# FLASH4.6.1 API

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## [Functions] source/numericalTools/Interpolate/Interpolate_cubic1DFd1d2

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NAME

```  Interpolate_cubic1DFd1d2
```

SYNOPSIS

```  Interpolate_cubic1DFd1d2 (real, intent (in) :: a (1:4),
real, intent (in) :: x)
```

DESCRIPTION

```  Calculates the function value and the rescaled 1st and 2nd derivative value for a
single [x] rescaled [0,1] coordinate and the 4 monocubic expansion coefficients.
The monocubic expansion reads, for one line, in terms of the rescaled [0,1] x coordinate:

3          i
F (x) = sum  a (i) x
i=0

The location index of the a (i) inside the 4-dimensional vector is:

location index of a (i)  =  1 + i

The rescaled derivatives are given by the general formula:

r    r    3                  i-r
d / dx  = sum (i)  * a (i) * x
i=r    r

where the Pochhammer symbols are defined as:

(i)  = i * (i-1) * (i-2) * ... * (i-r+1)
r

From the derivative formula we see, that the highest non-zero derivative is of
3-rd order.
```

ARGUMENTS

```  a (i) : the i-th monocubic expansion coefficient
x     : rescaled [0,1] x coordinate
```

NOTES

```  1) The function is defined as a real array of size 3:

Interpolate_cubic1DFd1d2 (1) = the function value
Interpolate_cubic1DFd1d2 (2) = the rescaled d/dx value
Interpolate_cubic1DFd1d2 (3) = the rescaled d2/dx2 value

2) The code checks, if the supplied coordinate [x] is rescaled.
```