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<title>Products of Polynomials - Untitled</title>
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<p>
<a name="Products-of-Polynomials"></a>
Next: <a rel="next" accesskey="n" href="Derivatives-and-Integrals.html#Derivatives-and-Integrals">Derivatives and Integrals</a>,
Previous: <a rel="previous" accesskey="p" href="Finding-Roots.html#Finding-Roots">Finding Roots</a>,
Up: <a rel="up" accesskey="u" href="Polynomial-Manipulations.html#Polynomial-Manipulations">Polynomial Manipulations</a>
<hr>
</div>
<h3 class="section">27.3 Products of Polynomials</h3>
<!-- ./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-2031"></a></var><br>
<blockquote><p>Convolve two vectors.
<p><code>y = conv (a, b)</code> returns a vector of length equal to
<code>length (a) + length (b) - 1</code>.
If <var>a</var> and <var>b</var> are polynomial coefficient vectors, <code>conv</code>
returns the coefficients of the product polynomial.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002ddeconv.html#doc_002ddeconv">deconv</a>, <a href="doc_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002droots.html#doc_002droots">roots</a>, <a href="doc_002dresidue.html#doc_002dresidue">residue</a>, <a href="doc_002dpolyval.html#doc_002dpolyval">polyval</a>, <a href="doc_002dpolyderiv.html#doc_002dpolyderiv">polyderiv</a>, <a href="doc_002dpolyinteg.html#doc_002dpolyinteg">polyinteg</a>.
</p></blockquote></div>
<!-- ./polynomial/convn.m -->
<p><a name="doc_002dconvn"></a>
<div class="defun">
— Function File: <var>c</var> = <b>convn</b> (<var>a, b, shape</var>)<var><a name="index-convn-2032"></a></var><br>
<blockquote><p>N-dimensional convolution of matrices <var>a</var> and <var>b</var>.
<p>The size of the output is determined by the <var>shape</var> argument.
This can be any of the following character strings:
<dl>
<dt>"full"<dd>The full convolution result is returned. The size out of the output is
<code>size (</code><var>a</var><code>) + size (</code><var>b</var><code>)-1</code>. This is the default behavior.
<br><dt>"same"<dd>The central part of the convolution result is returned. The size out of the
output is the same as <var>a</var>.
<br><dt>"valid"<dd>The valid part of the convolution is returned. The size of the result is
<code>max (size (</code><var>a</var><code>) - size (</code><var>b</var><code>)+1, 0)</code>.
</dl>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dconv.html#doc_002dconv">conv</a>, <a href="doc_002dconv2.html#doc_002dconv2">conv2</a>.
</p></blockquote></div>
<!-- ./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-2033"></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_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002droots.html#doc_002droots">roots</a>, <a href="doc_002dresidue.html#doc_002dresidue">residue</a>, <a href="doc_002dpolyval.html#doc_002dpolyval">polyval</a>, <a href="doc_002dpolyderiv.html#doc_002dpolyderiv">polyderiv</a>, <a href="doc_002dpolyinteg.html#doc_002dpolyinteg">polyinteg</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/conv2.cc -->
<p><a name="doc_002dconv2"></a>
<div class="defun">
— Loadable Function: y = <b>conv2</b> (<var>a, b, shape</var>)<var><a name="index-conv2-2034"></a></var><br>
— Loadable Function: y = <b>conv2</b> (<var>v1, v2, M, shape</var>)<var><a name="index-conv2-2035"></a></var><br>
<blockquote>
<p>Returns 2D convolution of <var>a</var> and <var>b</var> where the size
of <var>c</var> is given by
<dl>
<dt><var>shape</var>= 'full'<dd>returns full 2-D convolution
<br><dt><var>shape</var>= 'same'<dd>same size as a. 'central' part of convolution
<br><dt><var>shape</var>= 'valid'<dd>only parts which do not include zero-padded edges
</dl>
<p>By default <var>shape</var> is 'full'. When the third argument is a matrix
returns the convolution of the matrix <var>M</var> by the vector <var>v1</var>
in the column direction and by vector <var>v2</var> in the row direction
</p></blockquote></div>
<!-- ./polynomial/polygcd.m -->
<p><a name="doc_002dpolygcd"></a>
<div class="defun">
— Function File: <var>q</var> = <b>polygcd</b> (<var>b, a, tol</var>)<var><a name="index-polygcd-2036"></a></var><br>
<blockquote>
<p>Find 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.
Tolerance defaults to
<pre class="example"> sqrt(eps).
</pre>
<p>Note that this is an unstable
algorithm, so don't try it on large polynomials.
<p>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>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002dpolyinteg.html#doc_002dpolyinteg">polyinteg</a>, <a href="doc_002dpolyderiv.html#doc_002dpolyderiv">polyderiv</a>, <a href="doc_002dpolyreduce.html#doc_002dpolyreduce">polyreduce</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>, <a href="doc_002dfilter.html#doc_002dfilter">filter</a>, <a href="doc_002dpolyval.html#doc_002dpolyval">polyval</a>, <a href="doc_002dpolyvalm.html#doc_002dpolyvalm">polyvalm</a>.
</p></blockquote></div>
<!-- ./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-2037"></a></var><br>
<blockquote><p>Compute 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>
— Function File: [<var>b</var>, <var>a</var>] = <b>residue</b> (<var>r, p, k</var>)<var><a name="index-residue-2038"></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-2039"></a></var><br>
<blockquote><p>Compute 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 script mpoles.m.
<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.m 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_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_002dmpoles.html#doc_002dmpoles">mpoles</a>, <a href="doc_002dpolyval.html#doc_002dpolyval">polyval</a>, <a href="doc_002dpolyderiv.html#doc_002dpolyderiv">polyderiv</a>, <a href="doc_002dpolyinteg.html#doc_002dpolyinteg">polyinteg</a>.
</p></blockquote></div>
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