NAME
Interpolate_cubic3DFd1
SYNOPSIS
Interpolate_cubic3DFd1 (real, intent (in) :: a (1:64),
real, intent (in) :: x,
real, intent (in) :: y,
real, intent (in) :: z)
DESCRIPTION
Calculates the function value and all rescaled 1st derivative values for a triple
[x,y,z] of rescaled [0,1] coordinates and the 64 tricubic expansion coefficients.
The tricubic expansion reads, for one cube, in terms of rescaled [0,1] x,y,z coordinates:
3 3 3 i j k
F (x,y,z) = sum sum sum a (i,j,k) x y z
i=0 j=0 k=0
The order of the supplied expansion coefficients a (i,j,k) must be such, that the
k index has the highest ranking, followed by the j index and the i index. The overall
location index of the a (i,j,k) inside the 64-dimensional vector is given by the
following formula:
location index of (i,j,k) = 1 + i + 4j + 16k
The rescaled derivatives are given by the general formula:
r+s+t r s t 3 3 3 i-r j-s k-t
d / dx dy dz = sum sum sum (i) * (j) * (k) * a (i,j,k) * x y z
i=r j=s k=t r s t
where the Pochhammer symbols are defined as:
(i) = i * (i-1) * (i-2) * ... * (i-r+1)
r
The rescaled derivatives are therefore also sums of appropriate expansion coefficients
times monomial products. From the general formula we see, that the highest non-zero
derivative is of 9-th order.
Since this function is (potentially) called many times from external applications,
efficiency is key here and intermediate common summation terms are reused as much as
possible. The strategy is partial summation and reduction at each index summation stage.
The individual x-,y- and z-coordinate cubic polynomial sections are always evaluated
using the Horner scheme to minimize accumulation of computation rounding errors.
ARGUMENTS
a (i) : the i-th tricubic expansion coefficient x : rescaled [0,1] x coordinate y : rescaled [0,1] y coordinate z : rescaled [0,1] z coordinate
NOTES
1) The function is defined as a real array of size 4:
Interpolate_cubic3DFd1 (1) = the function value
Interpolate_cubic3DFd1 (2) = the rescaled d/dx value
Interpolate_cubic3DFd1 (3) = the rescaled d/dy value
Interpolate_cubic3DFd1 (4) = the rescaled d/dz value
2) The code checks, if the supplied triple [x,y,z] is rescaled.