Procedure

     LGRINT ( Lagrange polynomial interpolation )

     DOUBLE PRECISION FUNCTION LGRINT ( N, XVALS, YVALS, WORK, 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

Declarations

     IMPLICIT NONE

     INTEGER               N
     DOUBLE PRECISION      XVALS ( * )
     DOUBLE PRECISION      YVALS ( * )
     DOUBLE PRECISION      WORK  ( * )
     DOUBLE PRECISION      X

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.

     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.

     WORK     is a work space array of the same dimension as
              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

     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. The function will return the
         value 0.D0.

     2)  If N is less than 1, the error SPICE(INVALIDSIZE) is
         signaled. The function will return the value 0.D0.

     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 LGRINT should be 1.D0, 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
              IMPLICIT NONE

              DOUBLE PRECISION      LGRINT
              DOUBLE PRECISION      ANSWER
              DOUBLE PRECISION      XVALS (4)
              DOUBLE PRECISION      YVALS (4)
              DOUBLE PRECISION      WORK  (4)
              INTEGER               N

              N         =   4

              XVALS(1)  =  -1.D0
              XVALS(2)  =   0.D0
              XVALS(3)  =   1.D0
              XVALS(4)  =   3.D0

              YVALS(1)  =  -2.D0
              YVALS(2)  =  -7.D0
              YVALS(3)  =  -8.D0
              YVALS(4)  =  26.D0

              ANSWER    =   LGRINT ( N, XVALS, YVALS, WORK, 2.D0 )

              WRITE (*,*) 'ANSWER = ', ANSWER
              END


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


         ANSWER =    1.0000000000000000

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)

Version

    SPICELIB Version 1.1.0, 19-MAY-2021 (JDR)

        Added IMPLICIT NONE statement.

        Updated the header to comply with NAIF standard.
        Added "IMPLICIT NONE" to code example.

    SPICELIB Version 1.0.1, 10-JAN-2014 (NJB)

        Updated description of the workspace array: now the array WORK
        is not described as being allowed to coincide with the input
        YVALS. Such overlap would be a violation of the ANSI Fortran
        77 standard. Corrected several spelling errors in header
        documentation.

    SPICELIB Version 1.0.0, 16-AUG-1993 (NJB)