 -4 -4 -4 -4 -4 -4 -4 4 -4 U, O - -4 -4 - -4 - -4 It r 6i Ln 0 -O co n rt(Mrn o C'or-4 c Io ioc^ 00 I- a', om^ -- N ^ m U) -4 - (n a, cCIo m mh- 1 o mooir~~o rooow f~t-ijr~o' oooomo ONOFj C(4 co r4 O I PO< OO oo oor Ln C) a, z I m oo,t LA Ln 0 - o4 co ooorC'IO U) o U) D Oo a, NF o i 000 1 ^I,,- P N m a, all P- O oC LO n M — ~ r -4 -A - N N- o ~~- cl o Ul u\ o A N I o l m I, N0 0.. C4 4 S 0 0 0 0 0 0 0 0 0 0 4 t-4 -4 0 -4 P-4 4 -4 N N N c m r( cc oo w Co Co Cc co ~???U,??- r no P- 1 I co(4 0 -4 ~j0 \ 4 I & r-co -4co m -- o 0 l- % 1 0 4 P4 0 L I4 I nI I i m oWW 1 ww wWW w0 WWW m~ (4 ~ ww LI)W aw DL 0 0f-0 NNma, 0m- W W 1 nC1 mr 4 ~..:~~~ ~ 0 0 00_0. ~.: -4__:14:~: _ Z C~~~j Ln 10 co (700 -- O0 e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ U~~~~~ OD~~~~~~~~~~~~~~~~~~~

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u0 O0 0000 0000 0000 0000 0000 0000 000 0000 00000 W LU LLJU W LJ WWJ LLJ WWW L W J UJ W WWUJLU LU UI LUJW W uLLJ WW UI LLJ J UI UWWWJ UJ L U4WJW U U ON O L L)n CM o Cr\ o U- ro -r-n rN-. —4-I \ OD no (-i n -Moo -4 0 0 N CA s C or -oo in N o co - rOr-~r r~ ~e It ~ Ln r- ~ ~ n ~ n ~ ~ ~ r-~ a ~- ~ ~ e ~4 ~ ~ m ~ ~ I c ~ t In ~ n ~ ~ ~ o r I4-4rf r- 3 QQC a —a.O a) VI - e 4 T- 4 Q a' -Ot O4 -o fn r-Mr 3: -0r N N0 Ln ( CV 4 N M N N e CC ~ I m0 10 Ln N 0 n tv) a re ~ ~t ~ OD ~ ~ LU00O 0000 0000 0000 0000 0000 0000 0000 0000 0000 W-J LUW WUWW UW W W WWLU uW WWWJ W U WW L LJUJ WWUJW WWW W WULJW W ui LW UJ W Q 0 ( N.-4 CC)-4'Ln N r —Cu -4lA41 a nc 0 Ln t-r o c7I 1-4 Ln 3 O rj u CV u. - 10 r 0- aV m <^ N CC) 10 -- C<M fI- ~-, CTCIC r - <M - - o CC) m j N U O Ul VI L o I ~ r — I- in rsj N in It oo'O 0r N0co m Or r-4 0 N. O4 r 0o 00 NOO m o o-4 4fOL- O.-O4 oCuN r A Cr —r4r o —W -4 M Pco- a, 1 r- j ~( - r ~ ~ 9c ooo oo or. co a 6 — 6 oo f COj co 0no Ocs 4 c C)OD J rO- rs 0000 re4 r4 -4 100 9 % 00 0000 eq N It' -6- Ln -4,-44 NN- rN-4 6 r( C rf -4- Ln - 0 M N It 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 m 4OlI 0 ^-OO L^ t N N " fn -0 0 r (0) 0 m -I It0 >0- 00 C Ur (In -i o f ItO -O 4re 0 0 00 00o O- ri rru- O 4-.O 0.orM (aJrr- 4 l- 0 LAN 00r 4000 0000 0000 0000 0000 0000 0000 c000 0000 0000 o-oo or-oo or ooo or-oo or- ooo oooo - L o oi r r - C oIor ) r- Ln o omI-or - 0D-or COoor — o ^ ro-o4o L 00 D-o4- N mco-o U)r — 0-oror- O LAO 0t Cf 0-? 0 0- OtAO4 014 -4 0 (NJ -40-4 0 0L 0- 0L4a0(V- 0 9 )(00Nua ~ re ~??o???????? N 00N Of 0 QONLA ooUt ooCCii Z'- Ln?? 0 C en - a, n I- N I n N 0 CCU, Ln Na, 0000 00000000 0000<000 0000 000 0000 00000000 I I I 1 1 I I I 1I 1I I 1I 1I 1I 1I 1I 1I 1I 1I 1I 1I 1I (I 1I 1 1I I I 1I t I I I lI I I I I - f T N -4 0 m It r4 c co N r0 P- o o omm m. mIt % u u, ru- m a u N I It or N co r It f\1 Ln 10 a, 4 UC00a a, O0UI C~ i??? U??~???c???? U??? c Uc??o n c L m sia 0 00 00 00 o co 0 c o co cooo0 00

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0 0 0000 0000 0000 0000 0000 0000 0000 0000 0000 U0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 W UL WJ ui UJWW WJ U UWW U W UWJW WW WW UJ UJ Uj ui U JWW L WUUJUJ W WWW U UJ W LUJ W U- 000 IT00f 0 00IT L00 00 0 00 0 -'-0 0 ("-4-4 —')l0 O (M- < l. I,I -4~ N Qr' WO000 0000 0000 0000 0000 0000 0000 0000 0000 0000 WW W WUW UW WWU W WWUW WUJ U.W WWWUJ WWUUIW W WUJi UJWUJ UJUjWWW CNJ-*- co A r-JO'a 0\ ON - OD r- jun OU) o r c o 0 or-LC Ln r' 0 Aa1 L rl O " r-ejfc rj r- 4 a, " n -- cv f- OD cc 0 t - r4- -4NIr- r0- oC r- o'0 -tNr\ - ( o t- -4 aN O n - -o -1-u-9 - r - Ln' 9 00 l a- -o4 rnoor- oo -4 4 ~ 4 oo or-4 ~o r I- N o or —4 0 4 ~ -O o o- oon r ~ - N o 4 oo o ~- ~r-I-t t 000- 000- 000' 000' 0000 0000 0000 0000 0000 0000 OCOO Co C3 Co C Co Co Cu3 Co L WW' U) k UJ LLJ W W L J U I UJ UJ J L LLJ W WW W ~ W LU LU I J iWW~ WW W UJ LW UW LLJ W J uWW 11W 0000 U00 0000 0000 u000 U Q0 OOO 0000 0000 0000 0WW 0W; W W W W ~ W W ~ W ~ W W ~ W W ~ W

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Unclassified Security Classification DOCUMENT CONTROL DATA - R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION The University of Michigan Radiation Laboratory Unclassified Department of Electrical Engineering 2b. GROUP Ann Arbor, Michigan 3. REPORT TITLE Tables of Prolate Spheroidal Functions for m = 0. Volume I 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Scientific Report No. 1 5. AUTHOR(S) (Last name, first name, initial) Hunter, H. E., Kirk, D. B., Senior, T. B. A and Wittenberg, E. R. 6. REPORT DATE 7a. TOTAL NO. OF PAGES.7b. NO. OF REFS April 1965 288 11 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) AF 19(628)-4328 713 -T b. PROJECT NO. 5635 c. Task 563502 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) d. AFCRL 65 283 (I) 10. AVA ILABILITY/LIMITATION NOTICES Qualified Department of Defense agencies, their contractors, or other government agencies may obtain copies of this report from DDC. All other persons and organizations should apply to CFSTI, Sills Bldg., Springfield, Va. 22151 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Cambridge Research Labs. OAR USAF, L. G Hanscom Field, Bedford, Mass 01731 13. ABSTRACT The eigenvalues Xon(c) and normalization constants Non(c) are tabulated for c=0.1(0.1)10.0 and n ranging from 0 up to a maximum of 20. The radial functions of the first and second kinds, and their first derivatives, are given for the same c and n for t=1- 0000500, 1- 0050378, 1- 0206207 and 1 1547005, corresponding to prolate spheroids of length-towidth radios 100:1, 10:1, 5:1 and 2:1 respectively. Values of the angular functions are contained in a companion report (7133-2-T). All computations were carried out on The University of Michigan IBM 7090 computer. The program, which can also be used for the determination of the functions of orders 1 and 2, is described in some detail. DD D 1AN64 1473 Unclassified Security Classification

Unclassified Security Classification 14. LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Tables Spheroidal Functions INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statements of the contractor, subcontractor, grantee, Department of De- such as: fense activity or other organization (corporate author) issuing (1) "Qualified requesters may obtain copies of this the report. report from DDC. " 2a. REPORT SECURITY CLASSIFICATION: Enter the over- (2) "Foreign announcement and dissemination of this all security classification of the report. Indicate whether is not authorized "Restricted Data" is included. Marking is to be in accordance with appropriate security regulations. (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC 2b. GROUP: Automatic downgrading is specified in DoD Di- users shall request through rective 5200.10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional _.' markings have been used for Group 3 and Group 4 as author- (4) "U. S. military agencies may obtain copies of this ized. report directly from DDC. Other qualified users 3. REPORT TITLE: Enter the complete report title in all shall request through capital letters. Titles in all cases should be unclassified.,, If a meaningful title cannot be selected without classifica- tion, show title classification in all capitals in parenthesis (5) "All distribution of this report is controlled. Qualimmediately following the title. ified DDC users shall request through 4. DESCRIPTIVE NOTES: If appropriate, enter the type of." report, e.g., interim, progress, summary, annual, or final. If the report has been furnished to the Office of Technical Give the inclusive dates when a specific reporting period is Services, Department of Commerce, for sale to the public, indicovered. cate this fact and enter the price, if known. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on 11. SUPPLEMENTARY NOTES: Use for additional explanaor in the report. Enter last name, first name, middle initial. tory notes. If military, show rank and branch of service. The name of the principal author is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (pay6. REPORT DATE: Enter the date of the report as day, ing for) the research and development. Include address. month, year; or month, year. If more than one date appears on the report, use date of publication. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though 7a. TOTAL NUMBER OF PAGES: The total page count it may also appear elsewhere in the body of the technical reshould follow normal pagination procedures, i.e., enter the port. If additional space is required, a continuation sheet shall number of pages containing information. be attached. 7b. NUMBER OF REFERENCES: Enter the total number of It is highly desirable that the abstract of classified reports references cited in the report. be unclassified. Each paragraph of the abstract shall end with 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the military security classification of the inthe applicable number of the contract or grant under which formation in the paragraph, represented as (TS), (S), (C), or (U). the report was written. There is no limitation on the length of the abstract. How8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate ever, the suggested length is from 150 to 225 words. military department identification, such as project number, military department identification, such as project number, 14. KEY WORDS: Key words are technically meaningful terms subproject number, system numbers, task number, etc. subprjectumbe, sy m n, eor short phrases that characterize a report and may be used as 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offi- index entries for cataloging the report. Key words must be cial report number by which the document will be identified selected so that no security classification is required. Identiand controlled by the originating activity. This number must fiers, such as equipment model designation, trade name, military be unique to this report. project code name, geographic location, may be used as key 9b. OTHER REPORT NUMBER(S): If the report has been words but will be followed by an indication of technical conassigned any other repcrt numbers (either by theeoriginator text. The assignment of links, rules, and weights is optional. or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those Unclassified Security Classification