Procedure

     QDERIV ( Quadratic derivative )

     SUBROUTINE QDERIV ( NDIM, F0, F2, DELTA, DFDT )

Abstract

     Estimate the derivative of a function by finding the derivative
     of a quadratic approximating function. This derivative estimate
     is equivalent to that found by computing the average of forward
     and backward differences.

Required_Reading

     None.

Keywords

     MATH
     UTILITY

Declarations

     IMPLICIT NONE

     INTEGER               NDIM
     DOUBLE PRECISION      F0    ( NDIM )
     DOUBLE PRECISION      F2    ( NDIM )
     DOUBLE PRECISION      DELTA
     DOUBLE PRECISION      DFDT  ( NDIM )

Brief_I/O

     VARIABLE  I/O  DESCRIPTION
     --------  ---  -------------------------------------------------
     NDIM       I   Dimension of function to be differentiated.
     F0         I   Function values at left endpoint.
     F2         I   Function values at right endpoint.
     DELTA      I   Separation of abscissa points.
     DFDT       O   Derivative vector.

Detailed_Input

     NDIM     is the dimension of the function to be
              differentiated. The derivative of each
              function component will be found.

     F0       is an array of NDIM function values at a point on
              the real line; we'll refer to this point as X0.

     F2       is an array of NDIM function values at a second
              point on the real line; we'll refer to this point
              as X2. The points X0 and X2 must satisfy

                 X2 = X0 + 2 * DELTA


     DELTA    is one half of the difference between X2 and X0:

                 DELTA = ( X2 - X0 ) / 2

              DELTA may be negative but must be non-zero.

Detailed_Output

     DFDT     is an N-dimensional vector representing an estimate
              of the derivative of the input function at the
              midpoint X1 of the interval between X0 and X2.

              The Ith component of DFDT is

                 ( 1 / (2*DELTA) ) * ( F2(I) - F0(I) )

              We may regard this estimate as the derivative
              at X1 of a parabola fitted to the points

                  ( X0, F0(I) ),  ( X2, F2(I) )

              We may also regard this derivative as the average
              of the forward and backward first-order
              differences of the input function defined by
              F0(I), F2(I), and DELTA.

Parameters

     None.

Exceptions

     1)  If DELTA is zero, the error SPICE(DIVIDEBYZERO) is signaled.

     2)  If NDIM is less than 1, this routine will fail in a
         system-dependent manner.

Files

     None.

Particulars

     This routine estimates the derivative of a vector-valued function
     using the average of forward and backward differences.

     The derivative estimate computed by this routine is equivalent to
     that obtained by fitting each component of the function with a
     parabola at the points

        (X0, f(X0)), (X1, f(X1)), (X2, f(X2))

     where

         X0  =  X1 - DELTA
         X2  =  X1 + DELTA

     and finding the derivative of the parabolas at X1.

Examples

     The numerical results shown for this example may differ across
     platforms. The results depend on the SPICE kernels used as
     input, the compiler and supporting libraries, and the machine
     specific arithmetic implementation.


     1) Estimate the derivative of x**2 at x = 2.

        Example code begins here.


              PROGRAM QDERIV_EX1
              IMPLICIT NONE

              DOUBLE PRECISION     DELTA
              DOUBLE PRECISION     DFDT  (1)
              DOUBLE PRECISION     F0    (1)
              DOUBLE PRECISION     F2    (1)
              INTEGER              N

              N     = 1
              DELTA = 1.D-3
              F0(1) = ( 2.D0 - DELTA ) ** 2.D0
              F2(1) = ( 2.D0 + DELTA ) ** 2.D0

              CALL QDERIV ( N, F0, F2, DELTA, DFDT )

              WRITE ( *, '(1X,A,E25.16)'  ) '4 - DFDT(1) = ',
             .                               4 - DFDT(1)
              END


        When this program was executed on a Mac/Intel/gfortran/64-bit
        platform, the output was:


         4 - DFDT(1) =    0.4547473508864641E-12


        Note that the difference displayed is platform-dependent, but
        should be on the order of 1.E-12.

Restrictions

     None.

Literature_References

     None.

Author_and_Institution

     N.J. Bachman       (JPL)
     J. Diaz del Rio    (ODC Space)

Version

    SPICELIB Version 1.1.0, 05-AUG-2020 (JDR)

        Changed input argument name "N" to "NDIM" for consistency with
        other routines.

        Edited the header to comply with NAIF standard.

    SPICELIB Version 1.0.0, 18-DEC-2004 (NJB)