*DECK CQCBK SUBROUTINE CQCBK (LUN, KPRINT, IPASS) C***BEGIN PROLOGUE CQCBK C***SUBSIDIARY C***PURPOSE Quick check for SLATEC subroutine C CBESK C***LIBRARY SLATEC C***CATEGORY C10B4 C***TYPE COMPLEX (CQCBK-C, ZQCBK-Z) C***KEYWORDS QUICK CHECK, CBESK 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 CQCBK (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 CQCBK is a quick check routine for the complex K Bessel function C generated by subroutine CBESK. C C CQCBK generates sequences of I and K Bessel functions from C CBESI and CBESK 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 CBESI, CBESK, I1MACH, R1MACH C***REVISION HISTORY (YYMMDD) C 830501 DATE WRITTEN C 890831 Revised to meet new SLATEC standard C C***END PROLOGUE CQCBK C C*Internal Notes: C Machine constants are defined by functions I1MACH and R1MACH. 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 REAL R1MACH EXTERNAL I1MACH, R1MACH C C Declare local variables. C COMPLEX CONE, CSGN, CV, CW, CY, W, Y, Z, ZN REAL 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, T, TOL, TS, XNU, XX 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), W(20), XNU(20), * Y(20) C C***FIRST EXECUTABLE STATEMENT CQCBK IF (KPRINT.GE.2) THEN WRITE (LUN,99999) 99999 FORMAT (' QUICK CHECK ROUTINE FOR THE K BESSEL FUNCTION FROM ', * 'CBESK'/) ENDIF 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----------------------------------------------------------------------- R1M4 = R1MACH(4) TOL = MAX(R1M4,1.0E-18) ATOL = 100.0E0*TOL AA = -LOG10(R1M4) K1 = I1MACH(12) K2 = I1MACH(13) R1M5 = R1MACH(5) K = MIN(ABS(K1),ABS(K2)) ELIM = 2.303E0*(K*R1M5-3.0E0) AB = AA*2.303E0 ALIM = ELIM + MAX(-AB,-41.45E0) DIG = MIN(AA,18.0E0) FNUL = 10.0E0 + 6.0E0*(DIG-3.0E0) RL = 1.2E0*DIG + 3.0E0 SLAK = 3.0E0+4.0E0*(-LOG10(TOL)-7.0E0)/11.0E0 SLAK = MAX(SLAK,3.0E0) ERTOL = TOL*10.0E0**SLAK RM = 0.5E0*(ALIM + ELIM) RM = MIN(RM,200.0E0) RM = MAX(RM,RL+10.0E0) 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,6E12.4/) ENDIF C----------------------------------------------------------------------- C Set other constants needed in the tests. C----------------------------------------------------------------------- CONE = (1.0E0,0.0E0) HPI = 2.0E0*ATAN(1.0E0) 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) = 0.0E0 XNU(2) = 1.0E0 XNU(3) = 2.0E0 XNU(4) = 0.5E0*FNUL XNU(5) = FNUL + 1.1E0 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) = 0.0E0 XNU(2) = 0.6E0 XNU(3) = 1.3E0 XNU(4) = 2.0E0 XNU(5) = 0.5E0*FNUL XNU(6) = FNUL + 1.1E0 ENDIF I = 2 EPS = 0.01E0 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 CSGN = CMPLX(COS(ARG),SIN(ARG)) IF (MOD(IFNU,2).EQ.1) THEN CSGN = -CSGN 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)+2.0E0*(IR-1))/3.0E0 GO TO 80 60 CONTINUE R = (2.0E0*(4-IR)+R2*(IR-1))/3.0E0 GO TO 80 70 CONTINUE IF (R2.GE.RM) GO TO 170 R = (R2*(4-IR)+RM*(IR-1))/3.0E0 80 CONTINUE C-------------- end switch DO 150 IT = 1,ITL CT = COS(T(IT)) ST = SIN(T(IT)) IF (ABS(CT).LT.ATOL) CT = 0.0E0 IF (ABS(ST).LT.ATOL) ST = 0.0E0 Z = CMPLX(R*CT,R*ST) CALL CBESI(Z, FNU, KODE, N1, W, 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.0.0E0) THEN C------------------ Z is in the right half plane CALL CBESK(Z, FNU, KODE, N1, Y, NZ1, IERR) CV = CONE/Z IF (KODE.EQ.2) THEN C-------------------- Adjust Wronskian due to scaled I and K functions XX = REAL(Z) AXX = ABS(XX) ZN = CMPLX(-AXX,0.0E0) CV = ZN + Z CV = EXP(CV)/Z ENDIF ELSE C------------------ Z is in the left half plane ZN = -Z CALL CBESK(ZN, FNU, KODE, N1, Y, NZ1, IERR) ZN = CSGN C------------------ CSGN set near top of DO 180 loop IF (ST.GT.0.0E0 .OR. (ST.EQ.0.0E0.AND.CT.LT.0.0E0)) * ZN = CONJG(ZN) DO 90 KK = 1,N1 Y(KK) = Y(KK)*ZN ZN = -ZN 90 CONTINUE CV = CONE/Z IF (KODE.EQ.2) THEN C-------------------- Adjust Wronskian due to scaled I and K functions XX = REAL(Z) AXX = ABS(XX) ZN = CMPLX(-AXX,0.0E0) CV = ZN - Z CV = EXP(CV)/Z ENDIF ENDIF MFLG = 0 DO 130 I = 1,N CW = W(I)*Y(I+1) CY = W(I+1)*Y(I) CY = CY + CW - CV ER = ABS(CY)/ABS(CV) 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 = ', E12.4/) WRITE (LUN,99994) 99994 FORMAT (' INPUT TO CBESK Z, FNU, KODE, N') ENDIF IF (KPRINT.GE.3) THEN WRITE (LUN,99993) 99993 FORMAT (' ERROR TEST ON THE WRONSKIAN OF ', * 'CBESI(Z,FNU) AND CBESK(Z,FNU)') WRITE (LUN,99992) 99992 FORMAT (' RESULTS FROM CBESK NZ1, Y(KK)'/, * ' RESULTS FROM CBESI 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) Z, FNU, KODE, N 99990 FORMAT (' INPUT: Z=',2E12.4,4X,'FNU=',E12.4, * 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)=',4E12.4) KK = MAX(NZ1,NZ2) + 1 KK = MIN(N,KK) WRITE (LUN,99988) NZ1, Y(KK), NZ2, W(KK) 99988 FORMAT (' RESULTS: NZ1=',I3,4X,'Y(KK)=',2E12.4, * /11X,'NZ2=',I3,4X,'W(KK)=',2E12.4) 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 CBESK 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 (/' ****** CBESK PASSED ALL TESTS ******'/) ENDIF IF (IPASS.EQ.0.AND.KPRINT.GE.1) THEN WRITE (LUN,99983) 99983 FORMAT (/' ****** CBESK FAILED SOME TESTS ******'/) ENDIF RETURN END