*DECK EXINT SUBROUTINE EXINT (X, N, KODE, M, TOL, EN, NZ, IERR) C***BEGIN PROLOGUE EXINT C***PURPOSE Compute an M member sequence of exponential integrals C E(N+K,X), K=0,1,...,M-1 for N .GE. 1 and X .GE. 0. C***LIBRARY SLATEC C***CATEGORY C5 C***TYPE SINGLE PRECISION (EXINT-S, DEXINT-D) C***KEYWORDS EXPONENTIAL INTEGRAL, SPECIAL FUNCTIONS C***AUTHOR Amos, D. E., (SNLA) C***DESCRIPTION C C EXINT computes M member sequences of exponential integrals C E(N+K,X), K=0,1,...,M-1 for N .GE. 1 and X .GE. 0. The C exponential integral is defined by C C E(N,X)=integral on (1,infinity) of EXP(-XT)/T**N C C where X=0.0 and N=1 cannot occur simultaneously. Formulas C and notation are found in the NBS Handbook of Mathematical C Functions (ref. 1). C C The power series is implemented for X .LE. XCUT and the C confluent hypergeometric representation C C E(A,X) = EXP(-X)*(X**(A-1))*U(A,A,X) C C is computed for X .GT. XCUT. Since sequences are computed in C a stable fashion by recurring away from X, A is selected as C the integer closest to X within the constraint N .LE. A .LE. C N+M-1. For the U computation, A is further modified to be the C nearest even integer. Indices are carried forward or C backward by the two term recursion relation C C K*E(K+1,X) + X*E(K,X) = EXP(-X) C C once E(A,X) is computed. The U function is computed by means C of the backward recursive Miller algorithm applied to the C three term contiguous relation for U(A+K,A,X), K=0,1,... C This produces accurate ratios and determines U(A+K,A,X), and C hence E(A,X), to within a multiplicative constant C. C Another contiguous relation applied to C*U(A,A,X) and C C*U(A+1,A,X) gets C*U(A+1,A+1,X), a quantity proportional to C E(A+1,X). The normalizing constant C is obtained from the C two term recursion relation above with K=A. C C Description of Arguments C C Input C X X .GT. 0.0 for N=1 and X .GE. 0.0 for N .GE. 2 C N order of the first member of the sequence, N .GE. 1 C (X=0.0 and N=1 is an error) C KODE a selection parameter for scaled values C KODE=1 returns E(N+K,X), K=0,1,...,M-1. C =2 returns EXP(X)*E(N+K,X), K=0,1,...,M-1. C M number of exponential integrals in the sequence, C M .GE. 1 C TOL relative accuracy wanted, ETOL .LE. TOL .LE. 0.1 C ETOL = single precision unit roundoff = R1MACH(4) C C Output C EN a vector of dimension at least M containing values C EN(K) = E(N+K-1,X) or EXP(X)*E(N+K-1,X), K=1,M C depending on KODE C NZ underflow indicator C NZ=0 a normal return C NZ=M X exceeds XLIM and an underflow occurs. C EN(K)=0.0E0 , K=1,M returned on KODE=1 C IERR error flag C IERR=0, normal return, computation completed C IERR=1, input error, no computation C IERR=2, error, no computation C algorithm termination condition not met C C***REFERENCES M. Abramowitz and I. A. Stegun, Handbook of C Mathematical Functions, NBS AMS Series 55, U.S. Dept. C of Commerce, 1955. C D. E. Amos, Computation of exponential integrals, ACM C Transactions on Mathematical Software 6, (1980), C pp. 365-377 and pp. 420-428. C***ROUTINES CALLED I1MACH, PSIXN, R1MACH C***REVISION HISTORY (YYMMDD) C 800501 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C 900326 Removed duplicate information from DESCRIPTION section. C (WRB) C 910408 Updated the REFERENCES section. (WRB) C 920207 Updated with code with a revision date of 880811 from C D. Amos. Included correction of argument list. (WRB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE EXINT REAL A,AA,AAMS,AH,AK,AT,B,BK,BT,CC,CNORM,CT,EM,EMX,EN, 1 ETOL,FNM,FX,PT,P1,P2,S,TOL,TX,X,XCUT,XLIM,XTOL,Y, 2 YT,Y1,Y2 REAL R1MACH,PSIXN INTEGER I,IC,ICASE,ICT,IERR,IK,IND,IX,I1M,JSET,K,KK,KN,KODE,KS,M, 1 ML,MU,N,ND,NM,NZ INTEGER I1MACH DIMENSION EN(*), A(99), B(99), Y(2) C***FIRST EXECUTABLE STATEMENT EXINT IERR = 0 NZ = 0 ETOL = MAX(R1MACH(4),0.5E-18) IF (X.LT.0.0E0) IERR = 1 IF (N.LT.1) IERR = 1 IF (KODE.LT.1 .OR. KODE.GT.2) IERR = 1 IF (M.LT.1) IERR = 1 IF (TOL.LT.ETOL .OR. TOL.GT.0.1E0) IERR = 1 IF (X.EQ.0.0E0 .AND. N.EQ.1) IERR = 1 IF (IERR.NE.0) RETURN I1M = -I1MACH(12) PT = 2.3026E0*R1MACH(5)*I1M XLIM = PT - 6.907755E0 BT = PT + (N+M-1) IF (BT.GT.1000.0E0) XLIM = PT - LOG(BT) C XCUT = 2.0E0 IF (ETOL.GT.2.0E-7) XCUT = 1.0E0 IF (X.GT.XCUT) GO TO 100 IF (X.EQ.0.0E0 .AND. N.GT.1) GO TO 80 C----------------------------------------------------------------------- C SERIES FOR E(N,X) FOR X.LE.XCUT C----------------------------------------------------------------------- TX = X + 0.5E0 IX = TX C----------------------------------------------------------------------- C ICASE=1 MEANS INTEGER CLOSEST TO X IS 2 AND N=1 C ICASE=2 MEANS INTEGER CLOSEST TO X IS 0,1, OR 2 AND N.GE.2 C----------------------------------------------------------------------- ICASE = 2 IF (IX.GT.N) ICASE = 1 NM = N - ICASE + 1 ND = NM + 1 IND = 3 - ICASE MU = M - IND ML = 1 KS = ND FNM = NM S = 0.0E0 XTOL = 3.0E0*TOL IF (ND.EQ.1) GO TO 10 XTOL = 0.3333E0*TOL S = 1.0E0/FNM 10 CONTINUE AA = 1.0E0 AK = 1.0E0 IC = 35 IF (X.LT.ETOL) IC = 1 DO 50 I=1,IC AA = -AA*X/AK IF (I.EQ.NM) GO TO 30 S = S - AA/(AK-FNM) IF (ABS(AA).LE.XTOL*ABS(S)) GO TO 20 AK = AK + 1.0E0 GO TO 50 20 CONTINUE IF (I.LT.2) GO TO 40 IF (ND-2.GT.I .OR. I.GT.ND-1) GO TO 60 AK = AK + 1.0E0 GO TO 50 30 S = S + AA*(-LOG(X)+PSIXN(ND)) XTOL = 3.0E0*TOL 40 AK = AK + 1.0E0 50 CONTINUE IF (IC.NE.1) GO TO 340 60 IF (ND.EQ.1) S = S + (-LOG(X)+PSIXN(1)) IF (KODE.EQ.2) S = S*EXP(X) EN(1) = S EMX = 1.0E0 IF (M.EQ.1) GO TO 70 EN(IND) = S AA = KS IF (KODE.EQ.1) EMX = EXP(-X) GO TO (220, 240), ICASE 70 IF (ICASE.EQ.2) RETURN IF (KODE.EQ.1) EMX = EXP(-X) EN(1) = (EMX-S)/X RETURN 80 CONTINUE DO 90 I=1,M EN(I) = 1.0E0/(N+I-2) 90 CONTINUE RETURN C----------------------------------------------------------------------- C BACKWARD RECURSIVE MILLER ALGORITHM FOR C E(N,X)=EXP(-X)*(X**(N-1))*U(N,N,X) C WITH RECURSION AWAY FROM N=INTEGER CLOSEST TO X. C U(A,B,X) IS THE SECOND CONFLUENT HYPERGEOMETRIC FUNCTION C----------------------------------------------------------------------- 100 CONTINUE EMX = 1.0E0 IF (KODE.EQ.2) GO TO 130 IF (X.LE.XLIM) GO TO 120 NZ = M DO 110 I=1,M EN(I) = 0.0E0 110 CONTINUE RETURN 120 EMX = EXP(-X) 130 CONTINUE IX = X+0.5E0 KN = N + M - 1 IF (KN.LE.IX) GO TO 140 IF (N.LT.IX .AND. IX.LT.KN) GO TO 170 IF (N.GE.IX) GO TO 160 GO TO 340 140 ICASE = 1 KS = KN ML = M - 1 MU = -1 IND = M IF (KN.GT.1) GO TO 180 150 KS = 2 ICASE = 3 GO TO 180 160 ICASE = 2 IND = 1 KS = N MU = M - 1 IF (N.GT.1) GO TO 180 IF (KN.EQ.1) GO TO 150 IX = 2 170 ICASE = 1 KS = IX ML = IX - N IND = ML + 1 MU = KN - IX 180 CONTINUE IK = KS/2 AH = IK JSET = 1 + KS - (IK+IK) C----------------------------------------------------------------------- C START COMPUTATION FOR C EN(IND) = C*U( A , A ,X) JSET=1 C EN(IND) = C*U(A+1,A+1,X) JSET=2 C FOR AN EVEN INTEGER A. C----------------------------------------------------------------------- IC = 0 AA = AH + AH AAMS = AA - 1.0E0 AAMS = AAMS*AAMS TX = X + X FX = TX + TX AK = AH XTOL = TOL IF (TOL.LE.1.0E-3) XTOL = 20.0E0*TOL CT = AAMS + FX*AH EM = (AH+1.0E0)/((X+AA)*XTOL*SQRT(CT)) BK = AA CC = AH*AH C----------------------------------------------------------------------- C FORWARD RECURSION FOR P(IC),P(IC+1) AND INDEX IC FOR BACKWARD C RECURSION C----------------------------------------------------------------------- P1 = 0.0E0 P2 = 1.0E0 190 CONTINUE IF (IC.EQ.99) GO TO 340 IC = IC + 1 AK = AK + 1.0E0 AT = BK/(BK+AK+CC+IC) BK = BK + AK + AK A(IC) = AT BT = (AK+AK+X)/(AK+1.0E0) B(IC) = BT PT = P2 P2 = BT*P2 - AT*P1 P1 = PT CT = CT + FX EM = EM*AT*(1.0E0-TX/CT) IF (EM*(AK+1.0E0).GT.P1*P1) GO TO 190 ICT = IC KK = IC + 1 BT = TX/(CT+FX) Y2 = (BK/(BK+CC+KK))*(P1/P2)*(1.0E0-BT+0.375E0*BT*BT) Y1 = 1.0E0 C----------------------------------------------------------------------- C BACKWARD RECURRENCE FOR C Y1= C*U( A ,A,X) C Y2= C*(A/(1+A/2))*U(A+1,A,X) C----------------------------------------------------------------------- DO 200 K=1,ICT KK = KK - 1 YT = Y1 Y1 = (B(KK)*Y1-Y2)/A(KK) Y2 = YT 200 CONTINUE C----------------------------------------------------------------------- C THE CONTIGUOUS RELATION C X*U(B,C+1,X)=(C-B)*U(B,C,X)+U(B-1,C,X) C WITH B=A+1 , C=A IS USED FOR C Y(2) = C * U(A+1,A+1,X) C X IS INCORPORATED INTO THE NORMALIZING RELATION C----------------------------------------------------------------------- PT = Y2/Y1 CNORM = 1.0E0 - PT*(AH+1.0E0)/AA Y(1) = 1.0E0/(CNORM*AA+X) Y(2) = CNORM*Y(1) IF (ICASE.EQ.3) GO TO 210 EN(IND) = EMX*Y(JSET) IF (M.EQ.1) RETURN AA = KS GO TO (220, 240), ICASE C----------------------------------------------------------------------- C RECURSION SECTION N*E(N+1,X) + X*E(N,X)=EMX C----------------------------------------------------------------------- 210 EN(1) = EMX*(1.0E0-Y(1))/X RETURN 220 K = IND - 1 DO 230 I=1,ML AA = AA - 1.0E0 EN(K) = (EMX-AA*EN(K+1))/X K = K - 1 230 CONTINUE IF (MU.LE.0) RETURN AA = KS 240 K = IND DO 250 I=1,MU EN(K+1) = (EMX-X*EN(K))/AA AA = AA + 1.0E0 K = K + 1 250 CONTINUE RETURN 340 CONTINUE IERR = 2 RETURN END