c06fq computes the discrete Fourier transforms of m Hermitian sequences, each containing n complex data values. This method is designed to be particularly efficient on vector processors.


public static void c06fq(
	int m,
	int n,
	double[] x,
	string init,
	double[] trig,
	out int ifail
Visual Basic
Public Shared Sub c06fq ( _
	m As Integer, _
	n As Integer, _
	x As Double(), _
	init As String, _
	trig As Double(), _
	<OutAttribute> ByRef ifail As Integer _
Visual C++
static void c06fq(
	int m, 
	int n, 
	array<double>^ x, 
	String^ init, 
	array<double>^ trig, 
	[OutAttribute] int% ifail
static member c06fq : 
        m : int * 
        n : int * 
        x : float[] * 
        init : string * 
        trig : float[] * 
        ifail : int byref -> unit 


Type: System..::..Int32
On entry: m, the number of sequences to be transformed.
Constraint: m1.
Type: System..::..Int32
On entry: n, the number of data values in each sequence.
Constraint: n1.
Type: array<System..::..Double>[]()[][]
An array of size [m×n]
On entry: the data must be stored in x as if in a two-dimensional array of dimension 1:m,0:n-1; each of the m sequences is stored in a row of the array in Hermitian form. If the n data values zjp are written as xjp+iyjp, then for 0jn/2, xjp is contained in x[p-1,j], and for 1jn-1/2, yjp is contained in x[p,n-j]. (See also [] in the C06 class Chapter Introduction.)
On exit: the components of the m discrete Fourier transforms, stored as if in a two-dimensional array of dimension 1:m,0:n-1. Each of the m transforms is stored as a row of the array, overwriting the corresponding original sequence. If the n components of the discrete Fourier transform are denoted by x^kp, for k=0,1,,n-1, then the mn elements of the array x contain the values
Type: System..::..String
On entry: indicates whether trigonometric coefficients are to be calculated.
Calculate the required trigonometric coefficients for the given value of n, and store in the array trig.
init="S" or "R"
The required trigonometric coefficients are assumed to have been calculated and stored in the array trig in a prior call to one of c06fpc06fq or c06fr. The method performs a simple check that the current value of n is consistent with the values stored in trig.
Constraint: init="I", "S" or "R".
Type: array<System..::..Double>[]()[][]
An array of size [2×n]
On entry: if init="S" or "R", trig must contain the required trigonometric coefficients that have been previously calculated. Otherwise trig need not be set.
On exit: contains the required coefficients (computed by the method if init="I").
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).


Given m Hermitian sequences of n complex data values zjp, for j=0,1,,n-1 and p=1,2,,m, c06fq simultaneously calculates the Fourier transforms of all the sequences defined by
x^kp=1nj=0n-1zjp×exp-i2πjkn,  k=0,1,,n-1​ and ​p=1,2,,m.
(Note the scale factor 1n in this definition.)
The transformed values are purely real (see also the C06 class).
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term
To compute this form, this method should be preceded by forming the complex conjugates of the z^kp; that is xk=-xk, for k=n/2+1×m+1,,m×n.
The method uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special coding is provided for the factors 2, 3, 4, 5 and 6. This method is designed to be particularly efficient on vector processors, and it becomes especially fast as m, the number of transforms to be computed in parallel, increases.


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

Error Indicators and Warnings

Errors or warnings detected by the method:
On entry,m<1.
On entry,n<1.
On entry,init"I", "S" or "R".
Not used at this Mark.
On entry,init="S" or "R", but the array trig and the current value of n are inconsistent.
An unexpected error has occurred in an internal call. Check all method calls and array dimensions. Seek expert help.
An error occured, see message report.
Negative dimension for array value
Invalid Parameters value


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).

Parallelism and Performance


Further Comments

The time taken by c06fq is approximately proportional to nmlogn, but also depends on the factors of n. c06fq 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.


See Also