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<h3 class="section">28.3 Products of Polynomials</h3>
<!-- conv scripts/polynomial/conv.m -->
<p><a name="doc_002dconv"></a>
<div class="defun">
— Function File: <b>conv</b> (<var>a, b</var>)<var><a name="index-conv-2710"></a></var><br>
— Function File: <b>conv</b> (<var>a, b, shape</var>)<var><a name="index-conv-2711"></a></var><br>
<blockquote><p>Convolve two vectors <var>a</var> and <var>b</var>.
<p>The output convolution is a vector with length equal to
<code>length (</code><var>a</var><code>) + length (</code><var>b</var><code>) - 1</code>.
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>The optional <var>shape</var> argument may be
<dl>
<dt><var>shape</var> = "full"<dd>Return the full convolution. (default)
<br><dt><var>shape</var> = "same"<dd>Return the central part of the convolution with the same size as <var>a</var>.
</dl>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002ddeconv.html#doc_002ddeconv">deconv</a>, <a href="doc_002dconv2.html#doc_002dconv2">conv2</a>, <a href="doc_002dconvn.html#doc_002dconvn">convn</a>, <a href="doc_002dfftconv.html#doc_002dfftconv">fftconv</a>.
</p></blockquote></div>
<!-- convn src/DLD-FUNCTIONS/conv2.cc -->
<p><a name="doc_002dconvn"></a>
<div class="defun">
— Loadable Function: <var>C</var> = <b>convn</b> (<var>A, B</var>)<var><a name="index-convn-2712"></a></var><br>
— Loadable Function: <var>C</var> = <b>convn</b> (<var>A, B, shape</var>)<var><a name="index-convn-2713"></a></var><br>
<blockquote><p>Return the n-D convolution of <var>A</var> and <var>B</var>. The size of the result
is determined by the optional <var>shape</var> argument which takes the following
values
<dl>
<dt><var>shape</var> = "full"<dd>Return the full convolution. (default)
<br><dt><var>shape</var> = "same"<dd>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(</code><var>B</var><code>)/2] + 1)</code>.
<br><dt><var>shape</var> = "valid"<dd>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>.
</dl>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dconv2.html#doc_002dconv2">conv2</a>, <a href="doc_002dconv.html#doc_002dconv">conv</a>.
</p></blockquote></div>
<!-- deconv scripts/polynomial/deconv.m -->
<p><a name="doc_002ddeconv"></a>
<div class="defun">
— Function File: <b>deconv</b> (<var>y, a</var>)<var><a name="index-deconv-2714"></a></var><br>
<blockquote><p>Deconvolve two vectors.
<p><code>[b, r] = deconv (y, a)</code> solves for <var>b</var> and <var>r</var> such that
<code>y = conv (a, b) + r</code>.
<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.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dconv.html#doc_002dconv">conv</a>, <a href="doc_002dresidue.html#doc_002dresidue">residue</a>.
</p></blockquote></div>
<!-- conv2 src/DLD-FUNCTIONS/conv2.cc -->
<p><a name="doc_002dconv2"></a>
<div class="defun">
— Loadable Function: <b>conv2</b> (<var>A, B</var>)<var><a name="index-conv2-2715"></a></var><br>
— Loadable Function: <b>conv2</b> (<var>v1, v2, m</var>)<var><a name="index-conv2-2716"></a></var><br>
— Loadable Function: <b>conv2</b> (<var><small class="dots">...</small>, shape</var>)<var><a name="index-conv2-2717"></a></var><br>
<blockquote><p>Return the 2-D convolution of <var>A</var> and <var>B</var>. The size of the result
is determined by the optional <var>shape</var> argument which takes the following
values
<dl>
<dt><var>shape</var> = "full"<dd>Return the full convolution. (default)
<br><dt><var>shape</var> = "same"<dd>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(</code><var>B</var><code>)/2] + 1)</code>.
<br><dt><var>shape</var> = "valid"<dd>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>.
</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.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dconv.html#doc_002dconv">conv</a>, <a href="doc_002dconvn.html#doc_002dconvn">convn</a>.
</p></blockquote></div>
<!-- polygcd scripts/polynomial/polygcd.m -->
<p><a name="doc_002dpolygcd"></a>
<div class="defun">
— Function File: <var>q</var> = <b>polygcd</b> (<var>b, a</var>)<var><a name="index-polygcd-2718"></a></var><br>
— Function File: <var>q</var> = <b>polygcd</b> (<var>b, a, tol</var>)<var><a name="index-polygcd-2719"></a></var><br>
<blockquote>
<p>Find the greatest common divisor of two polynomials. 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.
The tolerance <var>tol</var> defaults to <code>sqrt(eps)</code>.
<p><strong>Caution:</strong> This is a numerically unstable algorithm and should not
be used on large polynomials.
<p>Example code:
<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>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002droots.html#doc_002droots">roots</a>, <a href="doc_002dconv.html#doc_002dconv">conv</a>, <a href="doc_002ddeconv.html#doc_002ddeconv">deconv</a>, <a href="doc_002dresidue.html#doc_002dresidue">residue</a>.
</p></blockquote></div>
<!-- residue scripts/polynomial/residue.m -->
<p><a name="doc_002dresidue"></a>
<div class="defun">
— Function File: [<var>r</var>, <var>p</var>, <var>k</var>, <var>e</var>] = <b>residue</b> (<var>b, a</var>)<var><a name="index-residue-2720"></a></var><br>
— Function File: [<var>b</var>, <var>a</var>] = <b>residue</b> (<var>r, p, k</var>)<var><a name="index-residue-2721"></a></var><br>
— Function File: [<var>b</var>, <var>a</var>] = <b>residue</b> (<var>r, p, k, e</var>)<var><a name="index-residue-2722"></a></var><br>
<blockquote><p>The first calling form computes the partial fraction expansion for the
quotient of the polynomials, <var>b</var> and <var>a</var>.
<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>
<p class="noindent">where M 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 N-1
representing the direct contribution, and the <var>e</var> vector specifies
the multiplicity of the m-th residue's pole.
<p>For 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>
<p class="noindent">which represents the following partial fraction expansion
<pre class="example"> s^2 + s + 1 -2 7 3
------------------- = ----- + ------- + -----
s^3 - 5s^2 + 8s - 4 (s-2) (s-2)^2 (s-1)
</pre>
<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>If the multiplicity, <var>e</var>, is not explicitly specified the multiplicity is
determined by the function <code>mpoles</code>.
<p>For 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>
<p>Alternatively the multiplicity may be defined explicitly, for 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>
<p class="noindent">which represents the following partial fraction expansion
<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>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmpoles.html#doc_002dmpoles">mpoles</a>, <a href="doc_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002droots.html#doc_002droots">roots</a>, <a href="doc_002dconv.html#doc_002dconv">conv</a>, <a href="doc_002ddeconv.html#doc_002ddeconv">deconv</a>.
</p></blockquote></div>
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