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<div id="content">
<h1 class="title">Vandermonde matrices</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#sec-1">Intro</a></li>
<li><a href="#sec-2">Setup</a></li>
<li><a href="#sec-3">Code</a></li>
</ul>
</div>
</div>
<div id="outline-container-sec-1" class="outline-2">
<h2 id="sec-1">Intro</h2>
<div class="outline-text-2" id="text-1">
<p>
For context and to see what motivates the need for Vandermonde matrices, see the linear algebra system developed in <i>the parent directory</i>.
The code is based off of the theories developed in <i>Matrix Analysis & Applied Linear Algebra</i>
</p>
<p>
<code>pages 185-186</code> introduces the Vandermonde matix which provides us with a way to solve for polynomials that fit a set of given points.
</p>
<p>
Polynomials are equations of the form:
</p>
\begin{equation}
y=a_{1}+a_{2}x+a_{3}x^{2}+a_{4}x^{3}+...
\end{equation}
<p>
It's a typial function of the form <b>y=f(x)</b> that takes inputs <code>x</code> and returns outputs <code>y</code> and it's immediately apparant that this is non linear. Given a polynomial (ie. given it's <b>a<sub>n</sub></b> factors) and given a value for <code>y</code> there can be multiple solutions for <code>x</code>. At first blush this doesn't seem related to the problems we're tackling with matrices.
</p>
<p>
The critical insight that we need to understand how to use Vandermonde matrices is that it's linear in terms of it's own polynomial factors. We are ultimately interested in finding a polynomial <i>that fits</i>, so what we want to do is actually solve for the <code>a_{n}</code> polynomial factors. In other words we aren't taking <b>y=f(x)</b> and solving for <b>x</b>. We are taking a corresponding <b>y=f(a)</b> and solving for the polynomial factors <b>a<sub>n</sub></b>. The <b>x</b>'s are baked into the <b>f()</b>.
</p>
<p>
To see how this is built up we simply take the polynomial function and a series of points/measurements <code>[ (x_1,y_1) (x_2,y_2) (x_3,y_3) ... ]</code> and write out the polynomial equations out.
</p>
\begin{equation}
y_1=a_{1}+a_{2}x_1+a_{3}x_{1}^{2}+a_{4}x_{1}^{3}+...\\
y_2=a_{1}+a_{2}x_2+a_{3}x_{2}^{2}+a_{4}x_{2}^{3}+...\\
y_3=a_{1}+a_{2}x_3+a_{3}x_{3}^{2}+a_{4}x_{3}^{3}+...\\
...
\end{equation}
<p>
This is now a standard matrix problem with the <code>a</code>'s being the unknown. The <code>x</code>'s look nonlinear, but they're actually known values and not something we are solving for. Writen out in matrix form we get
</p>
\begin{equation}
\begin{bmatrix}
1 & x_1 & x_{1}^2 & x_{1}^3 ..\\
1 & x_2 & x_{2}^2 & x_{2}^3 ..\\
1 & x_3 & x_{3}^2 & x_{3}^3 ..\\
...\\
\end{bmatrix}
\begin{bmatrix}
a_1\\
a_2\\
a_3\\
a_4\\
...\\
\end{bmatrix}
=
\begin{bmatrix}
y_1\\
y_2\\
y_3\\
...\\
\end{bmatrix}
\end{equation}
<p>
Where all the values of the x-matrix are known. (the x-matrix is the actual <i>Vandermonde matrix</i> - <b>V</b>)
</p>
<p>
Th next trick is that we can adjust he number of terms in the polynomial (the number of exponents of x and the the number of <b>a<sub>n</sub></b>'s) so that the matrix is always square - and in the general case non-singular. Now since we know how to solve <b>Ax=b</b> using the <b>LU decomposition</b> we can solve for all the polynomial factors <b>a<sub>n</sub></b> in this equivalent system <b>Va=y</b> - where the <b>V</b> is the matrix of exponents of <code>x</code>.
</p>
<p>
Once we solve for the <b>a<sub>n</sub></b>'s we can then reconstruct the non-linear polynomial equation we started looking at in the beginning
</p>
\begin{equation}
y=a_{1}+a_{2}x+a_{3}x^{2}+a_{4}x^{3}+...
\end{equation}
<p>
It will not only hold true for all our <code>(x,y)</code> points but also for all other values of <code>x</code> we want to test - so we will have in the end fit a polynomial curve through all our points.
</p>
</div>
</div>
<div id="outline-container-sec-2" class="outline-2">
<h2 id="sec-2">Setup</h2>
<div class="outline-text-2" id="text-2">
<p>
To start we import ELisp linear algebra function we've developed
</p>
<div class="org-src-container">
<pre class="src src-emacs-lisp">(load-file <span style="color: #ff1f8b;">"matrix.el"</span>)
</pre>
</div>
</div>
</div>
<div id="outline-container-sec-3" class="outline-2">
<h2 id="sec-3">Code</h2>
<div class="outline-text-2" id="text-3">
<p>
Next we build the <i>Vandermonde matrix</i> which is the matrix of <code>x</code>'s and their exponents
</p>
<div class="org-src-container">
<pre class="src src-emacs-lisp">(<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">matrix-vandermonde</span> (list-of-xs number-of-points)
<span style="color: #cc0000;">"Build a Vandermonde matrix of the appropriate rank from a LIST-OF-Xs"</span>
(<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">matrix-build-polynomial-list</span> (x degree)
<span style="color: #cc0000;">"Build a list of (X,X^2,X^3,..,X^DEGREE)"</span>
(<span style="color: #00af00;">cond</span>
((zerop degree)
'(1))
(t
(cons
(expt x degree)
(matrix-build-polynomial-list x (1- degree))
))))
(<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">matrix-vandermonde-data</span> (list-of-xs degree)
<span style="color: #cc0000;">"Builds the data vector of the Vandermonde matrix"</span>
(<span style="color: #00af00;">cond</span>
((null list-of-xs)
'())
(t
(append
(reverse
(matrix-build-polynomial-list
(car list-of-xs)
degree))
(matrix-vandermonde-data
(cdr list-of-xs)
degree)))))
(matrix-from-data-list
number-of-points
number-of-points
(matrix-vandermonde-data
list-of-xs
(1- number-of-points))))
</pre>
</div>
<p>
Then given a set of points, we can fit a polynomial to them by using our input solver to solve for the polynomial factors.
</p>
<div class="org-src-container">
<pre class="src src-emacs-lisp">(<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">matrix-fit-polynomial</span> (x-coordinates y-coordinates)
<span style="color: #cc0000;">"Given a list of x and y coordinates, solve for a polynomial that fits them using a Vandermonde matrixs. The result is a vector of factors 'a' that should be used in the standard order: a_1+a_2*x+a_3*x^2+a_4*x^3+... etc"</span>
(<span style="color: #00af00;">let*</span> ((number-of-points (length x-coordinates))
(vandermonde-matrix
(matrix-vandermonde x-coordinates number-of-points))
(PLU (matrix-PLU-decomposition vandermonde-matrix)))
(matrix-solve-for-input
PLU
(matrix-from-data-list
number-of-points
1
y-coordinates))))
</pre>
</div>
<p>
To see it work we can try feeding in some random points and see what kind of polynomial we get
</p>
<div class="org-src-container">
<pre class="src src-emacs-lisp" id="edata">(matrix-data
(matrix-fit-polynomial
'(1.0 2.0 3.0 4.0 5.0)
'(1.9 2.5 1.8 2.8 4.2)))
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="right" />
<col class="right" />
<col class="right" />
<col class="right" />
<col class="right" />
</colgroup>
<tbody>
<tr>
<td class="right">-7.299999999999994</td>
<td class="right">17.008333333333322</td>
<td class="right">-9.920833333333327</td>
<td class="right">2.2916666666666656</td>
<td class="right">-0.17916666666666659</td>
</tr>
</tbody>
</table>
<p>
Then we take these factors and stick them into gnuplot to get a quick plot
</p>
<div class="org-src-container">
<pre class="src src-gnuplot"><span style="color: #ff8700;">f(x) =</span> -7.299 + 17.00833*x + -9.920833*x**2 + 2.29166*x**3 + -0.179166*x**4
<span style="color: #1f5bff;">set</span> xrange[<span style="color: #1f5bff;">0:6</span>]
<span style="color: #1f5bff;">set</span> yrange[<span style="color: #1f5bff;">-1:5</span>]
<span style="color: #00af00;">plot</span> f(x)
</pre>
</div>
<div class="figure">
<p><img src="polynomial-fit.png" alt="polynomial-fit.png" />
</p>
</div>
<p>
I don't have the original points plotted here, but by visual inspection you can see that the curve passes through all of the points we started with.
</p>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: George Kontsevich</p>
<p class="date">Created: 2018-09-15 Sat 17:34</p>
<p class="creator"><a href="http://www.gnu.org/software/emacs/">Emacs</a> 25.2.2 (<a href="http://orgmode.org">Org</a> mode 8.2.10)</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>