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cspice_lgresp

Table of contents
Abstract
I/O
Parameters
Examples
Particulars
Exceptions
Files
Restrictions
Required_Reading
Literature_References
Author_and_Institution
Version
Index_Entries

Abstract


   CSPICE_LGRESP evaluates a Lagrange interpolating polynomial for a
   specified set of coordinate pairs whose first components are equally
   spaced, at a specified abscissa value.

I/O


   Given:

      n        the number of points defining the polynomial.

               [1,1] = size(n); int32 = class(n)

               The array `yvals' contains `n' elements.

      first,
      step     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.

               [1,1] = size(first); double = class(first)
               [1,1] = size(step); double = class(step)

               The set of abscissa values is

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

               `step' must be non-zero.

      yvals    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.

               [n,1] = size(yvals); double = class(yvals)

               The set of points

                  (  first  +  (i-1)*STEP,   yvals(i)  )

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

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

               [1,1] = size(x); double = class(x)

   the call:

      [lgresp] = cspice_lgresp( n, first, step, yvals, x )

   returns:

      lgresp   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'.

               [1,1] = size(lgresp); double = class(lgresp)

Parameters


   None.

Examples


   Any numerical results shown for these examples may differ between
   platforms as the results depend on the SPICE kernels used as input
   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 cspice_lgresp 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.


      function lgresp_ex1()

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

         yvals  = [ -2.0, -8.0, 26.0, 148.0 ]';

         answer =   cspice_lgresp( n, first, step, yvals, 2.0 );

         fprintf( 'ANSWER = %f\n', answer )


      When this program was executed on a Mac/Intel/Octave6.x/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 cspice_lgresp would still be 1.0.


      Example code begins here.


      function lgresp_ex2()

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

         yvals  = [ 148.0, 26.0, -8.0, -2.0 ]';

         answer =   cspice_lgresp( n, first, step, yvals, 2.0 );

         fprintf( 'ANSWER = %f\n', answer )


      When this program was executed on a Mac/Intel/Octave6.x/64-bit
      platform, the output was:


      ANSWER = 1.000000


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 Mice routine cspice_lgrint.

Exceptions


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

   4)  If any of the input arguments, `n', `first', `step', `yvals'
       or `x', is undefined, an error is signaled by the Matlab error
       handling system.

   5)  If any of the input arguments, `n', `first', `step', `yvals'
       or `x', is not of the expected type, or it does not have the
       expected dimensions and size, an error is signaled by the Mice
       interface.

   6)  If the number of elements in `yvals' is less than `n', an error
       is signaled by the Mice interface.

Files


   None.

Restrictions


   None.

Required_Reading


   MICE.REQ

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


   -Mice Version 1.0.0, 01-JUL-2021 (JDR)

Index_Entries


   interpolate function using Lagrange polynomial
   Lagrange interpolation


Fri Dec 31 18:44:25 2021