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Index Page lgrind_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

       lgrind_c (Lagrange polynomial interpolation with derivative)

       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
       both the value of the polynomial and its derivative.

#### Required_Reading

       None.

#### Keywords

       INTERPOLATION
       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 * 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)  If any two elements of the array `xvals' are equal, the error
           SPICE(DIVIDEBYZERO) 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 by a routine in the call tree of this routine.

       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

       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 cubic polynomial through the points

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

          and evaluate this polynomial at x = 2.

          The returned value of `p' should be 1.0, since the
          unique cubic polynomial that fits these points is

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

          The returned value of `dp' should be 16.0, since the
          derivative of f(x) is

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

          Example code begins here.

          /.
             Program lgrind_ex1
          ./
          #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 = %f %f\n", p, dp );

             return ( 0 );
          }

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

          P, DP = 1.000000 16.000000

          Note that we could also have 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]  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.

#### Author_and_Institution

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

#### Version

       -CSPICE Version 1.0.1, 01-NOV-2021 (JDR)

           Edited the header to comply with NAIF standard.

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

#### Index_Entries

       interpolate function using Lagrange polynomial
       Lagrange interpolation

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

    Fri Dec 31 18:41:09 2021