-
Notifications
You must be signed in to change notification settings - Fork 159
/
Copy pathbessels.f90
1343 lines (1145 loc) · 44.4 KB
/
bessels.f90
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
!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 results
use RangeUtils
use MpiUtils
use splines
implicit none
private
! Bessel functions and their second derivatives for interpolation
real(dl), dimension(:,:), allocatable :: ajl, ajlpr, ddajlpr
integer num_xx, kmaxfile, max_ix
Type(lSamples), save :: file_l
! 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
real(dl) file_acc
type(TRanges), save:: BessRanges
public ajl, ajlpr, ddajlpr, BessRanges, InitSpherBessels, xlimmin, xlimfrac
public USpherBesselWithDeriv, phi_recurs,phi_langer, bjl, Bessels_Free
contains
subroutine InitSpherBessels(lSamp, CP,max_bessels_l_index, max_bessels_etak)
Type(lSamples) lSamp
Type(CAMBParams) :: CP
integer, intent(in) :: max_bessels_l_index
real(dl), intent(in) :: max_bessels_etak
!See if already loaded with enough (and correct) lSamp%l values and k*eta values
if (lSamp%nl <= file_l%nl) then
if (allocated(ajl) .and. all(file_l%l(1:lSamp%nl)==lSamp%l(1:lSamp%nl) .and. (max_bessels_l_index <= max_ix)) &
.and. (int(min(max_bessels_etak,CP%Max_eta_k))+1 <= kmaxfile) &
.and. (abs(CP%Accuracy%BesselBoost*CP%Accuracy%AccuracyBoost - file_acc) < 1d-2)) return
end if
!Haven't made them before, so make them now
kmaxfile = int(min(CP%Max_eta_k,max_bessels_etak))+1
max_ix = min(max_bessels_l_index,lSamp%nl)
call GenerateBessels(lSamp, CP)
if (DebugMsgs .and. FeedbackLevel > 0) write(*,*) 'Calculated Bessels'
end subroutine InitSpherBessels
subroutine GenerateBessels(lSamp, CP)
Type(lSamples) lSamp
Type(CAMBParams) :: CP
real(dl) x
real(dl) xlim
integer i,j
if (DebugMsgs .and. FeedbackLevel > 0) write (*,*) 'Generating flat Bessels...'
file_l = lSamp
if (do_bispectrum) kmaxfile = kmaxfile*2
if (DebugMsgs .and. FeedbackLevel > 0) write (*,*) 'x_max bessels', kmaxfile
call BessRanges%Init()
call BessRanges%Add_delta(0._dl, 1._dl,0.01_dl/CP%Accuracy%BesselBoost)
call BessRanges%Add_delta(1._dl, 5._dl,0.1_dl/CP%Accuracy%BesselBoost)
call BessRanges%Add_delta(5._dl, 25._dl,0.2_dl/CP%Accuracy%BesselBoost)
file_acc = CP%Accuracy%BesselBoost*CP%Accuracy%AccuracyBoost
call BessRanges%Add_delta(25._dl, 150._dl,0.5_dl/file_acc)
call BessRanges%Add_delta(150._dl, real(kmaxfile,dl),0.8_dl/file_acc)
call BessRanges%GetArray(.false.)
num_xx = BessRanges%npoints
! The three arrays are always (de-)allocated together. Therefore checking
! one of them for allocation is sufficient.
if (allocated(ajl)) then
if (any(ubound(ajl) < [num_xx, max_ix])) deallocate(ajl, ajlpr, ddajlpr)
end if
if (.not. allocated(ajl)) then
allocate(ajl(1:num_xx,1:max_ix), ajlpr(1:num_xx,1:max_ix), &
ddajlpr(1:num_xx,1:max_ix))
end if
!$OMP PARALLEL DO DEFAULT(SHARED), SCHEDULE(STATIC), PRIVATE(i, x, xlim)
do j=1,max_ix
do i=1,num_xx
x=BessRanges%points(i)
xlim=xlimfrac*lSamp%l(j)
xlim=max(xlim,xlimmin)
xlim=lSamp%l(j)-xlim
if (x > xlim) then
if ((lSamp%l(j)==3).and.(x <=0.2) .or. (lSamp%l(j) > 3).and.(x < 0.5) .or. &
(lSamp%l(j)>5).and.(x < 1.0)) then
ajl(i,j)=0
else
!if ( lSamp%l(j) > 40000) then
! ajl(i,j) = phi_langer(lSamp%l(j),0,1._dl,x)
!else
call bjl(lSamp%l(j),x,ajl(i,j))
!end if
end if
else
ajl(i,j)=0
end if
end do
! get the interpolation matrix for bessel functions
call spline_def(BessRanges%points,ajl(:,j),num_xx,ajlpr(:,j))
call spline_def(BessRanges%points,ajlpr(:,j),num_xx,ddajlpr(:,j))
end do
!$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 BessRanges%Free()
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
error stop 'Can not evaluate Spherical Bessel Function with index l<0'
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
subroutine USpherBesselWithDeriv(closed,CP,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
Type(CAMBParams), intent(in) :: CP
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*CP%Accuracy%AccuracyBoost).OR.((.not.closed.or.(abs(Chi-const_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) call MpiStop("Bessel function index ell < 0")
if(beta<0._dl) call MpiStop("Wavenumber beta < 0")
if ((abs(K)/=1).and.(K/=0)) call MpiStop("K must be 1, 0 or -1")
if(K==1) then
ibeta=nint(beta)
if(ibeta<3) call MpiStop("Wavenumber beta < 3 for K=1")
if(ibeta<=l) call MpiStop("Wavenumber beta <= l")
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) call MpiStop("Bessel function index ell < 0")
if(beta<0._dl) call MpiStop("Wavenumber beta < 0")
if ((abs(K)/=1).and.(K/=0)) call MpiStop("K must be 1, 0 or -1")
if(K == 1) then
ibeta=nint(beta)
if(ibeta<3) call MpiStop("Wavenumber beta < 3 for K=1")
if(ibeta<=l) call MpiStop("Wavenumber beta <= l")
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
! Multiply factors together
wkb=airy_ai(arg2)*symm*ROOTPI*sqrt(arg*epsilon/anu/tmp)/Sin_K
phi_langer=wkb
end function phi_langer
!cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
! c
! Evaluates the exact solution to the integral giving the WKB c
! eikonal solution, \int^x sqrt(abs(Q(x))) dx c
! c
! In the open case, this integral costs 1 or 2 square roots, an atan c
! and a log; its evaluation will be roughly as expensive as the rest c
! of the Phi routine. An analytic fit cannot be substantially faster c
! because the dependence on alpha of the y-intercept of the linear c
! region of the integrand contains a log and an atan, so at best a fit c
! can only save the computation of the square roots. c
! c
! The integrals are very bland functions of chi and alpha and could c
! be precomputed and cached to save computation time; interpolation c
! on a relatively small number of points should be very accurate c
! c
! Note that for the closed case, the variable arg must be between 0 c
! and alpha; for arg > alpha, symmetry properties reduce the needed c
! integral to this case. c
! c
!cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
function qintegral(sin_K,alpha, K)
implicit none
real(dl) qintegral, sin_K
integer K
real(dl) alpha,exact,arg, root1, root2, dummyarg
real(dl), parameter :: PI=3.1415926536d0,ROOTPI=1.772453851d0,ROOT2PI=2.506628275d0, &
PIOVER2=1.570796327d0