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cosolver/FM/FMsamples/FprimeZM.f95
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FprimeZM.f95
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FUNCTION ZM_FPRIME(N,A,F,NF)
USE FMVALS
USE FMZM
IMPLICIT NONE
! This routine finds the Nth derivative of F(X,NF), evaluated at A.
! NF is passed on to function F to indicate which function to use in cases where several
! different functions may be defined there.
! F must be defined in an interval containing A, so that F can be sampled on both sides of A.
! N may be zero, so that in cases where F suffers cancellation error at A, an accurate
! function value is returned.
! ZM_FPRIME tries to return full accuracy for the derivative, by raising precision above
! the user's level and using difference formulas.
TYPE (ZM) :: A, ZM_FPRIME
TYPE (ZM), EXTERNAL :: F
INTEGER :: J, K, KWARN_SAVE, NDSAVE, N, NF
TYPE (ZM), SAVE :: D1, D2, F1, F2, X1
TYPE (FM), SAVE :: ERR, H, TOL, TOL2
CALL FM_ENTER_USER_FUNCTION(ZM_FPRIME)
! Raise precision slightly.
NDSAVE = NDIG
NDIG = NDIG + NGRD52
CALL ZM_EQU(A,X1,NDSAVE,NDIG)
KWARN_SAVE = KWARN
KWARN = 0
ERR = 1
D2 = 0
F1 = F(X1,NF)
IF (F1 /= 0) THEN
CALL FM_ULP(ABS(F1),TOL)
ELSE
TOL = EPSILON(TO_FM(1))
ENDIF
TOL = ABS(TOL)
! Check for a legal function value.
IF (IS_UNKNOWN(ABS(F1)) .OR. IS_OVERFLOW(ABS(F1)) .OR. IS_UNDERFLOW(ABS(F1)) .OR. N < 0) THEN
ZM_FPRIME = TO_ZM(' UNKNOWN + UNKNOWN i ')
GO TO 110
ENDIF
F2 = F1
! Loop at increasing precision until the difference formula is accurate.
DO J = 1, 100
NDIG = 2*NDIG
! Define the variables used below at the new higher precision.
CALL ZM_EQU(D2,D1,NDIG/2,NDIG)
CALL ZM_EQU(F2,F1,NDIG/2,NDIG)
CALL FM_EQU(TOL,TOL2,NDSAVE,NDIG)
CALL ZM_EQU(A,X1,NDSAVE,NDIG)
! Special case for N = 0.
IF (N == 0) THEN
F2 = F(X1,NF)
D2 = F2
IF (ABS(F2-F1) < TOL2) GO TO 110
CYCLE
ENDIF
F2 = F1
! Special case for N = 1.
IF (N == 1) THEN
IF (X1 /= 0) THEN
CALL FM_ULP(ABS(X1),H)
ELSE
H = EPSILON(TO_FM(1))
ENDIF
H = SQRT(ABS(H))
D2 = ( F(X1+H,NF) - F(X1-H,NF) ) / (2*H)
IF (ABS(D2-D1) < TOL2 .AND. J > 1) GO TO 110
CYCLE
ENDIF
! General case for even N > 1.
IF (MOD(N,2) == 0) THEN
IF (X1 /= 0) THEN
CALL FM_ULP(ABS(X1),H)
ELSE
H = EPSILON(TO_FM(1))
ENDIF
H = ABS(H)**(TO_FM(1)/(N+2))
D2 = (-1)**(N/2) * BINOMIAL(TO_FM(N),TO_FM(N/2)) * F(X1,NF)
DO K = 0, N/2-1
D2 = D2 + (-1)**K * BINOMIAL(TO_FM(N),TO_FM(K)) * &
( F(X1+(N/2-K)*H,NF) + F(X1-(N/2-K)*H,NF) )
ENDDO
D2 = D2 / H**N
IF (ABS(D2-D1) < TOL2 .AND. J > 1) GO TO 110
CYCLE
ENDIF
! General case for odd N > 1.
IF (MOD(N,2) == 1) THEN
IF (X1 /= 0) THEN
CALL FM_ULP(ABS(X1),H)
ELSE
H = EPSILON(TO_FM(1))
ENDIF
H = ABS(H)**(TO_FM(1)/(N+2))
D2 = 0
DO K = 0, N/2
D2 = D2 + (-1)**K * BINOMIAL(TO_FM(N-1),TO_FM(K)) * &
( F(X1+(N/2-K+1)*H,NF) - F(X1-(N/2-K+1)*H,NF) ) * &
TO_FM(N*(N+1-2*K)) / ((N-K)*(N+1-K))
ENDDO
D2 = D2 / (2*H**N)
IF (ABS(D2-D1) < TOL2 .AND. J > 1) GO TO 110
CYCLE
ENDIF
ENDDO
! Round and return.
110 CALL ZM_EQU(D2,ZM_FPRIME,NDIG,NDSAVE)
NDIG = NDSAVE
KWARN = KWARN_SAVE
CALL FM_EXIT_USER_FUNCTION(ZM_FPRIME)
END FUNCTION ZM_FPRIME
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