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   chbder_c ( Derivatives of a Chebyshev expansion ) 

   void chbder_c ( ConstSpiceDouble * cp,
                   SpiceInt           degp,
                   SpiceDouble        x2s[2],
                   SpiceDouble        x,
                   SpiceInt           nderiv,
                   SpiceDouble      * partdp,
                   SpiceDouble      * dpdxs )


   Return the value of a polynomial and its first `nderiv'
   derivatives, evaluated at the input `x', using the coefficients of
   the Chebyshev expansion of the polynomial.






   --------  ---  --------------------------------------------------
   cp         I   degp+1 Chebyshev polynomial coefficients.
   degp       I   Degree of polynomial.
   x2s        I   Transformation parameters of polynomial.
   x          I   Value for which the polynomial is to be evaluated.
   nderiv     I   The number of derivatives to compute.
   partdp    I-O  Workspace provided for computing derivatives.
   dpdxs      O   Array of the derivatives of the polynomial.


   cp          is an array of coefficients a polynomial with respect
               to the Chebyshev basis. The polynomial to be
               evaluated is assumed to be of the form:

                 cp[degp]*T(degp,s) + cp[degp-1]*T(degp-1,s) + ...

                                    + cp[1]*T(1,s) + cp[0]*T(0,s)

               where T(i,s) is the I'th Chebyshev polynomial
               evaluated at a number `s' whose double precision
               value lies between -1 and 1. The value of `s' is
               computed from the input variables x2s[0], x2s[1]
               and `x'.

   degp        is the degree of the Chebyshev polynomial to be

   x2s         is an array of two parameters. These parameters are
               used to transform the domain of the input variable `x'
               into the standard domain of the Chebyshev polynomial.
               x2s[0] should be a reference point in the domain of
               `x'; x2s[1] should be the radius by which points are
               allowed to deviate from the reference point and while
               remaining within the domain of `x'. The value of
               `x' is transformed into the value `s' given by

                  s = ( x - x2s[0] ) / x2s[1]

               Typically x2s[0] is the midpoint of the interval over
               which `x' is allowed to vary and x2s[1] is the radius
               of the interval.

               The main reason for doing this is that a Chebyshev
               expansion is usually fit to data over a span
               from A to B where A and B are not -1 and 1
               respectively. Thus to get the "best fit" the
               data was transformed to the interval [-1,1] and
               coefficients generated. These coefficients are
               not rescaled to the interval of the data so that
               the numerical "robustness" of the Chebyshev fit will
               not be lost. Consequently, when the "best fitting"
               polynomial needs to be evaluated at an intermediate
               point, the point of evaluation must be transformed
               in the same way that the generating points were

   x           is the value for which the polynomial is to be

   nderiv      is the number of derivatives to be computed by the
               routine. `nderiv' should be non-negative.

   partdp      is a work space used by the program to compute
               all of the desired derivatives. It should be declared
               in the calling program as

                  SpiceDouble       partdp[3 * (nderiv+1)]


   dpdxs       is an array containing the value of the polynomial and
               its derivatives evaluated at `x'.

               dpdxs[0] is the value of the polynomial to be evaluated.
               It is given by

                  cp[degp]*T(degp,s) + cp[degp-1]*T(degp-1,s) + ...

                                     + cp[1]*T(1,s) + cp[0]*T(0,s)

               where T(i,s) is the i'th Chebyshev polynomial
               evaluated  at a number s = ( x -  x2s[0] )/ x2s[1].

               dpdxs(i) is the value of the i'th derivative of the
               polynomial at `x' (`i' ranges from 1 to `nderiv'). It is
               given by

                  (1/x2s[1]^i) ( cp[degp]*T   (degp,s)

                               + cp[degp-1]*T   (degp-1,s)

                              + ...

                              + cp[1]*T   (1,s)

                              + cp[0]*T   (0,s) )

               where T(k,s) is the k'th Chebyshev polynomial and the
               superscript [i] indicates its i'th derivative,
               evaluated at the number s = ( x - x2s[0] )/x2s[1].




   Error free.

   1)  No tests are performed for exceptional values (`nderiv'
       negative, `degp' negative, etc.). This routine is expected to
       be used at a low level in ephemeris evaluations. For that
       reason it has been elected as a routine that will not
       participate in error handling.




   This routine computes the value of a Chebyshev polynomial
   expansion and the derivatives of the expansion with respect to `x'.
   The polynomial is given by

      cp[degp]*T(degp,s) + cp[degp-1]*T(degp-1,s) + ...

                         + cp[1]*T(1,s) + cp[0]*T(0,s)


      s  =  ( x - x2s[0] ) / x2s[1]


      T(i,s) is the i'th Chebyshev polynomial of the first kind
      evaluated at `s'.


   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) Depending upon the user's needs, there are 3 routines
      available for evaluating Chebyshev polynomials.

         chbval_c   for evaluating a Chebyshev polynomial when no
                    derivatives are desired.

         chbint_c   for evaluating a Chebyshev polynomial and its
                    first derivative.

         chbder_c   for evaluating a Chebyshev polynomial and a user
                    or application dependent number of derivatives.

      Of these 3 the one most commonly employed by SPICE software
      is chbint_c as it is used to interpolate ephemeris state
      vectors; this requires the evaluation of a polynomial
      and its derivative. When no derivatives are desired one
      should use chbval_c, or when more than one or an unknown
      number of derivatives are desired one should use chbder_c.

      The code example below illustrates how this routine might
      be used to obtain points for plotting a polynomial
      and its derivatives.

      Example code begins here.

         Program chbder_ex1
      #include <stdio.h>
      #include "SpiceUsr.h"

      int main( )

         Local variables.
         SpiceDouble      cp []  = {  1., 3., 0.5, 1., 0.5, -1., 1. };
         SpiceInt         degp   = 6;
         SpiceInt         nderiv = 3;
         SpiceDouble      x2s[]  = { .5, 3.};
         SpiceDouble      x      = 1.;

         Dimension partdp as 3 * (nderiv + 1)
         SpiceDouble      partdp[3 * 4];

         Dimension dpdxs as nderiv + 1.
         SpiceDouble      dpdxs [3+1];

         SpiceInt         i;

         chbder_c ( cp, degp, x2s, x, nderiv, partdp, dpdxs );

         printf( "Value of the polynomial at X=1: %10.6f\n",
                  dpdxs[0] );

         for ( i=1; i<=nderiv; i++ )
            printf( "   Derivative %i at X=1        : %10.6f\n",
                                                (int)i, dpdxs[i] );

         return ( 0 );

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

      Value of the polynomial at X=1:  -0.340878
         Derivative 1 at X=1        :   0.382716
         Derivative 2 at X=1        :   4.288066
         Derivative 3 at X=1        :  -1.514403


   1)  The user must be sure that the provided workspace is declared
       properly in the calling routine. The proper declaration is:

          SpiceInt         nderiv = the desired number of derivatives;
          SpiceDouble      partdp[3 * (nderiv + 1)];

       If for some reason a parameter is not passed to this routine
       in `nderiv', the user should make sure that the value of `nderiv'
       is not so large that the work space provided is inadequate.

   2)  One needs to be careful that the value

          (x-x2s[0]) / x2s[1]

       lies between -1 and 1. Otherwise, the routine may fail
       spectacularly (for example with a floating point overflow).

   3)  While this routine will compute derivatives of the input
       polynomial, the user should consider how accurately the
       derivatives of the Chebyshev fit, match the derivatives of the
       function it approximates.


   [1]  W. Press, B. Flannery, S. Teukolsky and W. Vetterling,
        "Numerical Recipes -- The Art of Scientific Computing,"
        chapter 5.4, "Recurrence Relations and Clenshaw's Recurrence
        Formula," p 161, Cambridge University Press, 1986.

   [2]  T. Rivlin, "The Chebyshev Polynomials," Wiley, 1974.

   [3]  R. Weast and S. Selby, "CRC Handbook of Tables for
        Mathematics," 4th Edition, CRC Press, 1976.


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


   -CSPICE Version 1.1.0, 01-NOV-2021 (JDR)

       Removed error tracing calls. The function is declared
       as error free, therefore these calls are not required.

       Updated the header to comply with NAIF standard and correct several
       typos. Reformatted example's output.

       Removed unnecessary comments from the function's body.

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


   derivatives of a chebyshev expansion
Fri Dec 31 18:41:02 2021