A fast and accurate DE-formula algorithm to evaluate Ambartsumian-Chandrarasekhar $H$-function for isotropic scattering

Abstract

In the present work, a compact, fast, and yet accurate algorithm is developed to calculate the numerical values of the Ambartsumian-Chandrasekhar’s $H$-function for isotropic scattering and its moments on the basis of the double exponential (DE) formula of Takahashi and Mori (RIMS, Kyoto Univ., 9:721, 1974). The main improvement made in the new method is an elimination of the iterative procedure for automatic adjustment of the step-size of integrations carried out with the DE-formula. Instead, a set of optimal values for the upper limit of integration $T_{\text{max}}$ and the number of division points $N_{\text{T}}$ to specify the step-size of the quadrature is predetermined for calculations of the $H$-function with a 15-digit accuracy, and also another for the evaluations of the moments with an accuracy of 14-digits or better. FORTRAN90 subroutines HFISCA for the $H$-function and HFMOMENT for the moments of arbitrary degrees are subsequently constructed (their source codes and a driver together with a sample set of output are shown in Appendix of this paper). Tables of sample calculations of the $H$-function and its moments of degree −1 through 6 carried out by these programs are also presented. The routines HFISCA and HFMOMENT should prove useful not only in astrophysical applications but also in other disciplines of science such as the electron transports in condensed matter and remote-sensing data analyses. A request for a copy of the Fortran 90 source code of the program can be made by writing to kawabata@rs.kagu.tus.ac.jp.

Appendix: a program to calculate \(H(\varpi _{0}, \mu )\) and its moments

Given below is a Fortran90 program set consisted of a main program MAIN and three function subprograms CALRSPI0, HFISCA, and HFMOMENTS.

The function subprogram HFISCA enables us to evaluate, by means of the double-exponential (DE) formula developed by Takahashi and Mori (1974), the numerical values of the Ambartsumian-Chandrasekhar’s $H$-function $H(\varpi _{0}, \mu )$ for isotropic scattering, with a 15-digit accuracy, for a given set of values for $\varpi _{0}$, $1-\varpi _{0}$, and $\mu$. The value of $1-\varpi _{0}$ may better be calculated as shown in MAIN by using the function subprogram CALRSPI0, which should help minimize the effect of the loss of significant digits in the case the value of $\varpi _{0}$ is close to unity.

The function subprogram HFMOMENT produces again by employing the DE-formula the numerical values of the $n$-th degree moment of the $H$-function ($n\ge -1$) defined by Eqs. (28a)-(28d) for given values of $\varpi _{0}$, $1-\varpi _{0}$, and $n$. It should be reminded that, for $n\ge 1$, we use the integral expression Eq. (28b) to improve numerical accuracy of integration:

$$ \alpha _{n}(\varpi _{0})=\frac{1}{n}\int _{0}^{1}\!\! \! H(\varpi _{0}, s^{1/n}) s^{1/n}ds \quad (n\ge 1), $$

(32)

although the usual forms of defining equations are employed for $n=-1$ and $n=0$:

$$ \alpha _{-1}^{\ast}(\varpi _{0})=\int _{0}^{1}\!\!\! \left [H(\varpi _{0}, \mu ^{\prime})-1\right ]\frac{1}{\mu ^{\prime}} d \mu ^{\prime}=2\ln H(\varpi _{0}, 1), $$

(33)

$$ \alpha _{0}(\varpi _{0})=\int _{0}^{1}\!\!\!H(\varpi _{0}, \mu ^{\prime}) d\mu ^{\prime}, $$

(34)

(Ivanov 1973: van de Hulst 1980). The source code of the program is as shown below:

PROGRAM MAIN; ! driver for routines HFISCA and HFMOMENTS IMPLICIT REAL*8 (A-H,O-Z) PARAMETER(NPI0=6, NAMU=6, NALPHA=4) REAL*8 PI0S(NPI0), RSPI0S(NPI0), AMUS(NAMU),HFS(NAMU,NPI0),ALPHAS(-1:NALPHA) CHARACTER*6 CHAR; DATA CHAR/'Alpha_'/ !!!!EXTERNAL HFISCA DATA PI0S/0.1D0, 0.3D0, 0.5D0, 0.7D0, 0.9D0, 1.D0/ DATA AMUS/0.1D0, 0.3D0, 0.5D0, 0.7D0, 0.9D0, 1.D0/ ! PI0=the single scattering albedo; RSPI0=1-PI0; AMU=cos(zenith angle)   WRITE(*,*) '(1) Sample Numerical Calculations of the H-function'   WRITE(*,'(1X,A,20(7X,F3.1,12X))') 'PI0 RSPI0/ AMU=',(AMUS(J),J=1,NAMU)   DO 100 I=1,NPI0      PI0=PI0S(I)      RSPI0=CALRSPI0(PI0)      RSPI0S(I)=RSPI0    DO J=1,NAMU      HFS(J,I)=HFISCA(PI0,RSPI0,AMUS(J))    END DO      WRITE(*,'(1X,F3.1,2X,F3.1,7X,10(F17.15,5X))') PI0,RSPI0,(HFS(J,I),J=1,NAMU) 100 END DO    WRITE(*,'(/1X,A)') '(2)The Moments of the H-function'    WRITE(*,'(1X,''PI0 RSPI0'',6X,6(4X,A,I2,10X))') (CHAR,J,J=-1,NALPHA)   DO I=1,NPI0      PI0=PI0S(I)      RSPI0=CALRSPI0(PI0)     DO N=-1,NALPHA       ALPHAS(N)=HFMOMENT(PI0,RSPI0,N)     END DO     WRITE(*,'(1X,F3.1,2X,F3.1,7X,10(F17.15,5X))') PI0,RSPI0,(ALPHAS(N),N=-1,NALPHA)   END DO    STOP    END DOUBLE PRECISION FUNCTION CALRSPI0(PI0) ! << Calculates 1-PI0 >> CHARACTER*20 SINTER; REAL*8 PI0       IF(PI0<0.99D0) THEN            CALRSPI0=1.D0-PI0       ELSE          WRITE(SINTER,'(F18.15)') 1.D0-PI0          READ(SINTER, '(F18.15)') CALRSPI0       END IF       RETURN       END FUNCTION HFISCA(PI0,RSPI0,AMU);  IMPLICIT REAL*8 (A-H,O-Z) !  <<Ambartsumian-Chandrasekhar H-Function for Isotropic Scattering>> ! [PI0=the single scattering albedo,RSPI0=1.D0-PI0,AMU=cos(zenith angle)] SAVE XIS,TANXQS,UIS,VIS,WIS,IFIRST PARAMETER(PI=3.141592653589793D0, PIHF=PI/2.D0, XTRANS=1.238798627279953D-1) PARAMETER(NT=111,TMAX=3.2215D0,H=TMAX/NT, A=0.D0, B=PIHF,BMA2=(B-A)/2.D0,COEF=PI*H/8.D0) REAL*8 WIS(0:NT),XIS(-NT:NT),TANXQS(-NT:NT),UIS(-NT:NT),VIS(-NT:NT) INTEGER :: IFIRST=0 !====================================================================================  FFN(TANXSQ,U,V)=DLOG(RSPI0+PI0*V)*U/(1.D0+AMUSQ*TANXSQ)  VFN(S)=S*(3.3333333333333333D-1+S*(2.2222222222222223D-2+S*(2.1164021164021165D-3    & &      +S*(2.1164021164021165D-4+S*(2.1377799155576935D-5+2.1644042808063972D-6*S))))) !==================================================================================== IF(IFIRST==0) THEN    ! === The Section for the DE-formula preparation ===   DO  I=NT,0,-1     T=H*I; SINHT=DSINH(T); SINHPHT=PIHF*SINHT; COSHPHT=DCOSH(SINHPHT)     WIS(I)=DCOSH(T)/COSHPHT**2; AIPRP=BMA2*DEXP(-SINHPHT)/COSHPHT     X=B-AIPRP; XIS(I)=X; TANX=DTAN(X); TANXQS(I)=TANX*TANX; UIS(I)=1.D0+TANXQS(I)     VIS(I)=1.D0-X/TANX J=-I; X=AIPRP; XIS(J)=X; TANX=DTAN(X); TANXQS(J)=TANX*TANX; UIS(J)=1.D0+TANXQS(J)       IF(X<XTRANS)THEN; VIS(J)=VFN(X*X); ELSE; VIS(J)=1.D0-X/TANX; ENDIF     END DO;  IFIRST=1   END IF              ! End of the DE-formula preparation ! ---<<  Calculations of H(PI0,AMU)  >>---   IF(PI0==0.D0.OR.AMU==0.D0) THEN;  ! == Return H(0,AMU)=1 or H(PI0,0)=1     HFISCA=1.D0; RETURN ;   ELSE                              !  Evaluate H(PI0, AMU) for PI0*AMU>0     AMUSQ=AMU*AMU     SUM=(FFN(TANXQS(NT),UIS(NT),VIS(NT))+FFN(TANXQS(-NT),UIS(-NT),VIS(-NT)))*WIS(NT)     DO I=NT-1,1,-1       J=-I       SUM=SUM+(FFN(TANXQS(I),UIS(I),VIS(I))+FFN(TANXQS(J),UIS(J),VIS(J)))*WIS(I)     END DO       HFISCA=DEXP(-AMU*COEF*(SUM+FFN(TANXQS(0),UIS(0),VIS(0))))   END IF RETURN END FUNCTION HFMOMENT(PI0,RSPI0,N);  IMPLICIT REAL*8 (A-H,O-Z) !  <<The N-th degree moment HFMOMENT of the H-function for a given albedo value  PI0>> SAVE XIS, WIS, IFIRST;  PARAMETER(PI=3.141592653589793D0,PIHF=PI/2.D0) PARAMETER(NT=109,TMAX=3.3834D0,H=TMAX/NT,A=0.D0, B=1.D0,BMA2=(B-A)/2.D0,COEF=H*BMA2*PIHF) REAL*8 WIS(0:NT), XIS(-NT: NT); INTEGER:: IFIRST=0 IF(N.EQ.-1) THEN;  HFMOMENT=2.D0*DLOG(HFISCA(PI0,RSPI0,1.D0)); RETURN; END IF IF(IFIRST.EQ.0) THEN       ! *** Section for the DE-formula preparation ***   DO  I=NT,0,-1     T=H*I; SINHT=DSINH(T); SINHPHT=PIHF*SINHT; COSHPHT=DCOSH(SINHPHT);     WIS(I)=DCOSH(T)/COSHPHT**2;     AIPRP=BMA2*DEXP(-SINHPHT)/COSHPHT;  XIS(I)=B-AIPRP; XIS(-I)=AIPRP;   END DO ;   IFIRST=1 END IF                     ! *** End of the DE-formula preparation      *** ! ===<<  Calculations of the H-function moments of the degree -1 through n  >>===    HFMOMENT=0.D0 DO 10 I=NT,1,-1    XI=XIS(I); XJ=XIS(-I); WI=WIS(I)    SELECT CASE(N)      CASE(0) ; HFMOMENT=HFMOMENT+(HFISCA(PI0,RSPI0,XI)+HFISCA(PI0,RSPI0,XJ))*WI      CASE DEFAULT         AN=1.D0/N; AMUI=XI**AN; AMUJ=XJ**AN         HFMOMENT=HFMOMENT+(AMUI*HFISCA(PI0,RSPI0,AMUI)+AMUJ*HFISCA(PI0,RSPI0,AMUJ))*WI    END SELECT 10 END DO    X0=XIS(0)    SELECT CASE(N)      CASE(0) ;        HFMOMENT=COEF*(HFMOMENT+HFISCA(PI0,RSPI0,X0))      CASE DEFAULT         AMU0=X0**AN;  HFMOMENT=COEF*(HFMOMENT+AMU0*HFISCA(PI0,RSPI0,AMU0))/N    END SELECT RETURN END

Our results are:

 (1) Sample Numerical Calculations of the H-function  PI0 RSPI0/ AMU=    0.1        0.3        0.5        0.7        0.9        1.0  0.1  0.9        1.0123781  1.0230056  1.0289223  1.0328465  1.0356742  1.0368156  0.3  0.7        1.0398749  1.0763650  1.0975591  1.1119712  1.1225365  1.1268444  0.5  0.5        1.0723688  1.1438895  1.1877351  1.2185599  1.2416937  1.2512596  0.7  0.3        1.1130318  1.2364193  1.3179451  1.3781356  1.4249566  1.4447461  0.9  0.1        1.1721431  1.3913503  1.5560338  1.6893476  1.8007874  1.8500985  1.0  0.0        1.2473504  1.6425223  2.0127788  2.3739749  2.7305877  2.9078105  (2)The Moments of the H-function  PI0 RSPI0       Alpha_-1   Alpha_ 0   Alpha_ 1   Alpha_ 2   Alpha_ 3   Alpha_ 4  0.1  0.9        0.0723082  1.0263340  0.5156106  0.3443583  0.2585057  0.2069185  0.3  0.7        0.2388423  1.0889332  0.5531211  0.3709842  0.2791061  0.2237053  0.5  0.5        0.4483014  1.1715729  0.6034843  0.4070236  0.3071195  0.2466008  0.7  0.3        0.7358672  1.2922213  0.6786678  0.4614199  0.3496751  0.2815281  0.9  0.1        1.2304778  1.5194939  0.8253157  0.5694486  0.4351136  0.3521620  1.0  0.0        2.1348008  2.0000000  1.1547005  0.8203525  0.6378183  0.5222273

It should be noted that actual output from the program MAIN shown above produces the values of $H$-function as well as those of its moments $\alpha ^{\ast}_{-1}(\varpi _{0})$ and $\alpha _{n}(\varpi _{0})\ \ (n=0, 1, \ldots , 4)$ to the 15-th decimal place, although the results displayed here are rounded to the 7-th decimal place. The header “Alpha$\_-1$” should be interpreted as being “$\alpha ^{\ast}_{-1}$”.