Procedure

     LGRESP ( Lagrange interpolation on equally spaced points )

     DOUBLE PRECISION FUNCTION LGRESP (  N,      FIRST,  STEP,
    .                                    YVALS,  WORK,   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

Declarations

     IMPLICIT NONE

     INTEGER               N
     DOUBLE PRECISION      FIRST
     DOUBLE PRECISION      STEP
     DOUBLE PRECISION      YVALS ( * )
     DOUBLE PRECISION      WORK  ( * )
     DOUBLE PRECISION      X

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.
     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 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-1)*STEP,   YVALS(I)  )

              where I ranges from 1 to N, define the Lagrange
              polynomial used for interpolation.

     WORK     is a work space array of the same dimension as 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 FIRST,
     STEP, and YVALS.

Parameters

     None.

Exceptions

     1)  If STEP is zero, the error SPICE(INVALIDSTEPSIZE) 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.

     For Lagrange interpolation on unequally spaced abscissa values,
     see the SPICELIB routine LGRINT.

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 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 LGRESP_EX1
              IMPLICIT NONE

              DOUBLE PRECISION      LGRESP
              DOUBLE PRECISION      ANSWER
              DOUBLE PRECISION      FIRST
              DOUBLE PRECISION      STEP
              DOUBLE PRECISION      YVALS (4)
              DOUBLE PRECISION      WORK  (4)
              INTEGER               N

              N         =   4
              FIRST     =  -1.D0
              STEP      =   2.D0

              YVALS(1)  =  -2.D0
              YVALS(2)  =  -8.D0
              YVALS(3)  =  26.D0
              YVALS(4)  = 148.D0

              ANSWER    =   LGRESP ( N,     FIRST, STEP,
             .                       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


     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 would still be 1.D0.


        Example code begins here.


              PROGRAM LGRESP_EX2
              IMPLICIT NONE

              DOUBLE PRECISION      LGRESP
              DOUBLE PRECISION      ANSWER
              DOUBLE PRECISION      FIRST
              DOUBLE PRECISION      STEP
              DOUBLE PRECISION      YVALS (4)
              DOUBLE PRECISION      WORK  (4)
              INTEGER               N

              N         =   4
              FIRST     =   5.D0
              STEP      =  -2.D0

              YVALS(1)  = 148.D0
              YVALS(2)  =  26.D0
              YVALS(3)  =  -8.D0
              YVALS(4)  =  -2.D0

              ANSWER    =   LGRESP ( N,     FIRST, STEP,
             .                       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-FEB-2021 (JDR)

        Added IMPLICIT NONE statement.

        Update "N" and "WORK" detailed descriptions to remove
        references to nonexistent arguments.

        Updated the header to comply with NAIF standard.
        Added IMPLICIT NONE to code example. Added second example
        providing code to solve the problem of example #1 using a
        negative step.

    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, 14-AUG-1993 (NJB)