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
|
*DECK ZBESH
SUBROUTINE ZBESH (ZR, ZI, FNU, KODE, M, N, CYR, CYI, NZ, IERR)
C***BEGIN PROLOGUE ZBESH
C***PURPOSE Compute a sequence of the Hankel functions H(m,a,z)
C for superscript m=1 or 2, real nonnegative orders a=b,
C b+1,... where b>0, and nonzero complex argument z. A
C scaling option is available to help avoid overflow.
C***LIBRARY SLATEC
C***CATEGORY C10A4
C***TYPE COMPLEX (CBESH-C, ZBESH-C)
C***KEYWORDS BESSEL FUNCTIONS OF COMPLEX ARGUMENT,
C BESSEL FUNCTIONS OF THE THIRD KIND, H BESSEL FUNCTIONS,
C HANKEL FUNCTIONS
C***AUTHOR Amos, D. E., (SNL)
C***DESCRIPTION
C
C ***A DOUBLE PRECISION ROUTINE***
C On KODE=1, ZBESH computes an N member sequence of complex
C Hankel (Bessel) functions CY(L)=H(M,FNU+L-1,Z) for super-
C script M=1 or 2, real nonnegative orders FNU+L-1, L=1,...,
C N, and complex nonzero Z in the cut plane -pi<arg(Z)<=pi
C where Z=ZR+i*ZI. On KODE=2, CBESH returns the scaled
C functions
C
C CY(L) = H(M,FNU+L-1,Z)*exp(-(3-2*M)*Z*i), i**2=-1
C
C which removes the exponential behavior in both the upper
C and lower half planes. Definitions and notation are found
C in the NBS Handbook of Mathematical Functions (Ref. 1).
C
C Input
C ZR - DOUBLE PRECISION real part of nonzero argument Z
C ZI - DOUBLE PRECISION imag part of nonzero argument Z
C FNU - DOUBLE PRECISION initial order, FNU>=0
C KODE - A parameter to indicate the scaling option
C KODE=1 returns
C CY(L)=H(M,FNU+L-1,Z), L=1,...,N
C =2 returns
C CY(L)=H(M,FNU+L-1,Z)*exp(-(3-2M)*Z*i),
C L=1,...,N
C M - Superscript of Hankel function, M=1 or 2
C N - Number of terms in the sequence, N>=1
C
C Output
C CYR - DOUBLE PRECISION real part of result vector
C CYI - DOUBLE PRECISION imag part of result vector
C NZ - Number of underflows set to zero
C NZ=0 Normal return
C NZ>0 CY(L)=0 for NZ values of L (if M=1 and
C Im(Z)>0 or if M=2 and Im(Z)<0, then
C CY(L)=0 for L=1,...,NZ; in the com-
C plementary half planes, the underflows
C may not be in an uninterrupted sequence)
C IERR - Error flag
C IERR=0 Normal return - COMPUTATION COMPLETED
C IERR=1 Input error - NO COMPUTATION
C IERR=2 Overflow - NO COMPUTATION
C (abs(Z) too small and/or FNU+N-1
C too large)
C IERR=3 Precision warning - COMPUTATION COMPLETED
C (Result has half precision or less
C because abs(Z) or FNU+N-1 is large)
C IERR=4 Precision error - NO COMPUTATION
C (Result has no precision because
C abs(Z) or FNU+N-1 is too large)
C IERR=5 Algorithmic error - NO COMPUTATION
C (Termination condition not met)
C
C *Long Description:
C
C The computation is carried out by the formula
C
C H(m,a,z) = (1/t)*exp(-a*t)*K(a,z*exp(-t))
C t = (3-2*m)*i*pi/2
C
C where the K Bessel function is computed as described in the
C prologue to CBESK.
C
C Exponential decay of H(m,a,z) occurs in the upper half z
C plane for m=1 and the lower half z plane for m=2. Exponential
C growth occurs in the complementary half planes. Scaling
C by exp(-(3-2*m)*z*i) removes the exponential behavior in the
C whole z plane as z goes to infinity.
C
C For negative orders, the formula
C
C H(m,-a,z) = H(m,a,z)*exp((3-2*m)*a*pi*i)
C
C can be used.
C
C In most complex variable computation, one must evaluate ele-
C mentary functions. When the magnitude of Z or FNU+N-1 is
C large, losses of significance by argument reduction occur.
C Consequently, if either one exceeds U1=SQRT(0.5/UR), then
C losses exceeding half precision are likely and an error flag
C IERR=3 is triggered where UR=MAX(D1MACH(4),1.0D-18) is double
C precision unit roundoff limited to 18 digits precision. Also,
C if either is larger than U2=0.5/UR, then all significance is
C lost and IERR=4. In order to use the INT function, arguments
C must be further restricted not to exceed the largest machine
C integer, U3=I1MACH(9). Thus, the magnitude of Z and FNU+N-1
C is restricted by MIN(U2,U3). In IEEE arithmetic, U1,U2, and
C U3 approximate 2.0E+3, 4.2E+6, 2.1E+9 in single precision
C and 4.7E+7, 2.3E+15 and 2.1E+9 in double precision. This
C makes U2 limiting in single precision and U3 limiting in
C double precision. This means that one can expect to retain,
C in the worst cases on IEEE machines, no digits in single pre-
C cision and only 6 digits in double precision. Similar con-
C siderations hold for other machines.
C
C The approximate relative error in the magnitude of a complex
C Bessel function can be expressed as P*10**S where P=MAX(UNIT
C ROUNDOFF,1.0E-18) is the nominal precision and 10**S repre-
C sents the increase in error due to argument reduction in the
C elementary functions. Here, S=MAX(1,ABS(LOG10(ABS(Z))),
C ABS(LOG10(FNU))) approximately (i.e., S=MAX(1,ABS(EXPONENT OF
C ABS(Z),ABS(EXPONENT OF FNU)) ). However, the phase angle may
C have only absolute accuracy. This is most likely to occur
C when one component (in magnitude) is larger than the other by
C several orders of magnitude. If one component is 10**K larger
C than the other, then one can expect only MAX(ABS(LOG10(P))-K,
C 0) significant digits; or, stated another way, when K exceeds
C the exponent of P, no significant digits remain in the smaller
C component. However, the phase angle retains absolute accuracy
C because, in complex arithmetic with precision P, the smaller
C component will not (as a rule) decrease below P times the
C magnitude of the larger component. In these extreme cases,
C the principal phase angle is on the order of +P, -P, PI/2-P,
C or -PI/2+P.
C
C***REFERENCES 1. M. Abramowitz and I. A. Stegun, Handbook of Mathe-
C matical Functions, National Bureau of Standards
C Applied Mathematics Series 55, U. S. Department
C of Commerce, Tenth Printing (1972) or later.
C 2. D. E. Amos, Computation of Bessel Functions of
C Complex Argument, Report SAND83-0086, Sandia National
C Laboratories, Albuquerque, NM, May 1983.
C 3. D. E. Amos, Computation of Bessel Functions of
C Complex Argument and Large Order, Report SAND83-0643,
C Sandia National Laboratories, Albuquerque, NM, May
C 1983.
C 4. D. E. Amos, A Subroutine Package for Bessel Functions
C of a Complex Argument and Nonnegative Order, Report
C SAND85-1018, Sandia National Laboratory, Albuquerque,
C NM, May 1985.
C 5. D. E. Amos, A portable package for Bessel functions
C of a complex argument and nonnegative order, ACM
C Transactions on Mathematical Software, 12 (September
C 1986), pp. 265-273.
C
C***ROUTINES CALLED D1MACH, I1MACH, ZABS, ZACON, ZBKNU, ZBUNK, ZUOIK
C***REVISION HISTORY (YYMMDD)
C 830501 DATE WRITTEN
C 890801 REVISION DATE from Version 3.2
C 910415 Prologue converted to Version 4.0 format. (BAB)
C 920128 Category corrected. (WRB)
C 920811 Prologue revised. (DWL)
C***END PROLOGUE ZBESH
C
C COMPLEX CY,Z,ZN,ZT,CSGN
DOUBLE PRECISION AA, ALIM, ALN, ARG, AZ, CYI, CYR, DIG, ELIM,
* FMM, FN, FNU, FNUL, HPI, RHPI, RL, R1M5, SGN, STR, TOL, UFL, ZI,
* ZNI, ZNR, ZR, ZTI, D1MACH, ZABS, BB, ASCLE, RTOL, ATOL, STI,
* CSGNR, CSGNI
INTEGER I, IERR, INU, INUH, IR, K, KODE, K1, K2, M,
* MM, MR, N, NN, NUF, NW, NZ, I1MACH
DIMENSION CYR(N), CYI(N)
EXTERNAL ZABS
C
DATA HPI /1.57079632679489662D0/
C
C***FIRST EXECUTABLE STATEMENT ZBESH
IERR = 0
NZ=0
IF (ZR.EQ.0.0D0 .AND. ZI.EQ.0.0D0) IERR=1
IF (FNU.LT.0.0D0) IERR=1
IF (M.LT.1 .OR. M.GT.2) IERR=1
IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
IF (N.LT.1) IERR=1
IF (IERR.NE.0) RETURN
NN = N
C-----------------------------------------------------------------------
C SET PARAMETERS RELATED TO MACHINE CONSTANTS.
C TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18.
C ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
C EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND
C EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR
C UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
C RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
C DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
C FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU
C-----------------------------------------------------------------------
TOL = MAX(D1MACH(4),1.0D-18)
K1 = I1MACH(15)
K2 = I1MACH(16)
R1M5 = D1MACH(5)
K = MIN(ABS(K1),ABS(K2))
ELIM = 2.303D0*(K*R1M5-3.0D0)
K1 = I1MACH(14) - 1
AA = R1M5*K1
DIG = MIN(AA,18.0D0)
AA = AA*2.303D0
ALIM = ELIM + MAX(-AA,-41.45D0)
FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0)
RL = 1.2D0*DIG + 3.0D0
FN = FNU + (NN-1)
MM = 3 - M - M
FMM = MM
ZNR = FMM*ZI
ZNI = -FMM*ZR
C-----------------------------------------------------------------------
C TEST FOR PROPER RANGE
C-----------------------------------------------------------------------
AZ = ZABS(ZR,ZI)
AA = 0.5D0/TOL
BB = I1MACH(9)*0.5D0
AA = MIN(AA,BB)
IF (AZ.GT.AA) GO TO 260
IF (FN.GT.AA) GO TO 260
AA = SQRT(AA)
IF (AZ.GT.AA) IERR=3
IF (FN.GT.AA) IERR=3
C-----------------------------------------------------------------------
C OVERFLOW TEST ON THE LAST MEMBER OF THE SEQUENCE
C-----------------------------------------------------------------------
UFL = D1MACH(1)*1.0D+3
IF (AZ.LT.UFL) GO TO 230
IF (FNU.GT.FNUL) GO TO 90
IF (FN.LE.1.0D0) GO TO 70
IF (FN.GT.2.0D0) GO TO 60
IF (AZ.GT.TOL) GO TO 70
ARG = 0.5D0*AZ
ALN = -FN*LOG(ARG)
IF (ALN.GT.ELIM) GO TO 230
GO TO 70
60 CONTINUE
CALL ZUOIK(ZNR, ZNI, FNU, KODE, 2, NN, CYR, CYI, NUF, TOL, ELIM,
* ALIM)
IF (NUF.LT.0) GO TO 230
NZ = NZ + NUF
NN = NN - NUF
C-----------------------------------------------------------------------
C HERE NN=N OR NN=0 SINCE NUF=0,NN, OR -1 ON RETURN FROM CUOIK
C IF NUF=NN, THEN CY(I)=CZERO FOR ALL I
C-----------------------------------------------------------------------
IF (NN.EQ.0) GO TO 140
70 CONTINUE
IF ((ZNR.LT.0.0D0) .OR. (ZNR.EQ.0.0D0 .AND. ZNI.LT.0.0D0 .AND.
* M.EQ.2)) GO TO 80
C-----------------------------------------------------------------------
C RIGHT HALF PLANE COMPUTATION, XN.GE.0. .AND. (XN.NE.0. .OR.
C YN.GE.0. .OR. M=1)
C-----------------------------------------------------------------------
CALL ZBKNU(ZNR, ZNI, FNU, KODE, NN, CYR, CYI, NZ, TOL, ELIM, ALIM)
GO TO 110
C-----------------------------------------------------------------------
C LEFT HALF PLANE COMPUTATION
C-----------------------------------------------------------------------
80 CONTINUE
MR = -MM
CALL ZACON(ZNR, ZNI, FNU, KODE, MR, NN, CYR, CYI, NW, RL, FNUL,
* TOL, ELIM, ALIM)
IF (NW.LT.0) GO TO 240
NZ=NW
GO TO 110
90 CONTINUE
C-----------------------------------------------------------------------
C UNIFORM ASYMPTOTIC EXPANSIONS FOR FNU.GT.FNUL
C-----------------------------------------------------------------------
MR = 0
IF ((ZNR.GE.0.0D0) .AND. (ZNR.NE.0.0D0 .OR. ZNI.GE.0.0D0 .OR.
* M.NE.2)) GO TO 100
MR = -MM
IF (ZNR.NE.0.0D0 .OR. ZNI.GE.0.0D0) GO TO 100
ZNR = -ZNR
ZNI = -ZNI
100 CONTINUE
CALL ZBUNK(ZNR, ZNI, FNU, KODE, MR, NN, CYR, CYI, NW, TOL, ELIM,
* ALIM)
IF (NW.LT.0) GO TO 240
NZ = NZ + NW
110 CONTINUE
C-----------------------------------------------------------------------
C H(M,FNU,Z) = -FMM*(I/HPI)*(ZT**FNU)*K(FNU,-Z*ZT)
C
C ZT=EXP(-FMM*HPI*I) = CMPLX(0.0,-FMM), FMM=3-2*M, M=1,2
C-----------------------------------------------------------------------
SGN = DSIGN(HPI,-FMM)
C-----------------------------------------------------------------------
C CALCULATE EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE
C WHEN FNU IS LARGE
C-----------------------------------------------------------------------
INU = FNU
INUH = INU/2
IR = INU - 2*INUH
ARG = (FNU-(INU-IR))*SGN
RHPI = 1.0D0/SGN
C ZNI = RHPI*COS(ARG)
C ZNR = -RHPI*SIN(ARG)
CSGNI = RHPI*COS(ARG)
CSGNR = -RHPI*SIN(ARG)
IF (MOD(INUH,2).EQ.0) GO TO 120
C ZNR = -ZNR
C ZNI = -ZNI
CSGNR = -CSGNR
CSGNI = -CSGNI
120 CONTINUE
ZTI = -FMM
RTOL = 1.0D0/TOL
ASCLE = UFL*RTOL
DO 130 I=1,NN
C STR = CYR(I)*ZNR - CYI(I)*ZNI
C CYI(I) = CYR(I)*ZNI + CYI(I)*ZNR
C CYR(I) = STR
C STR = -ZNI*ZTI
C ZNI = ZNR*ZTI
C ZNR = STR
AA = CYR(I)
BB = CYI(I)
ATOL = 1.0D0
IF (MAX(ABS(AA),ABS(BB)).GT.ASCLE) GO TO 135
AA = AA*RTOL
BB = BB*RTOL
ATOL = TOL
135 CONTINUE
STR = AA*CSGNR - BB*CSGNI
STI = AA*CSGNI + BB*CSGNR
CYR(I) = STR*ATOL
CYI(I) = STI*ATOL
STR = -CSGNI*ZTI
CSGNI = CSGNR*ZTI
CSGNR = STR
130 CONTINUE
RETURN
140 CONTINUE
IF (ZNR.LT.0.0D0) GO TO 230
RETURN
230 CONTINUE
NZ=0
IERR=2
RETURN
240 CONTINUE
IF(NW.EQ.(-1)) GO TO 230
NZ=0
IERR=5
RETURN
260 CONTINUE
NZ=0
IERR=4
RETURN
END
|
