NAG Library Routine Document
C06PFF
1 Purpose
C06PFF computes the discrete Fourier transform of one variable in a multivariate sequence of complex data values.
2 Specification
INTEGER 
NDIM, L, ND(NDIM), N, LWORK, IFAIL 
COMPLEX (KIND=nag_wp) 
X(N), WORK(LWORK) 
CHARACTER(1) 
DIRECT 

3 Description
C06PFF computes the discrete Fourier transform of one variable (the $l$th say) in a multivariate sequence of complex data values ${z}_{{j}_{1}{j}_{2}\cdots {j}_{m}}$, where ${j}_{1}=0,1,\dots ,{n}_{1}1\text{, \hspace{1em}}{j}_{2}=0,1,\dots ,{n}_{2}1$, and so on. Thus the individual dimensions are ${n}_{1},{n}_{2},\dots ,{n}_{m}$, and the total number of data values is $n={n}_{1}\times {n}_{2}\times \cdots \times {n}_{m}$.
The routine computes
$n/{n}_{l}$ onedimensional transforms defined by
where
${k}_{l}=0,1,\dots ,{n}_{l}1$. The plus or minus sign in the argument of the exponential terms in the above definition determine the direction of the transform: a minus sign defines the
forward direction and a plus sign defines the
backward direction.
(Note the scale factor of $\frac{1}{\sqrt{{n}_{l}}}$ in this definition.)
A call of C06PFF with ${\mathbf{DIRECT}}=\text{'F'}$ followed by a call with ${\mathbf{DIRECT}}=\text{'B'}$ will restore the original data.
The data values must be supplied in a onedimensional complex array using columnmajor storage ordering of multidimensional data (i.e., with the first subscript ${j}_{1}$ varying most rapidly).
This routine
calls
C06PRF to perform onedimensional discrete Fourier transforms. Hence, the routine
uses a variant of the fast Fourier transform (FFT) algorithm (see
Brigham (1974)) known as the Stockham selfsorting algorithm, which is described in
Temperton (1983).
4 References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Selfsorting mixedradix fast Fourier transforms J. Comput. Phys. 52 1–23
5 Parameters
 1: DIRECT – CHARACTER(1)Input
On entry: if the forward transform as defined in
Section 3 is to be computed, then
DIRECT must be set equal to 'F'.
If the backward transform is to be computed then
DIRECT must be set equal to 'B'.
Constraint:
${\mathbf{DIRECT}}=\text{'F'}$ or $\text{'B'}$.
 2: NDIM – INTEGERInput
On entry: $m$, the number of dimensions (or variables) in the multivariate data.
Constraint:
${\mathbf{NDIM}}\ge 1$.
 3: L – INTEGERInput
On entry: $l$, the index of the variable (or dimension) on which the discrete Fourier transform is to be performed.
Constraint:
$1\le {\mathbf{L}}\le {\mathbf{NDIM}}$.
 4: ND(NDIM) – INTEGER arrayInput
On entry: the elements of
ND must contain the dimensions of the
NDIM variables; that is,
${\mathbf{ND}}\left(i\right)$ must contain the dimension of the
$i$th variable.
Constraints:
 ${\mathbf{ND}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{NDIM}}$;
 ${\mathbf{ND}}\left({\mathbf{L}}\right)$ must have less than $31$ prime factors (counting repetitions).
 5: N – INTEGERInput
On entry: $n$, the total number of data values.
Constraint:
N must equal the product of the first
NDIM elements of the array
ND.
 6: X(N) – COMPLEX (KIND=nag_wp) arrayInput/Output
On entry: the complex data values. Data values are stored in
X using columnmajor ordering for storing multidimensional arrays; that is,
${z}_{{j}_{1}{j}_{2}\cdots {j}_{m}}$ is stored in
${\mathbf{X}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}+\cdots \right)$.
On exit: the corresponding elements of the computed transform.
 7: WORK(LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
The workspace requirements as documented for C06PFF may be an overestimate in some implementations.
On exit: the real part of
${\mathbf{WORK}}\left(1\right)$ contains the minimum workspace required for the current value of
N with this implementation.
 8: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which C06PFF is called.
Suggested value:
${\mathbf{LWORK}}\ge {\mathbf{N}}+{\mathbf{ND}}\left({\mathbf{L}}\right)+15$
 9: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry,  ${\mathbf{NDIM}}<1$. 
 ${\mathbf{IFAIL}}=2$

On entry,  ${\mathbf{L}}<1$ or ${\mathbf{L}}>{\mathbf{NDIM}}$. 
 ${\mathbf{IFAIL}}=3$

On entry,  ${\mathbf{DIRECT}}\ne \text{'F'}$ or $\text{'B'}$. 
 ${\mathbf{IFAIL}}=4$

On entry,  at least one of the first NDIM elements of ND is less than $1$. 
 ${\mathbf{IFAIL}}=5$

On entry,  N does not equal the product of the first NDIM elements of ND. 
 ${\mathbf{IFAIL}}=6$

On entry,  LWORK is too small. The minimum amount of workspace required is returned in ${\mathbf{WORK}}\left(1\right)$. 
 ${\mathbf{IFAIL}}=7$

On entry,  ${\mathbf{ND}}\left({\mathbf{L}}\right)$ has more than $30$ prime factors. 
 ${\mathbf{IFAIL}}=8$

An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
7 Accuracy
Some indication of accuracy can be obtained by performing a subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).
The time taken is approximately proportional to $n\times \mathrm{log}{n}_{l}$, but also depends on the factorization of ${n}_{l}$. C06PFF is faster if the only prime factors of ${n}_{l}$ are $2$, $3$ or $5$; and fastest of all if ${n}_{l}$ is a power of $2$.
9 Example
This example reads in a bivariate sequence of complex data values and prints the discrete Fourier transform of the second variable. It then performs an inverse transform and prints the sequence so obtained, which may be compared with the original data values.
9.1 Program Text
Program Text (c06pffe.f90)
9.2 Program Data
Program Data (c06pffe.d)
9.3 Program Results
Program Results (c06pffe.r)