NAG Library Function Document

nag_sum_fft_hermitian_3d (c06pzc)

+− 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

1 Purpose

nag_sum_fft_hermitian_3d (c06pzc) computes the three-dimensional inverse discrete Fourier transform of a trivariate Hermitian sequence of complex data values.

2 Specification

#include <nag.h>

#include <nagc06.h>

void	nag_sum_fft_hermitian_3d (Integer n1, Integer n2, Integer n3, const Complex y[], double x[], NagError *fail)

3 Description

nag_sum_fft_hermitian_3d (c06pzc) computes the three-dimensional inverse discrete Fourier transform of a trivariate Hermitian sequence of complex data values

z_{j_{1} j_{2} j_{3}}

, 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

The discrete Fourier transform is here defined by

{\hat{x}}_{k_{1} k_{2} k_{3}} = \frac{1}{\sqrt{n_{1} n_{2} n_{3}}} \sum_{j_{1} = 0}^{n_{1} - 1} \sum_{j_{2} = 0}^{n_{2} - 1} \sum_{j_{3} = 0}^{n_{3} - 1} z_{j_{1} j_{2} j_{3}} \times \exp (2 π i (\frac{j_{1} k_{1}}{n_{1}} + \frac{j_{2} k_{2}}{n_{2}} + \frac{j_{3} k_{3}}{n_{3}})),

where

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

. (Note the scale factor of

\frac{1}{\sqrt{n_{1} n_{2} n_{3}}}

in this definition.)

Because the input data satisfies conjugate symmetry (i.e.,

z_{k_{1} k_{2} k_{3}}

is the complex conjugate of

z_{(n_{1} - k_{1}) k_{2} k_{3}}

), the transformed values

{\hat{x}}_{k_{1} k_{2} k_{3}}

are real.

A call of nag_sum_fft_real_3d (c06pyc) followed by a call of nag_sum_fft_hermitian_3d (c06pzc) will restore the original data.

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 – IntegerInput: On entry: $n_{1}$ , the first dimension of the transform.
Constraint: $n1 \geq 1$ .
2: n2 – IntegerInput: On entry: $n_{2}$ , the second dimension of the transform.
Constraint: $n2 \geq 1$ .
3: n3 – IntegerInput: On entry: $n_{3}$ , the third dimension of the transform.
Constraint: $n3 \geq 1$ .
4: y[ $\dim$ ] – const ComplexInput: 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[ $n1 \times n2 \times n3$ ] – doubleOutput: 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.

7 Accuracy

Some indication of accuracy can be obtained by performing a forward transform using nag_sum_fft_real_3d (c06pyc) and a backward transform using nag_sum_fft_hermitian_3d (c06pzc), and comparing the results with the original sequence (in exact arithmetic they would be identical).

8 Parallelism and Performance

nag_sum_fft_hermitian_3d (c06pzc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_sum_fft_hermitian_3d (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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by nag_sum_fft_hermitian_3d (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}

. nag_sum_fft_hermitian_3d (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 nag_sum_fft_hermitian_3d (c06pzc). The total size of these arrays is approximately proportional to

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

10 Example

See Section 10 in nag_sum_fft_real_3d (c06pyc).

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NAG Library Manual

NAG Library Function Documentnag_sum_fft_hermitian_3d (c06pzc)