-
Notifications
You must be signed in to change notification settings - Fork 1
/
cbessel.f90
6033 lines (5545 loc) · 226 KB
/
cbessel.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
Module CBessel
! REMARK ON ALGORITHM 644, COLLECTED ALGORITHMS FROM ACM.
! THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE,
! VOL. 21, NO. 4, December, 1995, P. 388--393.
! Code converted using TO_F90 by Alan Miller
! Date: 2002-02-08 Time: 17:53:05
! Latest revision - 16 April 2002
implicit none
private
public :: cbesh, cbesi, cbesj, cbesk, cbesy, cairy, cbiry
! Oct 2011: replaced function gamln() with Fortran2008 built-in log_gamma()
! June 2013 consolidated common constants from individual routines
integer, parameter, private :: DP = selected_real_kind(15, 307)
real(DP), parameter, private :: HPI = 2.0_DP*atan(1.0_DP) !! PI/2 = 1.57079632679489662
real(DP), parameter, private :: PI = 4.0_DP*atan(1.0_DP) !! PI = 3.14159265358979324
real(DP), parameter, private :: THPI = 6.0_DP*atan(1.0_DP) !! 3*PI/2 = 4.71238898038469
real(DP), parameter, private :: RTPI = 1.0_DP/(8.0_DP*atan(1.0_DP)) !! 1/(2*PI) = 0.159154943091895
real(DP), parameter, private :: RTHPI = sqrt(8.0_DP*atan(1.0_DP))/2.0_DP !! sqrt(2*pi)/2 = 1.25331413731550
real(DP), parameter, private :: SPI = 3.0_DP/(2.0_DP*atan(1.0_DP)) ! 6/pi = 1.90985931710274
complex(DP), parameter, private :: CONE = (1.0_DP,0.0_DP), CTWO = (2.0_DP,0.0_DP)
complex(DP), parameter, private :: CZERO = (0.0_DP,0.0_DP), CI = (0.0_DP,1.0_DP)
real(DP), parameter, private :: ULP = 2.0_DP
CONTAINS
SUBROUTINE cbesh(z, fnu, kode, m, n, cy, nz, ierr)
!***BEGIN PROLOGUE CBESH
!***DATE WRITTEN 830501 (YYMMDD)
!***REVISION DATE 890801, 930101 (YYMMDD)
!***CATEGORY NO. B5K
!***KEYWORDS H-BESSEL FUNCTIONS,BESSEL FUNCTIONS OF COMPLEX ARGUMENT,
! BESSEL FUNCTIONS OF THIRD KIND,HANKEL FUNCTIONS
!***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
!***PURPOSE TO COMPUTE THE H-BESSEL FUNCTIONS OF A COMPLEX ARGUMENT
!***DESCRIPTION
! ON KODE=1, CBESH COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
! HANKEL (BESSEL) FUNCTIONS CY(J)=H(M,FNU+J-1,Z) FOR KINDS M=1
! OR 2, REAL, NONNEGATIVE ORDERS FNU+J-1, J=1,...,N, AND COMPLEX
! Z.NE.CMPLX(0.0E0,0.0E0) IN THE CUT PLANE -PI < ARG(Z) <= PI.
! ON KODE=2, CBESH COMPUTES THE SCALED HANKEL FUNCTIONS
! CY(I)=H(M,FNU+J-1,Z)*EXP(-MM*Z*I) MM=3-2M, I**2=-1.
! WHICH REMOVES THE EXPONENTIAL BEHAVIOR IN BOTH THE UPPER
! AND LOWER HALF PLANES. DEFINITIONS AND NOTATION ARE FOUND IN
! THE NBS HANDBOOK OF MATHEMATICAL FUNCTIONS (REF. 1).
! INPUT
! Z - Z=CMPLX(X,Y), Z.NE.CMPLX(0.,0.),-PI < ARG(Z) <= PI
! FNU - ORDER OF INITIAL H FUNCTION, FNU >= 0.0E0
! KODE - A PARAMETER TO INDICATE THE SCALING OPTION
! KODE= 1 RETURNS
! CY(J)=H(M,FNU+J-1,Z), J=1,...,N
! = 2 RETURNS
! CY(J)=H(M,FNU+J-1,Z)*EXP(-I*Z*(3-2M))
! J=1,...,N , I**2=-1
! M - KIND OF HANKEL FUNCTION, M=1 OR 2
! N - NUMBER OF MEMBERS OF THE SEQUENCE, N >= 1
! OUTPUT
! CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN
! VALUES FOR THE SEQUENCE
! CY(J)=H(M,FNU+J-1,Z) OR
! CY(J)=H(M,FNU+J-1,Z)*EXP(-I*Z*(3-2M)) J=1,...,N
! DEPENDING ON KODE, I**2=-1.
! NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW,
! NZ= 0 , NORMAL RETURN
! NZ > 0 , FIRST NZ COMPONENTS OF CY SET TO ZERO DUE TO UNDERFLOW,
! CY(J)=CMPLX(0.0,0.0) J=1,...,NZ WHEN Y > 0.0 AND M=1
! OR Y < 0.0 AND M=2. FOR THE COMPLEMENTARY HALF PLANES,
! NZ STATES ONLY THE NUMBER OF UNDERFLOWS.
! IERR -ERROR FLAG
! IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
! IERR=1, INPUT ERROR - NO COMPUTATION
! IERR=2, OVERFLOW - NO COMPUTATION, FNU+N-1 TOO
! LARGE OR ABS(Z) TOO SMALL OR BOTH
! IERR=3, ABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
! BUT LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION
! PRODUCE LESS THAN HALF OF MACHINE ACCURACY
! IERR=4, ABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTATION BECAUSE OF
! COMPLETE LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION
! IERR=5, ERROR - NO COMPUTATION,
! ALGORITHM TERMINATION CONDITION NOT MET
!***LONG DESCRIPTION
! THE COMPUTATION IS CARRIED OUT BY THE RELATION
! H(M,FNU,Z)=(1/MP)*EXP(-MP*FNU)*K(FNU,Z*EXP(-MP))
! MP=MM*HPI*I, MM=3-2*M, HPI=PI/2, I**2=-1
! FOR M=1 OR 2 WHERE THE K BESSEL FUNCTION IS COMPUTED FOR THE
! RIGHT HALF PLANE RE(Z) >= 0.0. THE K FUNCTION IS CONTINUED
! TO THE LEFT HALF PLANE BY THE RELATION
! K(FNU,Z*EXP(MP)) = EXP(-MP*FNU)*K(FNU,Z)-MP*I(FNU,Z)
! MP=MR*PI*I, MR=+1 OR -1, RE(Z) > 0, I**2=-1
! WHERE I(FNU,Z) IS THE I BESSEL FUNCTION.
! EXPONENTIAL DECAY OF H(M,FNU,Z) OCCURS IN THE UPPER HALF Z PLANE FOR
! M=1 AND THE LOWER HALF Z PLANE FOR M=2. EXPONENTIAL GROWTH OCCURS IN THE
! COMPLEMENTARY HALF PLANES. SCALING BY EXP(-MM*Z*I) REMOVES THE
! EXPONENTIAL BEHAVIOR IN THE WHOLE Z PLANE FOR Z TO INFINITY.
! FOR NEGATIVE ORDERS,THE FORMULAE
! H(1,-FNU,Z) = H(1,FNU,Z)*EXP( PI*FNU*I)
! H(2,-FNU,Z) = H(2,FNU,Z)*EXP(-PI*FNU*I)
! I**2=-1
! CAN BE USED.
! IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELEMENTARY
! FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS LARGE, LOSSES OF
! SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
! CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN LOSSES
! EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG IERR=3 IS
! TRIGGERED WHERE UR=R1MACH(4)=UNIT ROUNDOFF. ALSO IF EITHER IS LARGER
! THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS LOST AND IERR=4.
! IN ORDER TO USE THE INT FUNCTION, ARGUMENTS MUST BE FURTHER RESTRICTED
! NOT TO EXCEED THE LARGEST MACHINE INTEGER, U3=I1MACH(9).
! THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS RESTRICTED BY MIN(U2,U3).
! ON 32 BIT MACHINES, U1,U2, AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6,
! 2.1E+9 IN SINGLE PRECISION ARITHMETIC AND 1.3E+8, 1.8D+16, 2.1E+9 IN
! DOUBLE PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3
! LIMITING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN
! EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
! IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
! SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
! THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX BESSEL
! FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT ROUNDOFF,1.0E-18)
! IS THE NOMINAL PRECISION AND 10**S REPRESENTS THE INCREASE IN ERROR DUE
! TO ARGUMENT REDUCTION IN THE ELEMENTARY FUNCTIONS.
! HERE, S=MAX(1, ABS(LOG10(ABS(Z))), ABS(LOG10(FNU))) APPROXIMATELY
! (I.E. S=MAX(1,ABS(EXPONENT OF ABS(Z),ABS(EXPONENT OF FNU)) ).
! HOWEVER, THE PHASE ANGLE MAY HAVE ONLY ABSOLUTE ACCURACY.
! THIS IS MOST LIKELY TO OCCUR WHEN ONE COMPONENT (IN ABSOLUTE VALUE)
! IS LARGER THAN THE OTHER BY SEVERAL ORDERS OF MAGNITUDE.
! IF ONE COMPONENT IS 10**K LARGER THAN THE OTHER, THEN ONE CAN EXPECT
! ONLY MAX(ABS(LOG10(P))-K, 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER
! WAY, WHEN K EXCEEDS THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN
! THE SMALLER COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE
! ACCURACY BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
! COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE MAGNITUDE
! OF THE LARGER COMPONENT. IN THESE EXTREME CASES, THE PRINCIPAL PHASE
! ANGLE IS ON THE ORDER OF +P, -P, PI/2-P, OR -PI/2+P.
!***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ AND
! I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF COMMERCE, 1955.
! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
! BY D. E. AMOS, SAND83-0083, MAY 1983.
! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
! AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY 1983
! A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX ARGUMENT
! AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-1018, MAY, 1985
! A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX ARGUMENT
! AND NONNEGATIVE ORDER BY D. E. AMOS, ACM TRANS. MATH. SOFTWARE,
! VOL. 12, NO. 3, SEPTEMBER 1986, PP 265-273.
!***ROUTINES CALLED CACON,CBKNU,CBUNK,CUOIK,I1MACH,R1MACH
!***END PROLOGUE CBESH
COMPLEX (dp), INTENT(IN) :: z
REAL (dp), INTENT(IN) :: fnu
INTEGER, INTENT(IN) :: kode
INTEGER, INTENT(IN) :: m
INTEGER, INTENT(IN) :: n
COMPLEX (dp), INTENT(OUT) :: cy(n)
INTEGER, INTENT(OUT) :: nz
INTEGER, INTENT(OUT) :: ierr
COMPLEX (dp) :: zn, zt, csgn
REAL (dp) :: aa, alim, aln, arg, az, cpn, dig, elim, fmm, fn, fnul, &
rhpi, rl, r1m5, sgn, spn, tol, ufl, xn, xx, yn, yy, &
bb, ascle, rtol, atol
INTEGER :: i, inu, inuh, ir, k, k1, k2, mm, mr, nn, nuf, nw
!***FIRST EXECUTABLE STATEMENT CBESH
nz = 0
xx = REAL(z, KIND=dp)
yy = AIMAG(z)
ierr = 0
IF (abs(xx)*ULP < spacing(0.0_dp) .AND. abs(yy)*ULP < spacing(0.0_dp)) ierr = 1
IF (fnu < 0.0_dp) ierr = 1
IF (m < 1 .OR. m > 2) ierr = 1
IF (kode < 1 .OR. kode > 2) ierr = 1
IF (n < 1) ierr = 1
IF (ierr /= 0) RETURN
nn = n
!-----------------------------------------------------------------------
! SET PARAMETERS RELATED TO MACHINE CONSTANTS.
! TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18.
! ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
! EXP(-ELIM) < EXP(-ALIM)=EXP(-ELIM)/TOL AND
! EXP(ELIM) > EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR
! UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
! RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
! DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
! FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU
!-----------------------------------------------------------------------
tol = MAX(EPSILON(0.0_dp), 1.0D-18)
k1 = MINEXPONENT(0.0_dp)
k2 = MAXEXPONENT(0.0_dp)
r1m5 = LOG10( REAL( RADIX(0.0_dp), KIND=dp) )
k = MIN(ABS(k1), ABS(k2))
elim = 2.303_dp * (k*r1m5 - 3.0_dp)
k1 = DIGITS(0.0_dp) - 1
aa = r1m5 * k1
dig = MIN(aa, 18.0_dp)
aa = aa * 2.303_dp
alim = elim + MAX(-aa, -41.45_dp)
fnul = 10.0_dp + 6.0_dp * (dig - 3.0_dp)
rl = 1.2_dp * dig + 3.0_dp
fn = fnu + (nn-1)
mm = 3 - m - m
fmm = mm
zn = z * CMPLX(0.0_dp, -fmm, KIND=dp)
xn = REAL(zn, KIND=dp)
yn = AIMAG(zn)
az = ABS(z)
!-----------------------------------------------------------------------
! TEST FOR RANGE
!-----------------------------------------------------------------------
aa = 0.5_dp / tol
bb = HUGE(0) * 0.5_dp
aa = MIN(aa,bb)
IF (az <= aa) THEN
IF (fn <= aa) THEN
aa = SQRT(aa)
IF (az > aa) ierr = 3
IF (fn > aa) ierr = 3
!-----------------------------------------------------------------------
! OVERFLOW TEST ON THE LAST MEMBER OF THE SEQUENCE
!-----------------------------------------------------------------------
ufl = TINY(0.0_dp) * 1.0D+3
IF (az >= ufl) THEN
IF (fnu <= fnul) THEN
IF (fn > 1.0_dp) THEN
IF (fn <= 2.0_dp) THEN
IF (az > tol) GO TO 10
arg = 0.5_dp * az
aln = -fn * LOG(arg)
IF (aln > elim) GO TO 50
ELSE
CALL cuoik(zn, fnu, kode, 2, nn, cy, nuf, tol, elim, alim)
IF (nuf < 0) GO TO 50
nz = nz + nuf
nn = nn - nuf
!-----------------------------------------------------------------------
! HERE NN=N OR NN=0 SINCE NUF=0,NN, OR -1 ON RETURN FROM CUOIK
! IF NUF=NN, THEN CY(I)=CZERO FOR ALL I
!-----------------------------------------------------------------------
IF (nn == 0) GO TO 40
END IF
END IF
10 IF (.NOT.(xn < 0.0_dp .OR. (abs(xn)*ULP < spacing(0.0_dp) .AND. yn < 0.0_dp &
.AND. m == 2))) THEN
!-----------------------------------------------------------------------
! RIGHT HALF PLANE COMPUTATION, XN >= 0. .AND. (XN.NE.0. .OR.
! YN >= 0. .OR. M=1)
!-----------------------------------------------------------------------
CALL cbknu(zn, fnu, kode, nn, cy, nz, tol, elim, alim)
GO TO 20
END IF
!-----------------------------------------------------------------------
! LEFT HALF PLANE COMPUTATION
!-----------------------------------------------------------------------
mr = -mm
CALL cacon(zn, fnu, kode, mr, nn, cy, nw, rl, fnul, tol, elim, alim)
IF (nw < 0) GO TO 60
nz = nw
ELSE
!-----------------------------------------------------------------------
! UNIFORM ASYMPTOTIC EXPANSIONS FOR FNU > FNUL
!-----------------------------------------------------------------------
mr = 0
IF (.NOT.(xn >= 0.0_dp .AND. (abs(xn)*ULP >= spacing(0.0_dp) .OR. yn >= 0.0_dp &
.OR. m /= 2))) THEN
mr = -mm
IF (abs(xn)*ULP < spacing(0.0_dp) .AND. yn < 0.0_dp) zn = -zn
END IF
CALL cbunk(zn, fnu, kode, mr, nn, cy, nw, tol, elim, alim)
IF (nw < 0) GO TO 60
nz = nz + nw
END IF
!-----------------------------------------------------------------------
! H(M,FNU,Z) = -FMM*(I/HPI)*(ZT**FNU)*K(FNU,-Z*ZT)
! ZT=EXP(-FMM*HPI*I) = CMPLX(0.0,-FMM), FMM=3-2*M, M=1,2
!-----------------------------------------------------------------------
20 sgn = SIGN(hpi,-fmm)
!-----------------------------------------------------------------------
! CALCULATE EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE
! WHEN FNU IS LARGE
!-----------------------------------------------------------------------
inu = int(fnu)
inuh = inu / 2
ir = inu - 2 * inuh
arg = (fnu - (inu-ir)) * sgn
rhpi = 1.0_dp / sgn
cpn = rhpi * COS(arg)
spn = rhpi * SIN(arg)
! ZN = CMPLX(-SPN,CPN)
csgn = CMPLX(-spn, cpn, KIND=dp)
! IF (MOD(INUH,2).EQ.1) ZN = -ZN
IF (MOD(inuh,2) == 1) csgn = -csgn
zt = CMPLX(0.0_dp, -fmm, KIND=dp)
rtol = 1.0_dp / tol
ascle = ufl * rtol
DO i = 1, nn
! CY(I) = CY(I)*ZN
! ZN = ZN*ZT
zn = cy(i)
aa = REAL(zn, KIND=dp)
bb = AIMAG(zn)
atol = 1.0_dp
IF (MAX(ABS(aa),ABS(bb)) <= ascle) THEN
zn = zn * rtol
atol = tol
END IF
zn = zn * csgn
cy(i) = zn * atol
csgn = csgn * zt
END DO
RETURN
40 IF (xn >= 0.0_dp) RETURN
END IF
50 ierr = 2
nz = 0
RETURN
60 IF (nw == -1) GO TO 50
nz = 0
ierr = 5
RETURN
END IF
END IF
nz = 0
ierr = 4
RETURN
END SUBROUTINE cbesh
SUBROUTINE cbesi(z, fnu, kode, n, cy, nz, ierr)
!***BEGIN PROLOGUE CBESI
!***DATE WRITTEN 830501 (YYMMDD)
!***REVISION DATE 890801, 930101 (YYMMDD)
!***CATEGORY NO. B5K
!***KEYWORDS I-BESSEL FUNCTION,COMPLEX BESSEL FUNCTION,
! MODIFIED BESSEL FUNCTION OF THE FIRST KIND
!***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
!***PURPOSE TO COMPUTE I-BESSEL FUNCTIONS OF COMPLEX ARGUMENT
!***DESCRIPTION
! ON KODE=1, CBESI COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
! BESSEL FUNCTIONS CY(J)=I(FNU+J-1,Z) FOR REAL (dp), NONNEGATIVE
! ORDERS FNU+J-1, J=1,...,N AND COMPLEX Z IN THE CUT PLANE
! -PI < ARG(Z) <= PI. ON KODE=2, CBESI RETURNS THE SCALED FUNCTIONS
! CY(J)=EXP(-ABS(X))*I(FNU+J-1,Z) J = 1,...,N , X=REAL(Z)
! WITH THE EXPONENTIAL GROWTH REMOVED IN BOTH THE LEFT AND
! RIGHT HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND
! NOTATION ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL
! FUNCTIONS (REF.1)
! INPUT
! Z - Z=CMPLX(X,Y), -PI < ARG(Z) <= PI
! FNU - ORDER OF INITIAL I FUNCTION, FNU >= 0.0_dp
! KODE - A PARAMETER TO INDICATE THE SCALING OPTION
! KODE= 1 RETURNS
! CY(J)=I(FNU+J-1,Z), J=1,...,N
! = 2 RETURNS
! CY(J)=I(FNU+J-1,Z)*EXP(-ABS(X)), J=1,...,N
! N - NUMBER OF MEMBERS OF THE SEQUENCE, N >= 1
! OUTPUT
! CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN
! VALUES FOR THE SEQUENCE
! CY(J)=I(FNU+J-1,Z) OR
! CY(J)=I(FNU+J-1,Z)*EXP(-ABS(X)) J=1,...,N
! DEPENDING ON KODE, X=REAL(Z)
! NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW,
! NZ= 0 , NORMAL RETURN
! NZ > 0 , LAST NZ COMPONENTS OF CY SET TO ZERO
! DUE TO UNDERFLOW, CY(J)=CMPLX(0.0,0.0),
! J = N-NZ+1,...,N
! IERR - ERROR FLAG
! IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
! IERR=1, INPUT ERROR - NO COMPUTATION
! IERR=2, OVERFLOW - NO COMPUTATION, REAL(Z) TOO
! LARGE ON KODE=1
! IERR=3, ABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
! BUT LOSSES OF SIGNIFICANCE BY ARGUMENT
! REDUCTION PRODUCE LESS THAN HALF OF MACHINE
! ACCURACY
! IERR=4, ABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA-
! TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI-
! CANCE BY ARGUMENT REDUCTION
! IERR=5, ERROR - NO COMPUTATION,
! ALGORITHM TERMINATION CONDITION NOT MET
!***LONG DESCRIPTION
! THE COMPUTATION IS CARRIED OUT BY THE POWER SERIES FOR
! SMALL ABS(Z), THE ASYMPTOTIC EXPANSION FOR LARGE ABS(Z),
! THE MILLER ALGORITHM NORMALIZED BY THE WRONSKIAN AND A
! NEUMANN SERIES FOR IMTERMEDIATE MAGNITUDES, AND THE
! UNIFORM ASYMPTOTIC EXPANSIONS FOR I(FNU,Z) AND J(FNU,Z)
! FOR LARGE ORDERS. BACKWARD RECURRENCE IS USED TO GENERATE
! SEQUENCES OR REDUCE ORDERS WHEN NECESSARY.
! THE CALCULATIONS ABOVE ARE DONE IN THE RIGHT HALF PLANE AND
! CONTINUED INTO THE LEFT HALF PLANE BY THE FORMULA
! I(FNU,Z*EXP(M*PI)) = EXP(M*PI*FNU)*I(FNU,Z) REAL(Z) > 0.0
! M = +I OR -I, I**2=-1
! FOR NEGATIVE ORDERS,THE FORMULA
! I(-FNU,Z) = I(FNU,Z) + (2/PI)*SIN(PI*FNU)*K(FNU,Z)
! CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO INTEGERS, THE FUNCTION
! CHANGES RADICALLY. WHEN FNU IS A LARGE POSITIVE INTEGER,THE MAGNITUDE OF
! I(-FNU,Z) = I(FNU,Z) IS A LARGE NEGATIVE POWER OF TEN. BUT WHEN FNU IS NOT
! AN INTEGER, K(FNU,Z) DOMINATES IN MAGNITUDE WITH A LARGE POSITIVE POWER OF
! TEN AND THE MOST THAT THE SECOND TERM CAN BE REDUCED IS BY UNIT ROUNDOFF
! FROM THE COEFFICIENT. THUS, WIDE CHANGES CAN OCCUR WITHIN UNIT ROUNDOFF OF
! A LARGE INTEGER FOR FNU. HERE, LARGE MEANS FNU > ABS(Z).
! IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELEMENTARY
! FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS
! LARGE, LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
! CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN
! LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG
! IERR=3 IS TRIGGERED WHERE UR=EPSILON(0.0_dp)=UNIT ROUNDOFF. ALSO
! IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS
! LOST AND IERR=4. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS
! MUST BE FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE
! INTEGER, U3=HUGE(0). THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS
! RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3
! ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION
! ARITHMETIC AND 1.3E+8, 1.8D+16, 2.1E+9 IN DOUBLE PRECISION
! ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
! THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT
! TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
! IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
! SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
! THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX BESSEL
! FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P = MAX(UNIT ROUNDOFF,1.0E-18)
! IS THE NOMINAL PRECISION AND 10**S REPRESENTS THE INCREASE IN ERROR DUE TO
! ARGUMENT REDUCTION IN THE ELEMENTARY FUNCTIONS. HERE, S =
! MAX(1,ABS(LOG10(ABS(Z))), ABS(LOG10(FNU))) APPROXIMATELY
! (I.E. S = MAX(1,ABS(EXPONENT OF ABS(Z), ABS(EXPONENT OF FNU)) ).
! HOWEVER, THE PHASE ANGLE MAY HAVE ONLY ABSOLUTE ACCURACY.
! THIS IS MOST LIKELY TO OCCUR WHEN ONE COMPONENT (IN ABSOLUTE VALUE) IS
! LARGER THAN THE OTHER BY SEVERAL ORDERS OF MAGNITUDE.
! IF ONE COMPONENT IS 10**K LARGER THAN THE OTHER, THEN ONE CAN EXPECT ONLY
! MAX(ABS(LOG10(P))-K, 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K
! EXCEEDS THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
! COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY BECAUSE,
! IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER COMPONENT WILL NOT
! (AS A RULE) DECREASE BELOW P TIMES THE MAGNITUDE OF THE LARGER COMPONENT.
! IN THESE EXTREME CASES, THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF
! +P, -P, PI/2-P, OR -PI/2+P.
!***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ AND
! I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF COMMERCE, 1955.
! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
! BY D. E. AMOS, SAND83-0083, MAY 1983.
! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
! AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY 1983
! A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX ARGUMENT
! AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-1018, MAY 1985
! A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX ARGUMENT
! AND NONNEGATIVE ORDER BY D. E. AMOS, ACM TRANS. MATH. SOFTWARE,
! VOL. 12, NO. 3, SEPTEMBER 1986, PP 265-273.
!***ROUTINES CALLED CBINU,I1MACH,R1MACH
!***END PROLOGUE CBESI
COMPLEX (dp), INTENT(IN) :: z
REAL (dp), INTENT(IN) :: fnu
INTEGER, INTENT(IN) :: kode
INTEGER, INTENT(IN) :: n
COMPLEX (dp), INTENT(OUT) :: cy(n)
INTEGER, INTENT(OUT) :: nz
INTEGER, INTENT(OUT) :: ierr
COMPLEX (dp) :: csgn, zn
REAL (dp) :: aa, alim, arg, dig, elim, fnul, rl, r1m5, s1, s2, &
tol, xx, yy, az, fn, bb, ascle, rtol, atol
INTEGER :: i, inu, k, k1, k2, nn
!***FIRST EXECUTABLE STATEMENT CBESI
ierr = 0
nz = 0
IF (fnu < 0.0_dp) ierr = 1
IF (kode < 1 .OR. kode > 2) ierr = 1
IF (n < 1) ierr = 1
IF (ierr /= 0) RETURN
xx = REAL(z, KIND=dp)
yy = AIMAG(z)
!-----------------------------------------------------------------------
! SET PARAMETERS RELATED TO MACHINE CONSTANTS.
! TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18.
! ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
! EXP(-ELIM) < EXP(-ALIM)=EXP(-ELIM)/TOL AND
! EXP(ELIM) > EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR
! UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
! RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
! DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
! FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU.
!-----------------------------------------------------------------------
tol = MAX(EPSILON(0.0_dp), 1.0D-18)
k1 = MINEXPONENT(0.0_dp)
k2 = MAXEXPONENT(0.0_dp)
r1m5 = LOG10( REAL( RADIX(0.0_dp), KIND=dp) )
k = MIN(ABS(k1),ABS(k2))
elim = 2.303_dp * (k*r1m5 - 3.0_dp)
k1 = DIGITS(0.0_dp) - 1
aa = r1m5 * k1
dig = MIN(aa, 18.0_dp)
aa = aa * 2.303_dp
alim = elim + MAX(-aa, -41.45_dp)
rl = 1.2_dp * dig + 3.0_dp
fnul = 10.0_dp + 6.0_dp * (dig - 3.0_dp)
az = ABS(z)
!-----------------------------------------------------------------------
! TEST FOR RANGE
!-----------------------------------------------------------------------
aa = 0.5_dp / tol
bb = HUGE(0) * 0.5_dp
aa = MIN(aa,bb)
IF (az <= aa) THEN
fn = fnu + (n-1)
IF (fn <= aa) THEN
aa = SQRT(aa)
IF (az > aa) ierr = 3
IF (fn > aa) ierr = 3
zn = z
csgn = cone
IF (xx < 0.0_dp) THEN
zn = -z
!-----------------------------------------------------------------------
! CALCULATE CSGN=EXP(FNU*PI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE
! WHEN FNU IS LARGE
!-----------------------------------------------------------------------
inu = int(fnu)
arg = (fnu - inu) * pi
IF (yy < 0.0_dp) arg = -arg
s1 = COS(arg)
s2 = SIN(arg)
csgn = CMPLX(s1, s2, KIND=dp)
IF (MOD(inu,2) == 1) csgn = -csgn
END IF
CALL cbinu(zn, fnu, kode, n, cy, nz, rl, fnul, tol, elim, alim)
IF (nz >= 0) THEN
IF (xx >= 0.0_dp) RETURN
!-----------------------------------------------------------------------
! ANALYTIC CONTINUATION TO THE LEFT HALF PLANE
!-----------------------------------------------------------------------
nn = n - nz
IF (nn == 0) RETURN
rtol = 1.0_dp / tol
ascle = TINY(0.0_dp) * rtol * 1.0E+3
DO i = 1, nn
! CY(I) = CY(I)*CSGN
zn = cy(i)
aa = REAL(zn, KIND=dp)
bb = AIMAG(zn)
atol = 1.0_dp
IF (MAX(ABS(aa),ABS(bb)) <= ascle) THEN
zn = zn * rtol
atol = tol
END IF
zn = zn * csgn
cy(i) = zn * atol
csgn = -csgn
END DO
RETURN
END IF
IF (nz /= -2) THEN
nz = 0
ierr = 2
RETURN
END IF
nz = 0
ierr = 5
RETURN
END IF
END IF
nz = 0
ierr = 4
RETURN
END SUBROUTINE cbesi
SUBROUTINE cbesj(z, fnu, kode, n, cy, nz, ierr)
!***BEGIN PROLOGUE CBESJ
!***DATE WRITTEN 830501 (YYMMDD)
!***REVISION DATE 890801, 930101 (YYMMDD)
!***CATEGORY NO. B5K
!***KEYWORDS J-BESSEL FUNCTION,BESSEL FUNCTION OF COMPLEX ARGUMENT,
! BESSEL FUNCTION OF FIRST KIND
!***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
!***PURPOSE TO COMPUTE THE J-BESSEL FUNCTION OF A COMPLEX ARGUMENT
!***DESCRIPTION
! ON KODE=1, CBESJ COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
! BESSEL FUNCTIONS CY(I) = J(FNU+I-1,Z) FOR REAL (dp), NONNEGATIVE
! ORDERS FNU+I-1, I=1,...,N AND COMPLEX Z IN THE CUT PLANE
! -PI < ARG(Z) <= PI. ON KODE=2, CBESJ RETURNS THE SCALED FUNCTIONS
! CY(I) = EXP(-ABS(Y))*J(FNU+I-1,Z) I = 1,...,N , Y=AIMAG(Z)
! WHICH REMOVE THE EXPONENTIAL GROWTH IN BOTH THE UPPER AND
! LOWER HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND NOTATION
! ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL FUNCTIONS (REF. 1).
! INPUT
! Z - Z=CMPLX(X,Y), -PI < ARG(Z) <= PI
! FNU - ORDER OF INITIAL J FUNCTION, FNU >= 0.0_dp
! KODE - A PARAMETER TO INDICATE THE SCALING OPTION
! KODE= 1 RETURNS
! CY(I)=J(FNU+I-1,Z), I=1,...,N
! = 2 RETURNS
! CY(I)=J(FNU+I-1,Z)*EXP(-ABS(Y)), I=1,...
! N - NUMBER OF MEMBERS OF THE SEQUENCE, N >= 1
! OUTPUT
! CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN
! VALUES FOR THE SEQUENCE
! CY(I)=J(FNU+I-1,Z) OR
! CY(I)=J(FNU+I-1,Z)*EXP(-ABS(Y)) I=1,...,N
! DEPENDING ON KODE, Y=AIMAG(Z).
! NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW,
! NZ= 0 , NORMAL RETURN
! NZ > 0 , LAST NZ COMPONENTS OF CY SET TO ZERO
! DUE TO UNDERFLOW, CY(I)=CMPLX(0.0,0.0),
! I = N-NZ+1,...,N
! IERR - ERROR FLAG
! IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
! IERR=1, INPUT ERROR - NO COMPUTATION
! IERR=2, OVERFLOW - NO COMPUTATION, AIMAG(Z)
! TOO LARGE ON KODE=1
! IERR=3, ABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
! BUT LOSSES OF SIGNIFICANCE BY ARGUMENT
! REDUCTION PRODUCE LESS THAN HALF OF MACHINE ACCURACY
! IERR=4, ABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTATION BECAUSE
! OF COMPLETE LOSSES OF SIGNIFICANCE BY ARGUMENT
! REDUCTION
! IERR=5, ERROR - NO COMPUTATION,
! ALGORITHM TERMINATION CONDITION NOT MET
!***LONG DESCRIPTION
! THE COMPUTATION IS CARRIED OUT BY THE FORMULA
! J(FNU,Z)=EXP( FNU*PI*I/2)*I(FNU,-I*Z) AIMAG(Z) >= 0.0
! J(FNU,Z)=EXP(-FNU*PI*I/2)*I(FNU, I*Z) AIMAG(Z) < 0.0
! WHERE I**2 = -1 AND I(FNU,Z) IS THE I BESSEL FUNCTION.
! FOR NEGATIVE ORDERS,THE FORMULA
! J(-FNU,Z) = J(FNU,Z)*COS(PI*FNU) - Y(FNU,Z)*SIN(PI*FNU)
! CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO INTEGERS, THE FUNCTION
! CHANGES RADICALLY. WHEN FNU IS A LARGE POSITIVE INTEGER, THE MAGNITUDE
! OF J(-FNU,Z)=J(FNU,Z)*COS(PI*FNU) IS A LARGE NEGATIVE POWER OF TEN.
! BUT WHEN FNU IS NOT AN INTEGER, Y(FNU,Z) DOMINATES IN MAGNITUDE WITH A
! LARGE POSITIVE POWER OF TEN AND THE MOST THAT THE SECOND TERM CAN BE
! REDUCED IS BY UNIT ROUNDOFF FROM THE COEFFICIENT. THUS, WIDE CHANGES CAN
! OCCUR WITHIN UNIT ROUNDOFF OF A LARGE INTEGER FOR FNU. HERE, LARGE MEANS
! FNU > ABS(Z).
! IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELEMENTARY
! FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS LARGE, LOSSES OF
! SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF EITHER ONE
! EXCEEDS U1=SQRT(0.5/UR), THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY
! AND AN ERROR FLAG IERR=3 IS TRIGGERED WHERE UR = EPSILON(0.0_dp) = UNIT
! ROUNDOFF. ALSO IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE
! IS LOST AND IERR=4. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS MUST BE
! FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE INTEGER, U3 = HUGE(0).
! THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS RESTRICTED BY MIN(U2,U3).
! ON 32 BIT MACHINES, U1,U2, AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9
! IN SINGLE PRECISION ARITHMETIC AND 1.3E+8, 1.8D+16, 2.1E+9 IN DOUBLE
! PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
! THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT TO RETAIN,
! IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS IN SINGLE AND ONLY 7
! DIGITS IN DOUBLE PRECISION ARITHMETIC.
! SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
! THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX BESSEL
! FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P = MAX(UNIT ROUNDOFF, 1.0E-18)
! IS THE NOMINAL PRECISION AND 10**S REPRESENTS THE INCREASE IN ERROR DUE
! TO ARGUMENT REDUCTION IN THE ELEMENTARY FUNCTIONS. HERE,
! S = MAX(1,ABS(LOG10(ABS(Z))), ABS(LOG10(FNU))) APPROXIMATELY
! (I.E. S = MAX(1, ABS(EXPONENT OF ABS(Z), ABS(EXPONENT OF FNU)) ).
! HOWEVER, THE PHASE ANGLE MAY HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST
! LIKELY TO OCCUR WHEN ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN
! THE OTHER BY SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K
! LARGER THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K, 0)
! SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS THE EXPONENT
! OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER COMPONENT.
! HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY BECAUSE, IN COMPLEX
! ARITHMETIC WITH PRECISION P, THE SMALLER COMPONENT WILL NOT (AS A RULE)
! DECREASE BELOW P TIMES THE MAGNITUDE OF THE LARGER COMPONENT.
! IN THESE EXTREME CASES, THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P,
! -P, PI/2-P, OR -PI/2+P.
!***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ AND
! I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF COMMERCE, 1955.
! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
! BY D. E. AMOS, SAND83-0083, MAY 1983.
! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
! AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY 1983
! A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX ARGUMENT
! AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-1018, MAY 1985
! A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX ARGUMENT
! AND NONNEGATIVE ORDER BY D. E. AMOS, ACM TRANS. MATH. SOFTWARE,
! VOL. 12, NO. 3, SEPTEMBER 1986, PP 265-273.
!***ROUTINES CALLED CBINU,I1MACH,R1MACH
!***END PROLOGUE CBESJ
COMPLEX (dp), INTENT(IN) :: z
REAL (dp), INTENT(IN) :: fnu
INTEGER, INTENT(IN) :: kode
INTEGER, INTENT(IN) :: n
COMPLEX (dp), INTENT(OUT) :: cy(n)
INTEGER, INTENT(OUT) :: nz
INTEGER, INTENT(OUT) :: ierr
COMPLEX (dp) :: ci, csgn, zn
REAL (dp) :: aa, alim, arg, dig, elim, fnul, rl, r1, r1m5, r2, &
tol, yy, az, fn, bb, ascle, rtol, atol
INTEGER :: i, inu, inuh, ir, k1, k2, nl, k
!***FIRST EXECUTABLE STATEMENT CBESJ
ierr = 0
nz = 0
IF (fnu < 0.0_dp) ierr = 1
IF (kode < 1 .OR. kode > 2) ierr = 1
IF (n < 1) ierr = 1
IF (ierr /= 0) RETURN
!-----------------------------------------------------------------------
! SET PARAMETERS RELATED TO MACHINE CONSTANTS.
! TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18.
! ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
! EXP(-ELIM) < EXP(-ALIM)=EXP(-ELIM)/TOL AND
! EXP(ELIM) > EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR
! UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
! RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
! DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
! FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU.
!-----------------------------------------------------------------------
tol = MAX(EPSILON(0.0_dp), 1.0D-18)
k1 = MINEXPONENT(0.0_dp)
k2 = MAXEXPONENT(0.0_dp)
r1m5 = LOG10( REAL( RADIX(0.0_dp), KIND=dp) )
k = MIN(ABS(k1),ABS(k2))
elim = 2.303_dp * (k*r1m5 - 3.0_dp)
k1 = DIGITS(0.0_dp) - 1
aa = r1m5 * k1
dig = MIN(aa, 18.0_dp)
aa = aa * 2.303_dp
alim = elim + MAX(-aa, -41.45_dp)
rl = 1.2_dp * dig + 3.0_dp
fnul = 10.0_dp + 6.0_dp * (dig - 3.0_dp)
ci = CMPLX(0.0_dp, 1.0_dp, KIND=dp)
yy = AIMAG(z)
az = ABS(z)
!-----------------------------------------------------------------------
! TEST FOR RANGE
!-----------------------------------------------------------------------
aa = 0.5_dp / tol
bb = HUGE(0) * 0.5_dp
aa = MIN(aa,bb)
fn = fnu + (n-1)
IF (az <= aa) THEN
IF (fn <= aa) THEN
aa = SQRT(aa)
IF (az > aa) ierr = 3
IF (fn > aa) ierr = 3
!-----------------------------------------------------------------------
! CALCULATE CSGN=EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE
! WHEN FNU IS LARGE
!-----------------------------------------------------------------------
inu = int(fnu)
inuh = inu / 2
ir = inu - 2 * inuh
arg = (fnu - (inu-ir)) * hpi
r1 = COS(arg)
r2 = SIN(arg)
csgn = CMPLX(r1, r2, KIND=dp)
IF (MOD(inuh,2) == 1) csgn = -csgn
!-----------------------------------------------------------------------
! ZN IS IN THE RIGHT HALF PLANE
!-----------------------------------------------------------------------
zn = -z * ci
IF (yy < 0.0_dp) THEN
zn = -zn
csgn = CONJG(csgn)
ci = CONJG(ci)
END IF
CALL cbinu(zn, fnu, kode, n, cy, nz, rl, fnul, tol, elim, alim)
IF (nz >= 0) THEN
nl = n - nz
IF (nl == 0) RETURN
rtol = 1.0_dp / tol
ascle = TINY(0.0_dp) * rtol * 1.0E+3
DO i = 1, nl
! CY(I)=CY(I)*CSGN
zn = cy(i)
aa = REAL(zn, KIND=dp)
bb = AIMAG(zn)
atol = 1.0_dp
IF (MAX(ABS(aa),ABS(bb)) <= ascle) THEN
zn = zn * rtol
atol = tol
END IF
zn = zn * csgn
cy(i) = zn * atol
csgn = csgn * ci
END DO
RETURN
END IF
IF (nz /= -2) THEN
nz = 0
ierr = 2
RETURN
END IF
nz = 0
ierr = 5
RETURN
END IF
END IF
nz = 0
ierr = 4
RETURN
END SUBROUTINE cbesj
SUBROUTINE cbesk(z, fnu, kode, n, cy, nz, ierr)
!***BEGIN PROLOGUE CBESK
!***DATE WRITTEN 830501 (YYMMDD)
!***REVISION DATE 890801, 930101 (YYMMDD)
!***CATEGORY NO. B5K
!***KEYWORDS K-BESSEL FUNCTION,COMPLEX BESSEL FUNCTION,
! MODIFIED BESSEL FUNCTION OF THE SECOND KIND,
! BESSEL FUNCTION OF THE THIRD KIND
!***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
!***PURPOSE TO COMPUTE K-BESSEL FUNCTIONS OF COMPLEX ARGUMENT
!***DESCRIPTION
! ON KODE=1, CBESK COMPUTES AN N MEMBER SEQUENCE OF COMPLEX BESSEL FUNCTIONS
! CY(J)=K(FNU+J-1,Z) FOR REAL (dp), NONNEGATIVE ORDERS FNU+J-1, J=1,...,N
! AND COMPLEX Z.NE.CMPLX(0.0,0.0) IN THE CUT PLANE -PI < ARG(Z) <= PI.
! ON KODE=2, CBESK RETURNS THE SCALED K FUNCTIONS,
! CY(J)=EXP(Z)*K(FNU+J-1,Z) , J=1,...,N,
! WHICH REMOVE THE EXPONENTIAL BEHAVIOR IN BOTH THE LEFT AND RIGHT HALF
! PLANES FOR Z TO INFINITY. DEFINITIONS AND NOTATION ARE FOUND IN THE NBS
! HANDBOOK OF MATHEMATICAL FUNCTIONS (REF. 1).
! INPUT
! Z - Z=CMPLX(X,Y),Z.NE.CMPLX(0.,0.),-PI < ARG(Z) <= PI
! FNU - ORDER OF INITIAL K FUNCTION, FNU >= 0.0_dp
! N - NUMBER OF MEMBERS OF THE SEQUENCE, N >= 1
! KODE - A PARAMETER TO INDICATE THE SCALING OPTION
! KODE= 1 RETURNS
! CY(I)=K(FNU+I-1,Z), I=1,...,N
! = 2 RETURNS
! CY(I)=K(FNU+I-1,Z)*EXP(Z), I=1,...,N
! OUTPUT
! CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN
! VALUES FOR THE SEQUENCE
! CY(I)=K(FNU+I-1,Z), I=1,...,N OR
! CY(I)=K(FNU+I-1,Z)*EXP(Z), I=1,...,N
! DEPENDING ON KODE
! NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW.
! NZ= 0 , NORMAL RETURN
! NZ > 0 , FIRST NZ COMPONENTS OF CY SET TO ZERO
! DUE TO UNDERFLOW, CY(I)=CMPLX(0.0,0.0),
! I=1,...,N WHEN X >= 0.0. WHEN X < 0.0, NZ STATES
! ONLY THE NUMBER OF UNDERFLOWS IN THE SEQUENCE.
! IERR - ERROR FLAG
! IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
! IERR=1, INPUT ERROR - NO COMPUTATION
! IERR=2, OVERFLOW - NO COMPUTATION, FNU+N-1 IS
! TOO LARGE OR ABS(Z) IS TOO SMALL OR BOTH
! IERR=3, ABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE, BUT
! LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION PRODUCE
! LESS THAN HALF OF MACHINE ACCURACY
! IERR=4, ABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTATION BECAUSE OF
! COMPLETE LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION
! IERR=5, ERROR - NO COMPUTATION,
! ALGORITHM TERMINATION CONDITION NOT MET
!***LONG DESCRIPTION
! EQUATIONS OF THE REFERENCE ARE IMPLEMENTED FOR SMALL ORDERS DNU AND
! DNU+1.0 IN THE RIGHT HALF PLANE X >= 0.0. FORWARD RECURRENCE GENERATES
! HIGHER ORDERS. K IS CONTINUED TO THE LEFT HALF PLANE BY THE RELATION
! K(FNU,Z*EXP(MP)) = EXP(-MP*FNU)*K(FNU,Z)-MP*I(FNU,Z)
! MP=MR*PI*I, MR=+1 OR -1, RE(Z) > 0, I**2=-1
! WHERE I(FNU,Z) IS THE I BESSEL FUNCTION.
! FOR LARGE ORDERS, FNU > FNUL, THE K FUNCTION IS COMPUTED
! BY MEANS OF ITS UNIFORM ASYMPTOTIC EXPANSIONS.
! FOR NEGATIVE ORDERS, THE FORMULA
! K(-FNU,Z) = K(FNU,Z)
! CAN BE USED.
! CBESK ASSUMES THAT A SIGNIFICANT DIGIT SINH(X) FUNCTION IS AVAILABLE.
! IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELEMENTARY
! FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS LARGE, LOSSES OF
! SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
! CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN LOSSES EXCEEDING
! HALF PRECISION ARE LIKELY AND AN ERROR FLAG IERR=3 IS TRIGGERED WHERE
! UR = EPSILON(0.0_dp) = UNIT ROUNDOFF. ALSO IF EITHER IS LARGER THAN
! U2 = 0.5/UR, THEN ALL SIGNIFICANCE IS LOST AND IERR=4. IN ORDER TO USE
! THE INT FUNCTION, ARGUMENTS MUST BE FURTHER RESTRICTED NOT TO EXCEED THE
! LARGEST MACHINE INTEGER, U3=HUGE(0). THUS, THE MAGNITUDE OF Z AND FNU+N-1
! IS RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3 ARE
! APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION ARITHMETIC AND
! 1.3E+8, 1.8D+16, 2.1E+9 IN DOUBLE PRECISION ARITHMETIC RESPECTIVELY.
! THIS MAKES U2 AND U3 LIMITING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS
! THAT ONE CAN EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO
! DIGITS IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
! SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
! THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX BESSEL
! FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P = MAX(UNIT ROUNDOFF,1.0E-18)
! IS THE NOMINAL PRECISION AND 10**S REPRESENTS THE INCREASE IN ERROR DUE TO
! ARGUMENT REDUCTION IN THE ELEMENTARY FUNCTIONS. HERE, S =
! MAX(1,ABS(LOG10(ABS(Z))), ABS(LOG10(FNU))) APPROXIMATELY (I.E. S =
! MAX(1,ABS(EXPONENT OF ABS(Z),ABS(EXPONENT OF FNU)) ).
! HOWEVER, THE PHASE ANGLE MAY HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST
! LIKELY TO OCCUR WHEN ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE
! OTHER BY SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
! THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K, 0)
! SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS THE EXPONENT
! OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER COMPONENT. HOWEVER, THE
! PHASE ANGLE RETAINS ABSOLUTE ACCURACY BECAUSE, IN COMPLEX ARITHMETIC WITH
! PRECISION P, THE SMALLER COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P
! TIMES THE MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
! THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P, OR -PI/2+P.
!***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ AND
! I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF COMMERCE, 1955.
! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
! BY D. E. AMOS, SAND83-0083, MAY 1983.
! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
! AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY 1983.
! A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX ARGUMENT