C ELI2 10
C ..................................................................ELI2 20
C ELI2 30
C SUBROUTINE ELI2 ELI2 40
C ELI2 50
C PURPOSE ELI2 60
C COMPUTES THE GENERALIZED ELLIPTIC INTEGRAL OF SECOND KIND ELI2 70
C ELI2 80
C USAGE ELI2 90
C CALL ELI2(R,X,CK,A,B) ELI2 100
C ELI2 110
C DESCRIPTION OF PARAMETERS ELI2 120
C R - RESULT VALUE ELI2 130
C X - UPPER INTEGRATION BOUND (ARGUMENT OF ELLIPTIC ELI2 140
C INTEGRAL OF SECOND KIND) ELI2 150
C CK - COMPLEMENTARY MODULUS ELI2 160
C A - CONSTANT TERM IN NUMERATOR ELI2 170
C B - QUADRATIC TERM IN NUMERATOR ELI2 180
C ELI2 190
C REMARKS ELI2 200
C MODULUS K = SQRT(1.-CK*CK). ELI2 210
C SPECIAL CASES OF THE GENERALIZED ELLIPTIC INTEGRAL OF ELI2 220
C SECOND KIND ARE ELI2 230
C F(ATAN(X),K) OBTAINED WITH A=1., B=1. ELI2 240
C E(ATAN(X),K) OBTAINED WITH A=1., B=CK*CK. ELI2 250
C B(ATAN(X),K) OBTAINED WITH A=1., B=0. ELI2 260
C D(ATAN(X),K) OBTAINED WITH A=0., B=1. ELI2 270
C ELI2 280
C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED ELI2 290
C NONE ELI2 300
C ELI2 310
C METHOD ELI2 320
C DEFINITION ELI2 330
C R=INTEGRAL((A+B*T*T)/(SQRT((1+T*T)*(1+(CK*T)**2))*(1+T*T)), ELI2 340
C SUMMED OVER T FROM 0 TO X). ELI2 350
C EQUIVALENT IS THE DEFINITION ELI2 360
C R=INTEGRAL((A+(B-A)*(SIN(T))**2)/SQRT(1-(K*SIN(T))**2), ELI2 370
C SUMMED OVER T FROM 0 TO ATAN(X)). ELI2 380
C EVALUATION ELI2 390
C LANDENS TRANSFORMATION IS USED FOR CALCULATION. ELI2 400
C REFERENCE ELI2 410
C R. BULIRSCH, NUMERICAL CALCULATION OF ELLIPTIC INTEGRALS ANDELI2 420
C ELLIPTIC FUNCTIONS ELI2 430
C HANDBOOK SERIES OF SPECIAL FUNCTIONS ELI2 440
C NUMERISCHE MATHEMATIK VOL. 7, 1965, PP. 78-90. ELI2 450
C ELI2 460
C ..................................................................ELI2 470
C ELI2 480
SUBROUTINE ELI2(R,X,CK,A,B) ELI2 490
C TEST ARGUMENT ELI2 500
IF(X)2,1,2 ELI2 510
1 R=0. ELI2 520
RETURN ELI2 530
C TEST MODULUS ELI2 540
2 C=0. ELI2 550
D=0.5 ELI2 560
IF(CK)7,3,7 ELI2 570
3 R=SQRT(1.+X*X) ELI2 580
R=(A-B)*ABS(X)/R+B*ALOG(ABS(X)+R) ELI2 590
C TEST SIGN OF ARGUMENT ELI2 600
4 R=R+C*(A-B) ELI2 610
IF(X)5,6,6 ELI2 620
5 R=-R ELI2 630
6 RETURN ELI2 640
C INITIALIZATION ELI2 650
7 AN=(B+A)*0.5 ELI2 660
AA=A ELI2 670
R=B ELI2 680
ANG=ABS(1./X) ELI2 690
PIM=0. ELI2 700
ISI=0 ELI2 710
ARI=1. ELI2 720
GEO=ABS(CK) ELI2 730
C LANDEN TRANSFORMATION ELI2 740
8 R=AA*GEO+R ELI2 750
SGEO=ARI*GEO ELI2 760
AA=AN ELI2 770
AARI=ARI ELI2 780
C ARITHMETIC MEAN ELI2 790
ARI=GEO+ARI ELI2 800
C SUM OF SINE VALUES ELI2 810
AN=(R/ARI+AA)*0.5 ELI2 820
AANG=ABS(ANG) ELI2 830
ANG=-SGEO/ANG+ANG ELI2 840
PIMA=PIM ELI2 850
IF(ANG)10,9,11 ELI2 860
9 ANG=-1.E-8*AANG ELI2 870
10 PIM=PIM+3.1415927 ELI2 880
ISI=ISI+1 ELI2 890
11 AANG=ARI*ARI+ANG*ANG ELI2 900
P=D/SQRT(AANG) ELI2 910
IF(ISI-4)13,12,12 ELI2 920
12 ISI=ISI-4 ELI2 930
13 IF(ISI-2)15,14,14 ELI2 940
14 P=-P ELI2 950
15 C=C+P ELI2 960
D=D*(AARI-GEO)*0.5/ARI ELI2 970
IF(ABS(AARI-GEO)-1.E-4*AARI)17,17,16 ELI2 980
16 SGEO=SQRT(SGEO) ELI2 990
C GEOMETRIC MEAN ELI21000
GEO=SGEO+SGEO ELI21010
PIM=PIM+PIMA ELI21020
ISI=ISI+ISI ELI21030
GOTO 8 ELI21040
C ACCURACY WAS SUFFICIENT ELI21050
17 R=(ATAN(ARI/ANG)+PIM)*AN/ARI ELI21060
C=C+D*ANG/AANG ELI21070
GOTO 4 ELI21080
END ELI21090