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<a name="Functions-and-Variables-for-FFTPACK5"></a>
<div class="header">
<p>
Next: <a href="maxima_116.html#Functions-for-numerical-solution-of-equations" accesskey="n" rel="next">Functions for numerical solution of equations</a>, Previous: <a href="maxima_114.html#Functions-and-Variables-for-fast-Fourier-transform" accesskey="p" rel="previous">Functions and Variables for fast Fourier transform</a>, Up: <a href="maxima_112.html#Numerical" accesskey="u" rel="up">Numerical</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-FFTPACK5-1"></a>
<h3 class="section">22.3 Functions and Variables for FFTPACK5</h3>
<p><code>FFTPACK5</code> provides several routines to compute Fourier
transforms for both real and complex sequences and their inverses.
The forward transform is defined the same as for <code>fft</code>. The
major difference is the length of the sequence is not constrained to
be a power of two. In fact, any length is supported, but it is most
efficient when the length has the form <em>2^r*3^s*5^t</em>.
</p>
<p><code>load("fftpack5")</code> loads this function.
</p>
<a name="fftpack5_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002ffftpack5_005ffft"></a><dl>
<dt><a name="index-fftpack5_005ffft"></a>Function: <strong>fftpack5_fft</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>Like <code>fft</code> (<code><a href="maxima_114.html#fft">fft</a></code>), this computes the fast Fourier transform
of a complex sequence. However, the length of <var>x</var> is not limited
to a power of 2.
</p>
<p><code>load("fftpack5")</code> loads this function.
</p>
<p>Examples:
</p>
<p>Real data.
</p>
<div class="example">
<pre class="example">(%i1) load("fftpack5") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2 ,-3, -4] $
(%i4) L1 : fftpack5_fft(L);
(%o4) [0.0, 1.811 %i - 0.1036, 0.0, 0.3107 %i + 0.6036, 0.0,
0.6036 - 0.3107 %i, 0.0, (- 1.811 %i) - 0.1036]
(%i5) L2 : fftpack5_inverse_fft(L1);
(%o5) [1.0, 4.441e-16 %i + 2.0, 1.837e-16 %i + 3.0, 4.0 - 4.441e-16 %i,
- 1.0, (- 4.441e-16 %i) - 2.0, (- 1.837e-16 %i) - 3.0, 4.441e-16
%i - 4.0]
(%i6) lmax (abs (L2-L));
(%o6) 4.441e-16
(%i7) L : [1, 2, 3, 4, 5, 6]$
(%i8) L1 : fftpack5_fft(L);
(%o8) [3.5, (- 0.866 %i) - 0.5, (- 0.2887 %i) - 0.5, (- 1.48e-16 %i) - 0.5,
0.2887 %i - 0.5, 0.866
%%i - 0.5]
(%i9) L2 : fftpack5_inverse_fft (L1);
(%o9) [1.0 - 1.48e-16 %i, 3.701e-17 %i + 2.0, 3.0 - 1.48e-16 %i,
4.0 - 1.811e-16 %i, 5.0 - 1.48e-16 %i, 5.881e-16
%i + 6.0]
(%i10) lmax (abs (L2-L));
(%o10) 9.064e-16
</pre></div>
<p>Complex data.
</p>
<div class="example">
<pre class="example">(%i1) load("fftpack5") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
(%i4) L1 : fftpack5_inverse_fft (L);
(%o4) [4.0, 2.828 %i + 2.828, (- 2.0 %i) - 2.0, 4.0, 0.0,
(- 2.828 %i) - 2.828, 2.0 %i - 2.0, 4.0]
(%i5) L2 : fftpack5_fft(L1);
(%o5) [1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, (- 2.776e-17 %i) - 1.0, - 1.0,
1.0 - 1.0 %i, 1.0 %i + 1.0, 1.0 -
%2.776e-17 %i]
(%i6) lmax(abs(L2-L));
(%o6) 1.11e-16
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-fftpack5">Package fftpack5</a>
·</div></dd></dl>
<a name="fftpack5_005finverse_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002ffftpack5_005finverse_005ffft"></a><dl>
<dt><a name="index-fftpack5_005finverse_005ffft"></a>Function: <strong>fftpack5_inverse_fft</strong> <em>(<var>y</var>)</em></dt>
<dd>
<p>Computes the inverse complex Fourier transform, like
<code>inverse_fft</code>, but is not constrained to be a power of two.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-fftpack5">Package fftpack5</a>
·</div></dd></dl>
<a name="fftpack5_005freal_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002ffftpack5_005freal_005ffft"></a><dl>
<dt><a name="index-fftpack5_005freal_005ffft"></a>Function: <strong>fftpack5_real_fft</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>Computes the fast Fourier transform of a real-valued sequence <var>x</var>,
just like <code>real_fft</code>, except the length is not constrained to be
a power of two.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) fpprintprec : 4 $
(%i2) L : [1, 2, 3, 4, 5, 6] $
(%i3) L1 : fftpack5_real_fft(L);
(%o3) [3.5, (- 0.866 %i) - 0.5, (- 0.2887 %i) - 0.5, - 0.5]
(%i4) L2 : fftpack5_inverse_real_fft(L1, 6);
(%o4) [1.0, 2.0, 3.0, 4.0, 5.0, 6.0]
(%i5) lmax(abs(L2-L));
(%o5) 1.332e-15
(%i6) fftpack5_inverse_real_fft(L1, 7);
(%o6) [0.5, 2.083, 2.562, 3.7, 4.3, 5.438, 5.917]
</pre></div>
<p>The last example shows how important it to set the length correctly
for <code>fftpack5_inverse_real_fft</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-fftpack5">Package fftpack5</a>
·</div></dd></dl>
<a name="fftpack5_005finverse_005freal_005ffft"></a><a name="Item_003a-Numerical_002fdeffn_002ffftpack5_005finverse_005freal_005ffft"></a><dl>
<dt><a name="index-fftpack5_005finverse_005freal_005ffft"></a>Function: <strong>fftpack5_inverse_real_fft</strong> <em>(<var>y</var>, <var>n</var>)</em></dt>
<dd>
<p>Computes the inverse Fourier transform of <var>y</var>, which must have a
length of <code>floor(n/2) + 1</code>. The length of sequence produced by the
inverse transform must be specified by <var>n</var>. This is required
because the length of <var>y</var> does not uniquely determine <var>n</var>.
The last element of <var>y</var> is always real if <var>n</var> is even, but it
can be complex when <var>n</var> is odd.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-fftpack5">Package fftpack5</a>
·</div></dd></dl>
<a name="Item_003a-Numerical_002fnode_002fFunctions-for-numerical-solution-of-equations"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_116.html#Functions-for-numerical-solution-of-equations" accesskey="n" rel="next">Functions for numerical solution of equations</a>, Previous: <a href="maxima_114.html#Functions-and-Variables-for-fast-Fourier-transform" accesskey="p" rel="previous">Functions and Variables for fast Fourier transform</a>, Up: <a href="maxima_112.html#Numerical" accesskey="u" rel="up">Numerical</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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