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comparison libcruft/amos/cbknu.f @ 7789:82be108cc558

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First attempt at single precision tyeps * * * corrections to qrupdate single precision routines * * * prefer demotion to single over promotion to double * * * Add single precision support to log2 function * * * Trivial PROJECT file update * * * Cache optimized hermitian/transpose methods * * * Add tests for tranpose/hermitian and ChangeLog entry for new transpose code
author David Bateman <dbateman@free.fr>
date Sun, 27 Apr 2008 22:34:17 +0200
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7788:45f5faba05a2 7789:82be108cc558
1 SUBROUTINE CBKNU(Z, FNU, KODE, N, Y, NZ, TOL, ELIM, ALIM)
2 C***BEGIN PROLOGUE CBKNU
3 C***REFER TO CBESI,CBESK,CAIRY,CBESH
4 C
5 C CBKNU COMPUTES THE K BESSEL FUNCTION IN THE RIGHT HALF Z PLANE
6 C
7 C***ROUTINES CALLED CKSCL,CSHCH,GAMLN,I1MACH,R1MACH,CUCHK
8 C***END PROLOGUE CBKNU
9 C
10 COMPLEX CCH, CK, COEF, CONE, CRSC, CS, CSCL, CSH, CSR, CSS, CTWO,
11 * CZ, CZERO, F, FMU, P, PT, P1, P2, Q, RZ, SMU, ST, S1, S2, Y, Z,
12 * ZD, CELM, CY
13 REAL AA, AK, ALIM, ASCLE, A1, A2, BB, BK, BRY, CAZ, CC, DNU,
14 * DNU2, ELIM, ETEST, FC, FHS, FK, FKS, FNU, FPI, G1, G2, HPI, PI,
15 * P2I, P2M, P2R, RK, RTHPI, R1, S, SPI, TM, TOL, TTH, T1, T2, XX,
16 * YY, GAMLN, R1MACH, HELIM, ELM, XD, YD, ALAS, AS
17 INTEGER I, IDUM, IFLAG, INU, K, KFLAG, KK, KMAX, KODE, KODED, N,
18 * NZ, I1MACH, NW, J, IC, INUB
19 DIMENSION BRY(3), CC(8), CSS(3), CSR(3), Y(N), CY(2)
20 C
21 DATA KMAX / 30 /
22 DATA R1 / 2.0E0 /
23 DATA CZERO,CONE,CTWO /(0.0E0,0.0E0),(1.0E0,0.0E0),(2.0E0,0.0E0)/
24 C
25 DATA PI, RTHPI, SPI ,HPI, FPI, TTH /
26 1 3.14159265358979324E0, 1.25331413731550025E0,
27 2 1.90985931710274403E0, 1.57079632679489662E0,
28 3 1.89769999331517738E0, 6.66666666666666666E-01/
29 C
30 DATA CC(1), CC(2), CC(3), CC(4), CC(5), CC(6), CC(7), CC(8)/
31 1 5.77215664901532861E-01, -4.20026350340952355E-02,
32 2 -4.21977345555443367E-02, 7.21894324666309954E-03,
33 3 -2.15241674114950973E-04, -2.01348547807882387E-05,
34 4 1.13302723198169588E-06, 6.11609510448141582E-09/
35 C
36 XX = REAL(Z)
37 YY = AIMAG(Z)
38 CAZ = CABS(Z)
39 CSCL = CMPLX(1.0E0/TOL,0.0E0)
40 CRSC = CMPLX(TOL,0.0E0)
41 CSS(1) = CSCL
42 CSS(2) = CONE
43 CSS(3) = CRSC
44 CSR(1) = CRSC
45 CSR(2) = CONE
46 CSR(3) = CSCL
47 BRY(1) = 1.0E+3*R1MACH(1)/TOL
48 BRY(2) = 1.0E0/BRY(1)
49 BRY(3) = R1MACH(2)
50 NZ = 0
51 IFLAG = 0
52 KODED = KODE
53 RZ = CTWO/Z
54 INU = INT(FNU+0.5E0)
55 DNU = FNU - FLOAT(INU)
56 IF (ABS(DNU).EQ.0.5E0) GO TO 110
57 DNU2 = 0.0E0
58 IF (ABS(DNU).GT.TOL) DNU2 = DNU*DNU
59 IF (CAZ.GT.R1) GO TO 110
60 C-----------------------------------------------------------------------
61 C SERIES FOR CABS(Z).LE.R1
62 C-----------------------------------------------------------------------
63 FC = 1.0E0
64 SMU = CLOG(RZ)
65 FMU = SMU*CMPLX(DNU,0.0E0)
66 CALL CSHCH(FMU, CSH, CCH)
67 IF (DNU.EQ.0.0E0) GO TO 10
68 FC = DNU*PI
69 FC = FC/SIN(FC)
70 SMU = CSH*CMPLX(1.0E0/DNU,0.0E0)
71 10 CONTINUE
72 A2 = 1.0E0 + DNU
73 C-----------------------------------------------------------------------
74 C GAM(1-Z)*GAM(1+Z)=PI*Z/SIN(PI*Z), T1=1/GAM(1-DNU), T2=1/GAM(1+DNU)
75 C-----------------------------------------------------------------------
76 T2 = EXP(-GAMLN(A2,IDUM))
77 T1 = 1.0E0/(T2*FC)
78 IF (ABS(DNU).GT.0.1E0) GO TO 40
79 C-----------------------------------------------------------------------
80 C SERIES FOR F0 TO RESOLVE INDETERMINACY FOR SMALL ABS(DNU)
81 C-----------------------------------------------------------------------
82 AK = 1.0E0
83 S = CC(1)
84 DO 20 K=2,8
85 AK = AK*DNU2
86 TM = CC(K)*AK
87 S = S + TM
88 IF (ABS(TM).LT.TOL) GO TO 30
89 20 CONTINUE
90 30 G1 = -S
91 GO TO 50
92 40 CONTINUE
93 G1 = (T1-T2)/(DNU+DNU)
94 50 CONTINUE
95 G2 = 0.5E0*(T1+T2)*FC
96 G1 = G1*FC
97 F = CMPLX(G1,0.0E0)*CCH + SMU*CMPLX(G2,0.0E0)
98 PT = CEXP(FMU)
99 P = CMPLX(0.5E0/T2,0.0E0)*PT
100 Q = CMPLX(0.5E0/T1,0.0E0)/PT
101 S1 = F
102 S2 = P
103 AK = 1.0E0
104 A1 = 1.0E0
105 CK = CONE
106 BK = 1.0E0 - DNU2
107 IF (INU.GT.0 .OR. N.GT.1) GO TO 80
108 C-----------------------------------------------------------------------
109 C GENERATE K(FNU,Z), 0.0D0 .LE. FNU .LT. 0.5D0 AND N=1
110 C-----------------------------------------------------------------------
111 IF (CAZ.LT.TOL) GO TO 70
112 CZ = Z*Z*CMPLX(0.25E0,0.0E0)
113 T1 = 0.25E0*CAZ*CAZ
114 60 CONTINUE
115 F = (F*CMPLX(AK,0.0E0)+P+Q)*CMPLX(1.0E0/BK,0.0E0)
116 P = P*CMPLX(1.0E0/(AK-DNU),0.0E0)
117 Q = Q*CMPLX(1.0E0/(AK+DNU),0.0E0)
118 RK = 1.0E0/AK
119 CK = CK*CZ*CMPLX(RK,0.0)
120 S1 = S1 + CK*F
121 A1 = A1*T1*RK
122 BK = BK + AK + AK + 1.0E0
123 AK = AK + 1.0E0
124 IF (A1.GT.TOL) GO TO 60
125 70 CONTINUE
126 Y(1) = S1
127 IF (KODED.EQ.1) RETURN
128 Y(1) = S1*CEXP(Z)
129 RETURN
130 C-----------------------------------------------------------------------
131 C GENERATE K(DNU,Z) AND K(DNU+1,Z) FOR FORWARD RECURRENCE
132 C-----------------------------------------------------------------------
133 80 CONTINUE
134 IF (CAZ.LT.TOL) GO TO 100
135 CZ = Z*Z*CMPLX(0.25E0,0.0E0)
136 T1 = 0.25E0*CAZ*CAZ
137 90 CONTINUE
138 F = (F*CMPLX(AK,0.0E0)+P+Q)*CMPLX(1.0E0/BK,0.0E0)
139 P = P*CMPLX(1.0E0/(AK-DNU),0.0E0)
140 Q = Q*CMPLX(1.0E0/(AK+DNU),0.0E0)
141 RK = 1.0E0/AK
142 CK = CK*CZ*CMPLX(RK,0.0E0)
143 S1 = S1 + CK*F
144 S2 = S2 + CK*(P-F*CMPLX(AK,0.0E0))
145 A1 = A1*T1*RK
146 BK = BK + AK + AK + 1.0E0
147 AK = AK + 1.0E0
148 IF (A1.GT.TOL) GO TO 90
149 100 CONTINUE
150 KFLAG = 2
151 BK = REAL(SMU)
152 A1 = FNU + 1.0E0
153 AK = A1*ABS(BK)
154 IF (AK.GT.ALIM) KFLAG = 3
155 P2 = S2*CSS(KFLAG)
156 S2 = P2*RZ
157 S1 = S1*CSS(KFLAG)
158 IF (KODED.EQ.1) GO TO 210
159 F = CEXP(Z)
160 S1 = S1*F
161 S2 = S2*F
162 GO TO 210
163 C-----------------------------------------------------------------------
164 C IFLAG=0 MEANS NO UNDERFLOW OCCURRED
165 C IFLAG=1 MEANS AN UNDERFLOW OCCURRED- COMPUTATION PROCEEDS WITH
166 C KODED=2 AND A TEST FOR ON SCALE VALUES IS MADE DURING FORWARD
167 C RECURSION
168 C-----------------------------------------------------------------------
169 110 CONTINUE
170 COEF = CMPLX(RTHPI,0.0E0)/CSQRT(Z)
171 KFLAG = 2
172 IF (KODED.EQ.2) GO TO 120
173 IF (XX.GT.ALIM) GO TO 290
174 C BLANK LINE
175 A1 = EXP(-XX)*REAL(CSS(KFLAG))
176 PT = CMPLX(A1,0.0E0)*CMPLX(COS(YY),-SIN(YY))
177 COEF = COEF*PT
178 120 CONTINUE
179 IF (ABS(DNU).EQ.0.5E0) GO TO 300
180 C-----------------------------------------------------------------------
181 C MILLER ALGORITHM FOR CABS(Z).GT.R1
182 C-----------------------------------------------------------------------
183 AK = COS(PI*DNU)
184 AK = ABS(AK)
185 IF (AK.EQ.0.0E0) GO TO 300
186 FHS = ABS(0.25E0-DNU2)
187 IF (FHS.EQ.0.0E0) GO TO 300
188 C-----------------------------------------------------------------------
189 C COMPUTE R2=F(E). IF CABS(Z).GE.R2, USE FORWARD RECURRENCE TO
190 C DETERMINE THE BACKWARD INDEX K. R2=F(E) IS A STRAIGHT LINE ON
191 C 12.LE.E.LE.60. E IS COMPUTED FROM 2**(-E)=B**(1-I1MACH(11))=
192 C TOL WHERE B IS THE BASE OF THE ARITHMETIC.
193 C-----------------------------------------------------------------------
194 T1 = FLOAT(I1MACH(11)-1)*R1MACH(5)*3.321928094E0
195 T1 = AMAX1(T1,12.0E0)
196 T1 = AMIN1(T1,60.0E0)
197 T2 = TTH*T1 - 6.0E0
198 IF (XX.NE.0.0E0) GO TO 130
199 T1 = HPI
200 GO TO 140
201 130 CONTINUE
202 T1 = ATAN(YY/XX)
203 T1 = ABS(T1)
204 140 CONTINUE
205 IF (T2.GT.CAZ) GO TO 170
206 C-----------------------------------------------------------------------
207 C FORWARD RECURRENCE LOOP WHEN CABS(Z).GE.R2
208 C-----------------------------------------------------------------------
209 ETEST = AK/(PI*CAZ*TOL)
210 FK = 1.0E0
211 IF (ETEST.LT.1.0E0) GO TO 180
212 FKS = 2.0E0
213 RK = CAZ + CAZ + 2.0E0
214 A1 = 0.0E0
215 A2 = 1.0E0
216 DO 150 I=1,KMAX
217 AK = FHS/FKS
218 BK = RK/(FK+1.0E0)
219 TM = A2
220 A2 = BK*A2 - AK*A1
221 A1 = TM
222 RK = RK + 2.0E0
223 FKS = FKS + FK + FK + 2.0E0
224 FHS = FHS + FK + FK
225 FK = FK + 1.0E0
226 TM = ABS(A2)*FK
227 IF (ETEST.LT.TM) GO TO 160
228 150 CONTINUE
229 GO TO 310
230 160 CONTINUE
231 FK = FK + SPI*T1*SQRT(T2/CAZ)
232 FHS = ABS(0.25E0-DNU2)
233 GO TO 180
234 170 CONTINUE
235 C-----------------------------------------------------------------------
236 C COMPUTE BACKWARD INDEX K FOR CABS(Z).LT.R2
237 C-----------------------------------------------------------------------
238 A2 = SQRT(CAZ)
239 AK = FPI*AK/(TOL*SQRT(A2))
240 AA = 3.0E0*T1/(1.0E0+CAZ)
241 BB = 14.7E0*T1/(28.0E0+CAZ)
242 AK = (ALOG(AK)+CAZ*COS(AA)/(1.0E0+0.008E0*CAZ))/COS(BB)
243 FK = 0.12125E0*AK*AK/CAZ + 1.5E0
244 180 CONTINUE
245 K = INT(FK)
246 C-----------------------------------------------------------------------
247 C BACKWARD RECURRENCE LOOP FOR MILLER ALGORITHM
248 C-----------------------------------------------------------------------
249 FK = FLOAT(K)
250 FKS = FK*FK
251 P1 = CZERO
252 P2 = CMPLX(TOL,0.0E0)
253 CS = P2
254 DO 190 I=1,K
255 A1 = FKS - FK
256 A2 = (FKS+FK)/(A1+FHS)
257 RK = 2.0E0/(FK+1.0E0)
258 T1 = (FK+XX)*RK
259 T2 = YY*RK
260 PT = P2
261 P2 = (P2*CMPLX(T1,T2)-P1)*CMPLX(A2,0.0E0)
262 P1 = PT
263 CS = CS + P2
264 FKS = A1 - FK + 1.0E0
265 FK = FK - 1.0E0
266 190 CONTINUE
267 C-----------------------------------------------------------------------
268 C COMPUTE (P2/CS)=(P2/CABS(CS))*(CONJG(CS)/CABS(CS)) FOR BETTER
269 C SCALING
270 C-----------------------------------------------------------------------
271 TM = CABS(CS)
272 PT = CMPLX(1.0E0/TM,0.0E0)
273 S1 = PT*P2
274 CS = CONJG(CS)*PT
275 S1 = COEF*S1*CS
276 IF (INU.GT.0 .OR. N.GT.1) GO TO 200
277 ZD = Z
278 IF(IFLAG.EQ.1) GO TO 270
279 GO TO 240
280 200 CONTINUE
281 C-----------------------------------------------------------------------
282 C COMPUTE P1/P2=(P1/CABS(P2)*CONJG(P2)/CABS(P2) FOR SCALING
283 C-----------------------------------------------------------------------
284 TM = CABS(P2)
285 PT = CMPLX(1.0E0/TM,0.0E0)
286 P1 = PT*P1
287 P2 = CONJG(P2)*PT
288 PT = P1*P2
289 S2 = S1*(CONE+(CMPLX(DNU+0.5E0,0.0E0)-PT)/Z)
290 C-----------------------------------------------------------------------
291 C FORWARD RECURSION ON THE THREE TERM RECURSION RELATION WITH
292 C SCALING NEAR EXPONENT EXTREMES ON KFLAG=1 OR KFLAG=3
293 C-----------------------------------------------------------------------
294 210 CONTINUE
295 CK = CMPLX(DNU+1.0E0,0.0E0)*RZ
296 IF (N.EQ.1) INU = INU - 1
297 IF (INU.GT.0) GO TO 220
298 IF (N.EQ.1) S1=S2
299 ZD = Z
300 IF(IFLAG.EQ.1) GO TO 270
301 GO TO 240
302 220 CONTINUE
303 INUB = 1
304 IF (IFLAG.EQ.1) GO TO 261
305 225 CONTINUE
306 P1 = CSR(KFLAG)
307 ASCLE = BRY(KFLAG)
308 DO 230 I=INUB,INU
309 ST = S2
310 S2 = CK*S2 + S1
311 S1 = ST
312 CK = CK + RZ
313 IF (KFLAG.GE.3) GO TO 230
314 P2 = S2*P1
315 P2R = REAL(P2)
316 P2I = AIMAG(P2)
317 P2R = ABS(P2R)
318 P2I = ABS(P2I)
319 P2M = AMAX1(P2R,P2I)
320 IF (P2M.LE.ASCLE) GO TO 230
321 KFLAG = KFLAG + 1
322 ASCLE = BRY(KFLAG)
323 S1 = S1*P1
324 S2 = P2
325 S1 = S1*CSS(KFLAG)
326 S2 = S2*CSS(KFLAG)
327 P1 = CSR(KFLAG)
328 230 CONTINUE
329 IF (N.EQ.1) S1 = S2
330 240 CONTINUE
331 Y(1) = S1*CSR(KFLAG)
332 IF (N.EQ.1) RETURN
333 Y(2) = S2*CSR(KFLAG)
334 IF (N.EQ.2) RETURN
335 KK = 2
336 250 CONTINUE
337 KK = KK + 1
338 IF (KK.GT.N) RETURN
339 P1 = CSR(KFLAG)
340 ASCLE = BRY(KFLAG)
341 DO 260 I=KK,N
342 P2 = S2
343 S2 = CK*S2 + S1
344 S1 = P2
345 CK = CK + RZ
346 P2 = S2*P1
347 Y(I) = P2
348 IF (KFLAG.GE.3) GO TO 260
349 P2R = REAL(P2)
350 P2I = AIMAG(P2)
351 P2R = ABS(P2R)
352 P2I = ABS(P2I)
353 P2M = AMAX1(P2R,P2I)
354 IF (P2M.LE.ASCLE) GO TO 260
355 KFLAG = KFLAG + 1
356 ASCLE = BRY(KFLAG)
357 S1 = S1*P1
358 S2 = P2
359 S1 = S1*CSS(KFLAG)
360 S2 = S2*CSS(KFLAG)
361 P1 = CSR(KFLAG)
362 260 CONTINUE
363 RETURN
364 C-----------------------------------------------------------------------
365 C IFLAG=1 CASES, FORWARD RECURRENCE ON SCALED VALUES ON UNDERFLOW
366 C-----------------------------------------------------------------------
367 261 CONTINUE
368 HELIM = 0.5E0*ELIM
369 ELM = EXP(-ELIM)
370 CELM = CMPLX(ELM,0.0)
371 ASCLE = BRY(1)
372 ZD = Z
373 XD = XX
374 YD = YY
375 IC = -1
376 J = 2
377 DO 262 I=1,INU
378 ST = S2
379 S2 = CK*S2+S1
380 S1 = ST
381 CK = CK+RZ
382 AS = CABS(S2)
383 ALAS = ALOG(AS)
384 P2R = -XD+ALAS
385 IF(P2R.LT.(-ELIM)) GO TO 263
386 P2 = -ZD+CLOG(S2)
387 P2R = REAL(P2)
388 P2I = AIMAG(P2)
389 P2M = EXP(P2R)/TOL
390 P1 = CMPLX(P2M,0.0E0)*CMPLX(COS(P2I),SIN(P2I))
391 CALL CUCHK(P1,NW,ASCLE,TOL)
392 IF(NW.NE.0) GO TO 263
393 J=3-J
394 CY(J) = P1
395 IF(IC.EQ.(I-1)) GO TO 264
396 IC = I
397 GO TO 262
398 263 CONTINUE
399 IF(ALAS.LT.HELIM) GO TO 262
400 XD = XD-ELIM
401 S1 = S1*CELM
402 S2 = S2*CELM
403 ZD = CMPLX(XD,YD)
404 262 CONTINUE
405 IF(N.EQ.1) S1 = S2
406 GO TO 270
407 264 CONTINUE
408 KFLAG = 1
409 INUB = I+1
410 S2 = CY(J)
411 J = 3 - J
412 S1 = CY(J)
413 IF(INUB.LE.INU) GO TO 225
414 IF(N.EQ.1) S1 = S2
415 GO TO 240
416 270 CONTINUE
417 Y(1) = S1
418 IF (N.EQ.1) GO TO 280
419 Y(2) = S2
420 280 CONTINUE
421 ASCLE = BRY(1)
422 CALL CKSCL(ZD, FNU, N, Y, NZ, RZ, ASCLE, TOL, ELIM)
423 INU = N - NZ
424 IF (INU.LE.0) RETURN
425 KK = NZ + 1
426 S1 = Y(KK)
427 Y(KK) = S1*CSR(1)
428 IF (INU.EQ.1) RETURN
429 KK = NZ + 2
430 S2 = Y(KK)
431 Y(KK) = S2*CSR(1)
432 IF (INU.EQ.2) RETURN
433 T2 = FNU + FLOAT(KK-1)
434 CK = CMPLX(T2,0.0E0)*RZ
435 KFLAG = 1
436 GO TO 250
437 290 CONTINUE
438 C-----------------------------------------------------------------------
439 C SCALE BY EXP(Z), IFLAG = 1 CASES
440 C-----------------------------------------------------------------------
441 KODED = 2
442 IFLAG = 1
443 KFLAG = 2
444 GO TO 120
445 C-----------------------------------------------------------------------
446 C FNU=HALF ODD INTEGER CASE, DNU=-0.5
447 C-----------------------------------------------------------------------
448 300 CONTINUE
449 S1 = COEF
450 S2 = COEF
451 GO TO 210
452 310 CONTINUE
453 NZ=-2
454 RETURN
455 END