Procedure

     HRMINT ( Hermite polynomial interpolation  )

     SUBROUTINE HRMINT ( N,  XVALS,  YVALS,  X,  WORK,  F,  DF )

Abstract

     Evaluate a Hermite interpolating polynomial at a specified
     abscissa value.

Required_Reading

     None.

Keywords

     INTERPOLATION
     POLYNOMIAL

Declarations

     IMPLICIT NONE

     INTEGER               N
     DOUBLE PRECISION      XVALS (   N )
     DOUBLE PRECISION      YVALS ( 2*N )
     DOUBLE PRECISION      X
     DOUBLE PRECISION      WORK  ( 2*N, 2 )
     DOUBLE PRECISION      F
     DOUBLE PRECISION      DF

Brief_I/O

     VARIABLE  I/O  DESCRIPTION
     --------  ---  --------------------------------------------------
     N          I   Number of points defining the polynomial.
     XVALS      I   Abscissa values.
     YVALS      I   Ordinate and derivative values.
     X          I   Point at which to interpolate the polynomial.
     WORK      I-O  Work space array.
     F          O   Interpolated function value at X.
     DF         O   Interpolated function's derivative at X.

Detailed_Input

     N        is the number of points defining the polynomial.
              The arrays XVALS and YVALS contain N and 2*N
              elements respectively.

     XVALS    is an array of length N containing abscissa values.

     YVALS    is an array of length 2*N containing ordinate and
              derivative values for each point in the domain
              defined by XVALS. The elements

                 YVALS( 2*I - 1 )
                 YVALS( 2*I     )

              give the value and first derivative of the output
              polynomial at the abscissa value

                 XVALS(I)

              where I ranges from 1 to N.

     WORK     is a work space array. It is used by this routine
              as a scratch area to hold intermediate results.

     X        is the abscissa value at which the interpolating
              polynomial and its derivative are to be evaluated.

Detailed_Output

     F,
     DF       are the value and derivative at X of the unique
              polynomial of degree 2N-1 that fits the points and
              derivatives defined by XVALS and YVALS.

Parameters

     None.

Exceptions

     1)  If two input abscissas are equal, the error
         SPICE(DIVIDEBYZERO) is signaled.

     2)  If N is less than 1, the error SPICE(INVALIDSIZE) is
         signaled.

     3)  This routine does not attempt to ward off or diagnose
         arithmetic overflows.

Files

     None.

Particulars

     Users of this routine must choose the number of points to use
     in their interpolation method. The authors of Reference [1] have
     this to say on the topic:

        Unless there is solid evidence that the interpolating function
        is close in form to the true function f, it is a good idea to
        be cautious about high-order interpolation. We
        enthusiastically endorse interpolations with 3 or 4 points, we
        are perhaps tolerant of 5 or 6; but we rarely go higher than
        that unless there is quite rigorous monitoring of estimated
        errors.

     The same authors offer this warning on the use of the
     interpolating function for extrapolation:

        ...the dangers of extrapolation cannot be overemphasized:
        An interpolating function, which is perforce an extrapolating
        function, will typically go berserk when the argument x is
        outside the range of tabulated values by more than the typical
        spacing of tabulated points.

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) Fit a 7th degree polynomial through the points ( x, y, y' )

           ( -1,      6,       3 )
           (  0,      5,       0 )
           (  3,   2210,    5115 )
           (  5,  78180,  109395 )

        and evaluate this polynomial at x = 2.

        The returned value should be 141.0, and the returned
        derivative value should be 456.0, since the unique 7th degree
        polynomial that fits these constraints is

                     7       2
           f(x)  =  x   +  2x  + 5


        Example code begins here.


              PROGRAM HRMINT_EX1
              IMPLICIT NONE

              DOUBLE PRECISION      ANSWER
              DOUBLE PRECISION      DERIV
              DOUBLE PRECISION      XVALS (4)
              DOUBLE PRECISION      YVALS (8)
              DOUBLE PRECISION      WORK  (8,2)
              INTEGER               N

              N         =   4

              XVALS(1)  =      -1.D0
              XVALS(2)  =       0.D0
              XVALS(3)  =       3.D0
              XVALS(4)  =       5.D0

              YVALS(1)  =       6.D0
              YVALS(2)  =       3.D0
              YVALS(3)  =       5.D0
              YVALS(4)  =       0.D0
              YVALS(5)  =    2210.D0
              YVALS(6)  =    5115.D0
              YVALS(7)  =   78180.D0
              YVALS(8)  =  109395.D0

              CALL HRMINT ( N, XVALS, YVALS, 2.D0, WORK, ANSWER,
             .                                           DERIV )

              WRITE (*,*) 'ANSWER = ', ANSWER
              WRITE (*,*) 'DERIV  = ', DERIV

              END


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


         ANSWER =    141.00000000000000
         DERIV  =    456.00000000000000

Restrictions

     None.

Literature_References

     [1]  W. Press, B. Flannery, S. Teukolsky and W. Vetterling,
          "Numerical Recipes -- The Art of Scientific Computing,"
          chapters 3.0 and 3.1, Cambridge University Press, 1986.

     [2]  S. Conte and C. de Boor, "Elementary Numerical Analysis -- An
          Algorithmic Approach," 3rd Edition, p 64, McGraw-Hill, 1980.

Author_and_Institution

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

Version

    SPICELIB Version 1.2.2, 01-OCT-2021 (NJB) (JDR)

        Edited the header to comply with NAIF standard. Fixed
        a few more comment typos. Added IMPLICIT NONE to code
        example.

    SPICELIB Version 1.2.1, 28-JAN-2014 (NJB)

        Fixed a few comment typos.

    SPICELIB Version 1.2.0, 01-FEB-2002 (NJB) (EDW)

        Bug fix: declarations of local variables XI and XIJ
        were changed from DOUBLE PRECISION to INTEGER.
        Note: bug had no effect on behavior of this routine.

    SPICELIB Version 1.1.0, 28-DEC-2001 (NJB)

        Blanks following final newline were truncated to
        suppress compilation warnings on the SGI-N32 platform.

    SPICELIB Version 1.0.0, 01-MAR-2000 (NJB)