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HERMITE

HERMITE

## Purpose

To compute Hermite spline interpolation of a tabulated function.

## Explanation

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.

## Calling Sequence

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

## Input Parameters

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

## Optional Input Keyword

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.

## Output Parameter

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

## Example

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

## Notes

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

## Revision History

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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