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OpenLibm/slatec/dpsifn.f
Viral B. Shah c977aa998f Add Makefile.extras to build libopenlibm-extras. 
Replace amos with slatec
2012-12-31 16:37:05 -05:00

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Fortran
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*DECK DPSIFN
SUBROUTINE DPSIFN (X, N, KODE, M, ANS, NZ, IERR)
C***BEGIN PROLOGUE DPSIFN
C***PURPOSE Compute derivatives of the Psi function.
C***LIBRARY SLATEC
C***CATEGORY C7C
C***TYPE DOUBLE PRECISION (PSIFN-S, DPSIFN-D)
C***KEYWORDS DERIVATIVES OF THE GAMMA FUNCTION, POLYGAMMA FUNCTION,
C PSI FUNCTION
C***AUTHOR Amos, D. E., (SNLA)
C***DESCRIPTION
C
C The following definitions are used in DPSIFN:
C
C Definition 1
C PSI(X) = d/dx (ln(GAMMA(X)), the first derivative of
C the log GAMMA function.
C Definition 2
C K K
C PSI(K,X) = d /dx (PSI(X)), the K-th derivative of PSI(X).
C ___________________________________________________________________
C DPSIFN computes a sequence of SCALED derivatives of
C the PSI function; i.e. for fixed X and M it computes
C the M-member sequence
C
C ((-1)**(K+1)/GAMMA(K+1))*PSI(K,X)
C for K = N,...,N+M-1
C
C where PSI(K,X) is as defined above. For KODE=1, DPSIFN returns
C the scaled derivatives as described. KODE=2 is operative only
C when K=0 and in that case DPSIFN returns -PSI(X) + LN(X). That
C is, the logarithmic behavior for large X is removed when KODE=2
C and K=0. When sums or differences of PSI functions are computed
C the logarithmic terms can be combined analytically and computed
C separately to help retain significant digits.
C
C Note that CALL DPSIFN(X,0,1,1,ANS) results in
C ANS = -PSI(X)
C
C Input X is DOUBLE PRECISION
C X - Argument, X .gt. 0.0D0
C N - First member of the sequence, 0 .le. N .le. 100
C N=0 gives ANS(1) = -PSI(X) for KODE=1
C -PSI(X)+LN(X) for KODE=2
C KODE - Selection parameter
C KODE=1 returns scaled derivatives of the PSI
C function.
C KODE=2 returns scaled derivatives of the PSI
C function EXCEPT when N=0. In this case,
C ANS(1) = -PSI(X) + LN(X) is returned.
C M - Number of members of the sequence, M.ge.1
C
C Output ANS is DOUBLE PRECISION
C ANS - A vector of length at least M whose first M
C components contain the sequence of derivatives
C scaled according to KODE.
C NZ - Underflow flag
C NZ.eq.0, A normal return
C NZ.ne.0, Underflow, last NZ components of ANS are
C set to zero, ANS(M-K+1)=0.0, K=1,...,NZ
C IERR - Error flag
C IERR=0, A normal return, computation completed
C IERR=1, Input error, no computation
C IERR=2, Overflow, X too small or N+M-1 too
C large or both
C IERR=3, Error, N too large. Dimensioned
C array TRMR(NMAX) is not large enough for N
C
C The nominal computational accuracy is the maximum of unit
C roundoff (=D1MACH(4)) and 1.0D-18 since critical constants
C are given to only 18 digits.
C
C PSIFN is the single precision version of DPSIFN.
C
C *Long Description:
C
C The basic method of evaluation is the asymptotic expansion
C for large X.ge.XMIN followed by backward recursion on a two
C term recursion relation
C
C W(X+1) + X**(-N-1) = W(X).
C
C This is supplemented by a series
C
C SUM( (X+K)**(-N-1) , K=0,1,2,... )
C
C which converges rapidly for large N. Both XMIN and the
C number of terms of the series are calculated from the unit
C roundoff of the machine environment.
C
C***REFERENCES Handbook of Mathematical Functions, National Bureau
C of Standards Applied Mathematics Series 55, edited
C by M. Abramowitz and I. A. Stegun, equations 6.3.5,
C 6.3.18, 6.4.6, 6.4.9 and 6.4.10, pp.258-260, 1964.
C D. E. Amos, A portable Fortran subroutine for
C derivatives of the Psi function, Algorithm 610, ACM
C Transactions on Mathematical Software 9, 4 (1983),
C pp. 494-502.
C***ROUTINES CALLED D1MACH, I1MACH
C***REVISION HISTORY (YYMMDD)
C 820601 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890911 Removed unnecessary intrinsics. (WRB)
C 891006 Cosmetic changes to prologue. (WRB)
C 891006 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE DPSIFN
INTEGER I, IERR, J, K, KODE, M, MM, MX, N, NMAX, NN, NP, NX, NZ,
* FN
INTEGER I1MACH
DOUBLE PRECISION ANS, ARG, B, DEN, ELIM, EPS, FLN,
* FX, RLN, RXSQ, R1M4, R1M5, S, SLOPE, T, TA, TK, TOL, TOLS, TRM,
* TRMR, TSS, TST, TT, T1, T2, WDTOL, X, XDMLN, XDMY, XINC, XLN,
* XM, XMIN, XQ, YINT
DOUBLE PRECISION D1MACH
DIMENSION B(22), TRM(22), TRMR(100), ANS(*)
SAVE NMAX, B
DATA NMAX /100/
C-----------------------------------------------------------------------
C BERNOULLI NUMBERS
C-----------------------------------------------------------------------
DATA B(1), B(2), B(3), B(4), B(5), B(6), B(7), B(8), B(9), B(10),
* B(11), B(12), B(13), B(14), B(15), B(16), B(17), B(18), B(19),
* B(20), B(21), B(22) /1.00000000000000000D+00,
* -5.00000000000000000D-01,1.66666666666666667D-01,
* -3.33333333333333333D-02,2.38095238095238095D-02,
* -3.33333333333333333D-02,7.57575757575757576D-02,
* -2.53113553113553114D-01,1.16666666666666667D+00,
* -7.09215686274509804D+00,5.49711779448621554D+01,
* -5.29124242424242424D+02,6.19212318840579710D+03,
* -8.65802531135531136D+04,1.42551716666666667D+06,
* -2.72982310678160920D+07,6.01580873900642368D+08,
* -1.51163157670921569D+10,4.29614643061166667D+11,
* -1.37116552050883328D+13,4.88332318973593167D+14,
* -1.92965793419400681D+16/
C
C***FIRST EXECUTABLE STATEMENT DPSIFN
IERR = 0
NZ=0
IF (X.LE.0.0D0) IERR=1
IF (N.LT.0) IERR=1
IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
IF (M.LT.1) IERR=1
IF (IERR.NE.0) RETURN
MM=M
NX = MIN(-I1MACH(15),I1MACH(16))
R1M5 = D1MACH(5)
R1M4 = D1MACH(4)*0.5D0
WDTOL = MAX(R1M4,0.5D-18)
C-----------------------------------------------------------------------
C ELIM = APPROXIMATE EXPONENTIAL OVER AND UNDERFLOW LIMIT
C-----------------------------------------------------------------------
ELIM = 2.302D0*(NX*R1M5-3.0D0)
XLN = LOG(X)
41 CONTINUE
NN = N + MM - 1
FN = NN
T = (FN+1)*XLN
C-----------------------------------------------------------------------
C OVERFLOW AND UNDERFLOW TEST FOR SMALL AND LARGE X
C-----------------------------------------------------------------------
IF (ABS(T).GT.ELIM) GO TO 290
IF (X.LT.WDTOL) GO TO 260
C-----------------------------------------------------------------------
C COMPUTE XMIN AND THE NUMBER OF TERMS OF THE SERIES, FLN+1
C-----------------------------------------------------------------------
RLN = R1M5*I1MACH(14)
RLN = MIN(RLN,18.06D0)
FLN = MAX(RLN,3.0D0) - 3.0D0
YINT = 3.50D0 + 0.40D0*FLN
SLOPE = 0.21D0 + FLN*(0.0006038D0*FLN+0.008677D0)
XM = YINT + SLOPE*FN
MX = INT(XM) + 1
XMIN = MX
IF (N.EQ.0) GO TO 50
XM = -2.302D0*RLN - MIN(0.0D0,XLN)
ARG = XM/N
ARG = MIN(0.0D0,ARG)
EPS = EXP(ARG)
XM = 1.0D0 - EPS
IF (ABS(ARG).LT.1.0D-3) XM = -ARG
FLN = X*XM/EPS
XM = XMIN - X
IF (XM.GT.7.0D0 .AND. FLN.LT.15.0D0) GO TO 200
50 CONTINUE
XDMY = X
XDMLN = XLN
XINC = 0.0D0
IF (X.GE.XMIN) GO TO 60
NX = INT(X)
XINC = XMIN - NX
XDMY = X + XINC
XDMLN = LOG(XDMY)
60 CONTINUE
C-----------------------------------------------------------------------
C GENERATE W(N+MM-1,X) BY THE ASYMPTOTIC EXPANSION
C-----------------------------------------------------------------------
T = FN*XDMLN
T1 = XDMLN + XDMLN
T2 = T + XDMLN
TK = MAX(ABS(T),ABS(T1),ABS(T2))
IF (TK.GT.ELIM) GO TO 380
TSS = EXP(-T)
TT = 0.5D0/XDMY
T1 = TT
TST = WDTOL*TT
IF (NN.NE.0) T1 = TT + 1.0D0/FN
RXSQ = 1.0D0/(XDMY*XDMY)
TA = 0.5D0*RXSQ
T = (FN+1)*TA
S = T*B(3)
IF (ABS(S).LT.TST) GO TO 80
TK = 2.0D0
DO 70 K=4,22
T = T*((TK+FN+1)/(TK+1.0D0))*((TK+FN)/(TK+2.0D0))*RXSQ
TRM(K) = T*B(K)
IF (ABS(TRM(K)).LT.TST) GO TO 80
S = S + TRM(K)
TK = TK + 2.0D0
70 CONTINUE
80 CONTINUE
S = (S+T1)*TSS
IF (XINC.EQ.0.0D0) GO TO 100
C-----------------------------------------------------------------------
C BACKWARD RECUR FROM XDMY TO X
C-----------------------------------------------------------------------
NX = INT(XINC)
NP = NN + 1
IF (NX.GT.NMAX) GO TO 390
IF (NN.EQ.0) GO TO 160
XM = XINC - 1.0D0
FX = X + XM
C-----------------------------------------------------------------------
C THIS LOOP SHOULD NOT BE CHANGED. FX IS ACCURATE WHEN X IS SMALL
C-----------------------------------------------------------------------
DO 90 I=1,NX
TRMR(I) = FX**(-NP)
S = S + TRMR(I)
XM = XM - 1.0D0
FX = X + XM
90 CONTINUE
100 CONTINUE
ANS(MM) = S
IF (FN.EQ.0) GO TO 180
C-----------------------------------------------------------------------
C GENERATE LOWER DERIVATIVES, J.LT.N+MM-1
C-----------------------------------------------------------------------
IF (MM.EQ.1) RETURN
DO 150 J=2,MM
FN = FN - 1
TSS = TSS*XDMY
T1 = TT
IF (FN.NE.0) T1 = TT + 1.0D0/FN
T = (FN+1)*TA
S = T*B(3)
IF (ABS(S).LT.TST) GO TO 120
TK = 4 + FN
DO 110 K=4,22
TRM(K) = TRM(K)*(FN+1)/TK
IF (ABS(TRM(K)).LT.TST) GO TO 120
S = S + TRM(K)
TK = TK + 2.0D0
110 CONTINUE
120 CONTINUE
S = (S+T1)*TSS
IF (XINC.EQ.0.0D0) GO TO 140
IF (FN.EQ.0) GO TO 160
XM = XINC - 1.0D0
FX = X + XM
DO 130 I=1,NX
TRMR(I) = TRMR(I)*FX
S = S + TRMR(I)
XM = XM - 1.0D0
FX = X + XM
130 CONTINUE
140 CONTINUE
MX = MM - J + 1
ANS(MX) = S
IF (FN.EQ.0) GO TO 180
150 CONTINUE
RETURN
C-----------------------------------------------------------------------
C RECURSION FOR N = 0
C-----------------------------------------------------------------------
160 CONTINUE
DO 170 I=1,NX
S = S + 1.0D0/(X+NX-I)
170 CONTINUE
180 CONTINUE
IF (KODE.EQ.2) GO TO 190
ANS(1) = S - XDMLN
RETURN
190 CONTINUE
IF (XDMY.EQ.X) RETURN
XQ = XDMY/X
ANS(1) = S - LOG(XQ)
RETURN
C-----------------------------------------------------------------------
C COMPUTE BY SERIES (X+K)**(-(N+1)) , K=0,1,2,...
C-----------------------------------------------------------------------
200 CONTINUE
NN = INT(FLN) + 1
NP = N + 1
T1 = (N+1)*XLN
T = EXP(-T1)
S = T
DEN = X
DO 210 I=1,NN
DEN = DEN + 1.0D0
TRM(I) = DEN**(-NP)
S = S + TRM(I)
210 CONTINUE
ANS(1) = S
IF (N.NE.0) GO TO 220
IF (KODE.EQ.2) ANS(1) = S + XLN
220 CONTINUE
IF (MM.EQ.1) RETURN
C-----------------------------------------------------------------------
C GENERATE HIGHER DERIVATIVES, J.GT.N
C-----------------------------------------------------------------------
TOL = WDTOL/5.0D0
DO 250 J=2,MM
T = T/X
S = T
TOLS = T*TOL
DEN = X
DO 230 I=1,NN
DEN = DEN + 1.0D0
TRM(I) = TRM(I)/DEN
S = S + TRM(I)
IF (TRM(I).LT.TOLS) GO TO 240
230 CONTINUE
240 CONTINUE
ANS(J) = S
250 CONTINUE
RETURN
C-----------------------------------------------------------------------
C SMALL X.LT.UNIT ROUND OFF
C-----------------------------------------------------------------------
260 CONTINUE
ANS(1) = X**(-N-1)
IF (MM.EQ.1) GO TO 280
K = 1
DO 270 I=2,MM
ANS(K+1) = ANS(K)/X
K = K + 1
270 CONTINUE
280 CONTINUE
IF (N.NE.0) RETURN
IF (KODE.EQ.2) ANS(1) = ANS(1) + XLN
RETURN
290 CONTINUE
IF (T.GT.0.0D0) GO TO 380
NZ=0
IERR=2
RETURN
380 CONTINUE
NZ=NZ+1
ANS(MM)=0.0D0
MM=MM-1
IF (MM.EQ.0) RETURN
GO TO 41
390 CONTINUE
NZ=0
IERR=3
RETURN
END
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