HERMITE

       To compute Hermite spline interpolation of a tabulated function.

       Hermite interpolation computes the cubic polynomial that agrees with
       the tabulated function and its derivative at the two nearest
       tabulated points.   It may be preferable to Lagrangian interpolation
       (QUADTERP) when either (1) the first derivatives are known, or (2)
       one desires continuity of the first derivative of the interpolated
       values.    HERMITE() will numerically compute the necessary
       derivatives, if they are not supplied.

       F = HERMITE( XX, FF, X, [ FDERIV = ])

       XX - Vector giving tabulated X values of function to be interpolated
               Must be either monotonic increasing or decreasing
       FF - Tabuluated values of function, same number of elements as X
       X -  Scalar or vector giving the X values at which to interpolate

       FDERIV - function derivative values computed at XX.    If not supplied,
               then HERMITE() will compute the derivatives numerically.
               The FDERIV keyword is useful either when (1) the derivative
               values are (somehow) known to better accuracy than can be
               computed numerically, or (2) when HERMITE() is called repeatedly
               with the same tabulated function, so that the derivatives
               need be computed only once.

       F - Interpolated values of function, same number of points as X

       Interpolate the function 1/x at x = 0.45 using tabulated values
       with a spacing of 0.1
       IDL> x = findgen(20)*0.1 + 0.1
       IDL> y = 1/x
       IDL> print,hermite(x,y,0.45)
               This gives 2.2188 compared to the true value 1/0.45 = 2.2222
       IDL> yprime = -1/x^2      ;But in this case we know the first derivatives
       IDL> print,hermite(x,y,0.45,fderiv = yprime)
             == 2.2219            ;and so can get a more accurate interpolation

       The algorithm here is based on the FORTRAN code discussed by
       Hill, G. 1982, Publ Dom. Astrophys. Obs., 16, 67.   The original
       FORTRAN source is U.S. Airforce. Surveys in Geophysics No 272.
       HERMITE() will return an error if one tries to interpolate any values
       outside of the range of the input table XX

       None

       Written,    B. Dorman (GSFC) Oct 1993, revised April 1996
       Added FDERIV keyword,  W. Landsman (HSTX)  April 1996
       Test for out of range values  W. Landsman (HSTX) May 1996
       Converted to IDL V5.0   W. Landsman   September 1997
       Use VALUE_LOCATE instead of TABINV   W. Landsman   February 2001

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