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# center_finite_diff

Performs a centered finite difference operation on the rightmost dimension.

## Prototype

```	function center_finite_diff (
q        : numeric,
r        : numeric,
rCyclic  : logical,
opt      : integer
)

return_val [dimsizes(q)] :  numeric
```

## Arguments

q

A multi-dimensional array.

r

A scalar, one-dimensional, or multi-dimensional array containing the coordinates along which q is to be differenced. Does need not be equally spaced from a computational point of view.

• scalar: r assumed to be the (constant) distance between adjacent points.
• one-dimensional (and the same size as the rightmost dimension of q): applied to all dimensions of q.
• multi-dimensional: then it must be the same size as q.

rCyclic

True: q treated as cyclic in r and the end values and all the returned values will be calculated via centered differences. False: q NOT treated as cyclic in r and the end values will use a one-sided difference scheme for the end points. q should not include a cyclic point.

```   result(n) = (q(n+1)-q(n))/(r(n+1)-r(n))  for the initial value
result(m) = (q(m)-q(m-1))/(r(m)-r(m-1))  for the last value
```

opt

Reserved for future use. Currently not used. Set to an integer.

## Description

Performs a centered finite difference operation on the rightmost dimension. If missing values are present, the calculation will occur at all points possible, but coordinates which could not be used will set to missing.

```result(n) = (q(n+1)-q(n-1))/(r(n+1)-r(n-1))
```

Use center_finite_diff_n if the dimension to do the calculation on is not the rightmost dimension and reordering is not desired. This function can be significantly faster than center_finite_diff.

## Examples

Example 1

```   q = (/30,33,39,36,41,37/)
r = 2.                                 ; constant

dqdr = center_finite_diff (q,r,False,0)
```

Result:

```   dqdr(0) = (33-30)/2 = 1.5
dqdr(1) = (39-30)/4 = 2.25
dqdr(2) = (36-33)/4 = 0.75
dqdr(3) = (41-39)/4 = 0.5
dqdr(4) = (37-36)/4 = 0.25
dqdr(5) = (37-41)/2 = -2.0
```
Example 2:

```   theta = (/ 298,299,300,302,...,345,355,383/)   ; potential temp.
p     = (/1000,950,900,850,...,200,150,100/)   ; pressure (hPa)

dtdp  = center_finite_diff (theta,p,False,0)
```
Example 3

p is not equally spaced. Demonstrates defining the appropriate pressure levels when the r coordinate is not equally spaced. The end points are computed via a one-sided difference.

```   theta = (/ 297, 298,300,302,...,345,350,355,383/)   ; potential temp.
p     = (/1013,1000,900,850,...,200,175,150,100/)   ; pressure (hPa)

dtdp  = center_finite_diff (theta,p,False,0)

np    = dimsizes(p)
pPlot = new ( np, "float" )
; arithmetic mean
pPlot(0)      = (p(0)+p(1))*0.5                     ; set bottom
pPlot(np-1)   = (p(np-1)+p(np-2))*0.5               ; set top
pPlot(1:np-2) = (p(0:np-3) + p(2:np-1))*0.5         ; mid points

; or
; log (mass) wgted
pPlot(0)      = exp((log(p(0))+log(p(1)))*0.5)             ; set bottom
pPlot(np-1)   = exp((log(p(np-1))+log(p(np-2)))*0.5)       ; set top
pPlot(1:np-2) = exp((log(p(0:np-3)) + log(p(2:np-1)))*0.5) ; mid points
```
Example 4

Let T be four-dimensional with dimensions time,level,lat,lon. P is one-dimensional. Perform the finite differencing only in the vertical (level) dimension. This requires that T be reordered to put level in the rightmost dimension.

Note: in V5.2.0 or later you can use center_finite_diff_n to avoid having to reorder the data.

```   dTdP = center_finite_diff(T(time|:,lat|:,lon|:,lev|:),P,False,0)
;  returns dTdP (time,lat,lon,lev). this variable can be reordered
;  again to place it back in the original order

; In version 5.2.0 or later:
;   dTdP = center_finite_diff_n(T,P,False,0,1)
;  returns dTdP (time,lev,lat,lon)
```
Example 5

Now P is also four-dimensional and requires reordering:

```   dTdP = center_finite_diff (T(time|:,lat|:,lon|:,lev|:)  \
,P(time|:,lat|:,lon|:,lev|:), False,0)
; returns dTdP(time,lat,lon,lev)

; In version 5.2.0 or later:
;   dTdP = center_finite_diff_n(T,P,False,0,1)
;  returns dTdP (time,lev,lat,lon)
```
Example 6 Assume that the longitude coordinate variable associated with T in the examples above is cyclic and is equally spaced in degrees but not in physical space.
```   dlon = (lon(2)-lon(1))*0.0174533 ; convert to radians
; pre-allocate space
dTdX = new ( (/ntim,klev,nlat,mlon/), typeof(T), T@_FillValue)

do nl=0,nlat-1                      ; loop over each latitude
dX = 6378388.*cos(0.0174533*lat(nl))*dlon  ; constant at this latitude
dTdX(:,:,nl,:) = center_finite_diff (T(:,:,nl;,:), dX , True,0)
end do
; result: dTdX(time,lev,lat,lon)
```