TEST OF THE CONTINUED-FRACTION COULOMB & BESSEL PROGRAM - COUL90 WHEN LAMBDA IS REAL (IN GENERAL) = L + XM F IS REGULAR AT THE ORIGIN ( X = 0 ) WHILE G IS IRREGULAR ( => INFINITY AT X = 0 ) L F(ETA,X,LAMBDA) G(ETA,X,LAMBDA) FCP = D/DX (F) GCP = D/DX (G) ETA = -50.000 X = 5.000 XLMIN = .00 KFN = 0 (Coulomb) 0 1.5223698D-01 4.4168069D-01 2.0309104D+00 -6.7648537D-01 10 -3.6811436D-01 3.3158832D-01 1.3384675D+00 1.5108886D+00 50 4.2026691D-23 1.3218614D+21 3.8302803D-22 -1.1747060D+22 ETA = -50.000 X = 50.000 XLMIN = 10.00 KFN = 0 (Coulomb) 10 2.7712945D-01 -7.1051944D-01 -1.2206861D+00 -4.7876107D-01 20 7.5587629D-01 -1.5128101D-01 -2.5236060D-01 -1.2724604D+00 ETA = -4.000 X = 5.000 XLMIN = .00 KFN = 0 (Coulomb) 0 4.0786272D-01 6.7432704D-01 1.0982123D+00 -6.3611043D-01 5 2.8365783D-01 -8.7018930D-01 -1.0454500D+00 -3.1820253D-01 10 7.2198552D-02 5.2800349D+00 1.1008435D-01 -5.7999886D+00 ETA = -4.000 X = 50.000 XLMIN = 50.00 KFN = 0 (Coulomb) 50 1.5953399D+00 -5.1531014D-03 -8.6429971D-02 -6.2654651D-01 100 3.1417916D-19 9.3778882D+17 5.3748951D-19 -1.5785526D+18 ETA = 10.000 X = 5.000 XLMIN = .00 KFN = 0 (Coulomb) 0 1.7308286D-06 1.6659407D+05 3.1157507D-06 5.1875469D+07 5 1.3259851D-07 -1.6295734D+08 2.8209675D-07 3.2654019D+08 20 4.0665288D-18 -2.4660591D+18 1.8485566D-17 1.0741835D+19 ETA = 10.000 X = 50.000 XLMIN = 30.00 KFN = 0 (Coulomb) 30 -3.5263544D-01 -1.3973175D+00 -6.6443675D-01 2.0296003D-01 50 5.7672910D-03 1.3422682D+02 3.8956550D-03 -8.2724910D+01 ETA = 100.000 X = 1000.000 XLMIN = .00 KFN = 0 (Coulomb) 0 -1.6558131D-01 -1.0443260D+00 -9.3406327D-01 1.4816571D-01 50 -1.0272888D+00 2.5396390D-01 2.2685598D-01 9.1735326D-01 100 9.3627061D-01 -4.9855425D-01 -4.4316196D-01 -8.3208819D-01 JWKB examples Calculations of Bardin et al. ETA = 100.000 X = 1.000 XLMIN = .00 KFN = 0 (Coulomb) 0 9.0102596-126 3.9320513+123 1.2943113-124 5.5859932+124 5 1.0751085-126 6.3908806+122 1.6582972-125 4.6326737+123 10 4.7191442-129 3.3844536+124 8.4597027-128 -5.8362401+125 30 7.0999356-149 1.1771878+144 2.4183077-147 -3.9022194+145 ETA = 100.000 X = 5.000 XLMIN = .00 KFN = 0 (Coulomb) 0 1.6350771-110 4.8962244+108 1.0295883-109 3.0054621+109 20 4.7717873-116 1.7803490+112 3.5963104-115 -1.3182941+113 ETA = 100.000 X = 50.000 XLMIN = .00 KFN = 0 (Coulomb) 0 1.7046048D-54 1.6934825D+53 2.9638483D-54 2.6760024D+53 20 4.5380773D-56 2.8762506D+53 8.1083884D-56 -4.9466128D+53 ETA = 10.000 X = 5.000 XLMIN = 30.00 KFN = 0 (Coulomb) 30 3.2074547D-28 2.4584387D+26 2.0651670D-27 1.5439226D+27 50 2.1514626D-52 2.8354111D+48 2.2260079D-51 -2.8774860D+49 ETA = -4.000 X = 5.000 XLMIN = .00 KFN = 0 (Coulomb) 30 4.8599439D-01 9.4043056D-01 8.3918879D-01 -4.3375236D-01 50 8.0894931D-01 -6.8424456D-01 -6.0957761D-01 -7.2056410D-01 ETA = 500.000 X = .001 XLMIN = .00 KFN = 0 (Coulomb) 0 6.1474748D-05 7.5147671D+00 8.8101131D-02 1.0765072D+04 **** IEXP = 679 F,FP *10**(-IEXP) G,GP *10**(+IEXP) ETA = 500.000 X = .025 XLMIN = .00 KFN = 0 (Coulomb) 0 4.8701028D-04 5.1159152D+00 1.0267872D-01 1.0755187D+03 **** IEXP = 676 F,FP *10**(-IEXP) G,GP *10**(+IEXP) Other test examples of Coulomb Functions ETA = 2.000 X = 10.000 XLMIN = .00 KFN = 0 (Coulomb) 0 -1.0616086D+00 -3.9930665D-01 -2.9353045D-01 8.3156007D-01 1 -8.6573739D-01 -7.4689323D-01 -5.5578053D-01 6.7559897D-01 ETA = 4.000 X = 10.000 XLMIN = .00 KFN = 0 (Coulomb) 4 4.9467257D-01 1.4437861D+00 4.4133358D-01 -7.3343208D-01 ETA = 5.000 X = 10.000 XLMIN = .00 KFN = 0 (Coulomb) 1 7.3325383D-01 1.9037910D+00 3.1752784D-01 -5.3936760D-01 7 4.9467257D-01 1.4437861D+00 4.4133358D-01 -7.3343208D-01 ETA = 10.000 X = 10.000 XLMIN = .00 KFN = 0 (Coulomb) 7 2.5361712D-04 8.1896132D+02 6.4387143D-04 -1.8638103D+03 10 1.3284056D-06 1.2121253D+05 4.2827576D-06 -3.6199497D+05 ETA = 25.000 X = 10.000 XLMIN = .00 KFN = 0 (Coulomb) 0 1.5484284D-16 1.6141945D+15 3.1457931D-16 3.2793953D+15 1 1.4276487D-16 1.4882853D+15 2.9076832D-16 3.0311721D+15 ETA = 5.000 X = 5.000 XLMIN = .00 KFN = 0 (Coulomb) 1 1.4579731D-02 3.0904676D+01 1.7674398D-02 -3.1123925D+01 19 3.7299073D-09 3.5567050D+07 1.4478350D-08 -1.3004285D+08 20 2.8484532-138 3.8343757+133 7.4460903-137 -9.6973412+134 ETA = 4.000 X = 6.000 XLMIN = .00 KFN = 0 (Coulomb) 0 2.9444922D-01 3.0137838D+00 2.0917340D-01 -1.2552134D+00 ETA = 4.000 X = 7.000 XLMIN = .00 KFN = 0 (Coulomb) 0 5.5158292D-01 2.1164851D+00 3.0193453D-01 -6.5440758D-01 ETA = 4.000 X = 8.000 XLMIN = .00 KFN = 0 (Coulomb) 0 8.8245265D-01 1.5526082D+00 3.4666624D-01 -5.2327245D-01 ETA = 1.000 X = .600 XLMIN = .00 KFN = 0 (Coulomb) 0 1.0709689D-01 2.7924404D+00 2.5192544D-01 -2.7686444D+00 ETA = 1.000 X = 3.000 XLMIN = .00 KFN = 0 (Coulomb) 0 1.0840527D+00 6.2703951D-01 3.0191679D-01 -7.4782919D-01 ETA = 1.000 X = 6.000 XLMIN = .00 KFN = 0 (Coulomb) 0 -1.6718337D-01 -1.0908385D+00 -8.9250499D-01 1.5803711D-01 ETA = 1.995 X = 1.200 XLMIN = .00 KFN = 0 (Coulomb) 0 4.3442292D-02 7.4684279D+00 8.0757951D-02 -9.1354542D+00 ETA = 3.981 X = 2.400 XLMIN = .00 KFN = 0 (Coulomb) 0 5.6239980D-03 5.8254036D+01 9.4364234D-03 -8.0065863D+01 ETA = 18.564 X = 116.790 XLMIN = .00 KFN = 0 (Coulomb) 0 -2.1971337D-01 -1.0782022D+00 -8.9027280D-01 1.8253761D-01 30 1.0607165D+00 3.8873669D-01 3.0291709D-01 -8.3174444D-01 60 -1.2326340D+00 -1.8670756D-01 -1.1468983D-01 7.9389868D-01 80 -8.2521028D-01 1.2292313D+00 5.7141019D-01 3.6064109D-01 100 5.0265191D-01 4.2105318D+00 1.4494939D-01 -7.7526009D-01 120 6.1064182D-05 1.3251262D+04 3.8572049D-05 -8.0058663D+03 140 5.5999298D-11 1.0208731D+10 4.9481237D-11 -8.8368854D+09 160 1.4728687D-18 3.0907271D+17 1.6284453D-18 -3.3722761D+17 180 2.1472254D-27 1.7825709D+26 2.8186616D-27 -2.3171930D+26 200 2.6457290D-37 1.2556817D+36 3.9974338D-37 -1.8824663D+36 ETA = -4.000 X = 5.000 XLMIN = .00 KFN = 0 (Coulomb) 0 4.0786272D-01 6.7432704D-01 1.0982123D+00 -6.3611043D-01 1 -6.4225633D-01 -4.6720421D-01 -7.5891296D-01 1.0049456D+00 2 7.9895741D-01 -1.1512121D-01 -1.5779695D-01 -1.2288943D+00 3 -2.5686309D-01 7.8798992D-01 1.1432294D+00 3.8599059D-01 Cylindrical Bessel Functions --- Abramowicz & Stegun --- pages 407-408 CYL Bessels X = 1.000 XLMIN = .00 KFN = 2 (CylBess) 0 7.6519769D-01 8.8256964D-02 -4.4005059D-01 7.8121282D-01 10 2.6306151D-10 -1.2161801D+08 2.6186351D-09 1.2093999D+09 30 3.4828698D-42 -3.0481288D+39 1.0442990D-40 9.1391293D+40 40 1.1079159D-60 -7.1848748D+57 4.4303121D-59 2.8730286D+59 CYL Bessels X = 2.000 XLMIN = .00 KFN = 2 (CylBess) 0 2.2389078D-01 5.1037567D-01 -5.7672481D-01 1.0703243D-01 10 2.5153863D-07 -1.2918454D+05 1.2346503D-06 6.3136288D+05 30 3.6502563D-33 -2.9132238D+30 5.4635975D-32 4.3597778D+31 40 1.1960775D-48 -6.6615412D+45 2.3892360D-47 1.3305990D+47 CYL Bessels X = 5.000 XLMIN = .00 KFN = 2 (CylBess) 0 -1.7759677D-01 -3.0851763D-01 3.2757914D-01 -1.4786314D-01 10 1.4678026D-03 -2.5129110D+01 2.5846778D-03 4.2494337D+01 30 2.6711773D-21 -4.0285684D+18 1.5810272D-20 2.3821403D+19 40 8.7022416D-33 -9.2168166D+29 6.9085369D-32 7.3141198D+30 CYL Bessels X = 10.000 XLMIN = .00 KFN = 2 (CylBess) 0 -2.4593576D-01 5.5671167D-02 -4.3472746D-02 -2.4901542D-01 10 2.0748611D-01 -3.5981415D-01 8.4369579D-02 1.6051489D-01 30 1.5510961D-12 -7.2561423D+09 4.3964788D-12 2.0476167D+10 40 6.0308953D-21 -1.3628033D+18 2.3377115D-20 5.2734406D+18 CYL Bessels X = 50.000 XLMIN = .00 KFN = 2 (CylBess) 0 5.5812328D-02 -9.8064995D-02 9.7511828D-02 5.6795669D-02 10 -1.1384785D-01 5.7238972D-03 -4.4228912D-03 -1.1161457D-01 30 4.8434257D-02 -1.1645723D-01 9.2453375D-02 4.0581421D-02 40 -1.3817628D-01 -4.5308011D-02 3.1054826D-02 -8.1963148D-02 CYL Bessels X = 100.000 XLMIN = .00 KFN = 2 (CylBess) 0 1.9985850D-02 -7.7244313D-02 7.7145352D-02 2.0372312D-02 10 -5.4732177D-02 5.8331574D-02 -5.7763544D-02 -5.4753154D-02 30 8.1460130D-02 6.1388392D-03 -6.3037659D-03 7.7676036D-02 40 7.2701755D-02 4.0746852D-02 -3.7779097D-02 6.6392049D-02 Spherical Bessel Functions --- Abramowitz & Stegun --- pages 465-466 sph Bessels X = 1.000 XLMIN = .00 KFN = 1 (SphBess) The 3 sets of results are COUL90(KFN), SBESJY & (1/X) RICBES 0 8.4147098D-01 -5.4030231D-01 -3.0116868D-01 1.3817733D+00 0 8.4147098D-01 -5.4030231D-01 -3.0116868D-01 1.3817733D+00 0 8.4147098D-01 -5.4030231D-01 5.4030231D-01 8.4147098D-01 10 7.1165526D-11 -6.7221501D+08 7.0855571D-10 7.3588750D+09 10 7.1165526D-11 -6.7221501D+08 7.0855571D-10 7.3588750D+09 10 7.1165526D-11 -6.7221501D+08 7.7972124D-10 6.6866600D+09 30 5.5668313D-43 -2.9464285D+40 1.6691655D-41 9.1289331D+41 30 5.5668313D-43 -2.9464285D+40 1.6691655D-41 9.1289331D+41 30 5.5668313D-43 -2.9464285D+40 1.7248339D-41 8.8342902D+41 40 1.5382104D-61 -8.0284509D+58 6.1509880D-60 3.2906484D+60 40 1.5382104D-61 -8.0284509D+58 6.1509880D-60 3.2906484D+60 40 1.5382104D-61 -8.0284509D+58 6.3048090D-60 3.2103639D+60 sph Bessels X = 2.000 XLMIN = .00 KFN = 1 (SphBess) The 3 sets of results are COUL90(KFN), SBESJY & (1/X) RICBES 0 4.5464871D-01 2.0807342D-01 -4.3539777D-01 3.5061200D-01 0 4.5464871D-01 2.0807342D-01 -4.3539777D-01 3.5061200D-01 0 9.0929743D-01 4.1614684D-01 -4.1614684D-01 9.0929743D-01 10 6.8253009D-08 -3.5541472D+05 3.3528817D-07 1.9168920D+06 10 6.8253009D-08 -3.5541472D+05 3.3528817D-07 1.9168920D+06 10 1.3650602D-07 -7.1082944D+05 7.3882935D-07 3.4783693D+06 30 5.8366179D-34 -1.4073939D+31 8.7363798D-33 2.1766840D+32 30 5.8366179D-34 -1.4073939D+31 8.7363798D-33 2.1766840D+32 30 1.1673236D-33 -2.8147877D+31 1.8056421D-32 4.2126286D+32 40 1.6609788D-49 -3.7209293D+46 3.3179529D-48 7.6184788D+47 40 1.6609788D-49 -3.7209293D+46 3.3179529D-48 7.6184788D+47 40 3.3219576D-49 -7.4418586D+46 6.8020037D-48 1.4864865D+48 sph Bessels X = 5.000 XLMIN = .00 KFN = 1 (SphBess) The 3 sets of results are COUL90(KFN), SBESJY & (1/X) RICBES 0 -1.9178485D-01 -5.6732437D-02 9.5089408D-02 -1.8043837D-01 0 -1.9178485D-01 -5.6732437D-02 9.5089408D-02 -1.8043837D-01 0 -9.5892427D-01 -2.8366219D-01 2.8366219D-01 -9.5892427D-01 10 4.0734424D-04 -2.6656114D+01 7.2194238D-04 5.0954007D+01 10 4.0734424D-04 -2.6656114D+01 7.2194238D-04 5.0954007D+01 10 2.0367212D-03 -1.3328057D+02 4.0170561D-03 2.2811392D+02 30 4.2827302D-22 -7.7607176D+18 2.5354382D-21 4.7453795D+19 30 4.2827302D-22 -7.7607176D+18 2.5354382D-21 4.7453795D+19 30 2.1413651D-21 -3.8803588D+19 1.3105464D-20 2.2950826D+20 40 1.2103476D-33 -2.0557587D+30 9.6096080D-33 1.6726571D+31 40 1.2103476D-33 -2.0557587D+30 9.6096080D-33 1.6726571D+31 40 6.0517379D-33 -1.0278794D+31 4.9258388D-32 8.1577096D+31 sph Bessels X = 10.000 XLMIN = .00 KFN = 1 (SphBess) The 3 sets of results are COUL90(KFN), SBESJY & (1/X) RICBES 0 -5.4402111D-02 8.3907153D-02 -7.8466942D-02 -6.2792826D-02 0 -5.4402111D-02 8.3907153D-02 -7.8466942D-02 -6.2792826D-02 0 -5.4402111D-01 8.3907153D-01 -8.3907153D-01 -5.4402111D-01 10 6.4605154D-02 -1.7245367D-01 2.9030740D-02 7.7293250D-02 10 6.4605154D-02 -1.7245367D-01 2.9030740D-02 7.7293250D-02 10 6.4605154D-01 -1.7245367D+00 3.5491255D-01 6.0047883D-01 30 2.5120574D-13 -6.9083186D+09 7.1272058D-13 2.0207736D+10 30 2.5120574D-13 -6.9083186D+09 7.1272058D-13 2.0207736D+10 30 2.5120574D-12 -6.9083186D+10 7.3784115D-12 1.9516904D+11 40 8.4356716D-22 -1.5103049D+18 3.2711521D-21 5.9978186D+18 40 8.4356716D-22 -1.5103049D+18 3.2711521D-21 5.9978186D+18 40 8.4356716D-21 -1.5103049D+19 3.3555089D-20 5.8467881D+19 sph Bessels X = 50.000 XLMIN = .00 KFN = 1 (SphBess) The 3 sets of results are COUL90(KFN), SBESJY & (1/X) RICBES 0 -5.2474971D-03 -1.9299321D-02 1.9404271D-02 -4.8615107D-03 0 -5.2474971D-03 -1.9299321D-02 1.9404271D-02 -4.8615107D-03 0 -2.6237485D-01 -9.6496603D-01 9.6496603D-01 -2.6237485D-01 10 -1.5039221D-02 1.3524688D-02 -1.2916299D-02 -1.4981567D-02 10 -1.5039221D-02 1.3524688D-02 -1.2916299D-02 -1.4981567D-02 10 -7.5196107D-01 6.7623438D-01 -6.6085415D-01 -7.3555364D-01 30 -1.4946735D-03 -2.2412268D-02 1.7807044D-02 -6.0465145D-04 30 -1.4946735D-03 -2.2412268D-02 1.7807044D-02 -6.0465145D-04 30 -7.4733673D-02 -1.1206134D+00 8.8885752D-01 -5.2644841D-02 40 -2.6063370D-02 4.9787972D-05 9.7438668D-04 -1.5349071D-02 40 -2.6063370D-02 4.9787972D-05 9.7438668D-04 -1.5349071D-02 40 -1.3031685D+00 2.4893986D-03 2.2655965D-02 -7.6740377D-01 sph Bessels X = 100.000 XLMIN = .00 KFN = 1 (SphBess) The 3 sets of results are COUL90(KFN), SBESJY & (1/X) RICBES 0 -5.0636564D-03 -8.6231887D-03 8.6738253D-03 -4.9774245D-03 0 -5.0636564D-03 -8.6231887D-03 8.6738253D-03 -4.9774245D-03 0 -5.0636564D-01 -8.6231887D-01 8.6231887D-01 -5.0636564D-01 10 -1.9565786D-04 1.0025777D-02 -9.9685241D-03 -2.9539408D-04 10 -1.9565786D-04 1.0025777D-02 -9.9685241D-03 -2.9539408D-04 10 -1.9565786D-02 1.0025777D+00 -9.9704807D-01 -1.9513631D-02 30 8.7006285D-03 -5.4129293D-03 5.0636694D-03 8.3431577D-03 30 8.7006285D-03 -5.4129293D-03 5.0636694D-03 8.3431577D-03 30 8.7006285D-01 -5.4129293D-01 5.1506756D-01 8.2890284D-01 40 1.0434109D-02 -7.0484204D-04 5.2990162D-04 9.5481567D-03 40 1.0434109D-02 -7.0484204D-04 5.2990162D-04 9.5481567D-03 40 1.0434109D+00 -7.0484204D-02 6.3424270D-02 9.5411083D-01 Bessels with order 1/3 --- Watson table III --- pages 714-729 CYL Bessels X = .240 XLMIN = .33 KFN = 2 (CylBess) 0 5.4640874D-01 -1.3761797D+00 7.0949514D-01 3.0676478D+00 CYL Bessels X = .500 XLMIN = .33 KFN = 2 (CylBess) 0 6.7283083D-01 -8.4062783D-01 3.1979029D-01 1.4928194D+00 CYL Bessels X = 1.000 XLMIN = .33 KFN = 2 (CylBess) 0 7.3087640D-01 -2.7880164D-01 -5.5285175D-02 8.9212536D-01 CYL Bessels X = 2.000 XLMIN = .33 KFN = 2 (CylBess) 0 4.4293982D-01 3.4319997D-01 -4.5613892D-01 3.6520317D-01 CYL Bessels X = 4.000 XLMIN = .33 KFN = 2 (CylBess) 0 -3.5542737D-01 1.7941677D-01 -1.3603948D-01 -3.7911312D-01 CYL Bessels X = 5.000 XLMIN = .33 KFN = 2 (CylBess) 0 -3.0642046D-01 -1.8192321D-01 2.1289098D-01 -2.8912607D-01 CYL Bessels X = 8.000 XLMIN = .33 KFN = 2 (CylBess) 0 2.5977616D-01 1.0958779D-01 -1.2590638D-01 2.5321672D-01 CYL Bessels X = 10.000 XLMIN = .33 KFN = 2 (CylBess) 0 -1.8614517D-01 1.7020112D-01 -1.6102288D-01 -1.9477113D-01 CYL Bessels X = 16.000 XLMIN = .33 KFN = 2 (CylBess) 0 -1.0416268D-01 1.7008276D-01 -1.6687528D-01 -1.0950301D-01 Blanch test of recurrence through a zero of J5 near x = 8.7714838 CYL Bessels X = 8.771 XLMIN = .00 KFN = 2 (CylBess) 0 -3.1676251D-02 2.6731905D-01 -2.6594508D-01 -4.6917089D-02 1 2.6594508D-01 4.6917089D-02 -6.1995532D-02 2.6197023D-01 2 9.2314813D-02 -2.5662141D-01 2.4489623D-01 1.0542975D-01 4 -2.4543421D-01 1.4447911D-01 -1.1192370D-01 -2.2982823D-01 5 3.9171188D-09 2.9571405D-01 -2.4543421D-01 -2.4086473D-02 6 2.4543422D-01 1.9265205D-01 -1.6788554D-01 1.6393334D-01 10 1.0782585D-01 -6.0328567D-01 7.1170366D-02 2.7490887D-01 14 2.9266487D-03 -1.0025114D+01 3.7342185D-03 1.2007720D+01 18 1.9900428D-05 -1.0185338D+03 3.5986249D-05 1.8052442D+03 20 1.1173628D-06 -1.5857119D+04 2.3037888D-06 3.2260687D+04 CYL Bessels X = 1.000 XLMIN = 100.00 KFN = 2 (CylBess) 100 8.4561605-189 -3.7644396+185 8.4557419-187 -3.7597925+187 Examples of ERROR conditions ARB 7/3/91 COUL90 ERROR! IFAIL= 2 ETA = -500.000 X = .001 XLMIN = .00 KFN = 0 (Coulomb) IFAIL = 2 NL,KFN = 1 0 .0010 -500.0000 .0000 0