Index of Functions: A B C D E F G H I J K L M N O P Q R S T U V W X
Index Page lgrint_c
Table of contents
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
       lgrint_c ( Lagrange polynomial interpolation )

       SpiceDouble lgrint_c ( SpiceInt            n,
                              ConstSpiceDouble    xvals  [],
                              ConstSpiceDouble    yvals  [],
                              SpiceDouble         x         )
#### Abstract
       Evaluate a Lagrange interpolating polynomial for a specified
       set of coordinate pairs, at a specified abscissa value.
#### 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.
       x          I   Point at which to interpolate the polynomial.

       The function returns the value at `x' of the unique polynomial of
       degree n-1 that fits the points in the plane defined by `xvals' and
       `yvals'.
#### 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.

       x           is the abscissa value at which the interpolating
                   polynomial is to be evaluated.
#### Detailed_Output
       The function returns the value at `x' of the unique polynomial of
       degree n-1 that fits the points in the plane defined by `xvals' and
       `yvals'.
#### 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. The function will return the value 0.0.

       2)  If `n' is less than 1, the error SPICE(INVALIDSIZE) is signaled. The
           function will return the value 0.0.

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

       4)  If memory cannot be allocated to create the temporary variable
           required for the execution of the underlying Fortran routine,
           the error SPICE(MALLOCFAILED) is signaled. The function
           returns the value result.
#### 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 lgrint_c should be 1.0, since the
          unique cubic polynomial that fits these points is

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

          Example code begins here.

          /.
             Program lgrint_ex1
          ./
          #include <stdio.h>
          #include "SpiceUsr.h"

          int main( )
          {

             /.
             Local variables.
             ./
             SpiceDouble          answer;
             SpiceDouble          xvals  [4];
             SpiceDouble          yvals  [4];
             SpiceInt             n;

             n         =   4;

             xvals[0]  =  -1.0;
             xvals[1]  =   0.0;
             xvals[2]  =   1.0;
             xvals[3]  =   3.0;

             yvals[0]  =  -2.0;
             yvals[1]  =  -7.0;
             yvals[2]  =  -8.0;
             yvals[3]  =  26.0;

             answer    =   lgrint_c ( n, xvals, yvals, 2.0 );

             printf( "ANSWER = %f\n", answer );

             return ( 0 );
          }

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

          ANSWER = 1.000000
#### 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
       J. Diaz del Rio     (ODC Space)
#### Version
       -CSPICE Version 1.0.0, 04-AUG-2021 (JDR)
#### Index_Entries
       interpolate function using Lagrange polynomial
       Lagrange interpolation
#### Link to routine lgrint_c source file lgrint_c.c

    Fri Dec 31 18:41:09 2021