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<a name="Products-of-Polynomials"></a>
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Next: <a href="Derivatives-_002f-Integrals-_002f-Transforms.html#Derivatives-_002f-Integrals-_002f-Transforms" accesskey="n" rel="next">Derivatives / Integrals / Transforms</a>, Previous: <a href="Finding-Roots.html#Finding-Roots" accesskey="p" rel="prev">Finding Roots</a>, Up: <a href="Polynomial-Manipulations.html#Polynomial-Manipulations" accesskey="u" rel="up">Polynomial Manipulations</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Products-of-Polynomials-1"></a>
<h3 class="section">28.3 Products of Polynomials</h3>
<a name="XREFconv"></a><dl>
<dt><a name="index-conv"></a><em></em> <strong>conv</strong> <em>(<var>a</var>, <var>b</var>)</em></dt>
<dt><a name="index-conv-1"></a><em></em> <strong>conv</strong> <em>(<var>a</var>, <var>b</var>, <var>shape</var>)</em></dt>
<dd><p>Convolve two vectors <var>a</var> and <var>b</var>.
</p>
<p>When <var>a</var> and <var>b</var> are the coefficient vectors of two polynomials, the
convolution represents the coefficient vector of the product polynomial.
</p>
<p>The size of the result is determined by the optional <var>shape</var> argument
which takes the following values
</p>
<dl compact="compact">
<dt><var>shape</var> = <code>"full"</code></dt>
<dd><p>Return the full convolution. (default)
The result is a vector with length equal to
<code>length (<var>a</var>) + length (<var>b</var>) - 1</code>.
</p>
</dd>
<dt><var>shape</var> = <code>"same"</code></dt>
<dd><p>Return the central part of the convolution with the same size as <var>a</var>.
</p>
</dd>
<dt><var>shape</var> = <code>"valid"</code></dt>
<dd><p>Return only the parts which do not include zero-padded edges.
The size of the result is
<code>max (size (<var>a</var>) - size (<var>b</var>) + 1, 0)</code>.
</p></dd>
</dl>
<p><strong>See also:</strong> <a href="#XREFdeconv">deconv</a>, <a href="#XREFconv2">conv2</a>, <a href="#XREFconvn">convn</a>, <a href="Signal-Processing.html#XREFfftconv">fftconv</a>.
</p></dd></dl>
<a name="XREFconvn"></a><dl>
<dt><a name="index-convn"></a><em><var>C</var> =</em> <strong>convn</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt><a name="index-convn-1"></a><em><var>C</var> =</em> <strong>convn</strong> <em>(<var>A</var>, <var>B</var>, <var>shape</var>)</em></dt>
<dd><p>Return the n-D convolution of <var>A</var> and <var>B</var>.
</p>
<p>The size of the result is determined by the optional <var>shape</var> argument
which takes the following values
</p>
<dl compact="compact">
<dt><var>shape</var> = <code>"full"</code></dt>
<dd><p>Return the full convolution. (default)
</p>
</dd>
<dt><var>shape</var> = <code>"same"</code></dt>
<dd><p>Return central part of the convolution with the same size as <var>A</var>.
The central part of the convolution begins at the indices
<code>floor ([size(<var>B</var>)/2] + 1)</code>.
</p>
</dd>
<dt><var>shape</var> = <code>"valid"</code></dt>
<dd><p>Return only the parts which do not include zero-padded edges.
The size of the result is <code>max (size (A) - size (B) + 1, 0)</code>.
</p></dd>
</dl>
<p><strong>See also:</strong> <a href="#XREFconv2">conv2</a>, <a href="#XREFconv">conv</a>.
</p></dd></dl>
<a name="XREFdeconv"></a><dl>
<dt><a name="index-deconv"></a><em><var>b</var> =</em> <strong>deconv</strong> <em>(<var>y</var>, <var>a</var>)</em></dt>
<dt><a name="index-deconv-1"></a><em>[<var>b</var>, <var>r</var>] =</em> <strong>deconv</strong> <em>(<var>y</var>, <var>a</var>)</em></dt>
<dd><p>Deconvolve two vectors (polynomial division).
</p>
<p><code>[<var>b</var>, <var>r</var>] = deconv (<var>y</var>, <var>a</var>)</code> solves for <var>b</var> and
<var>r</var> such that <code><var>y</var> = conv (<var>a</var>, <var>b</var>) + <var>r</var></code>.
</p>
<p>If <var>y</var> and <var>a</var> are polynomial coefficient vectors, <var>b</var> will
contain the coefficients of the polynomial quotient and <var>r</var> will be
a remainder polynomial of lowest order.
</p>
<p><strong>See also:</strong> <a href="#XREFconv">conv</a>, <a href="#XREFresidue">residue</a>.
</p></dd></dl>
<a name="XREFconv2"></a><dl>
<dt><a name="index-conv2"></a><em></em> <strong>conv2</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt><a name="index-conv2-1"></a><em></em> <strong>conv2</strong> <em>(<var>v1</var>, <var>v2</var>, <var>m</var>)</em></dt>
<dt><a name="index-conv2-2"></a><em></em> <strong>conv2</strong> <em>(…, <var>shape</var>)</em></dt>
<dd><p>Return the 2-D convolution of <var>A</var> and <var>B</var>.
</p>
<p>The size of the result is determined by the optional <var>shape</var> argument
which takes the following values
</p>
<dl compact="compact">
<dt><var>shape</var> = <code>"full"</code></dt>
<dd><p>Return the full convolution. (default)
</p>
</dd>
<dt><var>shape</var> = <code>"same"</code></dt>
<dd><p>Return the central part of the convolution with the same size as <var>A</var>.
The central part of the convolution begins at the indices
<code>floor ([size(<var>B</var>)/2] + 1)</code>.
</p>
</dd>
<dt><var>shape</var> = <code>"valid"</code></dt>
<dd><p>Return only the parts which do not include zero-padded edges.
The size of the result is <code>max (size (A) - size (B) + 1, 0)</code>.
</p></dd>
</dl>
<p>When the third argument is a matrix, return the convolution of the matrix
<var>m</var> by the vector <var>v1</var> in the column direction and by the vector
<var>v2</var> in the row direction.
</p>
<p><strong>See also:</strong> <a href="#XREFconv">conv</a>, <a href="#XREFconvn">convn</a>.
</p></dd></dl>
<a name="XREFpolygcd"></a><dl>
<dt><a name="index-polygcd"></a><em><var>q</var> =</em> <strong>polygcd</strong> <em>(<var>b</var>, <var>a</var>)</em></dt>
<dt><a name="index-polygcd-1"></a><em><var>q</var> =</em> <strong>polygcd</strong> <em>(<var>b</var>, <var>a</var>, <var>tol</var>)</em></dt>
<dd>
<p>Find the greatest common divisor of two polynomials.
</p>
<p>This is equivalent to the polynomial found by multiplying together all the
common roots. Together with deconv, you can reduce a ratio of two
polynomials.
</p>
<p>The tolerance <var>tol</var> defaults to <code>sqrt (eps)</code>.
</p>
<p><strong>Caution:</strong> This is a numerically unstable algorithm and should not
be used on large polynomials.
</p>
<p>Example code:
</p>
<div class="example">
<pre class="example">polygcd (poly (1:8), poly (3:12)) - poly (3:8)
⇒ [ 0, 0, 0, 0, 0, 0, 0 ]
deconv (poly (1:8), polygcd (poly (1:8), poly (3:12))) - poly (1:2)
⇒ [ 0, 0, 0 ]
</pre></div>
<p><strong>See also:</strong> <a href="Miscellaneous-Functions.html#XREFpoly">poly</a>, <a href="Finding-Roots.html#XREFroots">roots</a>, <a href="#XREFconv">conv</a>, <a href="#XREFdeconv">deconv</a>, <a href="#XREFresidue">residue</a>.
</p></dd></dl>
<a name="XREFresidue"></a><dl>
<dt><a name="index-residue"></a><em>[<var>r</var>, <var>p</var>, <var>k</var>, <var>e</var>] =</em> <strong>residue</strong> <em>(<var>b</var>, <var>a</var>)</em></dt>
<dt><a name="index-residue-1"></a><em>[<var>b</var>, <var>a</var>] =</em> <strong>residue</strong> <em>(<var>r</var>, <var>p</var>, <var>k</var>)</em></dt>
<dt><a name="index-residue-2"></a><em>[<var>b</var>, <var>a</var>] =</em> <strong>residue</strong> <em>(<var>r</var>, <var>p</var>, <var>k</var>, <var>e</var>)</em></dt>
<dd><p>The first calling form computes the partial fraction expansion for the
quotient of the polynomials, <var>b</var> and <var>a</var>.
</p>
<p>The quotient is defined as
</p>
<div class="example">
<pre class="example">B(s) M r(m) N
---- = SUM ------------- + SUM k(i)*s^(N-i)
A(s) m=1 (s-p(m))^e(m) i=1
</pre></div>
<p>where <em>M</em> is the number of poles (the length of the <var>r</var>, <var>p</var>,
and <var>e</var>), the <var>k</var> vector is a polynomial of order <em>N-1</em>
representing the direct contribution, and the <var>e</var> vector specifies the
multiplicity of the m-th residue’s pole.
</p>
<p>For example,
</p>
<div class="example">
<pre class="example">b = [1, 1, 1];
a = [1, -5, 8, -4];
[r, p, k, e] = residue (b, a)
⇒ r = [-2; 7; 3]
⇒ p = [2; 2; 1]
⇒ k = [](0x0)
⇒ e = [1; 2; 1]
</pre></div>
<p>which represents the following partial fraction expansion
</p>
<div class="example">
<pre class="example"> s^2 + s + 1 -2 7 3
------------------- = ----- + ------- + -----
s^3 - 5s^2 + 8s - 4 (s-2) (s-2)^2 (s-1)
</pre></div>
<p>The second calling form performs the inverse operation and computes the
reconstituted quotient of polynomials, <var>b</var>(s)/<var>a</var>(s), from the
partial fraction expansion; represented by the residues, poles, and a direct
polynomial specified by <var>r</var>, <var>p</var> and <var>k</var>, and the pole
multiplicity <var>e</var>.
</p>
<p>If the multiplicity, <var>e</var>, is not explicitly specified the multiplicity
is determined by the function <code>mpoles</code>.
</p>
<p>For example:
</p>
<div class="example">
<pre class="example">r = [-2; 7; 3];
p = [2; 2; 1];
k = [1, 0];
[b, a] = residue (r, p, k)
⇒ b = [1, -5, 9, -3, 1]
⇒ a = [1, -5, 8, -4]
where mpoles is used to determine e = [1; 2; 1]
</pre></div>
<p>Alternatively the multiplicity may be defined explicitly, for example,
</p>
<div class="example">
<pre class="example">r = [7; 3; -2];
p = [2; 1; 2];
k = [1, 0];
e = [2; 1; 1];
[b, a] = residue (r, p, k, e)
⇒ b = [1, -5, 9, -3, 1]
⇒ a = [1, -5, 8, -4]
</pre></div>
<p>which represents the following partial fraction expansion
</p>
<div class="example">
<pre class="example"> -2 7 3 s^4 - 5s^3 + 9s^2 - 3s + 1
----- + ------- + ----- + s = --------------------------
(s-2) (s-2)^2 (s-1) s^3 - 5s^2 + 8s - 4
</pre></div>
<p><strong>See also:</strong> <a href="Finding-Roots.html#XREFmpoles">mpoles</a>, <a href="Miscellaneous-Functions.html#XREFpoly">poly</a>, <a href="Finding-Roots.html#XREFroots">roots</a>, <a href="#XREFconv">conv</a>, <a href="#XREFdeconv">deconv</a>.
</p></dd></dl>
<hr>
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