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Index Page chbval_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

       chbval_c ( Value of a Chebyshev polynomial expansion )

       void chbval_c ( ConstSpiceDouble    cp     [],
                       SpiceInt            degp,
                       ConstSpiceDouble    x2s    [2],
                       SpiceDouble         x,
                       SpiceDouble       * p          )

#### Abstract

       Return the value of a polynomial evaluated at the input `x' using
       the coefficients for the Chebyshev expansion of the polynomial.

#### Required_Reading

       None.

#### Keywords

       INTERPOLATION
       MATH
       POLYNOMIAL

#### Brief_I/O

       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       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.
       p          O   Value of the polynomial at `x'.

#### Detailed_Input

       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
                   evaluated.

       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
                   transformed.

       x           is the value for which the polynomial is to be
                   evaluated.

#### Detailed_Output

       p           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]

#### Parameters

       None.

#### Exceptions

       Error free.

       1)  No tests are performed for exceptional values (`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.

#### Files

       None.

#### Particulars

       This routine computes the value `p' 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

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

       and

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

#### 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) 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.

          Example code begins here.

          /.
             Program chbval_ex1
          ./
          #include <stdio.h>
          #include "SpiceUsr.h"

          int main( )
          {

             /.
             Local variables.
             ./
             SpiceDouble          x;
             SpiceDouble          p;

             SpiceInt             degp;

             /.
             Set the coefficients of the polynomial and its
             transformation parameters
             ./
             SpiceDouble          cp     [7] = { 1.0,  3.0, 0.5, 1.0,
                                                 0.5, -1.0, 1.0     };
             SpiceDouble          x2s    [2] = { 0.5, 3.0 };

             degp   = 6;
             x      = 1.0;

             chbval_c ( cp, degp, x2s, x, &p );

             printf( "Value of the polynomial at x=1: %10.6f\n", p );

             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

#### Restrictions

       1)  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).

#### Literature_References

       [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," Wisley, 1974.

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

#### Author_and_Institution

       J. Diaz del Rio     (ODC Space)

#### Version

       -CSPICE Version 1.0.0, 16-JUL-2021 (JDR)

#### Index_Entries

       value of a chebyshev polynomial expansion

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

    Fri Dec 31 18:41:02 2021