*DECK PSI FUNCTION PSI (X) C***BEGIN PROLOGUE PSI C***PURPOSE Compute the Psi (or Digamma) function. C***LIBRARY SLATEC (FNLIB) C***CATEGORY C7C C***TYPE SINGLE PRECISION (PSI-S, DPSI-D, CPSI-C) C***KEYWORDS DIGAMMA FUNCTION, FNLIB, PSI FUNCTION, SPECIAL FUNCTIONS C***AUTHOR Fullerton, W., (LANL) C***DESCRIPTION C C PSI(X) calculates the psi (or digamma) function for real argument X. C PSI(X) is the logarithmic derivative of the gamma function of X. C C Series for PSI on the interval 0. to 1.00000D+00 C with weighted error 2.03E-17 C log weighted error 16.69 C significant figures required 16.39 C decimal places required 17.37 C C Series for APSI on the interval 0. to 2.50000D-01 C with weighted error 5.54E-17 C log weighted error 16.26 C significant figures required 14.42 C decimal places required 16.86 C C***REFERENCES (NONE) C***ROUTINES CALLED COT, CSEVL, INITS, R1MACH, XERMSG C***REVISION HISTORY (YYMMDD) C 770401 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C 900727 Added EXTERNAL statement. (WRB) C 920618 Removed space from variable names. (RWC, WRB) C***END PROLOGUE PSI DIMENSION PSICS(23), APSICS(16) LOGICAL FIRST EXTERNAL COT SAVE PSICS, APSICS, PI, NTPSI, NTAPSI, XBIG, DXREL, FIRST DATA PSICS( 1) / -.0380570808 35217922E0 / DATA PSICS( 2) / .4914153930 2938713E0 / DATA PSICS( 3) / -.0568157478 21244730E0 / DATA PSICS( 4) / .0083578212 25914313E0 / DATA PSICS( 5) / -.0013332328 57994342E0 / DATA PSICS( 6) / .0002203132 87069308E0 / DATA PSICS( 7) / -.0000370402 38178456E0 / DATA PSICS( 8) / .0000062837 93654854E0 / DATA PSICS( 9) / -.0000010712 63908506E0 / DATA PSICS(10) / .0000001831 28394654E0 / DATA PSICS(11) / -.0000000313 53509361E0 / DATA PSICS(12) / .0000000053 72808776E0 / DATA PSICS(13) / -.0000000009 21168141E0 / DATA PSICS(14) / .0000000001 57981265E0 / DATA PSICS(15) / -.0000000000 27098646E0 / DATA PSICS(16) / .0000000000 04648722E0 / DATA PSICS(17) / -.0000000000 00797527E0 / DATA PSICS(18) / .0000000000 00136827E0 / DATA PSICS(19) / -.0000000000 00023475E0 / DATA PSICS(20) / .0000000000 00004027E0 / DATA PSICS(21) / -.0000000000 00000691E0 / DATA PSICS(22) / .0000000000 00000118E0 / DATA PSICS(23) / -.0000000000 00000020E0 / DATA APSICS( 1) / -.0204749044 678185E0 / DATA APSICS( 2) / -.0101801271 534859E0 / DATA APSICS( 3) / .0000559718 725387E0 / DATA APSICS( 4) / -.0000012917 176570E0 / DATA APSICS( 5) / .0000000572 858606E0 / DATA APSICS( 6) / -.0000000038 213539E0 / DATA APSICS( 7) / .0000000003 397434E0 / DATA APSICS( 8) / -.0000000000 374838E0 / DATA APSICS( 9) / .0000000000 048990E0 / DATA APSICS(10) / -.0000000000 007344E0 / DATA APSICS(11) / .0000000000 001233E0 / DATA APSICS(12) / -.0000000000 000228E0 / DATA APSICS(13) / .0000000000 000045E0 / DATA APSICS(14) / -.0000000000 000009E0 / DATA APSICS(15) / .0000000000 000002E0 / DATA APSICS(16) / -.0000000000 000000E0 / DATA PI / 3.1415926535 8979324E0/ DATA FIRST /.TRUE./ C***FIRST EXECUTABLE STATEMENT PSI IF (FIRST) THEN NTPSI = INITS (PSICS, 23, 0.1*R1MACH(3)) NTAPSI = INITS (APSICS, 16, 0.1*R1MACH(3)) C XBIG = 1.0/SQRT(R1MACH(3)) DXREL = SQRT (R1MACH(4)) ENDIF FIRST = .FALSE. C Y = ABS(X) IF (Y.GE.2.0) GO TO 30 C C PSI(X) FOR -2. .LT. X .LT. 2. C N = X IF (X.LT.0.) N = N - 1 Y = X - N N = N - 1 PSI = CSEVL (2.*Y-1., PSICS, NTPSI) IF (N.EQ.0) RETURN C N = -N IF (X .EQ. 0.) CALL XERMSG ('SLATEC', 'PSI', 'X IS 0', 2, 2) IF (X .LT. 0. .AND. X+N-2 .EQ. 0.) CALL XERMSG ('SLATEC', 'PSI', + 'X IS A NEGATIVE INTEGER', 3, 2) IF (X .LT. (-0.5) .AND. ABS((X-AINT(X-0.5))/X) .LT. DXREL) + CALL XERMSG ('SLATEC', 'PSI', + 'ANSWER LT HALF PRECISION BECAUSE X TOO NEAR NEGATIVE INTEGER', + 1, 1) C DO 20 I=1,N PSI = PSI - 1.0/(X+I-1) 20 CONTINUE RETURN C C PSI(X) FOR ABS(X) .GE. 2. C 30 AUX = 0. IF (Y.LT.XBIG) AUX = CSEVL (8./Y**2-1., APSICS, NTAPSI) IF (X.LT.0.) PSI = LOG(ABS(X)) - 0.5/X + AUX - PI*COT(PI*X) IF (X.GT.0.) PSI = LOG(X) - 0.5/X + AUX RETURN C END