c06 Chapter Contents
c06 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_sum_fft_cosine (c06rfc)

## 1  Purpose

nag_sum_fft_cosine (c06rfc) computes the discrete Fourier cosine transforms of $m$ sequences of real data values. The elements of each sequence and its transform are stored contiguously.

## 2  Specification

 #include #include
 void nag_sum_fft_cosine (Integer m, Integer n, double x[], NagError *fail)

## 3  Description

Given $m$ sequences of $n+1$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, nag_sum_fft_cosine (c06rfc) simultaneously calculates the Fourier cosine transforms of all the sequences defined by
 $x^ k p = 2n 12 x0p + ∑ j=1 n-1 xjp × cos jk πn + 12 -1k xnp , k= 0, 1, …, n ​ and ​ p= 1, 2, …, m .$
(Note the scale factor $\sqrt{\frac{2}{n}}$ in this definition.)
This transform is also known as type-I DCT.
Since the Fourier cosine transform defined above is its own inverse, two consecutive calls of nag_sum_fft_cosine (c06rfc) will restore the original data.
The transform calculated by this function can be used to solve Poisson's equation when the derivative of the solution is specified at both left and right boundaries (see Swarztrauber (1977)).
The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, described in Temperton (1983), together with pre- and post-processing stages described in Swarztrauber (1982). Special coding is provided for the factors $2$, $3$, $4$ and $5$.

## 4  References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501
Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

## 5  Arguments

1:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
2:    $\mathbf{n}$IntegerInput
On entry: one less than the number of real values in each sequence, i.e., the number of values in each sequence is $n+1$.
Constraint: ${\mathbf{n}}\ge 1$.
3:    $\mathbf{x}\left[\left({\mathbf{n}}+1\right)×{\mathbf{m}}\right]$doubleInput/Output
On entry: the $m$ data sequences to be transformed. The $\left(n+1\right)$ data values of the $\mathit{p}$th sequence to be transformed, denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}\left[\left(p-1\right)×\left({\mathbf{n}}+1\right)+j\right]$.
On exit: the $m$ Fourier cosine transforms, overwriting the corresponding original sequences. The $\left(n+1\right)$ components of the $\mathit{p}$th Fourier cosine transform, denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}\left[\left(p-1\right)×\left({\mathbf{n}}+1\right)+k\right]$.
4:    $\mathbf{fail}$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.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 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).

## 8  Parallelism and Performance

nag_sum_fft_cosine (c06rfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.

The time taken by nag_sum_fft_cosine (c06rfc) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_cosine (c06rfc) is fastest if the only prime factors of $n$ are $2$, $3$ and $5$, and is particularly slow if $n$ is a large prime, or has large prime factors. This function internally allocates a workspace of order $\mathit{O}\left(n\right)$ double values.

## 10  Example

This example reads in sequences of real data values and prints their Fourier cosine transforms (as computed by nag_sum_fft_cosine (c06rfc)). It then calls nag_sum_fft_cosine (c06rfc) again and prints the results which may be compared with the original sequence.

### 10.1  Program Text

Program Text (c06rfce.c)

### 10.2  Program Data

Program Data (c06rfce.d)

### 10.3  Program Results

Program Results (c06rfce.r)