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Index Page lgresp_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

       lgresp_c ( Lagrange interpolation on equally spaced points )

       SpiceDouble lgresp_c ( SpiceInt            n,
                              SpiceDouble         first,
                              SpiceDouble         step,
                              ConstSpiceDouble    yvals  [],
                              SpiceDouble         x         )

#### Abstract

       Evaluate a Lagrange interpolating polynomial for a specified
       set of coordinate pairs whose first components are equally
       spaced, 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.
       first      I   First abscissa value.
       step       I   Step size.
       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 `first',
       `step', and `yvals'.

#### Detailed_Input

       n           is the number of points defining the polynomial. The
                   array `yvals' contains `n' elements.

       first,
       step        are, respectively, a starting abscissa value and a
                   step size that define the set of abscissa values
                   at which a Lagrange interpolating polynomial is to
                   be defined. The set of abscissa values is

                      first   +   i * step,     i = 0, ..., n-1

                   `step' must be non-zero.

       yvals       is an array of ordinate values that, together with
                   the abscissa values defined by `first' and `step',
                   define `n' ordered pairs belonging to the graph of
                   a function. The set of points

                      (  first  +  i*step,   yvals(i)  )

                   where `i' ranges from 0 to n-1, define the Lagrange
                   polynomial used for interpolation.

       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 `first',
       `step', and `yvals'.

#### Parameters

       None.

#### Exceptions

       1)  If `step' is zero, the error SPICE(INVALIDSTEPSIZE) 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.

       For Lagrange interpolation on unequally spaced abscissa values,
       see the CSPICE routine lgrint_c.

#### Examples

       The numerical results shown for these examples 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 )
              (  1,  -8 )
              (  3,  26 )
              (  5, 148 )

          and evaluate this polynomial at x = 2.

          The returned value of lgresp_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 lgresp_ex1
          ./
          #include <stdio.h>
          #include "SpiceUsr.h"

          int main( )
          {

             SpiceDouble          answer;
             SpiceDouble          first;
             SpiceDouble          step;
             SpiceDouble          yvals  [4];
             SpiceInt             n;

             n         =   4;
             first     =  -1.0;
             step      =   2.0;

             yvals[0]  =  -2.0;
             yvals[1]  =  -8.0;
             yvals[2]  =  26.0;
             yvals[3]  = 148.0;

             answer    =   lgresp_c ( n, first, step, 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

       2) Solve the same problem using a negative step. In order to
          find the solution, set the elements of `yvals' in reverse order.

          The returned value of lgresp_c would still be 1.0.

          Example code begins here.

          /.
             Program lgresp_ex2
          ./
          #include <stdio.h>
          #include "SpiceUsr.h"

          int main( )
          {

             SpiceDouble          answer;
             SpiceDouble          first;
             SpiceDouble          step;
             SpiceDouble          yvals  [4];
             SpiceInt             n;

             n         =   4;
             first     =   5.0;
             step      =  -2.0;

             yvals[0]  = 148.0;
             yvals[1]  =  26.0;
             yvals[2]  =  -8.0;
             yvals[3]  =  -2.0;

             answer    =   lgresp_c ( n, first, step, 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 lgresp_c source file lgresp_c.c

    Fri Dec 31 18:41:09 2021