function hermite,xx,ff,x, FDERIV = fderiv ;+ ; NAME: ; 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 - Tabulated 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 ; PROCEDURES CALLED: ; 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 ;- On_error,2 if N_Params() LT 3 then begin print,'Syntax: f = HERMITE( xx, ff, x, [FDERIV = ] )' return,0 endif n = N_elements(xx) ;Number of knot points m = N_elements(x) ;Number of points at which to interpolate l = value_locate(xx,x) ;Integer index of interpolation points bad = where( (l LT 0) or (l EQ n-1), Nbad) if Nbad GT 0 then message, 'ERROR - Valid interpolation range is ' + \$ strtrim(xx[0],2) + ' to ' + strtrim(xx[n-1],2) n1 = n - 1 n2 = n - 2 l1 = l + 1 l2 = l1 + 1 lm1 = l - 1 h1 = double(1./(xx[l] - xx[l1])) h2 = - h1 ; If derivatives were not supplied, then compute numeric derivatives at the ; two closest knot points if N_elements(fderiv) NE 0 then begin f2 = fderiv[l1] f1 = fderiv[l] endif else begin f1 = dblarr(m) f2 = dblarr(m) for i = 0,m-1 do begin if l[i] ne 0 then begin if l[i] lt n2 then begin f2[i] = (ff[l2[i]] - ff[l[i]])/(xx[l2[i]]-xx[l[i]]) endif else begin f2[i] = (ff[n1] - ff[n2])/(xx[n1] - xx[n2]) endelse f1[i] = ( ff[l1[i]] - ff[lm1[i]] )/( xx[l1[i]] - xx[lm1[i]] ) endif else begin f1[i] = (ff[1] - ff[0])/(xx[1] - xx[0]) f2[i] = (ff[2] - ff[0])/(xx[2] - xx[0]) endelse endfor endelse xl1 = x - xx[l1] xl = x - xx[l] s1 = xl1*h1 s2 = xl*h2 ; Now finally the Hermite interpolation formula f = (ff[l]*(1.-2.*h1*xl) + f1*xl)*s1*s1 + \$ (ff[l1]*(1.-2.*h2*xl1) + f2*xl1)*s2*s2 if m eq 1 then return,f[0] else return,f end