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Package fft (in fft.i) - Fast Fourier Transform functions

Index of documented functions or symbols:

fft

roll

### fft

```DOCUMENT fft(x, direction)
fft(x, ljdir, rjdir)
or fft(x, ljdir, rjdir, setup=workspace)
returns the complex Fast Fourier Transform of array X.
The DIRECTION determines which direction the transform is in --
e.g.- from time to frequency or vice-versa -- as follows:

DIRECTION    meaning
---------    -------
1        "forward" transform (coefficients of exp(+i * 2*pi*kl/N))
on every dimension of X
-1        "backward" transform (coefficients of exp(-i * 2*pi*kl/N))
on every dimension of X
[1,-1,1]     forward transform on first and third dimensions of X,
backward transform on second dimension of X (any other
dimensions remain untransformed)
[-1,0,0,1]   backward transform on first dimension of X, forward
transform on fourth dimension of X
etc.

The third positional argument, if present, allows the direction
of dimensions of X to be specified relative to the final dimension
of X, instead of relative to the first dimension of X.  In this
case, both LJDIR and RJDIR must be vectors of integers -- the
scalar form is illegal:

LJDIR    RJDIR      meaning
-----    -----      -------
[]        [1]       forward transform last dimension of X
[1]        []       forward transform first dimension of X
[]        [-1,-1]   backward transform last two dimensions of X,
leaving any other dimensions untransformed
[-1,0,0,1]    []       backward transform on first dimension of X,
forward transform on fourth dimension of X
[]      [-1,0,0,1]  backward transform on 4th to last dimension of X,
forward transform on last dimension of X
etc.

Note that the final element of RJDIR corresponds to the last dimension
of X, while the initial element of LJDIR corresponds to the first
dimension of X.

The explicit meaning of "forward" transform -- the coefficients of
exp(+i * 2*pi*kl/N) -- is:

result for j=1,...,n

result(j)=the sum from k=1,...,n of

x(k)*exp(-i*(j-1)*(k-1)*2*pi/n)

where i=sqrt(-1)

Note that the result is unnormalized.  Applying the "backward"
transform to the result of a "forward" transform returns N times
the original vector of length N.  Equivalently, applying either
the "forward" or "backward" transform four times in succession
yields N^2 times the original vector of length N.

Performing the transform requires some WORKSPACE, which can be
set up beforehand by calling fft_setup, if fft is to be called
more than once with arrays X of the same shape.  If no setup
keyword argument is supplied, the workspace allocation and setup
must be repeated for each call.
```

### fft_inplace

```DOCUMENT fft_inplace, x, direction
or fft_inplace, x, ljdir, rjdir
or fft_inplace, x, ljdir, rjdir, setup=workspace
is the same as the fft function, except that the transform is
performed "in_place" on the array X, which must be of type complex.```

### fft_setup

```DOCUMENT workspace= fft_setup(dimsof(x))
or workspace= fft_setup(dimsof(x), direction)
or workspace= fft_setup(dimsof(x), ljdir, rjdir)
allocates and sets up the workspace for a subsequent call to
fft(X, DIRECTION, setup=WORKSPACE)
or
fft(X, LJDIR, RJDIR, setup=WORKSPACE)
The DIRECTION or LJDIR, RJDIR arguments compute WORKSPACE only for
the dimensions which will actually be transformed.  If only the
dimsof(x) argument is supplied, then WORKSPACE will be enough to
transform any or all dimensions of X.  With DIRECTION or LJDIR, RJDIR
supplied, WORKSPACE will only be enough to compute the dimensions
which are actually to be transformed.  The WORKSPACE does not
depend on the sign of any element in the DIRECTION (or LJDIR, RJDIR),
so you can use the same WORKSPACE for both "forward" and "backward"
transforms.

Furthermore, as long as the length of any dimensions of the array
X to be transformed are present in WORKSPACE, it may be used in
a call to fft with the array.  Thus, if X were a 25-by-64 array,
and Y were a 64-vector, the following sequence is legal:
ws= fft_setup(dimsof(x));
xf= fft(x, 1, setup=ws);
yf= fft(y, -1, setup=ws);

The WORKSPACE required for a dimension of length N is 6*N+15 doubles.
```

### roll

```DOCUMENT roll(x, ljoff, rjoff)
or roll, x, ljoff, rjoff
or roll(x)
or roll, x

"rolls" selected dimensions of the array X.  The roll offsets
LJOFF and RJOFF (both optional) work in the same fashion as the
LJDIR and RJDIR arguments to the fft function:

A scalar LJDIR (and nil RJDIR) rolls all dimensions of X by
the specified offset.
Otherwise, the elements of the LJDIR vector [ljoff1, ljoff2, ...]
are used as the roll offsets for the first, second, etc.
dimensions of X.
Similarly, the elements of the RJDIR vector [..., rjoff1, rjoff0]
are matched to the final dimensions of X, so the next to last
dimension is rolled by rjoff1 and the last dimension by rjoff0.

As a special case (mostly for use with the fft function), if
both LJDIR and RJDIR are nil, every dimension is rolled by
half of its length.  Thus,
roll(x)
it equivalent to
roll(x, dimsof(x)(2:0)/2)

The result of the roll function is complex if X is complex, otherwise
double (i.e.- any other array type is promoted to type double).  If
roll is invoked as a subroutine, the operation is performed in place.
```