c06 Chapter Contents
c06 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_fft_multiple_cosine (c06hbc)

## 1  Purpose

nag_fft_multiple_cosine (c06hbc) computes the discrete Fourier cosine transforms of $m$ sequences of real data values.

## 2  Specification

 #include #include
 void nag_fft_multiple_cosine (Integer m, Integer n, double x[], const double trig[], 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 $\mathit{p}=1,2,\dots ,m$, this function simultaneously calculates the Fourier cosine transforms of all the sequences defined by
 $x ^ k p = 2 n 1 2 x 0 p + ∑ j=1 n-1 x j p cos jk π n + 1 2 -1 k x n p , for ​ k = 0 , 1 , … , n ; ​ p = 1 , 2 , … , m .$
(Note the scale factor $\sqrt{\frac{2}{n}}$ in this definition.)
The Fourier cosine transform defined above is its own inverse, and two consecutive calls of this function with the same data will restore the original data (but see Section 8).
The transform calculated by this function can be used to solve Poisson's equation when the solution is specified at both left and right boundaries (Swarztrauber (1977)).
The function uses a variant of the fast Fourier transform (FFT) algorithm (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, 5 and 6.

## 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:     mIntegerInput
On entry: the number of sequences to be transformed, $m$.
Constraint: ${\mathbf{m}}\ge 1$.
2:     nIntegerInput
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:     x[${\mathbf{m}}×\left({\mathbf{n}}+1\right)$]doubleInput/Output
On entry: the $m$ data sequences stored in x consecutively. If the $n+1$ data values of the $\mathit{p}$th sequence to be transformed are denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, then the first $m\left(n+1\right)$ elements of the array x must contain the values
 $x 0 1 , x 1 1 , … , x n 1 , x 0 2 , x 1 2 , … , x n 2 , … , x 0 m , x 1 m , … , x n m .$
On exit: the $m$ Fourier cosine transforms stored consecutively, overwriting the corresponding original sequence.
4:     trig[$2×{\mathbf{n}}$]const doubleInput
On entry: trigonometric coefficients as returned by a call of nag_fft_init_trig (c06gzc). nag_fft_multiple_cosine (c06hbc) makes a simple check to ensure that trig has been initialized and that the initialization is compatible with the value of n.
5:     failNagError *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_C06_NOT_TRIG
Value of n and trig array are incompatible or trig array not initialized.
NE_INT_ARG_LT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.

## 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).

The time taken is approximately proportional to $nm\mathrm{log}n$, but also depends on the factors of $n$. The function 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.

## 9  Example

This program reads in sequences of real data values and prints their Fourier cosine transforms (as computed by nag_fft_multiple_cosine (c06hbc)). It then calls nag_fft_multiple_cosine (c06hbc) again and prints the results which may be compared with the original sequence.

### 9.1  Program Text

Program Text (c06hbce.c)

### 9.2  Program Data

Program Data (c06hbce.d)

### 9.3  Program Results

Program Results (c06hbce.r)