# NAG C Library Function Document

## 1Purpose

nag_sum_fft_qtrsine (c06rgc) computes the discrete quarter-wave Fourier sine transforms of $m$ sequences of real data values. The elements of each sequence and its transform are stored contiguously.

## 2Specification

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

## 3Description

Given $m$ sequences of $n$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, nag_sum_fft_qtrsine (c06rgc) simultaneously calculates the quarter-wave Fourier sine transforms of all the sequences defined by
 $x^ k p = 1n ∑ j=1 n-1 xjp × sin j 2k-1 π2n + 12 -1 k-1 xnp , if ​ direct=Nag_ForwardTransform ,$
or its inverse
 $xkp = 2n ∑ j=1 n x^ j p × sin 2j-1 k π2n , if ​ direct=Nag_BackwardTransform ,$
where $k=1,2,\dots ,n$ and $p=1,2,\dots ,m$.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.)
A call of nag_sum_fft_qtrsine (c06rgc) with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ will restore the original data.
The two transforms are also known as type-III DST and type-II DST, respectively.
The transform calculated by this function can be used to solve Poisson's equation when the solution is specified at the left boundary, and the derivative of the solution is specified at the right boundary (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 $5$.

## 4References

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

## 5Arguments

1:    $\mathbf{direct}$Nag_TransformDirectionInput
On entry: indicates the transform, as defined in Section 3, to be computed.
${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$
Forward transform.
${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$
Inverse transform.
Constraint: ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ or $\mathrm{Nag_BackwardTransform}$.
2:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of real values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
4:    $\mathbf{x}\left[{\mathbf{n}}×{\mathbf{m}}\right]$doubleInput/Output
On entry: the $m$ data sequences to be transformed. The $\mathit{p}$th sequence to be transformed, denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}\left[\left(p-1\right)×{\mathbf{n}}+j-1\right]$.
On exit: the $m$ quarter-wave sine transforms, overwriting the corresponding original sequences. The $n$ components of the $\mathit{p}$th quarter-wave sine transform, denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=1,2,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}\left[\left(p-1\right)×{\mathbf{n}}+k-1\right]$.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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).

## 8Parallelism and Performance

nag_sum_fft_qtrsine (c06rgc) 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_qtrsine (c06rgc) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_qtrsine (c06rgc) 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. Workspace is internally allocated by this function. The total amount of memory allocated is $\mathit{O}\left(n\right)$ double values.

## 10Example

This example reads in sequences of real data values and prints their quarter-wave sine transforms as computed by nag_sum_fft_qtrsine (c06rgc) with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$. It then calls the function again with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ and prints the results which may be compared with the original data.

### 10.1Program Text

Program Text (c06rgce.c)

### 10.2Program Data

Program Data (c06rgce.d)

### 10.3Program Results

Program Results (c06rgce.r)