DERIV
This function computes the first, second or third derivative of a user-supplied function.
Function Return Value
DERIV — Estimate of the first (KORDER = 1), second (KORDER = 2) or third (KORDER = 3) derivative of FCN at X. (Output)
Required Arguments
FCN — User-supplied FUNCTION whose derivative at X will be computed. The form is FCN(X), where
X – Independent variable. (Input)
FCN – The function value. (Output)
FCN must be declared EXTERNAL in the calling program.
X — Point at which the derivative is to be evaluated. (Input)
Optional Arguments
KORDER — Order of the derivative desired (1, 2 or 3). (Input)
Default: KORDER = 1.
BGSTEP — Beginning value used to compute the size of the interval used in computing the derivative. (Input)
The interval used is the closed interval (X − 4 * BGSTEP, X + 4 * BGSTEP). BGSTEP must be positive.
Default: BGSTEP = .01.
TOL — Relative error desired in the derivative estimate. (Input)
Default: TOL = 1.e-2 for single precision and 1.d-4 for double precision.
FORTRAN 90 Interface
Generic: DERIV (FCN, X [, …])
Specific: The specific interface names are S_DERIV and D_DERIV.
FORTRAN 77 Interface
Single: DERIV (FCN, KORDER, X, BGSTEP, TOL)
Double: The double precision function name is DDERIV.
Description
DERIV produces an estimate to the first, second, or third derivative of a function. The estimate originates from first computing a spline interpolant to the input function using values within the interval (X − 4.0 * BGSTEP, X + 4.0 * BGSTEP), then differentiating the spline at X.
Comments
1. Informational errors
|
Type |
Code |
Description |
|
3 |
2 |
Roundoff error became dominant before estimates converged. Increase precision and/or increase BGSTEP. |
|
4 |
1 |
Unable to achieve desired tolerance in derivative estimation. Increase precision, increase TOL and/or change BGSTEP. If this error continues, the function may not have a derivative at X. |
2. Convergence is assumed when
for two successive derivative estimates D1 and D2.
3. The initial step size, BGSTEP, must be chosen small enough that FCN is defined and reasonably smooth in the interval (X − 4 * BGSTEP, X + 4 * BGSTEP), yet large enough to avoid roundoff problems.
Examples
In this example, we obtain the approximate first derivative of the function
f(x) = −2 sin(3x/2)
at the point x = 2.
USE DERIV_INT
USE UMACH_INT
IMPLICIT NONE
INTEGER KORDER, NCOUNT, NOUT
REAL BGSTEP, DERV, TOL, X
EXTERNAL FCN
! Get output unit number
CALL UMACH (2, NOUT)
!
X = 2.0
BGSTEP = 0.2
NCOUNT = 1
DERV = DERIV(FCN,X, BGSTEP=BGSTEP)
WRITE (NOUT,99999) DERV
99999 FORMAT (/, 1X, 'First derivative of FCN is ', 1PE10.3)
END
!
REAL FUNCTION FCN (X)
REAL X
REAL SIN
INTRINSIC SIN
FCN = -2.0*SIN(1.5*X)
RETURN
END
First derivative of FCN is 2.970E+00
In this example, we attempt to approximate in single precision the third derivative of the function
f(x) = 2x4 + 3x
at the point x = 0.75. Although the function is well-behaved near x = 0.75, finding derivatives is often computationally difficult on 32-bit machines. The difficulty is overcome in double precision.
USE IMSL_LIBRARIES
IMPLICIT NONE
INTEGER KORDER, NOUT
REAL BGSTEP, DERV, X, TOL
DOUBLE PRECISION DBGSTE, DDERV, DFCN, DTOL, DX
EXTERNAL DFCN, FCN
! Get output unit number
CALL UMACH (2, NOUT)
! Turn off stopping due to error
! condition
CALL ERSET (0, -1, 0)
!
X = 0.75
BGSTEP = 0.1
KORDER = 3
! In single precision, on a 32-bit
! machine, the following attempt
! produces an error message
DERV = DERIV(FCN, X, KORDER, BGSTEP,TOL)
! In double precision, we get good
! results
DX = 0.75D0
DBGSTE = 0.1D0
DTOL = 0.01D0
KORDER = 3
DDERV = DERIV(DFCN, DX,KORDER, DBGSTE, DTOL)
WRITE (NOUT,99999) DDERV
99999 FORMAT (/, 1X, 'The third derivative of DFCN is ', 1PD10.4)
END
!
REAL FUNCTION FCN (X)
REAL X
FCN = 2.0*X**4 + 3.0*X
RETURN
END
!
DOUBLE PRECISION FUNCTION DFCN (X)
DOUBLE PRECISION X
DFCN = 2.0D0*X**4 + 3.0D0*X
RETURN
END
*** FATAL ERROR 1 from DERIV. Unable to achieve desired tolerance.
*** Increase precision, increase TOL = 1.000000E-02 and/or change
*** BGSTEP = 1.000000E-01. If this error continues the function
*** may not have a derivative at X = 7.500000E-01
The third derivative of DFCN is 3.6000D+01
