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bessels.f90
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bessels.f90
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!CAMB spherical and hyperspherical Bessel function routines
!This version May 2006 - minor changes to bjl (http://cosmocoffee.info/viewtopic.php?t=530)
!Feb 2007: fixed for high l, uses Ranges
!Feb 2009: minor fix for non-flat compiled with non-smart IF evaluation
!Dec 2011: minor tweak to DoRecurs for smoother errors across flat for L~O(30)
!ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
!Flat bessel function module
module SpherBessels
use Precision
use ModelParams
use Ranges
implicit none
private
! Bessel functions and their second derivatives for interpolation
real(dl), dimension(:,:), allocatable :: ajl,ajlpr, ddajlpr
integer num_xx, kmaxfile, file_numl, file_l(lmax_arr)
! parameters for working out where the flat Bessel functions are small
! Both should increase for higher accuracy
! real(dl), parameter :: xlimmin=15._dl , xlimfrac = 0.05_dl
real(dl), parameter :: xlimmin=35._dl , xlimfrac = 0.05_dl
Type(Regions):: BessRanges
public ajl, ajlpr, ddajlpr, BessRanges, InitSpherBessels, xlimmin, xlimfrac
public USpherBesselWithDeriv, phi_recurs,phi_langer, bjl, Bessels_Free
contains
subroutine InitSpherBessels
! This subroutine reads the jl files from disk (or generates them if not on disk)
use lvalues
implicit none
!See if already loaded with enough (and correct) lSamp%l values and k*eta values
if (allocated(ajl) .and. (lSamp%l0 <= file_numl) .and. all(file_l(1:lSamp%l0)-lSamp%l(1:lSamp%l0)==0) &
.and. (int(min(max_bessels_etak,CP%Max_eta_k))+1 <= kmaxfile)) return
!Haven't made them before, so make them now
call GenerateBessels
if (DebugMsgs .and. FeedbackLevel > 0) write(*,*) 'Calculated Bessels'
end subroutine InitSpherBessels
subroutine GenerateBessels
use lvalues
real(dl) x
real(dl) xlim
integer i,j
integer max_ix
real(dl), parameter :: bessel_boost =1._dl
if (DebugMsgs .and. FeedbackLevel > 0) write (*,*) 'Generating flat Bessels...'
file_numl= lSamp%l0
file_l(1:lSamp%l0) = lSamp%l(1:lSamp%l0)
kmaxfile = int(min(CP%Max_eta_k,max_bessels_etak))+1
if (do_bispectrum) kmaxfile = kmaxfile*2
call Ranges_Init(BessRanges)
call Ranges_Add_delta(BessRanges,0._dl, 1._dl,0.01_dl/bessel_boost)
call Ranges_Add_delta(BessRanges,1._dl, 5._dl,0.1_dl/bessel_boost)
call Ranges_Add_delta(BessRanges,5._dl, 25._dl,0.2_dl/bessel_boost)
call Ranges_Add_delta(BessRanges,25._dl, 150._dl,0.5_dl/bessel_boost/AccuracyBoost)
call Ranges_Add_delta(BessRanges,150._dl, real(kmaxfile,dl),0.8_dl/bessel_boost/AccuracyBoost)
call Ranges_GetArray(bessRanges, .false.)
num_xx = BessRanges%npoints
max_ix = min(max_bessels_l_index,lSamp%l0)
if (allocated(ajl)) deallocate(ajl)
if (allocated(ajlpr)) deallocate(ajlpr)
if (allocated(ddajlpr)) deallocate(ddajlpr)
Allocate(ajl(1:num_xx,1:max_ix))
Allocate(ajlpr(1:num_xx,1:max_ix))
Allocate(ddajlpr(1:num_xx,1:max_ix))
!$OMP PARALLEL DO DEFAULT(SHARED),SCHEDULE(STATIC), PRIVATE(j,i,x,xlim)
do j=1,max_ix
do i=1,num_xx
x=BessRanges%points(i)
xlim=xlimfrac*lSamp%l(j)
xlim=max(xlim,xlimmin)
xlim=lSamp%l(j)-xlim
if (x > xlim) then
if ((lSamp%l(j)==3).and.(x <=0.2) .or. (lSamp%l(j) > 3).and.(x < 0.5) .or. &
(lSamp%l(j)>5).and.(x < 1.0)) then
ajl(i,j)=0
else
!if ( lSamp%l(j) > 40000) then
! ajl(i,j) = phi_langer(lSamp%l(j),0,1._dl,x)
!else
call bjl(lSamp%l(j),x,ajl(i,j))
!end if
end if
else
ajl(i,j)=0
end if
end do
! get the interpolation matrix for bessel functions
call spline(BessRanges%points,ajl(1,j),num_xx,spl_large,spl_large,ajlpr(1,j))
call spline(BessRanges%points,ajlpr(1,j),num_xx,spl_large,spl_large,ddajlpr(1,j))
end do
!$OMP END PARALLEL DO
end subroutine GenerateBessels
subroutine Bessels_Free
if (allocated(ajl)) deallocate(ajl)
if (allocated(ajlpr)) deallocate(ajlpr)
if (allocated(ddajlpr)) deallocate(ddajlpr)
call Ranges_Free(BessRanges)
end subroutine Bessels_Free
SUBROUTINE BJL(L,X,JL)
!!== MODIFIED SUBROUTINE FOR SPHERICAL BESSEL FUNCTIONS. ==!!
!!== CORRECTED THE SMALL BUGS IN PACKAGE CMBFAST&CAMB(for l=4,5, x~0.001-0.002)==!!
!!== CORRECTED THE SIGN OF J_L(X) FOR X<0 CASE ==!!
!!== WORKS FASTER AND MORE ACCURATE FOR LOW L, X<<L, AND L<<X cases ==!!
!!== [email protected] ==!!
IMPLICIT NONE
INTEGER L
real(dl) X,JL
real(dl) AX,AX2
real(dl),PARAMETER::LN2=0.6931471805599453094D0
real(dl),PARAMETER::ONEMLN2=0.30685281944005469058277D0
real(dl),PARAMETER::PID2=1.5707963267948966192313217D0
real(dl),PARAMETER::PID4=0.78539816339744830961566084582D0
real(dl),parameter::ROOTPI12 = 21.269446210866192327578D0
real(dl),parameter::GAMMA1 = 2.6789385347077476336556D0 !/* Gamma function of 1/3 */
real(dl),parameter::GAMMA2 = 1.3541179394264004169452D0 !/* Gamma function of 2/3 */
real(dl),PARAMETER::PI=3.141592653589793238463D0
real(dl) NU,NU2,BETA,BETA2,COSB
real(dl) sx,sx2
real(dl) cotb,cot3b,cot6b,secb,sec2b
real(dl) trigarg,expterm,L3
IF(L.LT.0)THEN
write(*,*) 'Can not evaluate Spherical Bessel Function with index l<0'
STOP
ENDIF
AX=DABS(X)
AX2=AX**2
IF(L.LT.7)THEN
IF(L.EQ.0)THEN
IF(AX.LT.1.D-1)THEN
JL=1.D0-AX2/6.D0*(1.D0-AX2/20.D0)
ELSE
JL=DSIN(AX)/AX
ENDIF
ELSEIF(L.EQ.1)THEN
IF(AX.LT.2.D-1)THEN
JL=AX/3.D0*(1.D0-AX2/10.D0*(1.D0-AX2/28.D0))
ELSE
JL=(DSIN(AX)/AX-DCOS(AX))/AX
ENDIF
ELSEIF(L.EQ.2)THEN
IF(AX.LT.3.D-1)THEN
JL=AX2/15.D0*(1.D0-AX2/14.D0*(1.D0-AX2/36.D0))
ELSE
JL=(-3.0D0*DCOS(AX)/AX-DSIN(AX)*(1.D0-3.D0/AX2))/AX
ENDIF
ELSEIF(L.EQ.3)THEN
IF(AX.LT.4.D-1)THEN
JL=AX*AX2/105.D0*(1.D0-AX2/18.D0*(1.D0-AX2/44.D0))
ELSE
JL=(DCOS(AX)*(1.D0-15.D0/AX2)-DSIN(AX)*(6.D0-15.D0/AX2)/AX)/AX
ENDIF
ELSEIF(L.EQ.4)THEN
IF(AX.LT.6.D-1)THEN
JL=AX2**2/945.D0*(1.D0-AX2/22.D0*(1.D0-AX2/52.D0))
ELSE
JL=(DSIN(AX)*(1.D0-(45.D0-105.D0/AX2)/AX2)+DCOS(AX)*(10.D0-105.D0/AX2)/AX)/AX
ENDIF
ELSEIF(L.EQ.5)THEN
IF(AX.LT.1.D0)THEN
JL=AX2**2*AX/10395.D0*(1.D0-AX2/26.D0*(1.D0-AX2/60.D0))
ELSE
JL=(DSIN(AX)*(15.D0-(420.D0-945.D0/AX2)/AX2)/AX-DCOS(AX)*(1.D0-(105.D0-945.0d0/AX2)/AX2))/AX
ENDIF
ELSE
IF(AX.LT.1.D0)THEN
JL=AX2**3/135135.D0*(1.D0-AX2/30.D0*(1.D0-AX2/68.D0))
ELSE
JL=(DSIN(AX)*(-1.D0+(210.D0-(4725.D0-10395.D0/AX2)/AX2)/AX2)+ &
DCOS(AX)*(-21.D0+(1260.D0-10395.D0/AX2)/AX2)/AX)/AX
ENDIF
ENDIF
ELSE
NU=0.5D0+L
NU2=NU**2
IF(AX.LT.1.D-40)THEN
JL=0.D0
ELSEIF((AX2/L).LT.5.D-1)THEN
JL=DEXP(L*DLOG(AX/NU)-LN2+NU*ONEMLN2-(1.D0-(1.D0-3.5D0/NU2)/NU2/30.D0)/12.D0/NU) &
/NU*(1.D0-AX2/(4.D0*NU+4.D0)*(1.D0-AX2/(8.D0*NU+16.D0)*(1.D0-AX2/(12.D0*NU+36.D0))))
ELSEIF((real(L,dl)**2/AX).LT.5.D-1)THEN
BETA=AX-PID2*(L+1)
JL=(DCOS(BETA)*(1.D0-(NU2-0.25D0)*(NU2-2.25D0)/8.D0/AX2*(1.D0-(NU2-6.25)*(NU2-12.25D0)/48.D0/AX2)) &
-DSIN(BETA)*(NU2-0.25D0)/2.D0/AX* (1.D0-(NU2-2.25D0)*(NU2-6.25D0)/24.D0/AX2*(1.D0-(NU2-12.25)* &
(NU2-20.25)/80.D0/AX2)) )/AX
ELSE
L3=NU**0.325
IF(AX .LT. NU-1.31*L3) then
COSB=NU/AX
SX = DSQRT(NU2-AX2)
COTB=NU/SX
SECB=AX/NU
BETA=DLOG(COSB+SX/AX)
COT3B=COTB**3
COT6B=COT3B**2
SEC2B=SECB**2
EXPTERM=( (2.D0+3.D0*SEC2B)*COT3B/24.D0 &
- ( (4.D0+SEC2B)*SEC2B*COT6B/16.D0 &
+ ((16.D0-(1512.D0+(3654.D0+375.D0*SEC2B)*SEC2B)*SEC2B)*COT3B/5760.D0 &
+ (32.D0+(288.D0+(232.D0+13.D0*SEC2B)*SEC2B)*SEC2B)*SEC2B*COT6B/128.D0/NU)*COT6B/NU) &
/NU)/NU
JL=DSQRT(COTB*COSB)/(2.D0*NU)*DEXP(-NU*BETA+NU/COTB-EXPTERM)
! /**************** Region 2: x >> l ****************/
ELSEIF (AX .GT. NU+1.48*L3) then
COSB=NU/AX
SX=DSQRT(AX2-NU2)
COTB=NU/SX
SECB=AX/NU
BETA=DACOS(COSB)
COT3B=COTB**3
COT6B=COT3B**2
SEC2B=SECB**2
TRIGARG=NU/COTB-NU*BETA-PID4 &
-((2.0+3.0*SEC2B)*COT3B/24.D0 &
+(16.D0-(1512.D0+(3654.D0+375.D0*SEC2B)*SEC2B)*SEC2B)*COT3B*COT6B/5760.D0/NU2)/NU
EXPTERM=( (4.D0+sec2b)*sec2b*cot6b/16.D0 &
-(32.D0+(288.D0+(232.D0+13.D0*SEC2B)*SEC2B)*SEC2B)*SEC2B*COT6B**2/128.D0/NU2)/NU2
JL=DSQRT(COTB*COSB)/NU*DEXP(-EXPTERM)*DCOS(TRIGARG)
! /***************** Region 3: x near l ****************/
ELSE
BETA=AX-NU
BETA2=BETA**2
SX=6.D0/AX
SX2=SX**2
SECB=SX**0.3333333333333333d0
SEC2B=SECB**2
JL=( GAMMA1*SECB + BETA*GAMMA2*SEC2B &
-(BETA2/18.D0-1.D0/45.D0)*BETA*SX*SECB*GAMMA1 &
-((BETA2-1.D0)*BETA2/36.D0+1.D0/420.D0)*SX*SEC2B*GAMMA2 &
+(((BETA2/1620.D0-7.D0/3240.D0)*BETA2+1.D0/648.D0)*BETA2-1.D0/8100.D0)*SX2*SECB*GAMMA1 &
+(((BETA2/4536.D0-1.D0/810.D0)*BETA2+19.D0/11340.D0)*BETA2-13.D0/28350.D0)*BETA*SX2*SEC2B*GAMMA2 &
-((((BETA2/349920.D0-1.D0/29160.D0)*BETA2+71.D0/583200.D0)*BETA2-121.D0/874800.D0)* &
BETA2+7939.D0/224532000.D0)*BETA*SX2*SX*SECB*GAMMA1)*DSQRT(SX)/ROOTPI12
ENDIF
ENDIF
ENDIF
IF(X.LT.0.AND.MOD(L,2).NE.0)JL=-JL
END SUBROUTINE BJL
! end module SpherBessels
!ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
! c
! Calculation of ultraspherical Bessel functions. c
! Fortran version of the c program hyperjl.c by Arthur Kosowsky. c
! WKB approx described in astro-ph/9805173 c
! c
! Modifications by Anthony Challinor and Antony Lewis c
! Minor modifications to correct K=1 case outside [0,pi], c
! the small chi approximations for lSamp%l=0 and lSamp%l=1, and c
! the quadratic approximation to Q(x) around Q(x)=0. c
! Bug fixed in downwards recursion (phi_recurs) c
! c
! The routine phi_recurs uses recursion relations to calculate c
! the functions, which is accurate but relatively slow. c
! ***NOT STABLE FOR K=1 or for all cases *** c
! c
! The routine phi_langer uses Langer's formula for a c
! uniform first-order asymptotic approximation in the open, closed c
! and flat cases. This approximation is EXCELLENT for all lSamp%l >= 3. c
! c
! The routine qintegral calculates the closed-form answer c
! to the eikonal integral used in the WKB approximation. c
! c
! The routine airy_ai returns the Airy function Ai(x) of the argument c
! passed. It employs a Pade-type approximation away from zero and c
! a Taylor expansion around zero. Highly accurate. c
! c
! The routines polevl and p1evl are auxiliary polynomial c
! evaluation routines used in the airy function calculation. c
! c
!ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
! module USpherBessels
! use Precision
! implicit none
! private
!public USpherBesselWithDeriv, phi_recurs,phi_langer
! contains
subroutine USpherBesselWithDeriv(closed,Chi,l,beta,y1,y2)
!returns y1=ujl*sinhChi and y2=diff(y1,Chi)
!aim for accuracy > 1% for all inputs
real(dl) Chi,beta,y1,y2,sinhChi,cothChi
real(dl) sin_K, cot_K
integer l,K
logical, intent(IN) :: closed
logical DoRecurs
if (closed) then
sin_K = sin(Chi)
cot_K= 1._dl/tan(Chi)
K=1
else
sin_K=sinh(Chi)
cot_K = 1._dl/tanh(Chi)
K=-1
end if
sinhChi = sin_K
cothChi = cot_K
DoRecurs = ((l<=45*AccuracyBoost).OR.((.not.closed.or.(abs(Chi-pi/2)>0.2d0)).and.(beta*l<750) &
.or.closed.and.(beta*l<4000)))
!Deep in the tails the closed recursion relation is not stable
!Added July 2003 to prevent problems with very nearly flat models
if (DoRecurs .and. closed) then
if (Chi < asin(sqrt(l*(l+1._dl))/beta) - 2/beta) then
if (phi_langer(l,K,beta,Chi) < 1e-7) then
call phi_small_closed_int(l,beta,chi,y1,y2)
return
end if
end if
end if
if (DoRecurs) then
!use recursive evaluation where WKB is poor and recurs is fast anyway.
y1=phi_recurs(l,K,beta,Chi)*sinhChi
y2=y1*(l+1)*cothChi
if (.not.closed.or.(l+1<nint(beta))) y2=y2 - &
sqrt(beta**2-(K*(l+1)**2))*phi_recurs(l+1,K,beta,Chi)*sinhChi
!of course we could get y2 much more quickly by modifying
!phi_recurs to return l and l+1 for each beta,Chi...
else !WKB approx
y1=phi_langer(l,K,beta,Chi)*sinhChi
y2=y1*(l+1)*cothChi
if (.not.closed.or.(l+1<nint(beta))) y2=y2 - &
sqrt(beta**2-(K*(l+1)**2))*phi_langer(l+1,K,beta,Chi)*sinhChi
end if
end subroutine USpherBesselWithDeriv
!Calculates y1,y2 (noramlized to a value near the turning point)
!by integrating up the differential equation and normalizing to phi_recurs
!in the region in which phi_recurs is stable
!This allows closed functions to be computed where chi << turning point
subroutine phi_small_closed_int(l,beta,chi,y1,y2)
integer, intent(IN) :: l
real(dl), intent(IN) :: beta, chi
real(dl) y1,y2
integer nsteps,i
real(dl) ap1,nu2,dydchi1,dydchi2,yt1,yt2,dyt1,dyt2,dym1,dym2
real(dl) x0, delchi,sh, h6,x
real(dl) y1_x,y2_x, tmp,xh,hh
nsteps = 200
ap1 = l*(l+1)
x0 = sqrt(ap1)/beta
nu2 = beta**2
if ((beta*chi)**2/l < 0.005) then
!Series solution
x = chi
sh = sin(x)
tmp=(ap1/sh**2 - nu2)
y1=1e-20
y2 = ((l+1)/x - (nu2-ap1/3)/(2*l+3)*x) * y1
else
x = max(1d-7,chi - 50._dl/l)
y1=1e-20
y2 = (l+1)*y1/x
delchi = (chi-x)/nSteps
h6=delchi/6
hh=delchi/2
sh = sin(x)
tmp=(ap1/sh**2 - nu2)
do i=1,nSteps
! One step in the ujl integration
! fourth-order Runge-Kutta method to integrate equation for ujl
dydchi1=y2 !deriv y1
dydchi2=tmp*y1 !deriv y2
xh=x+hh !midpoint of step
yt1=y1+hh*dydchi1 !y1 at midpoint
yt2=y2+hh*dydchi2 !y2 at midpoint
dyt1=yt2 !deriv y1 at mid
tmp=(ap1/sin(xh)**2 - nu2)
dyt2=tmp*yt1 !deriv y2 at mid
yt1=y1+hh*dyt1 !y1 at mid
yt2=y2+hh*dyt2 !y2 at mid
dym1=yt2 !deriv y1 at mid
dym2=tmp*yt1 !deriv y2 at mid
yt1=y1+delchi*dym1 !y1 at end
dym1=dyt1+dym1
yt2=y2+delchi*dym2 !y2 at end
dym2=dyt2+dym2
x=x+delchi !end point
sh=sin(x)
dyt1=yt2 !deriv y1 at end
tmp=(ap1/sh**2 - nu2)
dyt2=tmp*yt1 !deriv y2 at end
y1=y1+h6*(dydchi1+dyt1+2*dym1) !add up
y2=y2+h6*(dydchi2+dyt2+2*dym2)
if (y1 > 1d10 .or. y2> 1d10) then
y1=y1/1d10
y2=y2/1d10
end if
end do
end if
y1_x = y1; y2_x = y2
delchi = (x0 - chi)/nSteps
h6=delchi/6
hh=delchi/2
do i=1,nSteps
! One step in the ujl integration
! fourth-order Runge-Kutta method to integrate equation for ujl
dydchi1=y2 !deriv y1
dydchi2=tmp*y1 !deriv y2
xh=x+hh !midpoint of step
yt1=y1+hh*dydchi1 !y1 at midpoint
yt2=y2+hh*dydchi2 !y2 at midpoint
dyt1=yt2 !deriv y1 at mid
tmp=(ap1/sin(xh)**2 - nu2)
dyt2=tmp*yt1 !deriv y2 at mid
yt1=y1+hh*dyt1 !y1 at mid
yt2=y2+hh*dyt2 !y2 at mid
dym1=yt2 !deriv y1 at mid
dym2=tmp*yt1 !deriv y2 at mid
yt1=y1+delchi*dym1 !y1 at end
dym1=dyt1+dym1
yt2=y2+delchi*dym2 !y2 at end
dym2=dyt2+dym2
x=x+delchi !end point
sh=sin(x)
dyt1=yt2 !deriv y1 at end
tmp=(ap1/sh**2 - nu2)
dyt2=tmp*yt1 !deriv y2 at end
y1=y1+h6*(dydchi1+dyt1+2*dym1) !add up
y2=y2+h6*(dydchi2+dyt2+2*dym2)
if (y1 > 1d10 .or. y2 > 1d10) then
y1=y1/1d10
y2=y2/1d10
y1_x = y1_x/1d10
y2_x = y2_x/1d10
end if
end do
tmp = phi_recurs(l,1,beta,x0)*sin(x0) / y1
y1 = y1_x * tmp
y2 = y2_x * tmp
end subroutine phi_small_closed_int
!***********************************************************************
! *
! Calculates Phi(l,beta,chi) using recursion on l. *
! See Abbot and Schaefer, ApJ 308, 546 (1986) for needed *
! recursion relations and closed-form expressions for l=0,1. *
! (Note: Their variable y is the same as chi here.) *
! *
! When the flag direction is negative, downwards recursion on l *
! must be used because the upwards direction is unstable to roundoff *
! errors. The downwards recursion begins with arbitrary values and *
! continues downwards to l=1, where the desired l value is normalized *
! using the closed form solution for l=1. (See, e.g., Numerical *
! Recipes of Bessel functions for more detail) *
! *
!***********************************************************************
function phi_recurs(l, K, beta, chi)
!doesn't like values which give exponentially small phi
integer, intent(IN) :: l, K
real(dl), intent(IN) :: beta, chi
real(dl) phi_recurs
integer j, direction, lstart,ibeta
real(dl) ell, kay, arg, answer,beta2
real(dl) root_K
real(dl) phi0, phi1, phi_plus, phi_zero, phi_minus, b_zero, b_minus
real(dl), parameter :: ACC=40._dl, BIG=1.d10
real(dl) sin_K, cot_K
ell=dble(l)
! Test input values
if(l<0) then
write(*,*) "Bessel function index ell < 0"
stop
endif
if(beta<0._dl) then
write(*,*) "Wavenumber beta < 0"
stop
endif
if ((abs(K)/=1).and.(K/=0)) then
write(*,*) "K must be 1, 0 or -1"
stop
end if
if(K==1) then
ibeta=nint(beta)
if(ibeta<3) then
write(*,*) "Wavenumber beta < 3 for K=1"
stop
endif
if(ibeta<=l) then
write(*,*) "Wavenumber beta <= l"
stop
endif
endif
if (chi<1/BIG) then
phi_recurs=0
return
end if
kay = dble(K)
arg = beta * chi
beta2 = beta**2
if(K == 0) then
cot_K = 1._dl/chi
sin_K = chi
root_K = beta
else
root_K = sqrt(beta2 -kay*ell*ell)
if(K == -1) then
cot_K = 1._dl/tanh(chi)
sin_K = sinh(chi)
else
cot_K = 1._dl/tan(chi)
sin_K = sin(chi)
end if
endif
! Closed form solution for l=0
if (abs(chi) < 1.d-4) then
if (abs(arg)<1.d-4) then
phi0 = 1._dl-chi**2*(beta*beta-kay)/6._dl
else
phi0=sin(arg)/arg
end if
else
phi0 = sin(arg) / (beta * sin_K)
end if
if (l==0) then
phi_recurs=phi0
return
end if
! Closed form solution for l=1
if((abs(chi) < 1.d-4).and.(K/=0)) then
if(arg < 1.d-4) then
phi1 = chi*sqrt(beta*beta-kay)/3._dl
!beta2 * chi / (3._dl * sqrt(1._dl+ kay * beta2))
else
phi1 = (sin(arg)/arg-cos(arg))/(sqrt(beta*beta-kay)*chi)
!(sin(arg)/arg - cos(arg))/arg
end if
elseif ((abs(arg) < 1.d-4).and.(K == 0)) then
phi1 = arg / 3._dl
else
if (K /= 0 ) then
phi1 = sin(arg) * cot_K / (beta * sin_K) - cos(arg) / sin_K
phi1 = phi1/sqrt(beta2 - kay)
else
phi1 = (sin(arg)/arg - cos(arg))/arg
end if
end if
if(l==1) then
phi_recurs=phi1
return
end if
! Find recursion direction
! direction = +1 for upward recursion, -1 for downward
if(abs(cot_K) < root_K / ell) then
direction = 1
else
direction = -1
end if
! For K=1, must do upwards recursion:
! NOT STABLE for all values of chi
if(K==1) direction = 1
! Do upwards recursion on l
if(direction == 1)then
b_minus = sqrt(beta2 - kay)
phi_minus = phi0
phi_zero = phi1
do j=2,l
if(K == 0) then
phi_plus = ((2*j-1) * cot_K * phi_zero - beta*phi_minus)/ beta
else
b_zero = sqrt(beta2 - (K*j*j))
phi_plus = ((2*j-1) * cot_K * phi_zero - b_minus * phi_minus) / b_zero
b_minus = b_zero
end if
phi_minus = phi_zero
phi_zero = phi_plus
end do
phi_recurs=phi_plus
return
! Do downwards recursion on l
else
lstart = l + 2 * int(sqrt(ell*ACC))
b_zero = sqrt(beta2 - dble(K*lstart*lstart))
phi_plus = 0._dl
phi_zero = 1._dl
answer = 0._dl
do j= lstart - 2,1,-1
if(K == 0) then
phi_minus = ((2*j + 3) * cot_K * phi_zero - beta * phi_plus) / beta
else
b_minus = sqrt(beta2 - (K*(j+1)**2))
phi_minus = ((2*j + 3) * cot_K * phi_zero - b_zero * phi_plus) / b_minus
b_zero = b_minus
end if
phi_plus = phi_zero
phi_zero = phi_minus
if(j == l) answer = phi_minus
if((abs(phi_zero) > BIG).and.(j/=1)) then
phi_plus = phi_plus/BIG
phi_zero = phi_zero/BIG
answer = answer/BIG
end if
end do
! Normalize answer to previously computed phi1
answer = answer*phi1 / phi_minus
phi_recurs=answer
end if
end function phi_recurs
!ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
! c
! Calculates Phi(l,beta,chi) using the Langer uniform approximation c
! to the first-order WKB approximation. c
! See C.M. Bender and S.A. Orszag, Mathematical Methods for c
! Scientists and Engineers (McGraw-Hill, 1978; LC QA371.B43), c
! chapter 10. c
! c
! Differential equation for needed function can be cast into the c
! Schrodinger form \epsilon^2 y'' = Q(x) y c
! where \epsilon^2 = 1/l(l+1) and Q(x) depends on the parameter c
! alpha \equiv beta * epsilon. c
! c
! In the K= +1 case, the function is c
! determined by its value on the interval [0, pi/2] and the symmetry c
! conditions Phi(chi + pi) = (-1)^{beta - l - 1} Phi(chi), c
! Phi(pi - chi) = (-1)^{beta - l - 1} Phi(chi). c
! This interval contains one turning point, so the Langer formula c
! can be used. c
! Note that the second condition at chi = pi/2 gives an eigenvalue c
! condition on beta, which must corrected. For the lowest c
! eigenvalue(s), the region between the turning points is not large c
! enough for the asymptotic solution to be valid, so the functions c
! have a small discontinuity or discontinuous derivative at pi/2; c
! this behavior is corrected by employing a 4-term asymptotic c
! series around the regular point chi=pi/2. c
! The exact eigenvalue condition requires that beta must be an c
! integer >= 3 with beta > l. Note this implies alpha > 1. c
! c
!ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
function phi_langer(l,K,beta,chi)
integer l,K,ibeta,kay
real(dl) phi_langer
real(dl) ell,symm, anu, alpha2
real(dl) beta,chi,eikonal, wkb, arg, arg2, tmp
real(dl) epsilon, alpha, chi0, x, a, b,achi
real(dl) cot_K, sin_K
real(dl), parameter :: PI=3.1415926536d0,ROOTPI=1.772453851d0,ROOT2PI=2.506628275d0, &
PIOVER2=1.570796327d0
ell=dble(l)
achi=chi
symm=1._dl
!
! Test input values
!
if(l<0) then
write(*,*) "Bessel function index ell < 0"
stop
endif
if(beta<0._dl) then
write(*,*) "Wavenumber beta < 0"
stop
endif
if ((abs(K)/=1).and.(K/=0)) then
write(*,*) "K must be 1, 0 or -1"
stop
end if
if(K == 1) then
ibeta=nint(beta)
if(ibeta<3) then
write(*,*) "Wavenumber beta < 3 for K=1"
stop
endif
if(ibeta<=l) then
write(*,*) "Wavenumber beta <= l"
stop
endif
endif
kay=K
! For closed case, find equivalent chi in [0,pi/2]
!
if(K==1) then
achi=achi-2._dl*Pi*int(achi/2._dl/PI)
if(achi>PI) then
achi=2._dl*PI-achi
if(2*(l/2).eq.l) then
symm=symm
else
symm=-symm
endif
endif
if(achi>PI/2._dl) then
achi=PI-achi
if(2*((ibeta-l-1)/2).eq.(ibeta-l-1)) then
symm=symm
else
symm=-symm
endif
endif
endif
! Definitions
if(K == 0) then
sin_K = achi
else
if(K == -1) then
sin_K = sinh(achi)
else
sin_K = sin(achi)
end if
endif
! Closed form solution for l=0
!
if(l == 0) then
arg=beta*achi
if((abs(achi)<1.d-4).and.(K/=0)) then
if(abs(arg)<1.d-4) then
wkb=1._dl-achi*achi*(beta*beta-kay)/6._dl
else
wkb=sin(arg)/arg
endif
else if((abs(arg)<1.d-4).and.(K==0)) then
wkb=1._dl-arg*arg/6._dl
else
wkb=sin(arg)/(beta*sin_K)
endif
phi_langer=symm*wkb
return
endif
!
! Closed form solution for l=1
!
if(l==1) then
arg=beta*achi
if((abs(achi)<1.d-4).and.(K/=0)) then
if(abs(arg)<1.d-4) then
wkb=achi*sqrt(beta*beta-kay)/3._dl
else
wkb=(sin(arg)/arg-cos(arg))/(sqrt(beta*beta-kay)*achi)
endif
else if((abs(arg)<1.d-4).and.(K==0)) then
wkb=arg/3._dl
else
if(K/=0) then
if(K==1) then
cot_K=1._dl/tan(achi)
else
cot_K=1._dl/tanh(achi)
endif
wkb=sin(arg)*cot_K/(beta*sin_K)-cos(arg)/sin_K
wkb=wkb/sqrt(beta*beta-kay)
else
wkb=(sin(arg)/arg-cos(arg))/arg
endif
end if
phi_langer=symm*wkb
return
endif
!
! Closed form solution for K=1 and beta = l+1 (lowest eigenfunction)
!
if((K==1).and.(ibeta == (l+1))) then
wkb=(sin_K**ell)* &
sqrt(sqrt(2._dl*PI/(2._dl*ell+1._dl))*ell/((ell+1._dl)*(2._dl*ell+1._dl)))
wkb=wkb*(1+0.1875d0/ell-0.013671875/(ell*ell))
phi_langer=symm*wkb
return
endif
! Very close to 0, return 0 (exponentially damped)
!
if(abs(achi)<1.d-8) then
phi_langer=0._dl
return
endif
! For closed case, find corrected eigenvalue beta
!
if(K==1) then
anu=dble(ibeta)-1._dl/(8._dl*ell)+1._dl/(16._dl*ell*ell)
else
anu=beta
endif
!
! Evaluate epsilon using asymptotic form for large l
!
if(l<20) then
epsilon=1._dl/sqrt(ell*(ell+1._dl))
else
epsilon=1._dl/ell-0.5d0/(ell*ell)+0.375d0/(ell*ell*ell)
endif
alpha=epsilon*anu
!
! Calculate the turning point where Q(x)=0.
! Function in question has only a single simple turning point.
!
if(K==-1) chi0=log((1._dl+sqrt(1._dl+alpha*alpha))/alpha)
if(K==0) chi0=1._dl/alpha
if(K==1) chi0=asin(1._dl/alpha)
! Very close to chi0, use usual wkb form to avoid dividing by zero
!
if(abs(achi-chi0)<1.d-5) then
! Calculate coefficients of linear and quadratic terms in Q(x) expansion
! in the neighborhood of the turning point
! Q(chi)=a*(chi0-chi)+b*(chi0-chi)**2
alpha2=alpha*alpha
if(K==-1) then
a=2._dl*alpha2*sqrt(alpha2+1._dl)
b=3._dl*alpha2**2+2._dl*alpha2
endif
if(K==0) then
a=2._dl*alpha2*alpha
b=3._dl*alpha2**2
endif
if(K==1) then
a=2._dl*alpha2*sqrt(alpha2-1._dl)
b=3._dl*alpha2**2-2._dl*alpha2
endif
! Dependent variable x for which Q(x)=0 at x=0
! x>0 is the evanescent region
!
x=chi0-achi
!
! Argument of Airy function
!
arg=(x+b*x*x/(5._dl*a))/(epsilon*epsilon/a)**(0.333333333d0)
!
! Evaluate Airy function
!
wkb=airy_ai(arg)
!
! Rest of functional dependence
!
wkb=wkb*(1._dl-b*x/(5._dl*a))/sin_K
! Normalization factor:
wkb=wkb*symm*ROOTPI*((a*epsilon)**(-0.1666667d0))*sqrt(epsilon/anu)
phi_langer=wkb
return
endif
! Langer approximation.
!
! Transport factor:
!
tmp=sqrt(abs(1._dl/(sin_K*sin_K)-alpha*alpha))
! Eikonal factor
!
eikonal=qintegral(sin_K,alpha,K)
arg=(1.5d0*eikonal/epsilon)**(1._dl/3._dl)
arg2=arg*arg
if(achi>chi0) arg2=-arg2