NAG Library Routine Document
C06PZF
1 Purpose
C06PZF computes the three-dimensional inverse discrete Fourier transform of a trivariate Hermitian sequence of complex data values.
2 Specification
INTEGER |
N1, N2, N3, IFAIL |
REAL (KIND=nag_wp) |
X(N1*N2*N3) |
COMPLEX (KIND=nag_wp) |
Y((N1/2+1)*N2*N3) |
|
3 Description
C06PZF computes the three-dimensional inverse discrete Fourier transform of a trivariate Hermitian sequence of complex data values
z
j1
j2
j3
, for j1=0,1,…,n1-1, j2=0,1,…,n2-1 and j3=0,1,…,n3-1.
The discrete Fourier transform is here defined by
where
k1
=
0,1,…,
n1-1
,
k2
=
0,1,…,
n2-1
and
k3
=
0,1,…,
n3-1
. (Note the scale factor of
1
n1
n2
n3
in this definition.)
Because the input data satisfies conjugate symmetry (i.e.,
z
k1
k2
k3
is the complex conjugate of
z
n1
-
k1
k2
k3
), the transformed values
x^
k1
k2
k3
are real.
A call of
C06PYF followed by a call of C06PZF will restore the original data.
This routine calls
C06PQF and
C06PRF to perform multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in
Brigham (1974) and
Temperton (1983).
4 References
Brigham E O (1974)
The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms
J. Comput. Phys. 52 340–350
5 Parameters
- 1: N1 – INTEGERInput
On entry: n1, the first dimension of the transform.
Constraint:
N1≥1.
- 2: N2 – INTEGERInput
On entry: n2, the second dimension of the transform.
Constraint:
N2≥1.
- 3: N3 – INTEGERInput
On entry: n3, the third dimension of the transform.
Constraint:
N3≥1.
- 4: Y(
N1/2+1×N2×N3
) – COMPLEX (KIND=nag_wp) arrayInput
On entry: the Hermitian sequence of complex input dataset
z, where
z j1 j2 j3 is stored in
Y j3 × n1 /2+1 n2 + j2 × n1 /2+1 + j1 +1 , for
j1=0,1,…,n1/2,
j2=0,1,…,n2-1 and
j3=0,1,…,n3-1. That is, if
Y is regarded as a three-dimensional array of dimension
0:N1/2,0:N2-1,0:N3-1 , then
Yj1j2j3 must contain
z j1 j2 j3 .
- 5: X(
N1×N2×N3
) – REAL (KIND=nag_wp) arrayOutput
On exit: the real output dataset
x^, where
x^ k1 k2 k3 is stored in
X k3 × n1 n2 + k2 × n1 + k1 +1 , for
k1=0,1,…,n1-1,
k2=0,1,…,n2-1 and
k3=0,1,…,n3-1. That is, if
X is regarded as a three-dimensional array of dimension
0:N1-1,0:N2-1,0:N3-1 , then
Xk1k2k3 contains
x^ k1 k2 k3 .
- 6: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 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 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 -1 or 1 is used it is essential to test the value of IFAIL on exit.
On exit:
IFAIL=0 unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
- IFAIL=1
-
On entry, N1=value.
Constraint: N1≥1.
- IFAIL=2
-
On entry, N2=value.
Constraint: N2≥1.
- IFAIL=3
-
On entry, N3=value.
Constraint: N3≥1.
- IFAIL=4
-
An internal error has occurred in this function.
Check the function call and any array sizes.
If the call is correct then please contact
NAG for assistance.
- IFAIL=-999
-
Dynamic memory allocation failed.
7 Accuracy
Some indication of accuracy can be obtained by performing a forward transform using
C06PYF and a backward transform using C06PZF, and comparing the results with the original sequence (in exact arithmetic they would be identical).
8 Further Comments
The time taken by C06PZF is approximately proportional to
n1
n2
n3
log
n1
n2
n3
, but also depends on the factors of n1, n2 and n3. C06PZF is fastest if the only prime factors of n1, n2 and n3 are 2, 3 and 5, and is particularly slow if one of the dimensions is a large prime, or has large prime factors.
Workspace is internally allocated by C06PZF. The total size of these arrays is approximately proportional to
n1
n2
n3
.
9 Example