Index Page lgrind_c

A B C D E F G H I J K L M N O P Q R S T U V W X

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 lgrind_c ( SpiceInt            n,
                       ConstSpiceDouble  * xvals,
                       ConstSpiceDouble  * yvals,
                       SpiceDouble       * work,
                       SpiceDouble         x,
                       SpiceDouble       * p,
                       SpiceDouble       * dp )

#### Abstract

       Evaluate a Lagrange interpolating polynomial for a specified
       set of coordinate pairs, at a specified abscissa value.
       Return the value of both polynomial and derivative.

#### Required_Reading

       None.

#### Keywords

       INTERPOLATION
       MATH
       POLYNOMIAL

#### Brief_I/O

       Variable  I/O  Description
       --------  ---  --------------------------------------------------
       n          I   Number of points defining the polynomial.
       xvals      I   Abscissa values.
       yvals      I   Ordinate values.
       work      I-O  Work space array.
       x          I   Point at which to interpolate the polynomial.
       p          O   Polynomial value at x.
       dp         O   Polynomial derivative at x.

#### Detailed_Input

       n              is the number of points defining the polynomial.
                      The arrays xvals and yvals contain n elements.

       xvals,
       yvals          are arrays of abscissa and ordinate values that
                      together define N ordered pairs. The set of points

                         ( xvals[i], yvals[i] )

                      define the Lagrange polynomial used for
                      interpolation. The elements of xvals must be
                      distinct and in increasing order.

       work           is an n x 2 work space array, where n is the same
                      dimension as that of xvals and yvals. It is used
                      by this routine as a scratch area to hold
                      intermediate results.

       x              is the abscissa value at which the interpolating
                      polynomial is to be evaluated.

#### Detailed_Output

       p              is the value at x of the unique polynomial of
                      degree n-1 that fits the points in the plane
                      defined by xvals and yvals.

       dp             is the derivative at x of the interpolating
                      polynomial described above.

#### Parameters

       None.

#### Exceptions

       1)  The error SPICE(DIVIDEBYZERO) signals from a routine
           in the call tree if any two elements of the array
           xvals are equal.

       2)  The error SPICE(INVALIDSIZE) signals from a routine
           in the call tree if n is less than 1.

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

#### Files

       None.

#### Particulars

       Given a set of n distinct abscissa values and corresponding
       ordinate values, there is a unique polynomial of degree n-1, often
       called the `Lagrange polynomial', that fits the graph defined by
       these values. The Lagrange polynomial can be used to interpolate
       the value of a function at a specified point, given a discrete
       set of values of the function.

       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

       Example:

          Fit a cubic polynomial through the points

               ( -1, -2 )
               (  0, -7 )
               (  1, -8 )
               (  3, 26 )

          and evaluate this polynomial at x = 2.

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

          int main()
             {

             /.
             Local variables.
             ./
             SpiceDouble      p;
             SpiceDouble      dp;
             SpiceDouble      xvals [] = { -1., 0., 1., 3. };
             SpiceDouble      yvals [] = { -2., -7., -8., 26. };
             SpiceDouble      work  [4*2];
             SpiceInt         n = 4;

             lgrind_c ( n, xvals, yvals, work, 2., &p, &dp );

             printf( "p, dp = %lf %lf\n", p, dp);

             return(0);
            }

          The returned value of P should be 1., since the
          unique cubic polynomial that fits these points is

                         3       2
             f(x)   =   x   +  2x  - 4x  - 7

          The returned value of DP should be 16., since the
          derivative of f(x) is

              '          2
             f (x)  =  3x   +  4x  - 4

          We also could have invoked lgrind_c with the reference

             lgrind_c ( n, xvals, yvals, yvals, 2., &p, &dp );

          if we wished to; in this case yvals would have been
          modified on output.

#### Restrictions

       None.

#### Literature_References

       [1]  "Numerical Recipes---The Art of Scientific Computing" by
             William H. Press, Brian P. Flannery, Saul A. Teukolsky,
             William T. Vetterling (see sections 3.0 and 3.1).

#### Author_and_Institution

        N.J. Bachman    (JPL)
        E.D. Wright     (JPL)

#### Version

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

#### Index_Entries

       interpolate function using Lagrange polynomial
       Lagrange interpolation

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

    Wed Apr  5 17:54:38 2017