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Index Page hrmint_c

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Procedure Abstract Required_Reading Keywords Brief_I/O Detailed_Input | Detailed_Output Parameters Exceptions Files Particulars Examples | Restrictions Literature_References Author_and_Institution Version Index_Entries

#### Procedure

       hrmint_c ( Hermite polynomial interpolation  )

       void hrmint_c ( SpiceInt            n,
                       ConstSpiceDouble  * xvals,
                       ConstSpiceDouble  * yvals,
                       SpiceDouble         x,
                       SpiceDouble       * work,
                       SpiceDouble       * f,
                       SpiceDouble       * df )

#### Abstract

       Evaluate a Hermite interpolating polynomial at a specified
       abscissa value.

#### Required_Reading

       None.

#### Keywords

       INTERPOLATION
       POLYNOMIAL

#### 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    ]
                      yvals[ 2*i +1 ]

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

                      xvals[i]

                   where `i' ranges from 0 to n - 1.

       work        is a work space array. It is used by this routine
                   as a scratch area to hold intermediate results.
                   Generally sized at number of elements in yvals
                   times two.

       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 by a routine in the call tree of this routine.

       2)  If `n' is less than 1, the error SPICE(INVALIDSIZE) is
           signaled by a routine in the call tree of this routine.

       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
          ./

          #include <stdio.h>
          #include "SpiceUsr.h"

          int main( )
          {

             /.
             Local variables.
             ./
             SpiceDouble      answer;
             SpiceDouble      deriv;
             SpiceDouble      xvals [] = {-1., 0., 3., 5.};
             SpiceDouble      yvals [] = {6., 3., 5., 0.,
                                          2210., 5115., 78180., 109395.};
             SpiceDouble      work  [2*8];
             SpiceDouble      x = 2;
             SpiceInt         n = 4;

             hrmint_c ( n, xvals, yvals, x, work, &answer, &deriv );

             printf( "ANSWER = %f\n", answer );
             printf( "DERIV  = %f\n", deriv );

             return ( 0 );
          }

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

          ANSWER = 141.000000
          DERIV  = 456.000000

#### 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

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

#### Version

       -CSPICE Version 1.0.1, 22-FEB-2021 (JDR)

           Updated the header to comply with NAIF standard.

       -CSPICE Version 1.0.0, 24-AUG-2015 (EDW)

#### Index_Entries

       interpolate function using Hermite polynomial
       Hermite interpolation

#### Link to routine hrmint_c source file hrmint_c.c

    Fri Dec 31 18:41:08 2021