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Index Page hrmesp_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
       hrmesp_c ( Hermite polynomial interpolation, equal spacing )

       void hrmesp_c ( SpiceInt            n,
                       SpiceDouble         first,
                       SpiceDouble         step,
                       ConstSpiceDouble    yvals  [],
                       SpiceDouble         x,
                       SpiceDouble       * f,
                       SpiceDouble       * df        )
#### Abstract
       Evaluate, at a specified point, a Hermite interpolating polynomial
       for a specified set of equally spaced abscissa values and
       corresponding pairs of function and function derivative values.
#### Required_Reading
       None.
#### Keywords
       INTERPOLATION
       POLYNOMIAL
#### Brief_I/O
       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       n          I   Number of points defining the polynomial.
       first      I   First abscissa value.
       step       I   Step size.
       yvals      I   Ordinate and derivative values.
       x          I   Point at which to interpolate the polynomial.
       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 array `yvals' contains 2*n elements.

       first,
       step        are, respectively, a starting abscissa value and a
                   step size that define the set of abscissa values

                      first   +   i * step,     i = 0, ..., n-1

                   `step' must be non-zero.

       yvals       is an array of length 2*n containing ordinate and
                   derivative values for each point in the domain
                   defined by `first', `step', and `n'. The elements

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

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

                      first   +   i * step

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

       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 2*n-1 that fits the points and
                   derivatives defined by `first', `step', and `yvals'.
#### Parameters
       None.
#### Exceptions
       1)  If `step' is zero, the error SPICE(INVALIDSTEPSIZE) 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.

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

       4)  If memory cannot be allocated to create the temporary variable
           required for the execution of the underlying Fortran routine,
           the error SPICE(MALLOCFAILED) is signaled.
#### 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 )
             (  1,      8,      11 )
             (  3,   2210,    5115 )
             (  5,  78180,  109395 )

          and evaluate this polynomial at x = 2.

          The returned value of ANSWER 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 hrmesp_ex1
          ./
          #include <stdio.h>
          #include "SpiceUsr.h"

          int main( )
          {

             SpiceDouble          answer;
             SpiceDouble          deriv;
             SpiceDouble          first;
             SpiceDouble          step;
             SpiceDouble          yvals  [8];
             SpiceInt             n;

             n         =       4;

             yvals[0]  =       6.0;
             yvals[1]  =       3.0;
             yvals[2]  =       8.0;
             yvals[3]  =      11.0;
             yvals[4]  =    2210.0;
             yvals[5]  =    5115.0;
             yvals[6]  =   78180.0;
             yvals[7]  =  109395.0;

             first     =  -1.0;
             step      =   2.0;

             hrmesp_c ( n, first, step, yvals, 2.0, &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)
#### Version
       -CSPICE Version 1.0.0, 03-AUG-2021 (JDR)
#### Index_Entries
       interpolate function using Hermite polynomial
       Hermite interpolation
#### Link to routine hrmesp_c source file hrmesp_c.c

    Fri Dec 31 18:41:08 2021