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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

   void polyds_c ( ConstSpiceDouble    * coeffs,
                   SpiceInt              deg,
                   SpiceInt              nderiv,
                   SpiceDouble           t,
                   SpiceDouble         * p )

Abstract

   Compute the value of a polynomial and it's first
   n derivatives at the value t.

Required_Reading

   None.

Keywords

    INTERPOLATION
    MATH
    POLYNOMIAL


Brief_I/O

    VARIABLE  I/O  DESCRIPTION
    --------  ---  --------------------------------------------------
    coeffs     I   Coefficients of the polynomial to be evaluated.
    deg        I   Degree of the polynomial to be evaluated.
    nderiv     I   Number of derivatives to compute.
    t          I   Point to evaluate the polynomial and derivatives
    p          O   Value of polynomial and derivatives.

Detailed_Input

    coeffs     contains the coefficients of the polynomial that is
               to be evaluated. The first element of this array
               should be the constant term, the second element the
               linear coefficient, the third term the quadratic
               coefficient, and so on. The number of coefficients
               supplied should be one more than deg.

               f(x) = coeffs[0] + coeffs[1]*x + coeffs[2]*x^2 +
                      coeffs[3]*x^3 + ... + coeffs[deg]*x^deg

    deg        is the degree of the polynomial to be evaluated. deg
               should be one less than the number of coefficients
               supplied.

    nderiv     is the number of derivatives to compute. If nderiv
               is zero, only the polynomial will be evaluated. If
               nderiv = 1, then the polynomial and its first
               derivative will be evaluated, and so on. If the value
               of nderiv is negative, the routine returns
               immediately.

    t          is the point at which the polynomial and its
               derivatives should be evaluated.

Detailed_Output

    p          is an array containing the value of the polynomial and
               its derivatives evaluated at t. The first element of
               the array contains the value of p at t. The second
               element of the array contains the value of the first
               derivative of p at t and so on. The nderiv + 1'st
               element of the array contains the nderiv'th derivative
               of p evaluated at t.

Parameters

    None.

Exceptions

   Error free

   1) If nderiv is less than zero, the routine simply returns

   2) If the degree of the polynomial is less than 0, the routine
      simply returns.

Files

    None.

Particulars

    This routine uses the user supplied coefficients (coeffs)
    to evaluate a polynomial (having these coefficients) and its
    derivatives at the point t. The zero'th derivative of the
    polynomial is regarded as the polynomial itself.

Examples

   Example:
   
      For the polynomial
   
         f(x) = 1 + 3*x + 0.5*x^2 + x^3 + 0.5*x^4 - x^5 + x^6
   
      the coefficient set
   
      Degree  coeffs
      ------  ------
      0       1
      1       3
      2       0.5
      3       1
      4       0.5
      5      -1
      6       1
   
      Suppose t = 1.0
   
      We expect:
   
      Derivative Number     t = 1
      ------------------    -----
      f(x)         0        6
      f'(x)        1        10
      f''(x)       2        23
      f'''(x)      3        78


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

      int main()
         {

         /.
         Local variables.
         ./
         SpiceDouble      coeffs [] = { 1., 3., 0.5, 1., 0.5, -1., 1. };
         SpiceInt         deg    = 6;
         SpiceInt         nderiv = 3;
         SpiceDouble      t      = 1.;

         /. Dimension p as nderiv + 1. ./
         SpiceDouble      p [ 4 ];

         int              i;

         polyds_c ( coeffs, deg, nderiv, t, p );

         for ( i=0; i<=nderiv; i++ )
            {
            printf( "p = %lf\n", p[i] );
            }

         return(0);
         }

   The program outputs:

      p = 6.000000
      p = 10.000000
      p = 23.000000
      p = 78.000000

Restrictions

    Depending on the coefficients the user should be careful when
    taking high order derivatives. As the example shows, these
    can get big in a hurry. In general the coefficients of the
    derivatives of a polynomial grow at a rate greater
    than N! (N factorial).

Literature_References

    None.

Author_and_Institution

    W.L. Taber      (JPL)
    E.D. Wright     (JPL)

Version

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

Index_Entries

   compute a polynomial and its derivatives
Wed Apr  5 17:54:40 2017