/usr/share/gnudatalanguage/lib/deriv.pro is in libgnudatalanguage0 0.9.7-2.
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; NAME: DERIV
;
; PURPOSE:
; This function performs a numerical differentiation
; using a 3 points interpolation in Lagrangian way.
; The advantage of this procedure is we don't have to know the
; analytical equation of the "tabulated" data.
; Lagrangian interpolation means we "fit" the data with a polynomial
; which cross the data at collocation points. The fact we assume we
; have a polynomial give an explicit formula for the derivative.
;
; When only Y (data) are provided, a constant step of 1 is assumed
; between x-positions. Except at the edges, the formula is very simple.
;
; When X (positions) and Y (data) are provided, a more complex computation
; is donne.
;
; See below (in Section PROCEDURE) where you can found the formulae
;
; CATEGORY: Numerical analysis.
;
; CALLING SEQUENCE:
; result = DERIV, [x], y, [help=help], [test=test], [check=check]
;
; -- result = DERIV(data) (also : d=DERIV(Y))
; -- result = DERIV(positions, data) (also : d=DERIV(X,Y))
;
; INPUTS:
; -- When only one array (X) is provided, X contains the data to
; be differentiated ()
; -- When two arrays are provided
; X is the array containing the positions of data (colocations)
; Y is the array containing the data
;
; OPTIONAL INPUTS: none
;
; KEYWORD PARAMETERS:
; -- no_check : since we do extra checks, do not do them !
; (DESACTIVATED ! but we keep it for compatility issue)
; -- check : you can ask for extra tests on input
;
; -- Help : give a very simple summary of procedure then exit
; -- Test : will stop just before the return statement
; (useful for internal checking/debugging)
;
; OUTPUTS: Return the numerical derivative (Lagrangian formalism)
; Please note that the output type will be Float OR Double
;
; OPTIONAL OUTPUTS: none
;
; COMMON BLOCKS: none
;
; SIDE EFFECTS: none
;
; RESTRICTIONS: - We do check if the length of the 2 arrays (if 2)
; are the same (as in IDL)
; - We can check if the x(n+1)-x(n) is <> 0. (not
; done in IDL)
; - We can check if the X array is well sorted (not
; done in IDL)
; - We can check if Delta_x is constant or not (not
; done in IDL)
; - Neither IDL version or GDL version manage NaN values
;
; COMPATIBILITY:
; - this function is NOT FULLY compatible with the IDL one.
; NEVERTHELESS, it should give the same output with the same "normal" input(s).
; We DO have ADD some checks (zero steps, size of Y array, sorting,
; shortcut for constant step) which were missing in initial procedure.
; This will give clear warning when a problem is encountered.
;
; WARNING:
; - NaN values are maybe not fully managed now
;
; PROCEDURE:
; - in the IDL version, it is written that the procedure is described
; in Hildebrand, Introduction to Numerical Analysis, Mc Graw Hill,
; 1956, p. 82 (see IMCCE library, Bat A, Floor 2, Paris
; Observatory). But in that book, only the simple formulae for
; constant steps are explicitaly written. Nothing else in Chapter 3.
;
; - for the case with constant step, the procedure is obvious,
; except at the edges. We do use the formulae given in Hildebrand (p82)
; - when the X step is variable, a more complex computation is
; mandatory. We use the formulae written here:
; http://sonia_madani.club.fr/Cloaque/Arithmurgistan/Derivation/lagrange.html
;
; EXAMPLE: see also the accompagning program TEST_DERIV
; d=DERIV(REPLICATE(1.,nbp))
; d=DERIV(x_pos, y_data)
;
; TESTABILITY: the best way is to try on a pure slope !
;
; MODIFICATION HISTORY:
; - 17/03/2006 created by Alain Coulais (ARSC)
; - 09/10/2012 default is to be as tolerant as possible (/check if needed) !
;
;-
; LICENCE:
; Copyright (C) 2006, 2012 Alain Coulais
; This program is free software; you can redistribute it and/or modify
; it under the terms of the GNU General Public License as published by
; the Free Software Foundation; either version 2 of the License, or
; (at your option) any later version.
;
;-
function DERIV, x, y, help=help, test=test, no_check=no_check, check=check
;
ON_ERROR, 2
;
; When GDL behavior of {MESSAGE, .skip, .Continue} will be similar to
; IDL, the following flag can be switch to 1 !
;
flag_message=1
name_proc='% DERIV: '
;
if ((N_PARAMS() EQ 0) OR (N_PARAMS() GT 2)) then begin
if (flag_message EQ 1) then begin
MESSAGE, 'Incorrect number of arguments'
endif else begin
print, '% DERIV: Incorrect number of arguments'
help=1
endelse
endif
;
if KEYWORD_SET(help) then begin
print, 'function DERIV, [x,] y, $'
print, ' [help=help, test=test, no_check=no_check, check=check]'
return, -1
endif
;
nbp_x=N_ELEMENTS(x)
if (nbp_x LT 3) then begin
txt='Data must have at least 3 points !'
if (flag_message EQ 1) then MESSAGE, txt else print, name_proc+txt
return, -1
endif
;
if (SIZE(x,/type) EQ 7) then begin
txt='No STRING vector allowed !'
if (flag_message EQ 1) then MESSAGE, txt else print, name_proc+txt
return, -1
endif
;
; We check whether X and Y arrays have same size !
if (N_PARAMS() EQ 2) then begin
nbp_y=N_ELEMENTS(y)
if (nbp_x NE nbp_y) then begin
txt='X and Y vectors must have same size !'
if (flag_message EQ 1) then MESSAGE, txt else print, name_proc+txt
return, -1
endif
if (SIZE(y,/type) EQ 7) then begin
txt='No STRING vector allowed !'
if (flag_message EQ 1) then MESSAGE, txt else print, name_proc+txt
return, -1
endif
endif
;
; First case : we only have Data, no Position.
; That means that the data are equally spaced.
; We use the 3 formulae given by Hildebrand
;
if (N_PARAMS() EQ 1) then begin
;;
derivee=SHIFT(x,-1)-SHIFT(x,1)
;;
;; Specific computation for the 2 Edges
;; (if needed, the type conversion (Int/Long --> Float) will be
;; done here automatically)
;;
derivee[0]=-3.0*x[0]+4.0*x[1]-x[2]
derivee[nbp_x-1]=3.0*x[nbp_x-1]-4.0*x[nbp_x-2]+x[nbp_x-3]
;;
derivee=derivee/2.
endif
;
; Second case : we have Positions (X) and Data (Y)
; We use the formulae expressed on Web Site
;
if (N_PARAMS() EQ 2) then begin
if KEYWORD_SET(check) then begin
;; some checks : sorting, no-nul steps, no-constant steps, ...
;;
dx=x-SHIFT(x,1)
dx=dx[1:*]
dx_min=MIN(dx)
dx_max=MAX(dx)
;;
;; X locations must be sorted (by increasing order)
;;
if (dx_min LT 0.0) then begin
txt='X colocations MUST be sorted !'
if (flag_message EQ 1) then MESSAGE, txt else print, name_proc+txt
return, -1
endif
;;
;; X locations must be strictly different (to avoid zero division)
;;
if (dx_min EQ 0.0) then begin
txt='At least 2 X colocations are EQUAL !'
if (flag_message EQ 1) then MESSAGE, txt else print, name_proc+txt
return, -1
endif
;;
;; When All steps are equal, we can used "FAST" way ...
;;
if (dx_min EQ dx_max) then begin
if KEYWORD_SET(verbose) then print, name_proc+'Found constant STEP ... Algorithm changed'
derivee=DERIV(y)/dx_min
if KEYWORD_SET(test) then STOP
return, derivee
endif
;; after here, X are well conditionned !
;; x0 < x1 < x2
endif
;;
;; 3 useful expressions
;;
x0_x1=SHIFT(x,1)-x
x1_x2=x-SHIFT(x,-1)
x0_x2=SHIFT(x,1)-SHIFT(x,-1)
;;
derivee=SHIFT(y,1)*x1_x2/(x0_x1*x0_x2)
derivee=derivee+y*(1./x1_x2-1./x0_x1)
derivee=derivee-SHIFT(y,-1)*x0_x1/(x0_x2*x1_x2)
;;
;; The 2 Edges (see matrix formulae on Web Page)
;;
derivee[0]=y[0]*(1./x0_x1[1]+1./x0_x2[1])
derivee[0]=derivee[0]-y[1]*x0_x2[1]/(x0_x1[1]*x1_x2[1])
derivee[0]=derivee[0]+y[2]*x0_x1[1]/(x0_x2[1]*x1_x2[1])
;;
;; useful index
nm3=nbp_x-3
nm2=nbp_x-2
nm1=nbp_x-1
;;
derivee[nm1]=-y[nm3]*x1_x2[nm2]/(x0_x1[nm2]*x0_x2[nm2])
derivee[nm1]=derivee[nm1]+y[nm2]*x0_x2[nm2]/(x0_x1[nm2]*x1_x2[nm2])
derivee[nm1]=derivee[nm1]-y[nm1]*(1./x0_x2[nm2]+1./x1_x2[nm2])
endif
;
if KEYWORD_SET(test) then STOP
;
return, derivee
;
end
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