This function performs 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 Arguments

1: $n1$ – Integer Input: On entry: $n_{1}$ , the first dimension of the transform.

Constraint: $n1 \geq 1$ .
2: $n2$ – Integer Input: On entry: $n_{2}$ , the second dimension of the transform.

Constraint: $n2 \geq 1$ .
3: $n3$ – Integer Input: On entry: $n_{3}$ , the third dimension of the transform.

Constraint: $n3 \geq 1$ .
4: $x [n1 \times n2 \times n3]$ – const double Input: On entry: the real input dataset $x$ , where $x_{j_{1} j_{2} j_{3}}$ is stored in $x [j_{3} \times n_{1} n_{2} + j_{2} \times n_{1} + j_{1}]$ , for $j_{1} = 0, 1, \dots, n_{1} - 1$ , $j_{2} = 0, 1, \dots, n_{2} - 1$ and $j_{3} = 0, 1, \dots, n_{3} - 1$ .
5: $y [\dim]$ – Complex Output: Note: the dimension, dim, of the array y must be at least $(n1 / 2 + 1) \times n2 \times n3$ .

On exit: the complex output dataset $\hat{z}$ , where ${\hat{z}}_{k_{1} k_{2} k_{3}}$ is stored in $y [k_{3} \times (n_{1} / 2 + 1) n_{2} + k_{2} \times (n_{1} / 2 + 1) + k_{1}]$ , for $k_{1} = 0, 1, \dots, n_{1} / 2$ , $k_{2} = 0, 1, \dots, n_{2} - 1$ and $k_{3} = 0, 1, \dots, n_{3} - 1$ . Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
6: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n1 = ⟨ value ⟩$ .
Constraint: $n1 \geq 1$ .

On entry, $n2 = ⟨ value ⟩$ .
Constraint: $n2 \geq 1$ .

On entry, $n3 = ⟨ value ⟩$ .
Constraint: $n3 \geq 1$ .
NE_INTERNAL_ERROR: 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

Some indication of accuracy can be obtained by performing a forward transform using c06pyc and a backward transform using c06pzc, and comparing the results with the original sequence (in exact arithmetic they would be identical).

8 Parallelism and Performance

c06pyc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c06pyc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by c06pyc is approximately proportional to

n_{1} n_{2} n_{3} \log (n_{1} n_{2} n_{3})

, but also depends on the factors of

n_{1}

n_{2}

and

n_{3}

. c06pyc is fastest if the only prime factors of

n_{1}

n_{2}

and

n_{3}

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 c06pyc. The total size of these arrays is approximately proportional to

n_{1} n_{2} n_{3}

10 Example

This example reads in a trivariate sequence of real data values and prints their discrete Fourier transforms as computed by c06pyc. Inverse transforms are then calculated by calling c06pzc showing that the original sequences are restored.

c06py: FL CL CPP AD

NAG CL Interfacec06pyc (fft_​real_​3d)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
c06pyc (fft_real_3d)