*DECK ZQCBK SUBROUTINE ZQCBK (LUN, KPRINT, IPASS) C***BEGIN PROLOGUE ZQCBK C***SUBSIDIARY C***PURPOSE Quick check for SLATEC subroutine C ZBESK C***LIBRARY SLATEC C***CATEGORY C10B4 C***TYPE COMPLEX (CQCBK-C, ZQCBK-Z) C***KEYWORDS QUICK CHECK, ZBESK C***AUTHOR Amos, Don, (SNL) C Goudy, Sue, (SNL) C Walton, Lee, (SNL) C***DESCRIPTION C C *Usage: C C INTEGER LUN, KPRINT, IPASS C C CALL ZQCBK (LUN, KPRINT, IPASS) C C *Arguments: C C LUN :IN is the unit number to which output is to be written. C C KPRINT :IN controls the amount of output, as specified in the C SLATEC Guidelines. C C IPASS :OUT indicates whether the test passed or failed. C A value of one is good, indicating no failures. C C *Description: C C *** A DOUBLE PRECISION ROUTINE *** C C ZQCBK is a quick check routine for the complex K Bessel function C generated by subroutine ZBESK. C C ZQCBK generates sequences of I and K Bessel functions from C ZBESI and ZBESK and checks them against the Wronskian evaluation C in the (Z,FNU) space. C C***REFERENCES Abramowitz, M. and Stegun, I. A., Handbook C of Mathematical Functions, Dover Publications, C New York, 1964. C Amos, D. E., A Subroutine Package for Bessel C Functions of a Complex Argument and Nonnegative C Order, SAND85-1018, May, 1985. C***ROUTINES CALLED ZBESI, ZBESK, ZABS, ZDIV, ZEXP, I1MACH, D1MACH C***REVISION HISTORY (YYMMDD) C 830501 DATE WRITTEN C 890831 Revised to meet new SLATEC standard C 930122 Added ZEXP to EXTERNAL Statement. (RWC) C***END PROLOGUE ZQCBK C C*Internal Notes: C Machine constants are defined by functions I1MACH and D1MACH. C C The parameter MQC can have values 1 (the default) for a faster, C less definitive test or 2 for a slower, more definitive test. C C**End C C Set test complexity parameter. C INTEGER MQC PARAMETER (MQC=1) C C Declare arguments. C INTEGER LUN, KPRINT, IPASS C C Declare external functions. C INTEGER I1MACH DOUBLE PRECISION D1MACH, ZABS EXTERNAL I1MACH, D1MACH, ZABS, ZEXP C C Declare local variables. C DOUBLE PRECISION CONER,CONEI, CSGNR,CSGNI, CVR,CVI, CWR,CWI, * CYR,CYI, WR,WI, YR,YI, ZR,ZI, ZNR,ZNI DOUBLE PRECISION AA, AB, AER, ALIM, ARG, ATOL, AXX, CT, DIG, * ELIM, EPS, ER, ERTOL, FFNU, FILM, FNU, FNUL, HPI, PI, R, RL, * RM, R1M4, R1M5, R2, SLAK, ST, STI, STR, T, TOL, TS, XNU INTEGER I, ICASE, IERR, IFNU, IL, IR, IRB, IT, ITL, K, KDO, KEPS, * KK, KODE, K1, K2, LFLG, MFLG, N, NL, NU, NUL, NZ1, NZ2, N1 DIMENSION AER(20), KDO(20), KEPS(20), T(20), WR(20), WI(20), * XNU(20), YR(20), YI(20) C C***FIRST EXECUTABLE STATEMENT ZQCBK IF (KPRINT.GE.2) THEN WRITE (LUN,99999) 99999 FORMAT (' QUICK CHECK ROUTINE FOR THE K BESSEL FUNCTION FROM ', * 'ZBESK'/) ENDIF C----------------------------------------------------------------------- C Set parameters related to machine constants. C TOL is the approximate unit roundoff limited to 1.0D-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----------------------------------------------------------------------- R1M4 = D1MACH(4) TOL = MAX(R1M4,1.0D-18) ATOL = 100.0D0*TOL AA = -LOG10(R1M4) K1 = I1MACH(12) K2 = I1MACH(13) R1M5 = D1MACH(5) K = MIN(ABS(K1),ABS(K2)) ELIM = 2.303D0*(K*R1M5-3.0D0) AB = AA*2.303D0 ALIM = ELIM + MAX(-AB,-41.45D0) DIG = MIN(AA,18.0D0) FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0) RL = 1.2D0*DIG + 3.0D0 SLAK = 3.0D0+4.0D0*(-LOG10(TOL)-7.0D0)/11.0D0 SLAK = MAX(SLAK,3.0D0) ERTOL = TOL*10.0D0**SLAK RM = 0.5D0*(ALIM + ELIM) RM = MIN(RM,200.0D0) RM = MAX(RM,RL+10.0D0) R2 = MIN(RM,FNUL) IF (KPRINT.GE.2) THEN WRITE (LUN,99998) 99998 FORMAT (' PARAMETERS'/ * 5X,'TOL ',8X,'ELIM',8X,'ALIM',8X,'RL ',8X,'FNUL',8X,'DIG') WRITE (LUN,99997) TOL, ELIM, ALIM, RL, FNUL, DIG 99997 FORMAT (1X,6D12.4/) ENDIF C----------------------------------------------------------------------- C Set other constants needed in the tests. C----------------------------------------------------------------------- CONER = 1.0D0 CONEI = 0.0D0 HPI = 2.0D0*ATAN(1.0D0) PI = HPI + HPI C----------------------------------------------------------------------- C Generate angles for construction of complex Z to be used in tests. C----------------------------------------------------------------------- C KDO(K), K = 1,IL determines which of the IL angles in -PI to PI C are used to compute values of Z. C KDO(K) = 0 means that the index K will be used for one or two C values of Z, depending on the choice of KEPS(K) C = 1 means that the index K and the corresponding angle C will be skipped C KEPS(K), K = 1,IL determines which of the angles get incremented C up and down to put values of Z in regions where different C formulae are used. C KEPS(K) = 0 means that the angle will be used without change C = 1 means that the angle will be incremented up and C down by EPS C The angles to be used are stored in the T(I) array, I = 1,ITL. C----------------------------------------------------------------------- IF (MQC.NE.2) THEN NL = 2 IL = 5 DO 5 I = 1,IL KEPS(I) = 0 KDO(I) = 0 5 CONTINUE NUL = 5 XNU(1) = 0.0D0 XNU(2) = 1.0D0 XNU(3) = 2.0D0 XNU(4) = 0.5D0*FNUL XNU(5) = FNUL + 1.1D0 ELSE NL = 4 IL = 13 DO 6 I = 1,IL KDO(I) = 0 KEPS(I) = 0 6 CONTINUE KDO(2) = 1 KDO(6) = 1 KDO(8) = 1 KDO(12) = 1 KEPS(3) = 1 KEPS(4) = 1 KEPS(5) = 1 KEPS(9) = 1 KEPS(10) = 1 KEPS(11) = 1 NUL = 6 XNU(1) = 0.0D0 XNU(2) = 0.6D0 XNU(3) = 1.3D0 XNU(4) = 2.0D0 XNU(5) = 0.5D0*FNUL XNU(6) = FNUL + 1.1D0 ENDIF I = 2 EPS = 0.01D0 FILM = IL - 1 T(1) = -PI + EPS DO 30 K = 2,IL IF (KDO(K).EQ.0) THEN T(I) = PI*(-IL+2*K-1)/FILM IF (KEPS(K).NE.0) THEN TS = T(I) T(I) = TS - EPS I = I + 1 T(I) = TS + EPS ELSE I = I + 1 ENDIF ENDIF 30 CONTINUE ITL = I - 1 C----------------------------------------------------------------------- C Test values of Z IN -PI.lt.arg(Z).le.PI near formula boundaries. C----------------------------------------------------------------------- IF (KPRINT.GE.2) THEN WRITE (LUN,99996) 99996 FORMAT (' CHECKS IN THE (Z,FNU) SPACE') ENDIF LFLG = 0 DO 200 KODE = 1,2 DO 190 N = 1,NL N1 = N + 1 DO 180 NU = 1,NUL FNU = XNU(NU) IFNU = INT(FNU) FFNU = FNU - IFNU ARG = PI*FFNU CSGNR = COS(ARG) CSGNI = SIN(ARG) IF (MOD(IFNU,2).EQ.1) THEN CSGNR = -CSGNR CSGNI = -CSGNI ENDIF DO 170 ICASE = 1,3 IRB = MIN(2,ICASE) DO 160 IR = IRB,4 C-------------- switch (icase) GO TO (50, 60, 70), ICASE 50 CONTINUE R = (EPS*(4-IR)+2.0D0*(IR-1))/3.0D0 GO TO 80 60 CONTINUE R = (2.0D0*(4-IR)+R2*(IR-1))/3.0D0 GO TO 80 70 CONTINUE IF (R2.GE.RM) GO TO 170 R = (R2*(4-IR)+RM*(IR-1))/3.0D0 80 CONTINUE C-------------- end switch DO 150 IT = 1,ITL CT = COS(T(IT)) ST = SIN(T(IT)) IF (ABS(CT).LT.ATOL) CT = 0.0D0 IF (ABS(ST).LT.ATOL) ST = 0.0D0 ZR = R*CT ZI = R*ST CALL ZBESI(ZR, ZI, FNU, KODE, N1, WR, WI, NZ2, IERR) C---------------- Underflow? - skip test for this case. IF (NZ2.NE.0) GO TO 150 C----------------------------------------------------------------------- C In the left half plane, the analytic continuation formula for K C introduces an I function. The dominant terms in the Wronskian C I(FNU,Z)*I(FNU+1,Z) cancel out, giving losses of significance. C This cancellation can be done analytically to give a Wronskian in C terms of I in the left half plane and K in the right half plane. C----------------------------------------------------------------------- IF (ICASE.EQ.1.OR.CT.GE.0.0D0) THEN C------------------ Z is in the right half plane CALL ZBESK(ZR, ZI, FNU, KODE, N1, YR, YI, NZ1, IERR) CALL ZDIV(CONER, CONEI, ZR, ZI, CVR, CVI) IF (KODE.EQ.2) THEN C-------------------- Adjust Wronskian due to scaled I and K functions AXX = ABS(ZR) ZNR = -AXX ZNI = 0.0D0 CVR = ZNR + ZR CVI = ZNI + ZI CALL ZEXP(CVR, CVI, STR, STI) CALL ZDIV(STR, STI, ZR, ZI, CVR, CVI) ENDIF ELSE C------------------ Z is in the left half plane ZNR = -ZR ZNI = -ZI CALL ZBESK(ZNR, ZNI, FNU, KODE, N1, YR, YI, NZ1, * IERR) ZNR = CSGNR ZNI = CSGNI C------------------ CSGNR and CSGNI set near top of DO 180 loop IF (ST.GT.0.0D0 .OR. (ST.EQ.0.0D0.AND.CT.LT.0.0D0)) * ZNI = -ZNI DO 90 KK = 1,N1 STR = YR(KK)*ZNR - YI(KK)*ZNI YI(KK) = YR(KK)*ZNI + YI(KK)*ZNR YR(KK) = STR ZNR = -ZNR ZNI = -ZNI 90 CONTINUE CALL ZDIV(CONER, CONEI, ZR, ZI, CVR, CVI) IF (KODE.EQ.2) THEN C-------------------- Adjust Wronskian due to scaled I and K functions AXX = ABS(ZR) ZNR = -AXX ZNI = 0.0D0 CVR = ZNR - ZR CVI = ZNI - ZI CALL ZEXP(CVR, CVI, STR, STI) CALL ZDIV(STR, STI, ZR, ZI, CVR, CVI) ENDIF ENDIF MFLG = 0 DO 130 I = 1,N CWR = WR(I)*YR(I+1) - WI(I)*YI(I+1) CWI = WR(I)*YI(I+1) + WI(I)*YR(I+1) CYR = WR(I+1)*YR(I) - WI(I+1)*YI(I) CYI = WR(I+1)*YI(I) + WI(I+1)*YR(I) CYR = CYR + CWR - CVR CYI = CYI + CWI - CVI ER = ZABS(CYR,CYI)/ZABS(CVR,CVI) AER(I) = ER IF (ER.GT.ERTOL) THEN MFLG = 1 ENDIF 130 CONTINUE IF (MFLG.NE.0) THEN IF (LFLG.EQ.0) THEN IF (KPRINT.GE.2) THEN WRITE (LUN,99995) ERTOL 99995 FORMAT (/' CASES WHICH OR VIOLATE THE RELATIVE', * ' ERROR TEST WITH ERTOL = ',D12.4/) WRITE (LUN,99994) 99994 FORMAT (' INPUT TO ZBESK Z, FNU, KODE, N') ENDIF IF (KPRINT.GE.3) THEN WRITE (LUN,99993) 99993 FORMAT (' ERROR TEST ON THE WRONSKIAN OF ', * 'ZBESI(Z,FNU) AND ZBESK(Z,FNU)') WRITE (LUN,99992) 99992 FORMAT (' RESULTS FROM ZBESK NZ1, Y(KK)'/, * ' RESULTS FROM ZBESI NZ2, W(KK)') WRITE (LUN,99991) 99991 FORMAT (' TEST CASE INDICES IT, IR, ICASE'/) ENDIF ENDIF LFLG = LFLG + 1 IF (KPRINT.GE.2) THEN WRITE (LUN,99990) ZR, ZI, FNU, KODE, N 99990 FORMAT (' INPUT: Z=',2D12.4,4X,'FNU=',D12.4, * 4X,'KODE=',I3,4X,'N=',I3) ENDIF IF (KPRINT.GE.3) THEN WRITE (LUN,99989) (AER(K),K=1,N) 99989 FORMAT (' ERROR: AER(K)=',4D12.4) KK = MAX(NZ1,NZ2) + 1 KK = MIN(N,KK) WRITE (LUN,99988) NZ1, YR(KK), YI(KK), * NZ2, WR(KK), WI(KK) 99988 FORMAT (' RESULTS: NZ1=',I3,4X,'Y(KK)=',2D12.4, * /11X,'NZ2=',I3,4X,'W(KK)=',2D12.4) WRITE (LUN,99987) IT, IR, ICASE 99987 FORMAT (' CASE: IT=',I3,4X,'IR=',I3,4X, * 'ICASE=',I3/) ENDIF ENDIF 150 CONTINUE 160 CONTINUE 170 CONTINUE 180 CONTINUE 190 CONTINUE 200 CONTINUE IF (KPRINT.GE.2) THEN IF (LFLG.EQ.0) THEN WRITE (LUN,99986) 99986 FORMAT (' QUICK CHECKS OK') ELSE WRITE (LUN,99985) LFLG 99985 FORMAT (' ***',I5,' FAILURE(S) FOR ZBESK NEAR FORMULA ', * 'BOUNDARIES') ENDIF ENDIF IPASS = 0 IF (LFLG.EQ.0) THEN IPASS = 1 ENDIF IF (IPASS.EQ.1.AND.KPRINT.GE.2) THEN WRITE (LUN,99984) 99984 FORMAT (/' ****** ZBESK PASSED ALL TESTS ******'/) ENDIF IF (IPASS.EQ.0.AND.KPRINT.GE.1) THEN WRITE (LUN,99983) 99983 FORMAT (/' ****** ZBESK FAILED SOME TESTS ******'/) ENDIF RETURN END