NAG Toolbox: nag_sum_withdraw_fft_complex_2d_sep (c06fu)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{m}$ – int64int32nag_int scalar

$m$, the length of the first dimension of $Z$. Consider the matrix $Z$ with elements $Z_{ij}=z_{i+1j+1}$, where $z$ is the bivariate sequence defined in Description, then $m$ is the number of rows of $Z$.

Constraint: $\mathbf{m}\ge 1$.

2:     $\mathrm{n}$ – int64int32nag_int scalar

$n$, the length of the second dimension of $Z$. Consider the matrix $Z$ with elements $Z_{ij}=z_{i+1j+1}$, where $z$ is the bivariate sequence defined in Description, then $n$ is the number of columns of $Z$.

Constraint: $\mathbf{n}\ge 1$.

3:     $\mathrm{x}\left(\mathbf{m}\times \mathbf{n}\right)$ – double array

4:     $\mathrm{y}\left(\mathbf{m}\times \mathbf{n}\right)$ – double array

The real and imaginary parts of the complex data values must be stored in arrays x and y respectively. If x and y are regarded as two-dimensional arrays of dimension $\left(0:\mathbf{m}-1,0:\mathbf{n}-1\right)$, then $\mathbf{x}\left(j_1,j_2\right)$ and $\mathbf{y}\left(j_1,j_2\right)$ must contain the real and imaginary parts of $z_{j_1{j}_2}$.

5:     $\mathrm{init}$ – string (length ≥ 1)

Indicates whether trigonometric coefficients are to be calculated.

$\mathbf{init}=\text{'I'}$

Calculate the required trigonometric coefficients for the given values of $m$ and $n$, and store in the corresponding arrays trigm and trign.

$\mathbf{init}=\text{'S'}$ or $\text{'R'}$

The required trigonometric coefficients are assumed to have been calculated and stored in the arrays trigm and trign in a prior call to nag_sum_fft_complex_2d_sep (c06fu). The function performs a simple check that the current values of $m$ and $n$ are consistent with the corresponding values stored in trigm and trign.

Constraint: $\mathbf{init}=\text{'I'}$, $\text{'S'}$ or $\text{'R'}$.

6:     $\mathrm{trigm}\left(2\times \mathbf{m}\right)$ – double array

7:     $\mathrm{trign}\left(2\times \mathbf{n}\right)$ – double array

If $\mathbf{init}=\text{'S'}$ or $\text{'R'}$, trigm and trign must contain the required coefficients calculated in a previous call of the function. Otherwise trigm and trign need not be set.

If $m=n$ the same array may be supplied for trigm and trign.

Optional Input Parameters

None.

Output Parameters

1:     $\mathrm{x}\left(\mathbf{m}\times \mathbf{n}\right)$ – double array

2:     $\mathrm{y}\left(\mathbf{m}\times \mathbf{n}\right)$ – double array

The real and imaginary parts respectively of the corresponding elements of the computed transform.

3:     $\mathrm{trigm}\left(2\times \mathbf{m}\right)$ – double array

4:     $\mathrm{trign}\left(2\times \mathbf{n}\right)$ – double array

trigm and trign contain the required coefficients (computed by the function if $\mathbf{init}=\text{'I'}$).

5:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry,

$\mathbf{m}<1$.

   $\mathbf{ifail}=2$

On entry,

$\mathbf{n}<1$.

   $\mathbf{ifail}=3$

On entry,

$\mathbf{init}\ne \text{'I'}$, $\text{'S'}$ or $\text{'R'}$.

   $\mathbf{ifail}=4$

Not used at this Mark.

   $\mathbf{ifail}=5$

On entry,

$\mathbf{init}=\text{'S'}$ or $\text{'R'}$, but at least one of the arrays trigm and trign is inconsistent with the current value of m or n.

   $\mathbf{ifail}=6$

An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: c06fu_example

function c06fu_example


fprintf('c06fu example results\n\n');

m = int64(3);
n = int64(5);
x = [ 1.000     0.999     0.987     0.936     0.802;
      0.994     0.989     0.963     0.891     0.731;
      0.903     0.885     0.823     0.694     0.467];
y = [ 0.000    -0.040    -0.159    -0.352    -0.597;
     -0.111    -0.151    -0.268    -0.454    -0.682
     -0.430    -0.466    -0.568    -0.720    -0.884];

% transform, then inverse transform to restore data
init = 'Initial';
trigm = zeros(6,1);
trign = zeros(10,1);
[xt, yt, trigm, trign, ifail] = c06fu(m, n, x, y, init, trigm, trign);
init = 'Subsequent';
[xr, yr, trigm, trign, ifail] = c06fu(m, n, xt, -yt, init, trigm, trign);

% Display as complex matrices
z = x + i*y;
nd = [m,n];
zt = reshape(xt+i*yt,nd);
zr = reshape(xr-i*yr,nd);

matrix = 'general';
diag = ' ';
usefrm = 'Above';
format = 'F9.3';
labrow = 'None';
labcol = 'None';
ncols  = int64(80);
indent = int64(0);

title  = 'Original data:';
[ifail] = x04db(...
    matrix, diag, z, usefrm, format, title, labrow, labcol, ncols, indent);
disp(' ');
title = 'Discrete Fourier transform:';
[ifail] = x04db(...
    matrix, diag, zt, usefrm, format, title, labrow, labcol, ncols, indent);
disp(' ');
title = 'Original sequence as restored by inverse transform:';
[ifail] = x04db(...
    matrix, diag, zr, usefrm, format, title, labrow, labcol, ncols, indent);

c06fu example results

 Original data:
      1.000    0.999    0.987    0.936    0.802
      0.000   -0.040   -0.159   -0.352   -0.597

      0.994    0.989    0.963    0.891    0.731
     -0.111   -0.151   -0.268   -0.454   -0.682

      0.903    0.885    0.823    0.694    0.467
     -0.430   -0.466   -0.568   -0.720   -0.884
 
 Discrete Fourier transform:
      3.373    0.481    0.251    0.054   -0.419
     -1.519   -0.091    0.178    0.319    0.415

      0.457    0.055    0.009   -0.022   -0.076
      0.137    0.032    0.039    0.036    0.004

     -0.170   -0.037   -0.042   -0.038   -0.002
      0.493    0.058    0.008   -0.025   -0.083
 
 Original sequence as restored by inverse transform:
      1.000    0.999    0.987    0.936    0.802
      0.000   -0.040   -0.159   -0.352   -0.597

      0.994    0.989    0.963    0.891    0.731
     -0.111   -0.151   -0.268   -0.454   -0.682

      0.903    0.885    0.823    0.694    0.467
     -0.430   -0.466   -0.568   -0.720   -0.884