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: $y [\dim]$ – const Complex Input: Note: the dimension, dim, of the array y must be at least $(n1 / 2 + 1) \times n2 \times n3$ .

On entry: the Hermitian sequence of complex input dataset $z$ , where $z_{j_{1} j_{2} j_{3}}$ is stored in $y [j_{3} \times (n_{1} / 2 + 1) n_{2} + j_{2} \times (n_{1} / 2 + 1) + j_{1}]$ , for $j_{1} = 0, 1, \dots, n_{1} / 2$ , $j_{2} = 0, 1, \dots, n_{2} - 1$ and $j_{3} = 0, 1, \dots, n_{3} - 1$ .
5: $x [n1 \times n2 \times n3]$ – double Output: On exit: the real output dataset $\hat{x}$ , where ${\hat{x}}_{k_{1} k_{2} k_{3}}$ is stored in $x [k_{3} \times n_{1} n_{2} + k_{2} \times n_{1} + k_{1}]$ , for $k_{1} = 0, 1, \dots, n_{1} - 1$ , $k_{2} = 0, 1, \dots, n_{2} - 1$ and $k_{3} = 0, 1, \dots, n_{3} - 1$ .
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

Background information to multithreading can be found in the Multithreading documentation.

c06pzc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c06pzc 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 c06pzc 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}

. c06pzc 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 c06pzc. The total size of these arrays is approximately proportional to

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

10 Example

See c06pyc.

NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

C06 (Sum) Chapter Contents

C06 (Sum) Chapter Introduction

c06pz: FL CL CPP AD PY MB

NAG CL Interfacec06pzc (fft_​hermitian_​3d)

▸▿ Contents

1 Purpose

2 Specification

3 Description