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# ftcurvd

Calculates the derivatives of an interpolatory spline under tension.

## Prototype

```	function ftcurvd (
xi     : numeric,
yi     : numeric,
xo [*] : numeric
)

return_val  :  float or double
```

## Arguments

xi

An array containing the abscissae for the input function, with rightmost dimension npts. If xi is multi-dimensional, it must have the same dimension sizes as yi.

yi

An array of any dimensionality, whose rightmost dimension is npts, containing the functional values of the input function. That is, yi(...,k) is the functional value at xi(...,k) for k=0,npts-1.

xo

A 1D array of length nxo containing the abscissae for the interpolated values.

## Return value

An array of the same dimensionality as yi, but with the rightmost dimension replaced by nxo, containing the derivatives of the interpolated curve at the points specified by xo.

The output array will be of type double if any of the input is double, and float otherwise.

## Description

ftcurvd is in the Fitgrid package - a package containing 1D and 2D interpolators using cubic splines under tension.

There are some parameters that can alter the behavior of ftcurvd. These parameters all have reasonable default values. However, users may change any of these parameters by invoking ftsetp prior to calling ftcurvd. ftcurvd is called after all of the desired values for control parameters have been set.

Control parameters that apply to ftcurvd are: sig, sl1, sln, sf1.

The value for the parameter sig specifies the tension factor. Values near zero result in a cubic spline; large values (e.g. 50) result in nearly a polygonal line. A typical value is 1. (the default).

The values for sl1 and sln specify the slope of the curve at the first point and last point, respectively.

The value of sf1 controls whether to use the values for sl1 and sln, or compute those values internally. Specifically, sf1

= 0 if sl1 and sln are user-specified.
= 1 if sl1 is user-specified, but sln is internally calculated.
= 2 if sln is user-specified, but sl1 is internally calculated.
= 3 if sl1 and sln are internally calculated.
By default the slopes at the end points are computed internally.

You can extrapolate values with ftcurvd (that is, calculate interpolated values for abscissae outside of the domain of the input), but these values are, in general, unreliable.

## Examples

Example 1

```begin

xi = (/  0.00,   2.00,   5.00,   8.00,  10.00,  13.00,     \
15.00,  18.00,  21.00,  23.00,  30.00         /)
yi = (/  1.00,   0.81,   0.00,  -0.81,  -1.00,  -0.84,     \
-0.56,   0.04,   0.73,   1.18,   2.0          /)

npts = 201
xo   = fspan(0.,30.,npts)

yd = ftcurvd(xi, yi, xo)
end
```
Example 2

Assume p(time,lev,lat,lon) and y(time,lev,lat,lon) and that estimates of the vertical derivative dy/dp are desired. Further, assume that yo(time, LEV,lat,lon) and that LEV is a coordinate array of yo. You need to use dimension reordering so that the vertical coordinate is the rightmost dimension:

```    dydp = ftcurvd(p(time|:,lat|:,lon|:,lev|:), \
y(time|:,lat|:,lon|:,lev|:),yo&LEV)       ; ===> (time,lat,lon,LEV)
```

## Errors

= 1 if the number of input points is less than 2.
= 2 if xi values are not strictly increasing